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+{"text":"\\section{Introduction}\n\n\n\\Acp{SSM} have often been used to model the spread of a virus in a population, since they provide a practical and simple framework to describe the generation of epidemic time series \\citep{dureau2013ssm}. \n\\acp{SSM} are a latent variable representation of a dynamical systems evolving over specific domains (e.g. time, space, generations) \\citep{schon2018probabilistic}. The definition of an \\ac{SSM} involves the specification of a system of stochastic or deterministic equations that define the behaviour of the latent process, $ X_t $, where $ t $ indexes the domain (e.g. \\textit{time}), as well as the emission of observable quantities throughout the domain, denoted by $ Y_t $ \\citep{birrell2018evidence}. Thanks to their flexibility, \\ac{SSM} representations of observable phenomena have been widely used in many fields, from indoor positioning problems in engineering \\citep{solin2018modeling}, to environmental studies \\citep{anderson1996method}, to epidemic models \\citep{magpantay2016pertussis}. \n\nModelling epidemics as \\acp{SSM}, the latent process generally encapsulates: (i) a transmission process, describing the spread of infection through a population, (ii) a severity process, modelling the likelihood of infected individuals to become severe cases (e.g. symptomatic, in need of hospitalization, etc.), and (iii) a detection process that links processes (i) and (ii) to available observations. Moreover, (iv) other sources of randomness might affect the generation of epidemic data, including seasonal fluctuations in parameters and misclassification of cases unrelated to the pathogen of interest. \n\nWhile formulating a \\ac{SSM} to describe epidemic data, there are several model choices to make; a key choice is the level of stochasticity of the overall system. As highlighted in the four points above, there are many possible sources of randomness, each of them could be accounted for in several different ways or could be overlooked through the use of deterministic systems, a reasonable approximation when numbers are sufficiently large. A model with a higher level of stochasticity attempts to make only few deterministic approximations and, the more the noise is ignored, the lower the level of stochasticity of the model.\n\nSome models include complex dynamical components dynamic systems: \\cite{birrell2011bayesian}, for example, assumes that an age-structured deterministic discrete-time system could approximate the spread of the 2009 pandemic influenza; it accounts for delays between infection and detection of symptomatic cases and places most of the randomness in the observation model. A similar approach, with even more complex layers, is considered in other works including \\cite{weidemann2014bayesian}, analysing rotavirus data, and in recent works analysing COVID-19 data, e.g. \\cite{keeling2020fitting} and \\cite{birrell2021real}. Conversely, other models adopt higher levels of stochasticity and rely on simplifying assumptions on the dynamics. An example is the work of \\cite{lekone2006statistical}: here a discrete-time stochastic model is fitted to Ebola data and substantial, sometimes unrealistic, assumptions are made, including exponentially distributed time to events and full ascertainment of the cases. These kinds of models have been particularly utilised for small-size epidemics where the large-numbers deterministic approximation does not hold (e.g. \\cite{nishiura2011real}, \\cite{funk2018real}). \n\nHybrid models that exploit the strength from both approaches could potentially prove useful in many contexts. The transmission dynamics of large seasonal epidemics and of pandemics can be reliably approximated by deterministic systems, which, compared to their correspondent stochastic systems, allow greater model complexity (mixing heterogeneity, many compartments, etc.). In many contexts, however, the number of cases that become severe and detected might be smaller and subject to significant stochastic variation. Stochastic systems would be required to describe the severity process, marking a difference from many models listed above, where most of the randomness lies in the observational process. \n\nWhile a model of this type (with all stochastic relationships except for the deterministic transmission dynamics) has greater realism, it might be more challenging under an inferential perspective, especially when multiple data sources are considered. If individuals are detected at different levels of severity in multiple datasets, an evidence synthesis framework \\citep{presanis2013conflict, presanis2014synthesising} should be employed, where correlations and delays among these layers of severity (and their related data sources) are accounted for.\n\nThis paper addresses the problem of inference of epidemics from multiple dependent data by presenting a general framework for the inferential problem; it identifies and tackles the challenges that arise in this context, considering the specific case of a deterministic-transmission model with multiple layers of stochasticity in the severity and detection stage. \nSection \\ref{sec2} provides a background to the methods used for modelling and inference, specifically \\acp{SSM} and pseudo-marginal methods. Section \\ref{sec3} considers their application to the setting of epidemics, specifying a model that mirrors realistic situations; presents its problems and proposes an inferential routine. The aim of Section \\ref{sec4} is to compare the model presented in Section \\ref{sec3} to a similar model that, through a simplifying assumption does not account for the intrinsic dependence between datasets, inducing under-estimation of parameter uncertainty. Lastly Section \\ref{sec5} presents the analysis of real data on cases of seasonal influenza at different levels of severity, appropriately attributing stochasticity to each process. \n\n\n\n\n\\section{State-space models and their inference}\\label{sec2}\n\n\\acp{SSM} provide a flexible and general modelling framework for, among other problems, the inference of data arising from an epidemic. They are briefly outlined below, together with some useful tools to perform inference.\n\n\\subsection{Notation}\nIn this paper, Greek letters identify parameters (e.g. $ \\theta, \\lambda $) and their transformations ($ \\xi= f(\\theta)$). Parameters are \\acp{r.v.}, here distinguished from the system's \\acp{r.v.}, usually dependent on the parameters, denoted by uppercase Latin letters (e.g. $ X, Y $) and their instances which are lowercase ($ x, y $). Lower indexes indicate the domain (e.g. $ X_t $ is the \\ac{r.v.} $ X $ at time $ t $) and upper indices denote the states of the system ($ X^\\textsc{k} $, $ ^\\textsc{j}X^\\textsc{k} $ ). Lastly, the notation $ 1:r $ is used to define a running index (e.g. $ y_{1:t} $ is the set of observation $ y_1, y_2, \\dots, y_t $). \n\n\n\\subsection{State-space models}\nA \\ac{SSM} is a stochastic process that makes use of a latent variable representation to describe dynamical phenomena \\citep{schon2018probabilistic}. It has two components: a latent process, $\\left\\lbrace {X_t} \\right\\rbrace _{t\\geq0} $ representing the underlying states and their dynamics; and an observed process $\\left\\lbrace {Y_t} \\right\\rbrace _{t\\geq1} $. Here, the domain of the process is assumed to be discrete time, with intervals indexed by $ t $, although the methods reported are easily extended to other domains. \nA parameter-driven \\ac{SSM} can be defined through the state\nand the observation equations, as in Equations \\ref{eq1_1}, \\ref{eq1_2} and \\ref{eq1_3}, respectively\n\\begin{eqnarray}\n\\label{eq1_1}X_0|\\boldsymbol{\\theta} &\\sim&  p(x_0|\\boldsymbol{\\theta})\\\\\n\\label{eq1_2}{X_t}|({X_{t-1}}, \\boldsymbol{\\theta}) &\\sim& p({x_t}|{x_{t-1}},\\boldsymbol{\\theta})\\\\\n\\label{eq1_3}Y_t|({X_{t}}, \\boldsymbol{\\theta}) &\\sim & p(y_t|{x_{t}},\\boldsymbol{\\theta})\n\\end{eqnarray}\nfor  $ t=1, 2, \\dots, T $. These express the joint distribution of the initial state of the system, $ X_0 $, of the latent states, $ X_t$s, and of the observations, $ Y_t $s,  parametrised by a vector $ \\boldsymbol{\\theta} $ \\citep{brockwell2016introduction}. Equations \\ref{eq1_1} to \\ref{eq1_3} define a Markovian model. Markovianity is a common assumption for epidemic models and many other \\acp{SSM} that fall in the category of \\acp{POMP} \\citep{king2015statistical} or \\acp{HMM} \\citep{churchill2005hidden}. \n\n\nA Markovian \\ac{SSM} defines the following full probability model:\n\\begin{equation}\np({x_{0:T}}, y_{1:T}|{\\boldsymbol{\\theta}})= \\prod_{t=1}^{T}p(y_t|{x_t}, {\\boldsymbol{\\theta}})\\prod_{t=1}^{T}p({x_t}|{x_{t-1}}, {\\boldsymbol{\\theta}}) p({x_0}| {\\boldsymbol{\\theta}})\n\\label{eq2}\n\\end{equation}\nwhere, thanks to Markovianity and conditional independence, the joint density can be factorised into the state and observation densities.\n\nA \\ac{SSM} can also be represented through a graphical model where a graph $ \\mathcal{G}=(\\mathcal{V}, \\mathcal{E}) $ represents the conditional independence structure (edges $  \\mathcal{E} $) between \\acp{r.v.} (nodes $ \\mathcal{V} $). \nSome graphical models representing the state and observational process of an \\ac{SSM} are illustrated in Figure \\ref{f2}; the dependence on the parameter $ \\boldsymbol{\\theta} $ is omitted for simplicity.\n\nThis framework could account for many different modelling assumptions, including multidimensional state processes or deterministic relations, as exemplified in Figure \\ref{f2}. \n\n\n\\begin{figure}[h]\n\\begin{subfigure}[c]{6.42cm}\n\\centering\n\\hspace{-0.5cm}\\begin{tikzpicture} [node distance=1.5cm] \n\\node(x0)[state]{$ \\xi_0 $};\n\\node(x1)[state, right of=x0]{$ \\xi_1 $};\n\\node(x2)[state, right of=x1]{$ \\xi_2 $};\n\\node(xn)[state, right of=x2, draw=transparent]{\\dots};\n\\node(xT)[state, right of=xn]{$ \\xi_T $};\n\\draw [->, dashed] (x0) -- (x1);\n\\draw [->, dashed] (x1) -- (x2);\n\\draw [->, dashed] (x2) -- (xn);\n\\draw [->, dashed] (xn) -- (xT);\n\\node(y0)[obs, below of=x0, fill=transparent, draw=transparent]{ };\n\\node(y1)[obs, right of=y0]{$ y_1 $};\n\\node(y2)[obs, right of=y1]{$ y_2 $};\n\\node(yn)[obs, right of=y2, fill=transparent, draw=transparent]{\\dots};\n\\node(yT)[obs, right of=yn]{$ y_T $};\n\\draw [->] (x1) -- (y1);\n\\draw [->] (x2) -- (y2);\n\\draw [->] (xT) -- (yT);\n\\end{tikzpicture}\n\\caption{SSM assuming deterministic underlying relation: the states $ \\xi_{1:T} $ are evolving without stochasticity.}\n\\label{fig2b}\n\\end{subfigure}\\hfill\n\\begin{subfigure}[c]{6.42cm}\n\\centering\n\\hspace*{-1cm}\\begin{tikzpicture} [node distance=1.7cm] \\small\n\\node(x0)[state]{$ x^\\textsc{a}_0 $};\n\\node(x1)[state, right of=x0]{$ x^\\textsc{a}_1 $};\n\\node(x2)[state, right of=x1]{$ x^\\textsc{a}_2 $};\n\\node(xn)[state, right of=x2, draw=transparent]{\\dots};\n\\node(xT)[state, right of=xn]{$ x^\\textsc{a}_T $};\n\\draw [->] (x0) -- (x1);\n\\draw [->] (x1) -- (x2);\n\\draw [->] (x2) -- (xn);\n\\draw [->] (xn) -- (xT);\n\\node(xb0)[state, below of=x0, fill=transparent, draw=transparent]{ };\n\\node(xb1)[state, right of=xb0]{$ x^\\textsc{b}_1 $};\n\\node(xb2)[state, right of=xb1]{$ x^\\textsc{b}_2 $};\n\\node(xbn)[state, right of=xb2, fill=transparent, draw=transparent]{\\dots};\n\\node(xbT)[state, right of=xbn]{$ x^\\textsc{b}_T $};\n\\draw [->] (x1) -- (xb1);\n\\draw [->] (x2) -- (xb2);\n\\draw [->] (xT) -- (xbT);\n\\node(yb1)[obs, below of=xb1, xshift=0.8cm, yshift=0.8cm]{$ y^\\textsc{b}_1 $};\n\\node(yb2)[obs, right of=yb1]{$ y^\\textsc{b}_2 $};\n\\node(ybn)[obs, right of=yb2, fill=transparent, draw=transparent]{\\dots};\n\\node(ybT)[obs, right of=ybn]{$ y^\\textsc{b}_T $};\n\\draw [->] (xb1) -- (yb1);\n\\draw [->] (xb2) -- (yb2);\n\\draw [->] (xbT) -- (ybT);\n\\node(xc0)[state, below of=xb0, fill=transparent, draw=transparent]{ };\n\\node(xc1)[state, right of=xc0]{$ x^\\textsc{c}_1 $};\n\\node(xc2)[state, right of=xc1]{$ x^\\textsc{c}_2 $};\n\\node(xcn)[state, right of=xc2, fill=transparent, draw=transparent]{\\dots};\n\\node(xcT)[state, right of=xcn]{$ x^\\textsc{c}_T $};\n\\draw [->] (xb1) -- (xc1);\n\\draw [->] (xb2) -- (xc2);\n\\draw [->] (xbT) -- (xcT);\n\\node(yc1)[obs, below of=xc1, xshift=0.8cm, yshift=0.8cm]{$ y^\\textsc{c}_1 $};\n\\node(yc2)[obs, right of=yc1]{$ y^\\textsc{c}_2 $};\n\\node(ycn)[obs, right of=yc2, fill=transparent, draw=transparent]{\\dots};\n\\node(ycT)[obs, right of=ycn]{$ y^\\textsc{c}_T $};\n\\draw [->] (xc1) -- (yc1);\n\\draw [->] (xc2) -- (yc2);\n\\draw [->] (xcT) -- (ycT);\n\\end{tikzpicture}\n\\caption{SSM for multiple data streams: data $ y^\\textsc{b}_{1:T} $ and $ y^\\textsc{c}_{1:T} $ are related to processes $ x^\\textsc{b}_{1:T} $ and $ x^\\textsc{c}_{1:T} $, which themselves are influenced by the Markov process $ x^\\textsc{a}_{0:T} $.}\n\\label{fig2a}\n\\end{subfigure}\n\\caption{Some examples of graphical models for \\acp{SSM}. Grey nodes correspond to observed variables and white nodes are latent variables. Solid arrows express stochastic dependence among \\acp{r.v.}, while dashed arrows express deterministic links.}\n\\label{f2}\n\\end{figure}\n\n\\subsection{Inference in state-space models}\\label{sec2.3}\nObservations $\\left\\lbrace {Y_t} \\right\\rbrace _{t\\geq1} $ can be used to infer the state process $\\left\\lbrace {X_t} \\right\\rbrace _{t\\geq0} $, usually conditional on specific values of the static parameter $ \\boldsymbol{\\theta} $, or to estimate the static parameter $\\boldsymbol{\\theta}  $, usually marginally on the distribution of the state process. \n\\subsubsection{State inference}\nState inference could have multiple perspectives, for a full review of these see \\cite{lindsten2013backward}; in this paper the inference relies on the estimation of the filtering distribution,  $ p(x_{t}|y_{1:t}, \\boldsymbol{\\theta} ) $, for $ t=1, \\dots, T $, which is the state distribution at time $ t $, conditional on the data and parameters.\n\nThe filtering distribution, except for few cases (e.g. linear Gaussian \\acp{SSM} \\citep{kalman1960new}),  can be approximated recursively by a two-step procedure that alternates approximation of the state distribution given the observations (measurement update) and given the previous states (prediction update).\n\nThe sequential simulation of samples from the distributions $ X_t|y_{1:t}, \\boldsymbol{\\theta} $ and $ X_t|y_{1:t-1}, \\boldsymbol{\\theta} $ allows the approximation of the likelihood of the data $ y_{1:T} $, given a parameter value $ \\boldsymbol{\\theta} $ and marginally w.r.t. the state distribution $ X_{1:T} $ in Equation \\ref{eq5}: \n\\begin{equation}\\label{eq5}\np(y_{1:T}|\\boldsymbol{\\theta}) = \\prod_{t=1}^{T} p(y_t|y_{1:t-1}, \\boldsymbol{\\theta}) = \\prod_{t=1}^{T} \\int_{X_{0:t}}  p(y_t, x_{0:t}|y_{1:t-1}, \\boldsymbol{\\theta}) \\text{ d}x_{0:t}\n\\end{equation}\nThe most common among these simulation-based procedures is the \\ac{BPF} \\citep{arulampalam2002tutorial}, which uses the state and observation equations to iterative propose and weight samples to approximate the filtering distribution. \nThe \\ac{BPF} provides an unbiased estimator $ \\widehat{p}(y_{1:T}|\\boldsymbol{\\theta})  $ of the likelihood of the data given a parameter value $ \\boldsymbol{\\theta} $ which can be used to perform inference. \n\nA full derivation of Equation \\ref{eq5} and of the \\ac{BPF} is not necessary for the understanding of this paper, we leave curious readers to consult Chapter 4 of \\cite{corbella2019statistical} and \\cite{SMCschool}.\n\n\\subsubsection{Parameter inference}\nThere is not a unique way in which this approximated likelihood can be used to drive inference on the parameter $\\boldsymbol{\\theta}$. \nWithin the Bayesian framework, assumed here to enable a complex synthesis of available evidence, many iterative algorithms have been developed to sample from the posterior distribution of interest, $ \\boldsymbol{\\Theta}|y_{1:t} $, when only an approximation of the likelihood is available. \\cite{andrieu2010particle} review and summarise the algorithms used more frequently for parameter inference in \\acp{SSM}. Among the algorithms listed, {pseudo-marginal approaches}, previously introduced in \\citet{andrieu2009pseudo}, provide a simple way to integrate simulation-based approximation of the likelihood (such as the \\ac{BPF}) into \\ac{MCMC} algorithms for Bayesian inference.\n\nPseudo-marginal algorithms are aimed at exploring only the posterior distribution of the parameter, marginally from the distribution of the states, and they are based on the classical \\ac{MH} algorithm \\citep{metropolis1953equation, hastings1970monte}. Differently from the original \\ac{MH} algorithm, here the {unnormalised} posterior distribution is approximated by the product of the prior and a simulation-based approximation of the likelihood in the acceptance ratio (e.g. the \\ac{BPF}). Two pseudo-marginal algorithms are employed throughout this paper: \\ac{GIMH} \\citep{beaumont2003estimation} and \\ac{MCWM} \\citep{andrieu2009pseudo}. \n\n\\subsection{State-space models for epidemics}\nThe \\ac{SSM} methodology marries well with epidemic models: available data are only a partially-observed signal of latent variables which encapsulate all the processes involved in the data generation (e.g. transmission, severity, background noise). \nMore specifically, transmission models usually consists of stochastic or deterministic systems of equations that describe the flow of individuals between disjoint compartments according to their disease status: susceptible, infectious, recovered and immune, etc.\n\nEven the first epidemic models formulated \\citep{kermack1927contribution} could be seen from a \\ac{SSM} perspective: the state system describes the deterministic transmission dynamics via differential equations and the observational process consists of the detection likelihood. This approach has endured over time: the deterministic transmission dynamics enable the modelling of several complex aspects of an epidemic (e.g. heterogeneous populations, non-exponential transition time, see \\cite{keeling2011modeling}) and, when large populations are considered, they approximate well the correspondent stochastic system \\citep{diekmann2012mathematical}. However this kind of model relies on the unrealistic assumption that the only source of noise observed in the data is caused by the case ascertainment\/detection randomness. \n\nAnother stream of literature models explicitly the transitions among disease status as random variables, assuming a stochastic state process. Before the advent of simulation-based inference for \\acp{SSM}, the inference for these models heavily relied on simplifying model assumptions to derive closed forms of the likelihood \\citep{britton2010stochastic, andersson2012stochastic}.\n\nFinally in recent decades, \\ac{SSM} methodology has been used for epidemic dynamics more explicitly. Some notable works include \\cite{breto2009time, dukic2012tracking, dureau2013ssm, mckinley2014simulation, shubin2016revealing}. Even if, as illustrated in Section \\ref{sec2.3}, \\ac{SSM} inference is possible (both with pseudo-marginal methods and other algorithms), many simplifying assumptions are needed to allow computation under time constraints. These simplifications often include assuming independence between data stream, dropping transmission compartments, assuming full ascertainment of the cases, and discretizing time in large intervals. Moreover, some but not all parts of the state process of these models involve large numbers, and could be be reliably approximated by its deterministic counterpart.\n\nWithin the application of \\acp{SSM} to epidemics, there seem to be a lack of hybrid models that exploits the deterministic approximations of transmission dynamics that characterise large epidemics and pandemics while proper accounting for stochasticity in the latent states that cannot be approximated, e.g. the states that involve severe cases. In the remainder of the paper we will propose and discuss such a semi-stochastic model, with a particular focus on its use when multiple \\textit{dependent} data are available. \n\n\n\n\\section{Multiple data on epidemics}\\label{sec3}\nWhen a virus spreads in a population, multiple signals of its presence could be available in the form of time-series counts. Each of these counts is likely to be related to a specific level of severity (e.g. mild symptomatic cases, patients that require hospitalization) and could be affected by specific sources of noise and, possibly bias. Nevertheless, all data are linked to the underlying system that describes the transmission process in the population. Therefore, the joint analysis of multiple data allows better knowledge than separate analyses of single datasets, but introduces several challenges.\n\\subsection{Dependency between data streams at time $t$}\nLet $ X^\\textsc{0}_t $ be the process describing the number of new infections at time $ t $ of a specific epidemic in a given population. Infected individuals could experience a series of increasingly severe events in a pyramid perspective \\citep{presanis2009severity}, e.g. they might experience mild symptoms, be hospitalised, require intensive care, die; with cases in each severity layer being a subset of the cases in the previous layer as represented in Figure \\ref{fig3a}. The process describing the number of individuals experiencing an event $ \\textsc{j} $ among these severity layers is denoted by $ X^\\textsc{j}_t $, experiencing event $ \\textsc{k} $ by $ X^\\textsc{k}_t  $, etc. for $ t=1, \\dots,T $, with $\\textsc{j}=\\textsc{0} $ representing infection. \n\n\\begin{figure}[h]\n\\begin{subfigure}[c]{6.42cm}\n\\centering\n\\begin{tikzpicture}\n\\coordinate (A) at (-2.5,0) {};\n\\coordinate (B) at ( 2.5,0) {};\n\\coordinate (C) at (0,4) {};\n\\draw[name path=AC, blue] (A) -- (C);\n\\draw[name path=BC, blue] (B) -- (C);\n\\foreach \\y\/\\A in {\n0\/$ 0 $,\n1\/ $ \\textsc{j} $,\n2\/$ \\textsc{k} $, \n3\/$ \\textsc{l} $      } {\n\\path[name path=horiz, blue] (A|-0,\\y) -- (B|-0,\\y);\n\\draw[name intersections={of=AC and horiz,by=P},\nname intersections={of=BC and horiz,by=Q}, blue] (P) -- (Q)\nnode[midway,above,align=center,text width=\n\\dimexpr(3.5em-\\y em)*5\\relax, black] {\\A};\n}\n\\end{tikzpicture}\n\\caption{Pyramid representation of the severity states.}\n\\label{fig3a}\n\\end{subfigure}\n\\hspace*{1cm}\t\\begin{subfigure}[c]{6.42cm}\n\\centering\n\\begin{tikzpicture}[node distance=1.3cm]\n\\node (PL)[plate, xshift=-0.1cm , yshift=2.9cm]{};\n\\node (PL)[plate, xshift=-0.05cm , yshift=2.85cm]{};\n\\node (PL)[plate, yshift=2.8cm]{ };\n\\node (mu)[state, yshift=5cm, xshift=0cm] {$ {}^0x_{t}$};\n\\node (xh)[state, below of=mu, xshift=-1cm] {$ x^\\textsc{j}_{t}$};\n\\node (xic)[state, below of=xh] {$ x^\\textsc{k}_{t}$};\n\\node (xl)[state, below of=xic] {$ x^\\textsc{l}_{t}$};\n\\node (yic)[obs, below of=xic, xshift=2cm, yshift=1cm] {$ y^\\textsc{k}_{t}$};\n\\node (yh)[obs, below of=xh, xshift=2cm, yshift=1cm] {$ y^\\textsc{j}_{t}$};\n\\node (yl)[obs, below of=xl, xshift=2cm, yshift=1cm] {$ y^\\textsc{l}_{t}$};\n\\node (phiIC)[state, left of=xic, xshift=-1cm, yshift=1cm] {$ {}^\\textsc{j}\\theta^\\textsc{k}$};\n\\node (phiH)[state, left of=xh, xshift=-1cm, yshift=1cm] {$ {}^0\\theta^\\textsc{j}$};\n\\node (phil)[state, left of=xl, xshift=-1cm, yshift=1cm] {$ {}^\\textsc{k}\\theta^\\textsc{l}$};\n\\node (xiH)[state, right of=yh,  xshift=1cm, yshift=1cm] {$ \\zeta^\\textsc{j}$};\n\\node (xiIC)[state, right of=yic,  xshift=1cm, yshift=1cm] {$ \\zeta^\\textsc{k}$};\n\\node (xil)[state, right of=yl,  xshift=1cm, yshift=1cm] {$ \\zeta^\\textsc{l}$};\n\\draw [->, gray] (phiIC) -- node[anchor=south] { } (xic);\n\\draw [->, gray] (phiH) -- node[anchor=south] { } (xh);\n\\draw [->, gray] (phil) -- node[anchor=south] { } (xl);\n\\draw [->, gray] (xiH) -- node[anchor=south] { } (yh);\n\\draw [->, gray] (xiIC) -- node[anchor=south] { } (yic);\n\\draw [->, gray] (xil) -- node[anchor=south] { } (yl);\n\\draw [->, gray] (xh) -- node[anchor=south] { } (yh);\n\\draw [->, gray] (xic) -- node[anchor=south] { } (yic);\n\\draw [->, gray] (xl) -- node[anchor=south] { } (yl);\n\\draw [->, gray] (mu) -- node[anchor=south] { } (xh);\n\\draw [->, gray] (xh) -- node[anchor=south] { } (xic);\n\\draw [->, gray] (xic) -- node[anchor=south] { } (xl);\n\\node (nc)[namecompartment, below of=PL, yshift=-1.3cm, xshift=1.5cm] {\\tiny $t=2, \\dots, T$};\n\\end{tikzpicture}\n\\caption{\\ac{DAG} representation of a stochastic-severity model.\n}\n\\label{fig3b}\n\\end{subfigure}\n\\caption{Pyramid representation and \\ac{DAG} of a severity process.}\n\\label{f3}\n\\end{figure}\n\nA simple model would assume that the number of people in a severity category would be a subsample (e.g. a Binomial sample) of the number of people in the less-severe category with severity parameters $ {}^\\textsc{j}\\theta^\\textsc{k} $ encapsulating the risk of moving from severity state $ \\textsc{j} $ to $ \\textsc{k} $.\nThe observational processes would describe the number of detected cases in each specific severity layer $ Y^\\textsc{j}_{1:t} $, as, again a subsample of the cases in a specific layer $ \\textsc{j} $ with a layer-specific detection probability $ \\zeta^\\textsc{j} $ . Figure \\ref{fig3b} illustrates a model of this type comprising 3 events of increasing severity and the connection among multiple data can be easily spotted: i.e. not only they are linked through the underlying infection process, but also via the progression through severity states. In fact, while distributional properties could be used to write the likelihood of each of multiple datasets given the infection process $ X^\\textsc{0}_{1:t} $ and some parameters, these data are not independent, conditionally on the parameters: they share common severity processes $ X^\\textsc{j},X^\\textsc{k}, X^\\textsc{l} $, and therefore cannot be simply multiplied to obtain a joint likelihood of the data. \n\n\n\\subsubsection{Dependency between data streams across time}\nAnother challenge is given by delays between events: severe events usually do not happen simultaneously. More likely, there will be some time elapsing between events of increasing severity; this could lead to, for example, cases detected in severity layer $ \\textsc{j} $ at time $ t $, be present in data on layer $ \\textsc{k} $ at some time $ t+s , s\\geq 0$ with some chance. This introduces dependence not only between observations at different levels of severity at the same time, but also across time. \n\nTo include delays, a new notation is introduced: denote by $ {}^\\textsc{j}_tX^\\textsc{k} $ the number of people that will eventually experience severe event $ \\textsc{k} $, having already experienced event $ \\textsc{j} $ at $ t $; and by $ {}^\\textsc{j}_tX^\\textsc{k}_s $ the number of people that experience event $ \\textsc{k} $ at time $ s $, having already experienced event $ \\textsc{j} $ at $ t $. Thus, the subscript and superscript on the right side denote the time and type of final events and the ones on the left side denote the time and type of a previous event. When only the right superscript\/subscript is reported, e.g. $ X^\\textsc{k}_t $, as in the previous section, this denotes the number of people experiencing event $ \\textsc{k} $ at $ t $ irrespectively from the time of previous event, which can be obtained summing over the number of people experiencing event $ j $ at the previous event times: $ X^\\textsc{k}_s = \\sum_{t\\leq s}  {}^\\textsc{j}_tX^\\textsc{k}_s  $. \n\nA stochastic model for the time series $ {}^\\textsc{j}_{1:T}X^\\textsc{k} $ of the number of people that have had event $ \\textsc{j} $ at time $ t $ ($ t=1,\\dots, T $) and will experience event $ \\textsc{k} $ can be specified by, for example, a Binomial sample:\n\\begin{equation}\\label{eq9}\n\\left({}^\\textsc{j}_{t}X^\\textsc{k}|x^\\textsc{j}_t\\right) \\sim \\text{Bin} (x^\\textsc{j}_t, {}^\\textsc{j}\\theta^\\textsc{k}) \n\\end{equation}\nfor $ t=1, \\dots, T $. Denote by $ {}^\\textsc{j}f^\\textsc{k}_d({}^\\textsc{j}\\vartheta^\\textsc{k}) $ the probability that the delay experienced between event $ \\textsc{j} $ and $ \\textsc{k} $ is in the $ d $th interval of length $ \\delta $, $ [\\delta d; \\delta d+\\delta) $ for $ d=0,1,\\dots, D $, with $ D $ being the largest interval index for which the delay is relevant (i.e. $ {}^\\textsc{j}f^\\textsc{k}_d({}^\\textsc{j}\\vartheta^\\textsc{k})\\approx0 $ for $ d>D $). $ {}^\\textsc{j}f^\\textsc{k}_d({}^\\textsc{j}\\vartheta^\\textsc{k}) $ is often derived from the discretization of a parametric distribution with appropriate parameter vector $ {}^\\textsc{j}\\boldsymbol{\\vartheta}^\\textsc{k} $. To make the notation lighter, ${}^\\textsc{j}\\vartheta^\\textsc{k} $ is dropped, with $ {}^\\textsc{j}f^\\textsc{k}_d $ representing both the function and the parameters used to describe the delay from $ \\textsc{j} $ to $ \\textsc{k} $.\n\nUnder this discrete definition of the distribution of the time to event, and conditionally on $ {}^\\textsc{j}_tX^\\textsc{k} $, the introduction of stochastic delays could be allowed by defining \n$ {}^\\textsc{j}_tX^\\textsc{k}_s $, as a component of the Multinomial \\ac{r.v.}: \n\\begin{equation}\\label{eq10}\n{}^\\textsc{j}_tX^\\textsc{k}_{t:t+D}|{}^\\textsc{j}_tx^\\textsc{k} \\sim \\text {Multi} \\left( {}^\\textsc{j}_tx^\\textsc{k}, {}^\\textsc{j}f^\\textsc{k}_{0:D}\\right).\n\\end{equation}\nwith $ {}^\\textsc{j}f^\\textsc{k}_{0:D} = ({}^\\textsc{j}f^\\textsc{k}_{0}, {}^\\textsc{j}f^\\textsc{k}_{1}, \\dots, {}^\\textsc{j}f^\\textsc{k}_{D})$ the vector containing the probabilities for the waiting time between $ \\textsc{0} $ and $ \\textsc{j} $ as described above. The number of people that experience event $ \\textsc{k} $ at each time $ t=1, \\dots T $ can then be obtained by summing these stochastic terms, i.e.:\n\\begin{equation}\\label{eq11}\nX^\\textsc{k}_{t}= \\sum_{d=0}^{D} {}_{t-d}{^\\textsc{j}X^\\textsc{k}_{t}},\n\\end{equation}\nwhich can be recognised as a typical stochastic convolution to describe delays in epidemic models \\citep{brookmeyer1994aids}. The counts of events at a higher level of severity can be modelled likewise.\n\nThis form of delay structure, as many other stochastic delay formulations, introduces a dependence over time. This is evident in Equation \\ref{eq10}, where the number of people experiencing $ \\textsc{k} $ at time $ t $ depends on \\acp{r.v.} defined on the previous $ D $ intervals. \n\n\\subsection{A model for multiple data on severe influenza} \nInfluenza is monitored by many surveillance schemes in the UK, one of these is the Severe Acute Respiratory Infection (SARI)-Watch scheme \\citep{SARIWatch}, which has evolved from the \\ac{USISS}, the main source of data on severe influenza cases in the UK prior to the COVID-19 pandemic. According to the \\ac{USISS} protocol \\citep{health2011sourcesa}, all \\ac{NHS} trusts in England report, among other things, the weekly number of laboratory-confirmed influenza cases admitted to \\ac{IC} units. In addition to this mandatory scheme, a sentinel subgroup of NHS trusts in England is recruited every year to participate in the \\ac{USISS} sentinel scheme \\citep{health2011sourcesb, boddington2017developing}, which reports weekly numbers of laboratory-confirmed influenza cases hospitalised at all levels of care. Some individuals might be detected in both datasets, leading to a dependence.\n\n\\subsubsection{Parametrization and data generation}Denote by $ \\boldsymbol{\\theta} $ the set of parameters, composed of $ \\boldsymbol{\\theta}^T$, the parameters of a $SEIR$ transmission model, tracking the number of susceptible ($S$), exposed ($E$), infected ($I$) and removed ($R$), individuals, and $ \\boldsymbol{\\theta}^S $, the parameters of the severity and detection model. \n\n$ \\boldsymbol{\\theta}^T=\\left\\lbrace \\pi, \\iota, \\sigma, \\gamma, \\beta \\right\\rbrace   $  consists of the transmission rate $ \\beta $; the exit rates from compartments $ E $ and $ I $, $ \\sigma $ and $ \\gamma $ respectively; and the initial proportions immune, $\\pi$, and of infected\/infectious, $ \\iota $. The parameters $\\pi$ and $ \\iota $, together with $ \\sigma $, $ \\gamma $ and the known constant $ N $, the total size of the population, contribute to the formulation of the initial state of the epidemic. Seasonal influenza is a large epidemic that takes place every winter with high likelihood, hence a deterministic transmission model is assumed: the information contained in $ \\boldsymbol{\\theta}^T $, together with the known constants, provides the full time series of the number of new infections. Let the time be discretised in intervals of length $ \\delta $ so that the $ t $-th interval covers the time $ [t\\delta , t\\delta+\\delta) $ and the intervals are indexed by $ t=0,1,2, \\dots, T $, where $ t=0 $ coincides with the beginning of the data collection period and $ t=T $ is the end of the data collection period. Denote the number of susceptible individuals at the beginning of interval $ t $ by $ S_t $ and likewise for the other compartments $E,I,R $. Denote by $ \\xi^0_{1:T} $ the vector of the number of new infections in interval $ t=0,1,2,\\dots T $. \n\n$ \\boldsymbol{\\theta}^S=\\left\\lbrace {}^0\\theta^\\textsc{h}, {}^\\textsc{h}\\theta^\\textsc{ic}, {}^0f^\\textsc{h}, {}^\\textsc{h}f^\\textsc{ic}, \\zeta_t^\\textsc{h}, \\zeta_t^\\textsc{ic} \\right\\rbrace  $ includes severity and detection parameters: $ {}^0\\theta^\\textsc{h}$ is the probability of hospitalization given infection; $ {}^\\textsc{h}\\theta^\\textsc{ic} $ is the probability of \\ac{IC} admission given hospitalization; $ {}^0f^\\textsc{h}$ and $ {}^\\textsc{h}f^\\textsc{ic}   $ denote probabilities for the times from infection to hospitalisation and from hospital admission to \\ac{IC} admission, respectively; $ \\zeta_t^\\textsc{h} $ and $ \\zeta_t^\\textsc{ic} $ are the probability of detecting an hospitalised and \\ac{IC} case, respectively.\n\nThe transmission and severity compartments are reported in Figure \\ref{f4}. \n\n\\begin{figure}[h]\n\\begin{tikzpicture}[node distance=1.8cm]\n\\node (s)[compartment,  draw opacity=0.4]{\\color{gray}$S$};\n\\node (e)[compartment, draw opacity=0.4,  right of=s, xshift=0.1cm] {\\color{gray}$E$};\n\\node (i)[compartment,  draw opacity=0.4, right of=e, xshift=.1cm] {\\color{gray}$I$};\n\\node (r)[compartment,  draw opacity=0.4, right of=i, xshift=.1cm] {\\color{gray}$R$};\n\\draw [black, ->, thick, dashed] (s) -- node[anchor=south] {$\\xi^0$} (e);\n\\draw [gray, ->, thick, dashed] (e) -- node[anchor=south] {\\color{gray}$\\sigma$} (i);\n\\draw [gray, ->, thick, dashed] (i) -- node[anchor=south] {\\color{gray}$\\gamma$} (r);\n\\node (sympt)[compartment, below of=e, , xshift=1cm, yshift=.5cm] {Hospital};\n\\node (Hsent)[compartment, below of=sympt, xshift=-1.5cm] {H sentinel};\n\\node (yS)[compartment,  draw=black, fill=blue, left of=Hsent, xshift=-1cm] {\\color{white} H obs};\n\\node (Hnonsent)[compartment, right of=Hsent, xshift=1cm] {H non sentinel};\n\\node (ICUsent)[compartment, below of=Hsent] {IC sentinel};\n\\node (ICUnonsent)[compartment, below of=Hnonsent] {IC non sentinel};\n\\node (ICU)[compartment, below of=ICUnonsent, xshift=-1.5cm] {IC total};\n\\node (yM)[compartment,  draw=black, fill=blue, left of=ICU, xshift=-2.1cm] {\\color{white} IC obs };\n\\draw[thick, ->] (e.west) .. node[anchor=east] {$ {}^0\\theta^\\textsc{h} $} controls +(down:5mm) and +(left:20mm) .. (sympt.west);\n\\draw[thick, ->] (sympt.south) ..node[anchor=east] {$   $} controls +(down:0mm) and +(right:0mm) .. (Hsent.north);\n\\draw[thick, ->] (sympt.south) .. node[anchor=west] {$   $}controls +(up:0mm) and +(right:0mm) .. (Hnonsent.north);\n\\draw[thick, ->] (Hsent.south) .. node[anchor=east] {$   {}^\\textsc{h}\\theta^\\textsc{ic}  $}controls +(down:0mm) and +(right:0mm) .. (ICUsent.north);\n\\draw[thick, ->] (Hnonsent.south) .. node[anchor=west] {$  {}^\\textsc{h}\\theta^\\textsc{ic} $}controls +(down:0mm) and +(right:0mm) .. (ICUnonsent.north);\n\\draw[->, thick, dashed] (ICUnonsent.south) ..node[anchor=east] { } controls +(down:0mm) and +(right:0mm) .. (ICU.north);\n\\draw[->, thick, dashed] (ICUsent.south) .. controls +(down:0mm) and +(right:0mm) .. (ICU.north);\n\\draw[thick, ->] (Hsent.west) ..  node[anchor=south] {$ \\zeta_t^\\textsc{h} $} controls +(down:0mm) and +(right:0mm) .. (yS.east);\n\\draw[thick, ->] (ICU.west) ..node[anchor=south] {$\\zeta_t^\\textsc{ic}  $} controls +(down:0mm) and +(right:0mm) .. (yM.east);\n\\end{tikzpicture}\n\\caption{Flowchart of the possible severe events recorded in \\ac{USISS}. }\n\\label{f4}\n\\end{figure}\n\\subsubsection{\\ac{SSM} formulation}\nDescribing the model using \\ac{SSM} notation, the state process is composed of the distributions of: $  _t^0X^\\textsc{h}$ , the number of hospitalizations that were infected at each interval $ t $; $ X^\\textsc{h}_{t} $, the number of hospitalizations at $ t $, obtained via a convolution of $ _t^0X^\\textsc{h}$; the number of hospitalizations that eventually will be admitted to IC and have been hospitalised at $ t $, $ _t^\\textsc{h}X^\\textsc{ic} $; the number of IC admissions at $ t $, $ X^\\textsc{ic}_t$ obtained by the convolution of  $ _t^\\textsc{h}X^\\textsc{ic} $, for $ t=0,1,2,\\dots, T $. The state process is described as:\n\\begin{equation}\\label{eq12}\n\\begin{split}\n\\left(_t^0X^\\textsc{h}\\right) &\\sim \\text{Pois}({}^0\\xi\\cdot  {}^0\\theta^\\textsc{h} )\\\\\n\\left(_t^0X^\\textsc{h}_{t:t+D}|_t^0X^\\textsc{h}={_t^0x^\\textsc{h}} \\right)&\\sim \\text{Multi} (_t^0x^\\textsc{h}, {}^0f^\\textsc{h}_{0:D} ) , \\qquad\nX^\\textsc{h}_{t} = \\sum_{s=0}^{S} {_{t-s}{^0X}^\\textsc{h}_{t}}\\\\\n\\left(_t^\\textsc{h}X^\\textsc{ic}|X^\\textsc{h}_t={x^\\textsc{h}_t}\\right) &\\sim \\text{Bin}(x^\\textsc{h}_t, {}^\\textsc{h}\\theta^\\textsc{ic}  )\\\\\n\\left(_t^\\textsc{h}X^\\textsc{ic}_{t:t+D}|_t^\\textsc{h}X^\\textsc{ic}={_t^\\textsc{h}x^\\textsc{ic}}\\right) &\\sim \\text{Multi} (_t^\\textsc{h}x^\\textsc{ic},  {}^\\textsc{h}f_{0:D}^\\textsc{ic} ), \\qquad\nX^\\textsc{ic}_{t} = \\sum_{s=0}^{S} {_{t-s}{^\\textsc{h}X}^\\textsc{ic}_{t}}\\\\\n\\end{split}\n\\end{equation}\nfor $ t=0,1,\\dots, T $. Here $ {}^0f^\\textsc{h}_{0:D} $ and $ {}^\\textsc{h}f^\\textsc{ic}_{0:D} $ are vectors containing elements $ {}^0f^\\textsc{h}_{d} $ and $ {}^\\textsc{h}f^\\textsc{ic}_{d} $ denoting the probability of experiencing a delay of $ d $ weeks between infection and hospitalization and hospitalization and \\ac{IC} admission, respectively, for $ d=0, \\dots, D $. These are considered known and fixed. \n\nThe observational process, consists of the distributions of two datasets, $ y_{1:T}^\\textsc{h} $, the count of hospitalizations, and $ y_{1:T}^\\textsc{ic}  $, the count of IC admissions, conditional on the hidden states $ X_{1:T}^\\textsc{h} $ and $ X_{1:T}^\\textsc{ic}  $. These are  assumed to be Binomial with detection probabilities $ \\zeta^\\textsc{h}_t $ and $ \\zeta^\\textsc{ic}_t $ respectively:\n\\begin{equation}\\begin{split}\\label{eq13}\n\\left(Y_{t}^\\textsc{h}| X^\\textsc{h}_{t}= x^\\textsc{h}_{t} \\right) &\\sim \\text{Bin} (x^\\textsc{h}_{t}, \\zeta^\\textsc{h}_t)\\\\\n\\left(Y_{t}^\\textsc{ic}|X^\\textsc{ic}_{t}= x^\\textsc{ic}_{t}\\right)&\\sim \\text{Bin} (x^\\textsc{ic}_{t}, \\zeta^\\textsc{ic}_t)\n\\end{split}\n\\end{equation}\nfor $ t=0,1, 2, \\dots, T$. \n\n\\subsection{Inference} Two inferential methods are proposed here: the first does not account for the dependence among data while the second properly encapsulates the layers of stochasticity present in the model. \n\n\\subsubsection{The approximation under an independence assumption}\nThanks to the Poisson properties \\citep{kingman1992poisson}, several hidden states and the data distribute marginally according to a Poisson distribution:\n\\begin{equation}\\begin{split}\nX^\\textsc{h}_{t}&\\sim \\text{Pois} \\left( {}^0\\theta^\\textsc{h} \\cdot\\sum_{d=0}^{D}  \\xi^0_{t-d} \\cdot {}^0f^\\textsc{h}_{d} \\right) \\\\\nX^\\textsc{ic}_{t} &\\sim \\text{Pois}\\left({}^\\textsc{h}\\theta^\\textsc{ic}\\cdot {}^0\\theta^\\textsc{h} \\cdot\\sum_{d=0}^{D} \\sum_{g=0}^{d}  \\xi^0_{t-d} \\cdot {}^0f^\\textsc{h}_{d}\\cdot {}^\\textsc{h}f^\\textsc{ic}_{g}\\right) \\\\\nY_{t}^\\textsc{h} &\\sim \\text{Pois} \\left( \\zeta^\\textsc{h}_t\\cdot {}^0\\theta^\\textsc{h} \\cdot\\sum_{d=0}^{D}  \\xi^0_{t-d} \\cdot {}^0f^\\textsc{h}_{d} \\right) \\\\\nY_{t}^\\textsc{ic} &\\sim \\text{Pois}\\left(\\zeta^\\textsc{ic}_t\\cdot {}^\\textsc{h}\\theta^\\textsc{ic}\\cdot {}^0\\theta^\\textsc{h} \\cdot\\sum_{d=0}^{D} \\sum_{g=0}^{d}  \\xi^0_{t-d-g} \\cdot {}^0f^\\textsc{h}_{d}\\cdot {}^\\textsc{h}f^\\textsc{ic}_{g}\\right)\n\\end{split}\n\\label{eq14}\n\\end{equation}\nfor $ t=0,1, 2, \\dots T $.\n\nIgnoring the dependence between the two data streams, the joint likelihood is\n\\begin{equation}\\label{eq15}\np(y_{1:T}^\\textsc{h} | \\xi^0_{1:T} ,  {}^0\\theta^\\textsc{h} , \\zeta^\\textsc{h}_{1:T} )\\times p(y_{1:T}^\\textsc{ic} | \\xi^0_{1:T} , {}^\\textsc{h}\\theta^\\textsc{ic},  {}^0\\theta^\\textsc{h} , \\zeta^\\textsc{ic}_{1:T} ),\n\\end{equation}\nwhere each factor is the Poisson density of Equation \\ref{eq14}. \n\nThe independence assumption would substantially simplify inference since, with a likelihood available in closed form, there would be no need of a simulation-based estimation of the likelihood and, indeed, of pseudo-marginal methods. The aim of Section \\ref{sec4} is to assess if this approximation induces any error in terms of bias or uncertainty quantification. \n\\subsubsection{Exact inference via pseudo-marginal methods} To properly account for dependence and stochasticity, a simulation algorithm is proposed to approximate the joint likelihood of the hospitalization and IC data. \nThe joint probability distribution can be decomposed in two ways:\n\\begin{equation*}\n\\begin{split}\np(y_{1:T}^\\textsc{h}, y_{1:T}^\\textsc{ic}|  \\boldsymbol{\\theta})&=p(y_{1:T}^\\textsc{h}|  y_{1:T}^\\textsc{ic}, \\boldsymbol{\\theta})p( y_{1:T}^\\textsc{ic}|  \\boldsymbol{\\theta})\\\\\n&=p(y_{1:T}^\\textsc{ic}|  y_{1:T}^\\textsc{h}, \\boldsymbol{\\theta})p( y_{1:T}^\\textsc{h}|  \\boldsymbol{\\theta})\n\\end{split}\n\\end{equation*}\nwhere, in both cases, the second of the two factors is available in closed form (Equation \\ref{eq14}).\n\nTo approach the estimation of the other factor, state-inference methods for \\acp{SSM} can be used. The methods for inference described in Section \\ref{sec2} address the Markovian dependence across time. Here however, the time-dependence of the transmission process disappears thanks to the deterministic approximation. However, the dependence over the severity domain remains: the distributional assumptions of Equation \\ref{eq12} can be used to construct a simulation-based estimator of the likelihood that sequentially approximates severity states.\n\nAlgorithm \\ref{alg2} exploits the first decomposition, where $ p(y_{1:T}^\\textsc{ic}|  \\boldsymbol{\\theta}) $ is available in closed form and a solution is needed for $ p(y_{1:T}^\\textsc{h}|  y_{1:T}^\\textsc{ic}, \\boldsymbol{\\theta})$, which is obtained by approximating the $ T $-dimensional integral:\n\\begin{equation*}\\begin{split}\np(y_{1:T}^\\textsc{h}|  y_{1:T}^\\textsc{ic}, \\boldsymbol{\\theta})= &\\int_{X^\\textsc{h}_1}\\dots\\int_{X^\\textsc{h}_T} p(y_{1:T}^\\textsc{h}, X^\\textsc{h}_{1:T}| y_{1:T}^\\textsc{ic}, \\boldsymbol{\\theta})\\text{d}X^\\textsc{h}_1\\dots \\text{d}X^\\textsc{h}_T\\\\\n=&\\int_{X^\\textsc{h}_1}\\dots\\int_{X^\\textsc{h}_T} p(y_{1:T}^\\textsc{h}| X^\\textsc{h}_{1:T}, y_{1:T}^\\textsc{ic}, \\boldsymbol{\\theta})p( X^\\textsc{h}_{1:T}| y_{1:T}^\\textsc{ic}, \\boldsymbol{\\theta})\\text{d}X^\\textsc{h}_1\\dots \\text{d}X^\\textsc{h}_T\\\\\n=&\\int_{X^\\textsc{h}_1}\\dots\\int_{X^\\textsc{h}_T} p(y_{1:T}^\\textsc{h}| X^\\textsc{h}_{1:T}, \\boldsymbol{\\theta})p( X^\\textsc{h}_{1:T}| y_{1:T}^\\textsc{ic}, \\boldsymbol{\\theta})\\text{d}X^\\textsc{h}_1\\dots \\text{d}X^\\textsc{h}_T\n\\end{split}\n\\end{equation*}\nThe simulation of the hidden states is made simple  by the distribution chosen for the severity and detection process. \n\\begin{algorithm}[h]\n\\SetAlgoLined\n\\KwResult{$ \\widehat{p}(y_{1:T}^\\textsc{h},  y_{1:T}^\\textsc{ic}| \\boldsymbol{\\theta}) $}\n\\KwIn{fixed parameter $\\boldsymbol{\\theta} $, number of particels $ {N} $, data $ y_{1:T}^\\textsc{h} $,  $ y_{1:T}^\\textsc{ic} $}\ncompute\\\\\n$ p(y_{1:T}^\\textsc{ic}| \\boldsymbol{\\theta}) $ =  $ f(y_{1:T}^\\textsc{ic}|\\zeta^\\textsc{ic}_t\\cdot{}^\\textsc{h}\\theta^\\textsc{ic}\\cdot {}^0\\theta^\\textsc{h}\\cdot \\sum_{d\\text{=}0}^{D} \\sum_{g\\text{=}0}^{d}  \\xi^0_{t\\text{-}d\\text{-}g} \\cdot {}^0f^\\textsc{h}_{d}\\cdot {}^\\textsc{h}f^\\textsc{ic}_{g} ) $ with $ f(\\cdot) $ being a Poisson density\\\\\n\\For{$ n=1,\\dots, {N} $}{\n\\For{$ t=0, 1, \\dots T $}{\\vspace*{0.2cm}sample : $ {x^\\textsc{ic}_{t}}^{(n)}  \\sim$ Pois $ \\left(\\left(1-\\zeta^\\textsc{ic}_t\\right) [{}^\\textsc{h}\\theta^\\textsc{ic}\\cdot {}^0\\theta^\\textsc{h} \\sum_{d=0}^{D} \\sum_{g=0}^{d}  \\xi^0_{t-d-g} \\cdot {}^0f^\\textsc{h}_{d}\\cdot {}^\\textsc{h}f^\\textsc{ic}_{g} ]  \\right) +y_{T}^\\textsc{ic} $ \\\\\\vspace*{0.2cm}sample : $ {_{t-1}^\\textsc{h}x^\\textsc{ic}_{1}}^{(n)}\\text{, } \\dots\\text{, } {_{t-S}^\\textsc{h}x^\\textsc{ic}_{S}}^{(n)}\\sim \\text{Multi} \\left({x^\\textsc{ic}_{t}}^{(n)},  {}^\\textsc{h}f^\\textsc{ic}_{1:S} \\right) $\\\\\\vspace*{0.2cm}compute : \n${ _t^\\textsc{h}x^\\textsc{ic}}^{(n)}=\\sum_{s=1}^{S} {_{t}^\\textsc{h}x^\\textsc{ic}_{s}}^{(n)} $\\\\\n\\vspace*{0.2cm}sample : \n$ {x^\\textsc{h}_{t}}^{(n)}| {_{t}^\\textsc{h}x^\\textsc{ic}}^{(n)},\\boldsymbol{\\theta}\\sim \\text{Pois}\\left(\\left(1-{}^\\textsc{h}\\theta^\\textsc{ic} \\right)\\left[ {}^0\\theta^\\textsc{h}\\sum_{d=0}^{D}  \\xi^0_{t-d} \\cdot {}^0f^\\textsc{h}_{d} \\right] \\right) +{_{t}^\\textsc{h}x^\\textsc{ic}}^{(n)} $\n}compute : $ p\\left(y_{1:T}^\\textsc{h}|{x^\\textsc{h}_{1:T}}^{(n)}, \\boldsymbol{\\theta}\\right) $ = $g\\left(\ny_{1:T}^\\textsc{h}|{x^\\textsc{h}_{1:T}}^{(n)}, \\zeta^\\textsc{h}_t\\right) $\nwith $ g(\\cdot) $ being a Binomial density\n}\n$ \\widehat{p}\\left(y_{1:T}^\\textsc{h}, y_{1:T}^\\textsc{ic}|  \\boldsymbol{\\theta}\\right)= p\\left(y_{1:T}^\\textsc{ic}| \\boldsymbol{\\theta}\\right)\\cdot \\frac{1}{N}\\sum_{n=1}^{{N}}p\\left(y_{1:T}^\\textsc{h}|{x^\\textsc{h}_{1:T}}^{(n)}, \\boldsymbol{\\theta}\\right)$\n\\caption{Approximation of the likelihood ${p}(y_{1:T}^\\textsc{h}, y_{1:T}^\\textsc{ic}|  \\boldsymbol{\\theta})$}\n\\label{alg2}\n\\end{algorithm}\n\nA second algorithm uses the alternative factorization of the joint likelihood. While still feasible, this algorithm performs more poorly than Algorithm \\ref{alg2}, mainly because the \\ac{IC} admissions are more-completely observed and smaller in numbers, hence particle degradation is more likely to take place \\citep{brooks2011handbook}. \nSee the Supplementary information for the derivation of the two algorithms.\n\n\n\\section{Relevance of the dependence}\\label{sec4}\nA full simulation study is set up to assess whether (and in which situations) accounting for the dependence makes any difference {compared to} assuming the two datasets to be independent. Intuitively, a miss-specified model that does not account for dependencies would assume more independent information than is truly present in the dependent data, hence would achieve overly-confident results. Conversely, truly accounting for dependence should reflect more properly the unknowns and noise of the model and also help the estimation of those parameters that effectively link the dependent data streams. \n\n\\subsection{Simulation study set-up}\nThe simulated data are formulated to reflect a situation similar to the motivating \\ac{USISS} data on a smaller population, chosen  to reduce the computation time. The datasets are generated with some common parameters and some scenario-specific parameters. \n\nThe common parameters are:\n\\begin{equation*}\n\\left\\lbrace N=10000,\\beta=0.63, \\pi=0.3, \\iota=0.0001, \\sigma=\\frac{1}{4}, \\gamma=\\frac{1}{3.5}, {}^0f^\\textsc{h} \\sim \\text{Exp}(0.3), {}^\\textsc{h}f^\\textsc{ic}\\sim \\text{Exp}(0.4) \\right\\rbrace \n\\end{equation*}\nwhile the scenario specific parameters are reported in Table \\ref{t1}. Each of the severity and detection parameters can take either a small or a large value. The smaller leads to situations where the probability of being observed in both datasets is low, therefore data are less dependent; while when parameters take larger values, there is more overlap between datasets and therefore more dependence.\n\\begin{table}[h]\n\\caption{Parameters used to generate the datasets.}\n\\label{t1}\n\\centering\n\\begin{tabular}{c c c}\n&small dependence&large dependence\\\\\n\\hline\n${}^0\\theta^{\\textsc{h}}$&0.1&0.5\\\\\n${}^\\textsc{h}\\theta^{\\textsc{ic}}$&0.1&0.9\\\\\n$ \\zeta_t^\\textsc{h}=\\zeta^\\textsc{h}$&0.1&0.3\\\\\n$\\zeta_t^\\textsc{ic}=\\zeta^\\textsc{ic}$&0.1&0.9\\\\\n\\hline\n\\end{tabular}\n\\end{table}\nThe values have been chosen to be at the extremes of a realistic range: e.g., the probability of hospitalization and the detection of hospitalization in the {large dependence} case are smaller than the respective quantities for \\ac{IC} unit admission because usually more severe cases are better monitored, and hence have a higher detection.\n\nThe aim of the comparison is to assess whether a misspecified independent likelihood (Equation \\ref{eq13}) would affect the inference of the parameters, leading to different posterior distributions from the ones obtained with the \\ac{MC} approximation of the joint likelihood assuming  dependent data (Algorithm \\ref{alg2}). For this reason the results presented here are to be compared {within} each scenario: between the ones obtained with the independent misspecified model (abbreviated with \\textsc{miss ind}) and the ones obtained using the dependent joint model (abbreviated with \\textsc{joint dep}). \n\n\\subsection{Results of the simulation study} As outlined in detail below, the misspecified model leads to overly precise results in the estimation of the transmission parameters when the dependence is large. Furthermore, the parameter connecting the two severity states to which the data refer is estimated with less precision when the misspecified model is assumed.\n\n\\subsubsection{Comparison for transmission parameters}\nFor the parameter inference with the approximated joint likelihood, a \\ac{MCWM} algorithm with likelihood approximation via Algorithm \\ref{alg2} with $ N=2000 $ was chosen. \nResulting estimates were compared with those from the misspecified independent Poisson likelihood over 1000 synthetic datasets. 500 datasets were simulated using the smaller values of the parameters (left column of Table \\ref{t1}) and 500 datasets using the larger values (right column of Table \\ref{t1}). The only parameters inferred are the transmission parameters $ \\beta, \\iota $ and $ \\pi $, with the severity parameters being fixed at their true, scenario-specific, value.\n\nIn the case of {small dependence}, the results show that the posterior distributions obtained with the misspecified independent likelihood are very similar to the ones obtained with the approximated joint likelihood. To measure any discrepancy between the two estimation methods, the pairwise difference (PWD) in variance Var$ (\\hat{\\alpha}^{m} \\mid y^d) $ for $ m \\in \\{\\textsc{joint dep}, \\textsc{miss ind}\\} $ and $ \\alpha $ being one of the parameters, and in the width R${}_{95}(\\hat{\\alpha}^{m} \\mid y^d) $ of the 95\\% Credible Intervals (CrIs) was computed as:\n\\begin{equation*}\\begin{split}\n\\text{PWD}(\\text{Var}(\\alpha))^d&= \\text{Var}(\\widehat{\\alpha}^\\textsc{joint dep}|\\boldsymbol{y}^d)-\\text{Var}(\\widehat{\\alpha}^\\textsc{miss ind}|\\boldsymbol{y}^d) \\qquad \\alpha= \\beta, \\pi, \\dots ; d=1,2,\\dots, 500 \\\\\n\\text{PWD}(\\text{R}_{95}(\\alpha))^d&= \\text{R}_{95}(\\widehat{\\alpha}^\\textsc{joint dep}|\\boldsymbol{y}^d)-\\text{R}_{95}(\\widehat{\\alpha}^\\textsc{miss ind}|\\boldsymbol{y}^d) \\qquad \\alpha= \\beta, \\pi, \\dots ; d=1,2,\\dots, 500.\n\\end{split}\n\\end{equation*} \n\nThese quantities are reported in Figure \\ref{f5} (top) which shows an imperceptible difference in the posterior distributions and their precision-summaries between the two models. \n\nThe same analysis is run on the 500 datasets with a {large dependence}, with results reported in Figure \\ref{f5} (bottom). \n\\begin{figure}[h]\n\\hspace*{-1.5cm}\\includegraphics[scale=0.32,page=4]{ss_sec4.pdf}\\includegraphics[scale=0.32, page=10]{ss_sec4.pdf}\\\\\n\\hspace*{-1.5cm}\\includegraphics[scale=0.32,page=1]{ss_sec4.pdf}\\includegraphics[scale=0.32, page=7]{ss_sec4.pdf}\n\\caption{Histogram of the pairwise differences in variance and in 95\\% \\ac{CrI} length of the posterior distribution of the transmission parameters $ \\beta $ for the small dependence scenario (left) and big dependence scenario (right). The bottom plots report the posterior distribution estimated with the \\textsc{joint dep} (solid) and \\textsc{miss ind} (dashed) model for 5 randomly-selected simulated data (colors).}\n\\label{f5}\n\\end{figure}\nHere there is a notable difference between the results from the two models: the posterior distributions from the misspecified model that assumes independent data are less variable than the ones derived using the \\ac{MC} approximation of the joint dependent likelihood. \n\nThis result was expected, since the misspecified model, by assuming independent data, accounts for more information than is contained in the data. This leads to an overconfidence that can be detected in the underestimation of the posterior variance. Results are confirmed by the proportion of datasets for which the pairwise differences are less than or equal to 0 for each of the parameters (Table \\ref{t2}). When this quantity is close to 0.5, the variances of the estimates obtained with the two methods are similar within datasets; when this quantity is close to 1 it suggests that the variance of the estimates obtained using the misspecified independent likelihood is systematically larger than the variance of the estimates obtained with the joint likelihood; and when this quantity is close to 0 it highlights that the former variance is systematically smaller then the latter. \n\\begin{table}[h]\n\\caption{Proportion of datasets in which the pairwise difference of variance is smaller or equal to 0 for the three transmission parameters.} \n\\label{t2}\n\\centering\n\\begin{tabular}{c c c}\n{Parameter} & \\multicolumn{2}{c}{Proportion of ${\\textsc{pwd}(\\text{Var})}\\leq 0 $} \\\\\n\\hline\n&{Small dependence}&{Large dependence}\\\\\n\\hline\n$ {\\beta}$&0.392&0 \\\\\n$ {\\pi}$&0.390&0 \\\\\n$ {\\iota} $&0.378&0 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\nThe results strongly suggest a systematic difference in variability between the two methods in the large dependence scenario. In all the simulated datasets the variances of the posterior distributions of the parameters are smaller in the analysis using the misspecified independent model than in the approximate joint model.\n\n\\subsubsection{Comparison for transmission and severity parameters}\nThe same comparison is carried out in a context where inference is drawn both for the transmission and the severity parameters. Here, since more quantities are estimated and due to the high correlation of the parameters of epidemic models, a difference between the results from the two models may be more difficult to spot. Moreover, in this multi-parameter context, convergence is sometimes compromised, particularly in the large-dependence scenario. \nIn the distribution of the pairwise differences, neither for the transmission parameters nor for the newly estimated severity parameters, can a large difference be seen (figures reported in the Supplementary Information). \n\nFor the large dependence scenario, the only notable difference can be seen in the distribution of $ {}^\\textsc{h}\\theta^\\textsc{ic} $: the parameter that links the two datasets, since it defines the probability of \\ac{IC} admission conditional on hospitalization. When the two datasets are jointly analysed, they both contribute to the estimation of $ {}^\\textsc{h}\\theta^\\textsc{ic} $, with hospital data informing the Binomial size in Equation \\ref{eq13} and \\ac{IC} data informing the proportion of people in the more-severe state. When the two datasets are considered independently, the hospital data do not play any role in the inference of $ {}^\\textsc{h}\\theta^\\textsc{ic} $.The proportions of pairwise differences less than or equal to 0 confirm this: the variance of the posterior sample of the parameter $ {}^\\textsc{h}\\theta^\\textsc{ic} $ is always lower when inference is drawn with the approximation to the joint dependent likelihood compared to when the misspecified independent model is adopted {(Table \\ref{t3})}. \n\n\\begin{table}[!h]\n\\caption{Proportion of datasets in which the pairwise difference of variance is smaller or equal to 0 for the transmission and severity parameters. }\n\\label{t3}\n\\centering\n\\begin{tabular}{c c c} \n{Parameter} &  \\multicolumn{2}{c}{Proportion of $ {\\textsc{pwd}(\\text{Var})}\\leq 0 $ }\\\\\n\\hline\n&{Small dependence}&{Large dependence}\\\\\n\\hline\n$ {\\beta}$&0.498& 0.554\\\\\n$ {\\pi}$&0.464& 0.546\\\\\n$ {\\iota} $&0.348& 0.202\\\\\n$ {}^0\\theta^\\textsc{h} $&0.466&0.566\\\\\n$ {}^\\textsc{h}\\theta^\\textsc{ic} $&0.778& 1\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{Influential parameters}\nAs a final comparison, a further investigation into the main cause of the difference is {undertaken}. Starting from the small-dependence scenario, one at a time, each parameter of Table \\ref{t1} is allowed to take the larger value in simulating the 500 datasets.\n\nEstimates of the {five} parameters are then obtained according to the misspecified independent and the joint dependent model. The posterior distributions and the plots of the precision statistics are reported in the Supplementary Information. While a detectable difference in the results is observed when all the parameters affecting the level of dependence vary, the same cannot be said when each parameter increases alone. Differences are less evident, with the probability of detection in \\ac{IC} being the most influential parameter, as shown in {Table \\ref{t4}}, where each column corresponds to a scenario where all the parameters but the header of the column are assumed small. \n\n\\begin{table}[!h]\n\\caption{Proportion of datasets in which the pairwise difference of variance is smaller or equal to 0 for the transmission and severity parameters in the scenario with small dependence except for the respective column-name parameter.}\n\\label{t4}\n\\centering\n\\begin{tabular}{c c c c c}\nIncreased Parameter &\\hspace*{.5cm} $ {}^0\\theta^\\textsc{h} $ \\hspace*{.5cm}&\\hspace*{.5cm}$ {}^\\textsc{h}\\theta^\\textsc{ic} $\\hspace*{.5cm} &\\hspace*{.5cm} $ \\zeta^\\textsc{h} $\\hspace*{.5cm}&\\hspace*{.5cm}$ \\zeta^\\textsc{ic} $\\\\\n\\hline\n{Parameter} &\\multicolumn{4}{c}{Proportion of $ {\\textsc{pwd}(\\text{Var})}\\leq 0 $ }\\\\\n\\hline\n$ \\beta $&0.468   &0.454 &0.296 & 0.490 \\\\\n$ \\pi $ &0.450& 0.454  &0.214 &0.496\\\\\n$ \\iota $ &0.342&0.082 &0.052 &0.032\\\\\n$ {{}^0\\theta^\\textsc{h}}$&0.476& 0.458& 0.290 & 0.464\\\\\n$ {{{}^\\textsc{h}\\theta^\\textsc{ic}}} $&0.682&0.940 & 0.072& 0.994\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Case-study: influenza during the 2017\/18 season}\\label{sec5}\nThe \\ac{UKHSA}, formerly \\ac{PHE}, routinely collects several sources of data to monitor influenza cases. Many of these datasets have been used separately to provide information on the transmission (\\cite{birrell2011bayesian},  \\cite{baguelin2013assessing}, \\cite{corbella2018exploiting})  and severity \\citep{presanis2014synthesising} of influenza; nevertheless, a joint analysis of all the sources has never been performed. Here a joint model is fitted to multiple data from the 2017\/18 influenza season, with the aim of retrospectively characterising both severity and transmission, while appropriately attributing stochasticity to the various processes. Among other datasets, the \\ac{USISS} is analysed and the methodology proposed in Section \\ref{sec3} is used to jointly exploit hospitalisation and \\ac{IC} unit admissions data. \n\\subsection{Data}\nThe following datasets are included in the analysis:\n\\begin{itemize}\n\\item Confirmed influenza cases collected from week 40 of 2017 to week 20 of 2018 via the \\ac{USISS} comprising the weekly count of \\textit{\\ac{IC} admissions} from, in principle, all trusts in England \\citep{health2011sourcesa} and the weekly count of \\textit{hospitalizations} at all levels of care in a stratified sentinel sample of the trusts \\citep{health2011sourcesb};\n\\item Daily counts of\\textit{ \\acp{GP} consultations} for \\ac{ILI} from a sample of \\ac{GP} monitored  by EMIS \\citep{harcourt2012use} or The Phoenix Partnership \\citep{phoenix};\n\\item Observed proportion of respiratory swabs taken on a selected subset of \\ac{GP} patients consulting for \\ac{ILI} that test positive for influenza (\\textit{virological} positivity, \\cite{RCGPdata}). \n\\item Results from cross-sectional \\textit{serological} surveys that inform on the presence of antibodies in the population \\citep{Andre}. \n\\end{itemize}\n\n\\subsection{Model Specification}\n\nFigure \\ref{f5_2} provides an illustration of the data-generating processes which includes spread of the virus (transmission), probability of mild symptoms and severe outcomes (severity), background \\ac{ILI} cases and detection. \n\nThe model is specified in discrete time; intervals of length 1 day are denoted by the index $ u=1, 2, \\dots $, while 1-week intervals are indexed by $ t=1, 2, \\dots $ (e.g. daily \\ac{GP} consultations are denoted by $ y^\\textsc{g}_u $, while weekly hospitalizations are denoted by $ y^\\textsc{h}_t $). The remaining notation is similar to that of Section \\ref{sec4}, with, e.g.,  $^\\textsc{j}_r X_s^\\textsc{k} $ being the number of people that experienced event $\\textsc{j} $ at time $ {r} $ and subsequently event $ \\textsc{k} $ at time $ s $.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=1, trim={5cm, 17cm, 5cm, 4cm}, clip]{f2.pdf}\n\\caption{\nThe transmission model classifies the population into susceptible ($ S $), exposed ($ E_1, E_2 $), infectious ($ I_1, I_2 $) and removed ($ R $). The severity model defines the occurrence of: mild flu cases ${}^0X^\\textsc{f} $ with probability ${}^0\\theta^\\textsc{f} $, hospital cases ${}^0X^\\textsc{h} $ with probability ${}^0\\theta^\\textsc{h} $, and of \\ac{IC} cases conditional on hospitalization ${}^\\textsc{h}X^\\textsc{ic} $ with probability ${}^\\textsc{h}\\theta^\\textsc{ic} $. The detection process links cases to data defining the probability of reporting \\ac{GP} consultations ($ \\zeta_u^\\textsc{g} $), hospitalizations ($ \\zeta_u^\\textsc{h} $) and \\ac{IC} admissions ($ \\zeta_u^\\textsc{ic} $), respectively. The background process models the non-influenza \\ac{ILI} cases, $ X^\\textsc{b}_t $. }\n\\label{f5_2}\n\\end{figure}\nSome essential elements of the model are outlined here; the complete description of the model and the derivations of the data-distributions are provided in the Supplementary Information.\n\n\\subsubsection{Transmission and first severity layer}\nDenote by $ \\xi_{u}^0$ the number of new infections generated during day $ u $. A deterministic $ SEIR $ transmission model, is assumed so that $ \\xi_{u}^0$  is a function of the parameters $  \\pi, \\iota, \\beta, \\sigma, \\gamma, \\kappa  $ , representing the proportion of individuals initially immune; the proportion of initially infected\/infectious individuals; the transmission rate; the rate of becoming infectious; the recovery rate; and the school-closure effect, respectively. \n\nThe infection processes of individuals who will experience hospital admissions, $ {}^0_uX^\\textsc{h} $, and influenza-related \\ac{GP} consultations, $ {}^0_uX^\\textsc{f} $, are assumed to follow a time non-homogeneous Poisson process, i.e.:\n\\begin{equation}\n\\begin{split}\n\\left({}^0_uX^\\textsc{h}\\bigg|\\xi^0_u,{}^0\\theta^\\textsc{h}\\right)  &\\sim \\text{Pois}\\left({}^0\\theta^\\textsc{h}\\cdot\\xi^0_u\\right)  \\qquad \\text{for } u=0,1, \\dots, U\\\\\n\\left({}^0_uX^\\textsc{f}\\bigg|\\xi^0_u,{}^0\\theta^\\textsc{f} \\right) &\\sim \\text{Pois}\\left( {}^0\\theta^\\textsc{f} \\cdot\\xi^0_u\\right) \\qquad \\text{for } u=0,1, \\dots, U\n\\end{split}\n\\label{eq5_16}\n\\end{equation}\nwith $ {}^0\\theta^\\textsc{f} $ and $ {}^0\\theta^\\textsc{h} $ denoting the probability of being visited by a \\ac{GP} and being admitted to hospital, respectively. \n\n\\subsubsection{\\ac{GP}-consultations}\nThe inhomogeneous Poisson process of the people who will become mildly symptomatic and consult a \\ac{GP}, $ {}^0_uX^\\textsc{f} $ in Equation \\ref{eq5_16}, is the main component of the likelihood of the \\ac{GP} consultations data. This process too is affected by delays that can be modelled by a discrete \\ac{r.v.} modelling the days elapsing between infection and consulting a practice. \n\nIn addition to these individuals, background, non-influenza cases appear in \\ac{GP} consultation data; these endemic cases of other respiratory viruses and bacterial infections often follow a yearly seasonality, peaking around the same time as the seasonal influenza epidemic \\citep{paul2008multivariate}. The background seasonality pattern is modelled by a weekly-varying  sine-cosine oscillation, similar to \\cite{held2005statistical}. \n\nThe general process describing the number of people consulting a \\ac{GP} with \\ac{ILI} symptoms is then obtained by adding the endemic and epidemic processes. Virological data are used to disentangle the proportion of the former process out of the total.\n\nLastly, a Binomial emission is again chosen to model the observational process. However,  the probability of attending a \\ac{GP} practice is subject to weekly fluctuations, caused by the weekend closure of \\ac{GP} practices, hence a day-of the week distortion effect is included in the probability of detecting a \\ac{GP} consultation.\n\n\\subsubsection{Hospitalization and \\ac{IC} admissions}\nA model for dependent data on hospitalizations and \\ac{IC}-admissions data is illustrated in Section \\ref{sec3}. Here too the joint likelihood is factorised in the marginal distribution of the \\ac{IC} admissions $  Y_{1:T}^{\\textsc{ic}} $, from the Poisson distribution of Equation \\ref{eq14}, and the distribution of the hospitalizations $ Y_{1:T}^{\\textsc{h}} $ conditionally on \\ac{IC} data approximated via \\ac{MC} integration as proposed in Algorithm \\ref{alg2}.\n\n\n\\subsection{Results}\nThe joint distribution of the unknown parameter vector (including transmission, severity, background and detection parameters) is derived via \\ac{GIMH}. Weakly-informative priors are assumed for most of the parameters of interest, all the prior distributions used and the estimated posteriors are reported in the Supplementary Information. \n\nThe model presents a fair fit to the data (Figure \\ref{f6}) with all the observations being included in the 95\\% \\acp{CrI} of the posterior predictive distributions of the \\ac{GP} consultation, virology and hospitalization data. The model, however, struggles to fit \\ac{IC} data which might be in conflict with other sources of information in the model. These data, unlike \\ac{GP} and hospital data, don't suggest a second peak in infection in the latest weeks of winter. \n\n\\begin{figure}[!ht]\n\\includegraphics[scale=0.36, page=1]{GOF.pdf}\\includegraphics[scale=0.36, page=3]{GOF.pdf}\\\\\n\\includegraphics[scale=0.36, page=4]{GOF.pdf}\\includegraphics[scale=0.36, page=5]{GOF.pdf}\n\\caption{Median and 95\\% \\acp{CrI} (green) for the posterior predicted distribution of: \\ac{GP} data (a); virological data (b); hospitalizations (c) and \\ac{IC} admissions (d). Red points are the observed data.}\n\\label{f6}\t\n\\end{figure}\n\nThe model provides a comprehensive picture of influenza season 2017\/18. Transmission, in terms of daily number of new infections is inferred and background \\ac{ILI} cases are also described (Figure \\ref{f7}). Moreover, all the key severity parameters of interest are well identified and informed by the joint use of the data, including the case-hospitalization risk (Median 0.0032, 95\\% \\ac{CrI} 0.0022-0.0049) and the hospital-\\ac{IC} admission risk (Median 0.0667, 95\\% \\ac{CrI} 0.0574-0.078). \n\n\\begin{figure}[h]\n\\begin{subfigure}[c]{7.56cm}\n\\centering\n\\includegraphics[scale=0.35]{EpiCurve.pdf}\n\\caption{Median (red) and 95\\% \\acp{CrI} (green) of the daily number of new infections. The grey areas corresponding to periods of lower-transmission (school holidays). 20 randomly selected trajectories are also plotted as thin green lines.}\n\\label{f7a}\n\\end{subfigure}\\hfill\\begin{subfigure}[c]{6.13cm}\n\\centering\n\\includegraphics[scale=0.33]{BGrate.pdf}\n\\caption{Median (solid line) and 95\\% \\acp{CrI} (shaded area) of the prior (red) and posterior (green) for $ b_u $, the mean of the rate of the daily number of non-influenza \\ac{ILI} \\ac{GP} consultations.}\n\\label{f7b}\n\\end{subfigure}\n\\caption{Posterior results from the joint analysis of 2017\/18 data: transmission and background.}\n\\label{f7}\n\\end{figure}\n\nThe model in all presents many levels of sophistication. Firstly, \\ac{ILI} cases attributable to influenza are disentangled from the background \\ac{ILI}, which, while not being directly related to influenza, contributes to pressure on \\ac{GP} practices. Secondly, the model accounts for peculiarities of the \\ac{GP} data, such as the day-of-the-week variation, uncovering the true underlying transmission and severity process. Lastly and more importantly, exact inference from dependent data via \\ac{MC} methods enables the simultaneous use of information from hospital and \\ac{IC} surveillance.\n\n\\section{Discussion} \\label{sec6}\nThis work contributes to the state-of-the-art literature on the analysis of epidemic data, from a methodological\/computational perspective, providing new tools, applicable to a wide class of problems that includes dependent data; and from an application perspective, providing innovative results on the analysis of influenza from multiple sources.\n\\subsection{Methodological and computational perspective}\nSection \\ref{sec2} contains a concise review of \\acp{SSM} and their analysis, particularly focussed on the analysis of multiple epidemic data. There are other reviews of these methods in the literature (\\cite{kantas2015particle, schon2018probabilistic}), some of which even target epidemic applications (\\cite{mckinley2014simulation, dureau2013ssm, ionides2006inference}). However, these reviews mainly address temporal dependences and, to the knowledge of the authors, little is known about the dependence across other domains. In the context of epidemic analysis, where temporal dependence can be easily overcome through realistic deterministic approximations of transmission dynamics, it is key to consider the sources of stochasticity and dependence linked to other processes (in this case the severity process) and its implications. \n\nSections \\ref{sec3} and \\ref{sec4} embrace this challenge and highlight the possible complexities of such an analysis. Starting from a specific example, where there is dependence between data sources, an approach to the estimation of the static parameters of the system is proposed.\nKey aspects such as overlaps between datasets and delays between consecutive events are then revealed to affect results. It is shown via simulation that simplifying assumptions, which assume independence among data and avoid expensive integration, lead to overconfident results and misleadingly narrow interval estimates. \n\nOur general and comprehensive way to set up epidemic models are paired with a specific estimation routine for their static parameters: pseudo-marginal methods. These methods are robust and simple to set up since they rely on the standard \\ac{MH} algorithms, however they are expensive in two respects: firstly, a high number of simulations of the hidden states is required at each iteration of the static parameter to approximate its likelihood; and, secondly, the parameter space might be explored inefficiently due to their random walk behaviour. The first of these aspects might be addressed, for example, by converting to a \\ac{DA} \\ac{MCMC} perspective \\citep{o1999bayesian}, which only needs a single sample from the hidden states per \\ac{MCMC} iteration. \\ac{DA} approaches however, require tailored implementations for each different system considered, and therefore do not provide a general recipe applicable to a wide class of models. In terms of exploration of the parameter space, the unavailability of closed-form likelihood prevents the use of popular gradient based methods (e.g. \\ac{HMC} \\citep{neal2011mcmc} or the more recent \\ac{PDMP} sampler \\citep{bierkens2019zig}). Alternatively, a natural way to improve the estimation routine would be to consider \\ac{SMC} methods on the parameter space, which in this context can be thought of as \\ac{SMC}\\textsuperscript{2} \\citep{chopin2013smc2}.\n\n\\subsection{Application perspective}\nThe ability of estimating key transmission and severity parameters from epidemic data has never been more crucial. With the COVID-19 epidemic affecting almost every aspect of people's life, evidence-based decision-making has become an imperative aspect of public health policy. Uncertainty quantification around central estimates is key to understand both the extreme possible scenarios that could present in the near future and what further information is needed to increase knowledge and inform decisions. \n\nSection \\ref{sec5} of the paper proposes a complex joint analysis of multiple data on influenza where both mild and severe cases inform transmission and severity parameters. The model presents multiple levels of sophistication including: the use of dependent data; the use of data from both non-confirmed and confirmed cases; the disentangling of background \\ac{ILI} cases; the presence of delays between events. The key innovation of this model is that it matches a deterministic transmission dynamic with a stochastic severity and reporting process. In this sense it differentiates from other works that make joint use of multiple data but assume deterministic severity dynamics (e.g. \\cite{keeling2020fitting}). \n\n\\cite{shubin2016revealing} proposes a similar analysis of the 2009 pandemic in Finland. Even though his model presents many interesting aspects (e.g. the inclusion of both environmental and demographic stochasticity), unrealistic simplifications are made to allow for estimation in reasonable time, including the exclusion of an \\textit{exposed} compartment, the discretization of time into extremely-large intervals (one week), and the assumption of no time elapsing between events of increasing severity. We consider our approach as a valid alternative to \\cite{shubin2016revealing}, especially on a large population such as that of England, where the deterministic assumption is even more likely to hold. \n\nGoing forward, the model could be extended in, at least, two interesting ways: firstly, population heterogeneity could be considered, through the formulation of an age-specific model that makes use of contact patterns between age groups; and secondly, more overdispersion could be allowed in the severity process by, for example, replacing Poisson \\acp{r.v.} with Negative Binomial \\acp{r.v.}. These changes would almost certainly improve the fitting of the model to data.\n\nLastly, other interesting analyses could be performed starting from the proposed model. The analysis presented here is retrospective and aimed at a posterior estimation of key epidemiological parameters; it would be interesting to test the predictive ability of such a model so that it can be used in real time for decision-making on resource allocation for different healthcare facilities. Moreover, more insights could be gained from the quantification of the contribution to inference provided by each separate dataset from a value of information perspective \\cite{jackson2019value}.\n\n\n\\begin{acronym}\n\\acro{SSM}{state-space model}\n\\acro{r.v.}{random variable}\n\\acro{POMP}{partially observed Markov process}\n\\acrodefplural{POMP}{partially observed Markov processes}\n\\acro{HMM}{hidden Markov model}\n\\acro{BPF}{bootstrap particle filter}\n\\acro{MC}{Monte Carlo}\n\\acro{HMC}{Hamiltonian Monte Carlo}\n\\acro{SMC}{sequential Monte Carlo}\n\\acro{MH}{Metropolis Hastings}\n\\acro{MCMC}{Monte Carlo Markov chain}\n\\acro{GIMH}{grouped independence Metropolis Hastings}\n\\acro{MCWM}{Monte Carlo within Metropolis}\n\\acro{DAG}{directed acyclic graph}\n\\acro{USISS}{UK Severe Influenza Surveillance System}\n\\acro{NHS}{National Health Service}\n\\acro{IC}{Intensive Care}\n\\acro{CrI}{Credible Interval}\n\\acro{PHE}{Public Health England}\n\\acro{ILI}{influenza-like illness}\n\\acro{GP}{General Practitioner}\n\\acro{RCGP}{Royal College of General Practitioners}\n\\acro{ONS}{Office of National Statistics}\n\\acro{UKHSA}{UK Health Security Agency}\n\\acro{DA}{data-augmented}\n\\acro{PDMP}{Piecewise Deterministic Markov Process}\n\\end{acronym}\n\n\n\n\n\n\\section*{Acknowledgements}\nThe author would like to thank colleagues at Public Health England, particularly Dr Richard Pebody, Dr Andre Charlett and Dr Nikolaos Panagiotopoulos, for providing the data and prior information for the analysis in Section 6.\nWe are grateful to Dr Michail Shubin for insightful discussions on the earlier work on dependent data streams. Lastly, Dr Trevelyan J McKinley and Dr Paul Kirk provided useful feedback on an earlier version of this work for which we are thankful.\n\nAC is supported by Bayes4Health EPSRC Grant EP\/R018561\/1. All four authors were supported  by the MRC, programme grant MC\\_UU\\_00002\/11. \n\n\n\\begin{supplement}\n\\stitle{Supplementary Information}\n\\sdescription{This report contains further work on the study of dependence between data and supplementary information to the study of Section \\ref{sec5}. The former includes further comments on SSM inference, the derivations of Algorithm \\ref{alg2} and the results form the simulation study of Section \\ref{sec4}. The latter includes extensive explanation of the model, inferential methods and results of the analysis.}\n\\end{supplement}\n\\begin{supplement}\n\\stitle{Supplementary code and data}\n\\sdescription{Code for the simulation study in Section \\ref{sec4} and the analysis of Section \\ref{sec5} are available at \\href{https:\/\/github.com\/alicecorbella\/EpiDependentData}{https:\/\/github.com\/alicecorbella\/EpiDependentData}. The data analysed in Section \\ref{sec5} are based on routine health-care data, which cannot be made available to others by the study authors. Requests to access these non-publicly available data are handled by the \\href{https:\/\/www.gov.uk\/government\/publications\/accessing-public-health-england-data\/about-the-phe-odr-and-accessing-data}{Public Health England Office for Data Release} and its successor at the \\href{https:\/\/www.gov.uk\/government\/publications\/accessing-ukhsa-protected-data}{UK Health Security Agency}.}\n\\end{supplement}\n\n\n\\bibliographystyle{imsart-nameyear}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet us first recall the famous Goldbach conjecture in additive number\ntheory.\n\n\\begin{Conjecture}[Goldbach's Conjecture]\\label{Goldbach}\nAny even integer $n\\gs4$ can be written as the sum of two primes.\n\\end{Conjecture}\n\nThe number of primes not exceeding $n\\geqslant2$ is approximately\n$n\/\\log n$ by the prime number theorem. Hardy and Littlewood\nconjectured that the number of ways to write an even integer $n\\geqslant\n4$ as the sum of two primes is given asymptotically by\n$$\\frac {cn}{\\log^2n}\\prod_{p\\mid n}\\left(1+\\f1{p-2}\\right),$$\nwhere $c=2\\prod_p(1-(p-1)^{-2})=1.3203\\cdots$ is a constant  and $p$\nruns over odd primes. (Cf. \\cite[pp. 159-164]{Gu}.)\n\nGoldbach's conjecture remains open, and the best result in this\ndirection is Chen's theorem (cf. \\cite{C}): Each large\neven integer can be written as the sum of a prime and a product of at most two\nprimes.\n\nThose integers $T_x=x(x+1)\/2$ with $x\\in{\\Bbb N}=\\{0,1,2,\\ldots\\}$ are called triangular numbers.\nThere are less than $\\sqrt{2n}$ positive triangular numbers below an integer $n\\gs2$,\nso triangular numbers are more sparse than prime numbers.\nIn 2008 the author made the following conjecture.\n\n\\begin{Conjecture}[Sun \\cite{S09}]\\label{pT} {\\rm (i)} Each natural number $n\\not=216$\ncan be written in the form $p+T_x$ with $x\\in{\\Bbb N}$, where $p$ is zero or a prime.\n\n{\\rm (ii)} Any odd integer greater than $3$ can be written in the form $p+x(x+1)$, where $p$ is a prime and\n$x$ is positive integer.\n\\end{Conjecture}\n\nDouglas McNeil (University of London) (cf. \\cite{M2})\n has verified parts (i) and (ii) up to $10^{10}$ and $10^{12}$ respectively.\n The author \\cite{S-offer} would like to offer 1000 US dollars for the first positive solutions to both (i) and (ii),\n and \\$200 for the first explicit counterexample to (i) or (ii).\n\nPowers of two are even much more sparse than triangular numbers. In a letter to\nGoldbach, Euler posed the problem whether any odd integer $n>1$ can\nbe expressed in the form $p+2^a$, where $p$ is a prime and\n$a\\in{\\Bbb N}$. This question was reformulated by\nPolignac in 1849. By introducing covers of the integers by residue\nclasses, Erd\\H os \\cite{E50} showed that there exists an infinite\narithmetic progression of positive odd integers no term of which is\nof the form $p+2^a$. (See also Nathanson \\cite[pp. 204-208]{N}.) On\nthe basis of the work of Cohen and Selfridge \\cite{CS}, the author\n\\cite{S00} proved that  if\n$$x\\equiv 47867742232066880047611079\\ ({\\rm mod}\\ M)$$\nwith\n $$\\aligned M=&2\\times3\\times5\\times7\\times11\\times13\\times17\\times19\\times31\\times37\n \\\\&\\times41\\times61\\times73\\times97\n\\times109\\times151\\times241\\times257\\times331\n\\\\=&66483084961588510124010691590,\n\\endaligned$$ then $x$ is not of the form\n$\\pm p^a\\pm q^b$ where $p,q$ are primes and $a,b\\in{\\Bbb N}$.\n\nIn 1971 Crocker \\cite{Cr} proved that there are infinitely many positive odd integers\nnot of the form $p+2^a+2^b$ where $p$ is a prime and $a,b\\in{\\Bbb Z}^+=\\{1,2,3,\\ldots\\}$.\nHere are the first few such numbers greater than 5 recently found by Charles Greathouse (USA):\n$$6495105,\\ 848629545,\\ 1117175145,\\ 2544265305,\\ 3147056235,\\ 3366991695.$$\nNote that 1117175145 even cannot be written in the form $p+2^a+2^b$ with $p$ a prime and $a,b\\in{\\Bbb N}$.\n\nErd\\H os (cf. \\cite{E97}) asked whether there is a positive integer\n$k$ such that any odd number greater than 3 can be written the sum\nof an odd prime and at most $k$ positive powers of two. Gallagher\n\\cite{G} proved that for any $\\varepsilon>0$ there is a positive\ninteger $k=k(\\varepsilon)$ such that those positive odd integers not\nrepresentable as the sum of a prime and $k$ powers of two form a\nsubset of $\\{1,3,5,\\ldots\\}$ with lower asymptotic density at least\n$1-\\varepsilon$. In 1951 Linnik \\cite{L} showed that there exists a\npositive integer $k$ such that each large even number can be written\nas the sum of two primes and $k$ positive powers of two; Heath-Brown\nand Puchta \\cite{HP} proved that we can take $k=13$. (See also Pintz\nand Ruzsa \\cite{PR}.)\n\nIn March 2005  Georges Zeller-Meier \\cite{ZM} asked whether $2^{2^n-1}-2^n-1$ is composite for every $n=3,4,\\ldots$.\nClearly an affirmative answer follows from part (i) of our following theorem in the case $m=2$.\n\n\\begin{Theorem}\\label{p22} {\\rm (i)} Let $m\\equiv 2\\ ({\\rm mod}\\ 4)$ be an integer with $m+1$ a prime.\nThen, for each $n=3,4,\\ldots$, we have\n$$\\frac{m^{2^n-1}-1}{m-1}\\not=m^n+p^a,$$\nwhere $p$ is any prime and $a$ is any nonnegative integer.\n\n{\\rm (ii)} Let $m$ and $n$ be integers greater than one. Then\n$$\\frac{m^{2^n}-1}{m-1}\\not=p+m^a+m^b,$$\nwhere $p$ is any prime, $a,b\\in{\\Bbb N}$ and $a\\not=b$.\n\\end{Theorem}\n\n\\begin{Remark} In the case $m=2$, part (ii) of Theorem \\ref{p22} was observed by  A. Schinzel and Crocker independently\nin the 1960s, and this plays an important role in Crocker's result about\n$p+2^a+2^b$. In 2001 the author and Le \\cite {SL} proved that for\n$n=4,5,\\ldots$ we cannot write $2^{2^n-1}-1$ in the form\n$p^\\alpha+2^a+2^b$, where $p$ is a prime, $a,b,\\alpha\\in{\\Bbb N}$ and\n$a\\not=b$.\n\\end{Remark}\n\nFor any integer $m>1$, the sequence $\\{m^n\\}_{n\\gs0}$ is a\nfirst-order linear recurrence with earlier terms dividing all later\nterms. To seek for good representations of integers, we'd better\nturn resort to second-order linear recurrences whose general term\nusually does not divide all later terms.\n\nThe famous Fibonacci sequence $\\{F_n\\}_{n\\gs0}$ is defined as follows:\n$$F_0=0,\\ F_1=1,\\ \\text{and}\\ F_{n+1}=F_n+F_{n-1}\\ \\text{for}\\ n=1,2,3,\\ldots.$$\nHere are few initial Fibonacci numbers:\n$$F_0=0<F_1=F_2=1<F_3=2<F_4=3<F_5=5<F_6=8<F_7=13<F_8=21<\\cdots.$$\nIt is well known that\n$$F_n=\\f1{\\sqrt5}\\(\\(\\frac{1+\\sqrt5}2\\)^n-\\(\\frac{1-\\sqrt5}2\\)^n\\)\\ \\ \\ \\text{for all}\\ n\\in{\\Bbb N}.$$\nClearly $F_n<2^{n-1}$ for $n=2,3,\\ldots$, and\n$$F_n\\sim \\frac{\\varphi^n}{\\sqrt5}\\qquad\\ (n\\to+\\infty),$$\nwhere\n$$\\varphi=\\frac{1+\\sqrt5}2=1.618\\cdots.$$\nNote that $2\\mid F_n$ if and only if $3\\mid n$.\n\nIt is not known whether the positive integers not of the form $p+F_n$ with $p$ a prime and $n\\in{\\Bbb N}$\nform a subset of ${\\Bbb Z}^+$ with positive lower asymptotic density. However, Wu and Sun \\cite{WS} were able to construct\na residue class containing no integers of the form $p^a+F_{3n}\/2$ with $p$ a prime and $a,n\\in{\\Bbb N}$. Note that $u_n=F_{3n}\/2$\nis just half of an even Fibonacci number; also $u_0=0$, $u_1=1$, and $u_{n+1}=4u_n+u_{n-1}$ for $n=1,2,3,\\ldots$.\n\nOn December 23, 2008 the author \\cite{pFF} formulated the following conjecture.\n\\begin{Conjecture}[Conjecture on Sums of Primes and Fibonacci Numbers]\\label{pFF}\nAny integer $n>4$ can be written as the sum of an odd prime and two positive Fibonacci numbers.\nWe can require further that one of the two Fibonacci numbers is odd.\n\\end{Conjecture}\n\n\\begin{Remark}\\label{R-pFF} For a large integer $n$, there are about $\\log n\/\\log\\varphi$\nFibonacci numbers below $n$ but there are about $n\/\\log n$ primes\nbelow $n$. So, Fibonacci numbers are much more sparse than prime\nnumbers and hence the above conjecture looks more difficult than the\nGoldbach conjecture. D. McNeil (cf. \\cite{M2, M3}) has\nverified Conjecture \\ref{pFF} up to $10^{14}$. The author (cf. \\cite{S-offer}) would like\nto offer 5000 US dollars for the first positive solution published\nin a well-known mathematical journal and \\$250 for the first\nexplicit counterexample which can be rechecked by the author via\ncomputer. Note that Conjecture \\ref{pFF} implies that for any odd prime $p$ we can find an odd prime $q<p$ such that\n$p-q$ can be written as the sum of two odd Fibonacci numbers.\n\\end{Remark}\n\nRecall that the Pell sequence $\\{P_n\\}_{n\\gs0}$ is defined as follows.\n$$P_0=0,\\ P_1=1,\\ \\text{and}\\ P_{n+1}=2P_n+P_{n-1}\\ \\ \\text{for}\\ n=1,2,3,\\ldots.$$\nIt is well known that\n$$P_n=\\f1{2\\sqrt2}\\left((1+\\sqrt2)^n-(1-\\sqrt2)^n\\right)\\ \\ \\ \\text{for all}\\ n\\in{\\Bbb N}.$$\nClearly $P_n>2^{n}$ for $n=6,7,\\ldots$, and\n$$P_n\\sim \\frac{(1+\\sqrt2)^n}{2\\sqrt2}\\qquad\\ (n\\to+\\infty).$$\n\nOn Jan. 10, 2009, the author \\cite {FF} posed the following conjecture which is an analogue of Conjecture \\ref{pFF}.\n\\begin{Conjecture}[Conjecture on Sums of Primes and Pell Numbers]\\label{pP2P}\nAny integer $n>5$ can be written as the sum of an odd prime, a Pell number and twice a Pell number.\nWe can require further that the two Pell numbers are positive.\n\\end{Conjecture}\n\n\\begin{Remark}\\label{R-pP2P} D. McNeil (cf. \\cite{S-offer}) has verified Conjecture \\ref{pP2P} up to $5\\times10^{13}$ and found no counterexample.\nThe author (cf. \\cite{S-offer}) would like to offer 1000 US dollars for the first positive solution published in a well-known mathematical journal\nand \\$100 for the first explicit counterexample\nwhich can be rechecked by the author via computer.\n\\end{Remark}\n\n Soon after he learned Conjecture  \\ref{pP2P} from the author, Qing-Hu Hou (Nankai University) observed (without proof) that\n all the sums $P_s+2P_t\\ (s,t=1,2,3,\\ldots)$ are distinct.\n Clearly Hou's observation follows from our following theorem.\n\n\\begin{Theorem}\\label{uau} Let $a>1$ be an integer, and set\n$$u_0=0,\\ u_1=1, \\ \\text{and}\\ u_{i+1}=au_i+u_{i-1}\\ \\text{for}\\ i=1,2,3,\\ldots.$$\nThen no integer $x$ can be written as $u_m+au_n$ (with $m\\in{\\Bbb N}$ and $n\\in{\\Bbb Z}^+$) in at least two ways,\nexcept in the case $a=2$ and $x=u_0+au_2=u_2+au_1=4$.\n\\end{Theorem}\n\n\\begin{Remark}\\label{R-uau} Note that if $n\\in{\\Bbb Z}^+$ then $u_{n+1}+au_0=au_n+u_{n-1}$.\n\\end{Remark}\n\n\\begin{Corollary}\\label{P+2P} Let $k,l,m,n\\in{\\Bbb Z}^+$. Then $P_k+2P_l=P_m+2P_n$ if and only if $k=m$ and $l=n$.\n\\end{Corollary}\n\n\\begin{Remark}\\label{R-P+2P} In view of Corollary \\ref{P+2P},\nwe can assign an ordered pair $\\langle m,n\\rangle\\in{\\Bbb Z}^+\\times{\\Bbb Z}^+$\nthe code $P_m+2P_n$. Recall that a sequence $a_1<a_2<a_3<\\cdots$ of\npositive integers is called a Sidon sequence if all the sums of\npairs, $a_i+a_j$, are all distinct. An unsolved problem of Erd\\H os\n(cf. \\cite[p. 403]{Gu}) asks for a polynomial $P(x)\\in{\\Bbb Z}[x]$ such\nthat all the sums $P(m)+P(n)\\ (0\\leqslant m<n)$ are distinct.\n\\end{Remark}\n\n\nMotivated by Conjecture \\ref{pFF} and its variants, Qing-Hu Hou and Jiang Zeng (University of Lyon-I)\nformulated the following conjecture during their visit to the author in Jan. 2009.\n\n\\begin{Conjecture}[Hou and Zeng \\cite{HZ}]\\label{pFC}\nAny integer $n>4$ can be written as the sum of an odd prime, a positive Fibonacci number and a Catalan number.\n\\end{Conjecture}\n\n\\begin{Remark}\\label{R-pP2P} Catalan numbers are integers of the form\n$$C_n=\\f1{n+1}{2n\\choose n}={2n\\choose n}-{2n\\choose n+1}\\quad (n\\in{\\Bbb N}),$$\nwhich play important roles in combinatorics (see, e.g., Stanley\n\\cite[Chapter 6]{St}). They are also determined by $C_0=1$ and the\nrecurrence\n$$C_{n+1}=\\sum_{k=0}^nC_kC_{n-k}\\quad(n=0,1,2,\\ldots).$$\nBy Stirling's formula, $C_n\\sim 4^n\/(n\\sqrt{n\\pi})$ as $n\\to+\\infty$.\nD. McNeil \\cite{M3} has verified Conjecture \\ref{pFC} up to $3\\times10^{13}$ and found no counterexample.\nHou and Zeng  would like to offer 1000 US dollars for the first positive solution published in a well-known mathematical journal\nand \\$200 for the first explicit counterexample\nwhich can be rechecked by them via computer. Note that 3627586 cannot be written in the form $p+2F_s+C_t$ with $p$ a prime and $s,t\\in{\\Bbb N}$.\n\\end{Remark}\n\nThe Lucas sequence $\\{L_n\\}_{n\\gs0}$ is defined as follows.\n$$L_0=2,\\ L_1=1,\\ \\text{and}\\ L_{n+1}=L_n+L_{n-1}\\ (n=1,2,3,\\ldots).$$\nIt is known that\n$$L_n=2F_{n+1}-F_n=\\(\\frac{1+\\sqrt5}2\\)^n+\\(\\frac{1-\\sqrt5}2\\)^n$$\nfor every $n=0,1,2,3,\\ldots$.\n\nOn Jan. 16, 2009 the author (cf. \\cite{FF}) made the following conjecture which is\nsimilar to Conjecture \\ref{pFC}.\n\n\\begin{Conjecture}\\label{pLC}\nEach integer $n>4$ can be written as the sum of an odd prime, a\nLucas number and a Catalan number.\n\\end{Conjecture}\n\\begin{Remark}\\label{R-pLC} D. McNeil \\cite{M3} has verified Conjecture \\ref{pLC}\nup to $10^{13}$ and found no counterexample. Note that 1389082 cannot be written\nin the form $p+2L_s+C_t$ with $p$ a prime and $s,t\\in{\\Bbb N}$.\n\\end{Remark}\n\nRecall that there are infinitely many positive odd integers not of the form $p+2^a+2^b$ with $p$ a prime and $a,b\\in{\\Bbb Z}^+$.\nHowever, Crocker's trick in his proof of this result does not work for the form $p+2^a+k2^b$ with $p$ a prime and $a,b\\in{\\Bbb Z}^+$,\nwhere $k$ is an odd integer greater than one.\nOn Jan. 21, 2009 the author (cf. \\cite{FF}) made the following conjecture.\n\n\\begin{Conjecture}[Conjecture on Sums of Primes and Powers of Two]\\label{p222}\nAny odd integer greater than $8$ can be written as the sum of an odd prime and three positive powers of two.\nMoreover, we can write any odd integer $n>10$ in the form $p+2^a+3\\times 2^b=p+2^a+2^b+2^{b+1}$ with $p$ a prime and $a,b\\in{\\Bbb Z}^+$.\n\\end{Conjecture}\n\n\\begin{Remark}\\label{p2k2} The author verified Conjecture \\ref{p222} for odd integers below $10^7$. Later, on the request of the author,\nQing-Hu Hou and Charles Greathouse  continued the verification for odd integers below $2\\times 10^8$ and $10^{10}$ respectively.\n Note that if $k>61$ is odd then $2k+127$ cannot be written in the form $p+2^a+k2^b$\nwith $p$ an odd prime and $a,b\\in{\\Bbb Z}^+$ since $3+2+k2^2>2k+127$ and $127$ is not of the form $p+2^a$.\nFor $k\\in\\{3,5,\\ldots,61\\}\\setminus\\{47,51\\}$, the author (cf. \\cite{pFF}) checked odd integers below $10^8$\nand found no odd integer $n>2k+3$ not of the form\n$p+2^a+k2^b$ with $p$ an odd prime and $a,b\\in{\\Bbb Z}^+$.\n\\end{Remark}\n\n\n\n\n We are going to prove Theorems \\ref{p22} and \\ref{uau} in the next section.\nSection 3 is devoted to our discussion of Conjecture \\ref{pFF} and its variants.\n\n\\section{Proofs of Theorems \\ref{p22} and \\ref{uau}}\n\n\\medskip\n\\noindent{\\it Proof of Theorem \\ref{p22}}.\nFor $n=2,3,\\ldots$ we clearly have\n$$\\aligned(m-1)\\prod_{k=0}^{n-1}\\left(m^{2^k}+1\\right)=&\\left(m^{2^0}-1\\right)\\left(m^{2^0}+1\\right)\\left(m^{2^1}+1\\right)\\cdots\\left(m^{2^{n-1}}+1\\right)\n\\\\=&\\left(m^{2^1}-1\\right)\\left(m^{2^1}+1\\right)\\cdots\\left(m^{2^{n-1}}+1\\right)\n\\\\=&\\cdots=\\left(m^{2^{n-1}}-1\\right)\\left(m^{2^{n-1}}+1\\right)=m^{2^n}-1.\n\\endaligned$$\n\n(i) Fix an integer $n\\geqslant 3$. Write $n+1=2^kq$ with $k\\in{\\Bbb N}$, $q\\in{\\Bbb Z}^+$ and $2\\nmid q$. Since\n$$2^n=(1+1)^n\\geqslant 1+n+\\frac{n(n-1)}2>n+1,$$\nwe must have $0\\leqslant k\\leqslant n-1$. Thus $m^{2^k}+1$ divides both $(m^{2^n}-1)\/(m-1)$ and\n$m^{n+1}+1=(m^{2^k})^q+1$.\nSet\n$$d_n=\\frac{m^{2^n-1}-1}{m-1}-m^n.$$\nThen\n$$md_n=\\frac{m^{2^n}-m}{m-1}-m^{n+1}=\\frac{m^{2^n}-1}{m-1}-(m^{n+1}+1)$$\nand hence $m^{2^k}+1$ divides $d_n$.\n\nSuppose that $d_n$ is a prime power. By the above, we can write $d_n=p^a$, where $a\\in{\\Bbb N}$ and\n$p$ is a prime divisor of  $m^{2^k}+1$. As $m$ is even, $p$ is an odd prime.\nSince\n$$m^{p-1}\\eq1\\ ({\\rm mod}\\ p)\\ \\ \\text{and}\\ \\ m^{2^{k+1}}\\equiv(-1)^2=1\\ ({\\rm mod}\\ p),$$\nwe have\n$$m^{\\gcd(p-1,2^{k+1})}\\eq1\\ ({\\rm mod}\\ p).$$\nBut\n$$m^{2^k}\\equiv-1\\not\\eq1\\ ({\\rm mod}\\ p),$$\nso $p\\eq1\\ ({\\rm mod}\\ 2^{k+1})$.\nNote that\n$$p^a=\\frac{m^{2^n-1}-1}{m-1}-m^n=\\sum_{k=0}^{2^n-2}m^k-m^n\\equiv 1+m+m^2\\ ({\\rm mod}\\ m^3).$$\nIf $k>0$, then $p\\eq1\\ ({\\rm mod}\\ 2^2)$ and hence\n$$p^a\\eq1\\not\\equiv 1+m\\ ({\\rm mod}\\ 2^2),$$\nwhich contradicts the congruence $p^a\\eq1+m\\ ({\\rm mod}\\ m^2)$.\nSo $k=0$, $p\\mid m^{2^0}+1$ and hence $p=m+1$. (Recall that $m+1$ is a prime.)\nIt follows that $p^a$ is congruent to $1$ or $m+1$ modulo 8. Since $1+m+m^2\\not\\eq1, m+1\\ ({\\rm mod}\\ 8)$,\nwe get a contradiction.\nThis proves part (i).\n\n(ii) Let $a>b\\gs0$ be integers with $m^a+m^b<(m^{2^n}-1)\/(m-1)$. Clearly $2^n>a>b$. Write $a-b=2^kq$ with $k\\in{\\Bbb N}$, $q\\in{\\Bbb Z}^+$ and $2\\nmid q$.\nThen $0\\leqslant k<n$ and hence $d=m^{2^k}+1$ divides both $(m^{2^n}-1)\/(m-1)$ and $m^{a-b}+1=(m^{2^k})^q+1$. Thus\n$$\\frac{m^{2^n}-1}{m-1}-m^a-m^b$$\nis a multiple of $d$. Observe that\n$$\\aligned \\frac{m^{2^n}-1}{m-1}=&\\frac{m^{2^n-2}-1}{m-1}\\left(m^{2^n-2}+1\\right)\\left(m^{2^n-1}+1\\right)\n\\\\>&\\left(m^{2^n-2}+1\\right)\\left(m^{2^n-1}+1\\right)\\geqslant(m^b+1)(m^{a-b}+1)\\geqslant m^a+m^b+d.\n\\endaligned$$\nSo $d$ is a proper divisor of $D=(m^{2^n}-1)\/(m-1)-m^a-m^b$. This shows that $D$ cannot be a prime.\nWe are done. \\qed\n\n\n\\medskip\n\\noindent {\\it Proof of Theorem \\ref{uau}}. Observe that\n$$u_0=0<u_1=1<u_2=a<u_3<u_4<\\cdots.$$\nBy induction,\n$$u_{2i}\\equiv u_0=0\\ ({\\rm mod}\\ a)\\ \\text{and}\\ u_{2i+1}\\equiv u_1=1\\ ({\\rm mod}\\ a)\\quad\\text{for}\\ i=0,1,2,\\ldots.$$\nWe will make use of these simple properties.\n\nLet $k,m\\in{\\Bbb N}$ and $l,n\\in{\\Bbb Z}^+$ with $k\\leqslant m$. Below we discuss the\nequation $u_k+au_l=u_m+au_n$.\n\n{\\it Case} 1. $k=m$.\n\n In this case,\n $$u_k+au_l=u_m+au_n\\Rightarrow u_l=u_n\\Rightarrow l=n.$$\n\n{\\it Case} 2. $k=l<m$.\n\nIf $k=l<m-1$ then\n$$u_k+au_l<u_{m-2}+au_{m-1}=u_m<u_m+au_n.$$\nWhen $k=l=m-1$, as $u_m\\not\\equiv u_{m-1}\\ ({\\rm mod}\\ a)$ we have\n$$u_k+au_l=(a+1)u_{m-1}\\not=u_m+au_n.$$\n\n{\\it Case} 3. $l<k<m$.\n\nIn this case,\n$$u_k+au_l\\leqslant u_k+au_{k-1}<au_k+u_{k-1}=u_{k+1}\\leqslant u_m<u_m+au_n.$$\n\n{\\it Case} 4. $k<l<m$.\n\nIn this case,\n$$u_k+au_l\\leqslant au_l+u_{l-1}=u_{l+1}\\leqslant u_m<u_m+au_n.$$\n\n\n{\\it Case} 5. $k<m\\leqslant l$.\n\nSuppose that $u_k+au_l=u_m+au_n$. Then\n$$u_l>\\frac{u_k+au_l-u_m}a=u_n\\geqslant u_l-\\frac{u_m}a\\geqslant\\frac{a-1}au_l\\geqslant (a-1)u_{l-1}\\geqslant u_{l-1}.$$\nIt follows that\n$$k=0,\\ m=l=2, \\ \\text{and}\\ u_n=u_{l-1}=u_1=1.$$\nThus $au_2=u_2+au_1$, i.e., $a^2=a+a$ and hence $a=2$.\n\nCombining the above we have completed the proof. \\qed\n\n\\begin{Remark}\\label{F+F=F+F} By modifying the proof of Theorem \\ref{uau}, we can determine all the solutions\nof the equation $F_k+F_l=F_m+F_n$ with $k,l,m,n\\in{\\Bbb N}$.\n\\end{Remark}\n\n\\section{Discussion of Conjecture \\ref{pFF} and its variants}\n\nConcerning Conjecture \\ref{pFF}, we mention that there are very few natural numbers not representable\nas the sum of a prime $p\\eq5\\ ({\\rm mod}\\ 6)$ and two Fibonacci numbers.\nBjorn Poonen (MIT) informed the author that by a heuristic argument there should be infinitely\nmany positive integers not in the form $p+F_s+F_t$ if we require that the\nprime $p$ lies in a fixed residue classe with modulus greater than one.\nMcNeil \\cite{M1, M3} made a computer search to find natural numbers not representable as\nthe sum of a prime $p\\eq5\\ ({\\rm mod}\\ 6)$, an odd Fibonacci number and a positive Fibonacci number; he found that there\nare totally 729 such numbers in the interval $[0,10^{14}]$, 277 of which (such as 857530546) even cannot be written\nas the sum of a prime $p\\eq5\\ ({\\rm mod}\\ 6)$ and two Fibonacci numbers.\n\nIn 2008 the author (cf. \\cite{pFF, FF}) also made the following conjecture which is similar to Conjecture \\ref{pFF}.\n\\begin{Conjecture} \\label{pFFk}\n{\\rm (i)} Any integer $n>4$ can be written as the sum of an odd prime, a positive Fibonacci number\nand the square of a positive Fibonacci number. We can require further that one of the two Fibonacci numbers is odd.\n\n{\\rm (ii)} Each integer $n>4$ can be written as the sum of an odd prime, a positive Fibonacci number\nand the cube of a positive Fibonacci number. We can require further that one of the two Fibonacci numbers is odd.\n\\end{Conjecture}\n\n\\begin{Remark}\\label{R-pFFk} Note that 900068 cannot be written as the sum of a prime, a Fibonacci number and the fourth power\nof a Fibonacci number.\nAlso,\n$$F_n^3\\sim\\frac{\\varphi^{3n}}{(\\sqrt5)^3}=\\frac{(4.236\\cdots)^n}{5\\sqrt5}\\ \\quad(n\\to +\\infty).$$\n\\end{Remark}\n\nLet $k\\in\\{1,2,3\\}$. For $n\\in{\\Bbb Z}^+$ let $r_k(n)$ denote the number of ways to write $n$ as the sum of an odd prime, a positive Fibonacci number\nand the $k$th power of a positive Fibonacci number with one of the two Fibonacci numbers odd. That is,\n$$r_k(n)=|\\{\\langle p,s,t\\rangle:\\, p+F_s+F_t^k=n,\n\\ p\\ \\text{is an odd prime},\\ s,t\\ge2, \\ \\text{and}\\ 2\\nmid F_s\\ \\text{or}\\ 2\\nmid F_t\\}|.$$\nThe author has investigated values of the quotient\n$$s_k(n)=\\frac{r_k(n)}{\\log n}$$\nvia computer, and conjectured  that\n$$c_k=\\liminf_{n\\to+\\infty}s_k(n)>0.$$\nNumerical data suggest that $2<c_1<3$.\nIn fact, the author computed all values of $s_1(n)$ with $10^{50}\\leqslant n\\leqslant 10^{50}+4\\times10^4$, and here are the two smallest values:\n$$s_1(10^{50}+39030)=2.22359\\cdots\\ \\ \\text{and}\\ \\ s_1(10^{50}+5864)=2.29037\\cdots.$$\n\nHere is another variant of Conjecture \\ref{pFF} made by the author (cf. \\cite{pFF,FL}).\n\\begin{Conjecture} \\label{pFL}\n{\\rm (i)} Any integer $n>4$ can be written as the sum of an odd prime, an odd Lucas number and a positive Lucas number.\nFor $k=2,3$ we can write any integer $n>4$ in the form $p+L_s+L_t^k$, where $p$ is an odd prime, $s,t\\gs0$,\nand $L_s$ or $L_t$ is odd.\n\n{\\rm (ii)} Each integer $n>4$ can be written as the sum of an odd prime, a positive Fibonacci number\nand twice a positive Fibonacci number (or half of a positive Fibonacci number). We can also represent\nany integer $n>4$ as the sum of an odd prime, twice a positive Fibonacci number,\nand the square of a positive Fibonacci number.\n\n{\\rm (iii)} Any integer $n>4$ can be written in the form $p+F_s+L_t$\nwith $p$ an odd prime, $s>0$, and $F_s$ or $L_t$ odd.\n\\end{Conjecture}\n\n\n\\begin{Remark}\\label{R-pFL} The author verified Conjectures \\ref{pFFk} and \\ref{pFL} for $n\\leqslant 3\\times10^7$.\nQing-Hu Hou found that 17540144 cannot be written as the sum of a prime, a Lucas number and the fourth power\nof a Lucas number. McNeil (cf. \\cite{M2}) has verified the first assertions\nin parts (i) and (ii) of Conjectures  \\ref{pFFk} and \\ref{pFL} up to $10^{12}$. He (cf. \\cite{M3}) has also verified\npart (iii) of Conjecture \\ref{pFL} up to $10^{13}$, and found that 36930553345551 cannot be written as the sum of a prime,\na Fibonacci number and an even Lucas number.\n\\end{Remark}\n\nWhat about the representations $n=p+P_s+kP_t$ with $k\\in\\{1,3,4\\}$ related to\nConjecture \\ref{pP2P}? Note that 2176 cannot be written as the sum of a prime and two Pell numbers.\nMcNeil \\cite{M3} found that\n393185153350 cannot be written as the sum of a prime, a Pell number and three times a Pell number,\nand the smallest integer greater than 7 not representable as the sum of a prime, a Pell number\nand four times a Pell number is\n$$872377759846\\approx 8.7\\times 10^{11}.$$\nThe companion Pell sequence $\\{Q_n\\}_{n\\gs0}$ is defined by\n$$Q_0=Q_1=2\\ \\text{and}\\ Q_{n+1}=2Q_n+Q_{n-1}\\ (n=1,2,3,\\ldots).$$\nMcNeil \\cite{M3} found that the smallest integer greater than 5 not representable as the sum of a prime,\na Pell number and a companion\nPell number is 169421772576.\n\nMcNeil's counterexamples seem to suggest that Conjecture \\ref{pP2P} might also have large counterexamples.\nHowever, in the author's opinion, the large counterexamples to the representations $n=p+P_s+3P_t$\nand $n=p+P_s+4P_t$ hint that they are very close to the ``truth\" (Conjecture  \\ref{pP2P}).\nCorollary \\ref{P+2P} is also a good evidence to support Conjecture  \\ref{pP2P}.\nTo expel suspicion, the author has investigated the behavior of the representation function\n$$r(n)=|\\{\\langle p,s,t\\rangle:\\ p+P_s+2P_t=n \\ \\text{with}\\ p\\ \\text{a prime}\\ \\text{and}\\ s,t\\gs0\\}|.$$\nFor $n\\in[10^{50}, 10^{50}+10081]$ most values of $s(n)=r(n)\/\\log n$ lies in the interval $(1,2)$,\nthe smallest value of $s(n)$ with $n$ in the range is\n$$s(10^{50}+10045)=\\frac{76}{\\log(10^{50}+10045)}\\approx 0.66.$$\nThe author also computed the values of $s(n)$ with $n\\in[10^{200},10^{200}+100]$,\nthe smallest value and the largest value are\n$$s(10^{200}+33)=\\frac{443}{\\log(10^{200}+33)}\\approx 0.96$$\nand\n$$s(10^{200}+18)=\\frac{824}{\\log(10^{200}+18)}\\approx 1.79$$\nrespectively. The author conjectured that\n$$c=\\liminf_{n\\to+\\infty}s(n)\\in(0.6,1.2).$$\n\n\\bigskip\n\n\\noindent{\\bf Acknowledgment}. The author wishes to thank Dr. Douglas McNeil who has checked almost all conjectures\nmentioned in this paper (on the author's request) via his quite efficient and powerful computation.\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect:intro}\n\n\n\n\\begin{table*}\n\t\\centering\n\t\\caption{\n\tCharacteristics of our models, with initial pre-stellar core mass \n\t$M_{\\rm c}$, final protostellar age $t_{\\rm end}$, final stellar mass $M_{\\star}$, details of their corresponding accretion rate histories \\textcolor{black}{and of the initial conditions of the stellar evolution calculations}.   \n\t}\n\t\\begin{tabular}{lccccr}\n\t\\hline\n\t${\\rm {Models}}$             &  $M_{\\rm c}$ ($\\rm M_{\\odot}$)      &    $t_{\\rm end}$ ($\\rm kyr$)   &   $M_{\\star}$ ($\\rm M_{\\odot}$) & ${\\rm {Accretion}\\, \\rm {rate}}$ &   ${\\rm {Method}}$     \\\\ \n\t\\hline    \n\t{\\rm Run-100-hydro}     &  $100$                              &    $50.0$                      &  $33.3$     &  Episodic      &    Full hydrodynamical simulation with variable rate   \\\\  \n\t{\\rm Run-100-constant}  &  $100$                              &    $50.0$                      &  $33.3$     &  Constant      &    Hydrodynamical simulation (collapse) + constant analytic rate (disc)    \\\\ \n\t{\\rm Run-100-smoothed}    &  $100$                            &    $50.0$                      &  $33.3$     &  Smoothed      &    Hydrodynamical simulation (collapse) + smoothed burst-free rate (disc) \\\\\n\t\\textcolor{black}{{\\rm Run-100-compact}}    &  \\textcolor{black}{$100$}  &  \\textcolor{black}{$50.0$}   &  \\textcolor{black}{$33.3$}  &  \\textcolor{black}{Episodic}   &  \\textcolor{black}{As Run-100-hydro, with more compact initial conditions} \\\\ \n\n\t{\\rm Run-60-hydro}      &  $60$                               &    $65.2$                      &  $20.0$     &  Episodic      &    Full hydrodynamical simulation with variable rate   \\\\  \n\t\\hline    \n\t\\end{tabular}\n\\label{tab:models}\\\\\n\\end{table*}\n\n\n\nGravitationally-collapsing pre-stellar cores give birth to new young stars, which grow by mass \naccretion of surrounding molecular material. The rates at which protostars \ngain mass have been shown to exhibit such a diversity that the initial picture of isotropic \ncollapse~\\citep{larson_mnras_145_1969,shu_apj_214_1977} fails to explain the observed spread \nin accretion rates~\\citep{vorobyov_apj_704_2009}. Meanwhile, variations \nof the protostellar accretion rate can induce enormous changes in protostellar luminosity, the most extreme \nmanifestations of which are the so-called FO-Orionis and the very low luminosity \nobjects~\\citep{2017A&A...600A..36V}. \nNumerical simulations have demonstrated that this is possible thanks to the presence of a self-gravitating \ncircumstellar accretion disc prone to gravitational fragmentation~\\citep{vorobyov_apj_633_2005,vorobyov_apj_719_2010,vorobyov_apj_805_2015,machida_mnras_413_2011,zhao_mnras_473_2018,nayakshin_mnras_426_2012,2018arXiv180607675V}.\n\n\n\nThe disc fragmentation scenario equivalently applies to star formation in the high-mass regime. \nNumerical simulations predicted the formation of accretion discs around massive young stellar \nobjects~\\citep{1998MNRAS.298...93B,2002ApJ...569..846Y,peters_apj_711_2010,seifried_mnras_417_2011}, \ntogether with additional structures such as bipolar cavities filled with ionizing \nradiation generated by the UV feedback of the protostars~\\citep{harries_mnras_448_2015,klassen_apj_823_2016,harries_2017}. \nAccretion variability is a direct consequence of asymmetries in the accretion flow~\\citep{seifried_mnras_417_2011} and of the \ncoupling between the prostellar radiation feedback and its surrounding disc~\\citep{peters_apj_711_2010}. \nSuch variability is a generic feature of massive star formation in the sense that it is neither \nstopped by the radiation pressure in the bipolar \\hii regions~\\citep{peters_apj_725_2010} nor \nby disc fragmentation~\\citep{meyer_mnras_473_2018}.\n\\textcolor{black}{ \nOther studies on massive star formation also reported time-variabilities of the disc-to-star mass \ntransfer rate~\\citep{krumholz_sci_323_2009,rosen_mnras_463_2016}. \n}\nIn addition, self-gravitating discs around high-mass stars are subject to \nefficient gravitational instabilities, generating heavy spiral arms in which dense gaseous clumps form, \neventually leading to the formation of multiple hierarchical systems~\\citep{krumholz_apj_656_2007,peters_apj_711_2010}. \nThese circumstellar clumps can either rapidly migrate onto the stellar surface, trigger increases of the \nprotostellar accretion rate which aggravate the variability~\\citep{meyer_mnras_473_2018} and produce luminous \nbursts~\\citep{meyer_mnras_464_2017} or evolve to secondary low-mass stars which finally end up as low-mass \nspectroscopic companions to the MYSOs~\\citep{meyer_mnras_473_2018}. \n\\textcolor{black}{\nAccretion bursts are a feature of the formation of young massive stellar objects which seems common to \nmost massive protostars as it does not depend on the initial properties of their parent pre-stellar core~\\citep{2018arXiv181100574A}. \n}\n\\textcolor{black}{\nMoreover, although these eruptive phases represent \na small fraction ($\\sim \\%$) of their early formation time, MYSOs can acquire a substential fraction \nof their zero-age-main-sequence (ZAMS) mass via these flaring episods~\\citep{2018arXiv181100574A}.\n}\n\n\n\nFrom the point of view of observations, (variable) accretion \nflows~\\citep{keto_apj_637_2006,stecklum_2017a} and ionized, pulsed, collimated structures~\\citep{Cunningham_apj_692_2009,\ncesaroni_aa_509_2010,caratti_aa_573_2015,purser_mnras_460_2016,reiter_mnras_470_2017,burns_mnras_467_2017,arXiv180102211B,purser_mnras_475_2018,2018arXiv180311413S} underlined the scaled-up character of massive star formation with respect to low-mass stars~\\citep[see also][]{fuente_aa_366_2001,testi_2003,\ncesaroni_natur_444_2006,stecklum_2017b}. \nA growing number of observations of (Keplerian) discs around MYSOs have been  \nreported~\\citep{johnston_apj_813_2015,ilee_mnras_462_2016,forgan_mnras_463_2016,2018arXiv180410622G}, together \nwith evidences of a spiral filament feeding the candidate disc MM1-Main~\\citep{maud_467_mnras_2017}  \nand an infalling gaseous clump in the double-cored system G350.69-0.49~\\citep{chen_apj_835_2017}. \nInterestingly, a recent {\\it ALMA} view of the massive young object G023.01-00.41 exhibited a clear \ndisc-jet association~\\citep{2018arXiv180509842S}.  \nIn addition, some objects revealed the presence of high-mass proto-binary systems within a \ncircumbinary disc~\\citep{kraus_apj_835_2017}. \nFinally, several MYSOs experienced multi-wavelengths flares~\\citep{moscadelli_aa_600_2017,cesaroni_2018} \nin a fashion of the predictions of~\\citet{meyer_mnras_464_2017}, among which are the accretion-bursts \nof S255IR NIRS\\,3~\\citep{fujisawa_atel_2015,stecklum_ATel_2016,caratti_nature_2016} and\nfrom NGC6334I-MM1~\\citep{2017arXiv170108637H} experienced accretion-driven outbursts. \n\n\n\n\n\\begin{figure*}\n        \\begin{minipage}[b]{ 0.95\\textwidth}\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_rates.pdf}\n        \\end{minipage} \\\\                 \n        \\caption{ \n                 Accretion rate and \n                 protostellar mass evolution in our \n                 models with a pre-stellar core mass of $100\\, \\rm M_{\\odot}$ (a) and \n                 $60\\, \\rm M_{\\odot}$ (b), respectively. \n                 }      \n        \\label{fig:rate}  \n\\end{figure*}\n\n\n\nEarly stellar evolution calculations demonstrated the dominant influence of accretion on the internal  \nstellar structure prior to the ZAMS phase~\\citep{palla_apj_375_1991}. \nThe formation of a radiative barrier that turns the fully convective stellar embryo into a stable \nradiative core and an outer convective shell that causes the star to swell has been shown in the context of young \nintermediate-mass stars for a wide range of different constant accretion \nrates~\\citep{palla_apj_392_1992,palla_1993,beech_apjs_95_1994,bernasconi_aa_307_1996,bernasconi_aa_120_1996}. \nA monotonically increasing accretion rate yields similar results~\\citep{behrend_aa_373_2001}. \nCalculations with high-rate mass accretion exceeding $10^{-3}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$ equivalently \nshowed that disc-accreting young massive stellar objects have their pre-main-sequence evolution governed by \nthe accretion of circumstellar material, see~\\citet{bernasconi_aa_307_1996,norberg_aa_159_2000,behrend_aa_373_2001,hosokawa_apj_691_2009,hosokawa_apj_721_2010}. \nThese models reported a rapid swelling of the protostars up to radii $\\sim 1000\\, \\rm R_{\\odot}$ produced by the \nso-called luminosity wave~\\citep{larson_mnras_157_1972}, an internal redistribution of entropy following the abrupt \ndecrease of the opacity in the protostellar interior. \nInterestingly, for the constant accretion rates $\\ge\\, 10^{-2}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$, the protostars evolve to the \nred part of the Hertzsprung-Russell diagram~\\citep{hosokawa_apj_721_2010,haemmerle_585_aa_2016}. The authors postulate therein that the \\hii region \ngenerated by those stars can then disappear as their protostellar radius bloats, ultimately leading to lower-luminosity young massive stellar objects. \nSimultaneous hydrodynamical and stellar evolution calculations revealed \nthat variable-accreting MYSOs can experience a unique loop to the red before recovering their bluer characteristics \nand reach the ZAMS~\\citep{kuiper_apj_772_2013}. However, those 2.5-dimensional axisymmetric simulations do not \naccount for disc fragmentation physics and its influence on the protostellar variability~\\citep{meyer_mnras_464_2017} \nand the corresponding rates were burstsless. \nMore complex theoretical studies tackled the problem of the impact of various feedback mechanisms of MYSOs \nonto star formation efficiency, however, without considering their accretion variability~\\citep{2018arXiv180401132T}.  \n\n\n\nMotivated by the above arguments, we aim at investigating the effects of the repetitive \naccretion events responsible for luminous bursts on the evolution of pre-main-sequence young massive \nstellar objects. With the help of three-dimensional gravito-radiation-hydrodynamics models of \ncollapsing rotating massive pre-stellar cores, we first simulate the formation and evolution \nof fragmenting circumstellar accretion discs from which protostars gain mass~\\citep{meyer_mnras_473_2018}. \nFrom the disc models, we extract variable accretion rate histories, interspersed by episodic accretion bursts \ncaused by dense gaseous clumps that form in spiral arms and rapidly migrate onto the protostars~\\citep{meyer_mnras_464_2017}.  \nFinally, we calculate stellar evolution models with the Geneva stellar evolutionary code~\\citep{eggenberger_apss_316_2008,haemmerle_phd_2014} \nfed by our accretion rate histories. Using the method developed in the context of constant-accreting  \nMYSOs~\\citep{haemmerle_585_aa_2016,haemmerle_602_aap_2017} and accretion flows onto large-scale \\hii regions~\\citep{haemmerle_458_mnras_2016}, \nwe calculate the changes in the internal structure and the surface properties of MYSOs experiencing bursts. \n\n\n\nIn this paper, we investigate the effects of variable accretion on the evolution of MYSOs.  \nWe perform numerical gravito-radiation-hydrodynamics simulations and stellar evolution calculations of pre-main-sequence accreting \nmassive stellar objects to explore the effects of strong accretion bursts onto their internal as well as their surface properties \nand evolutionary path in the Hertzsprung-Russell diagram. This study is organized as follows. \nIn Section~\\ref{sect:methods}, we review the methods utilised to perform (i) gravito-radiation-hydrodynamical \nsimulations of the monolithic collapse of present-day rotating pre-stellar cores, from which we extract \naccretion rate histories, and (ii) stellar evolutionary calculations of MYSOs which accrete from their \ncircumstellar discs at pre-calculated time-variable rates. Our results are presented in Section~\\ref{sect:results}, \n\\textcolor{black}{\nthe effects of the initial conditions on the stellar excursions are investigated in Section~\\ref{sect:ic},  \n}\nand our findings further discussed in Section~\\ref{sect:discussion}. \nParticularly, we highlight that strong outbursts provoke rapid excursions towards colder regions of the \nHerztsprung-Russel diagram, which typically are not associated with such hot and young stellar objects. \nFinally, we discuss our results and conclude in Section~\\ref{sect:cc}.\n\n\n\n\n\n\n\\section{Modelling}\n\\label{sect:methods}\n\n\n\nIn the following paragraphs, we introduce the reader to the method employed to carry out our \ngravito-radiation-hydrodynamical simulations of high-mass star formation, from which we extract time-dependent \nprotostellar accretion rate histories. Furthermore, we detail how the outputs of the hydrodynamical \nmodels are subsequently used as boundary conditions for stellar evolution calculations. \n\n\n\n\\subsection{Hydrodynamical simulations}\n\\label{sect:hydro}\n\n\n\nOur three-dimensional midplane-symmetric hydrodynamical simulations are initialized with a rigidly-rotating \nspherically symmetric, pre-stellar core of density distribution $\\rho(r)\\propto r^{\\beta_{\\rho}}$, \nwith $\\beta_{\\rho}=-3\/2$~\\citep{butler_apj_754_2012,butler_apj_782_2014} and $r$ is the radial coordinate.  \nThe inner edge of the core is made of a semi-permeable sink cell centered onto the origin of the \ncomputational domain and the outer edge of the core, at $R_{\\rm c}=0.1\\, \\rm pc$, is assigned to the outflow boundary conditions. \nWe map the domain $[\\textcolor{black}{r_{\\rm in}},R_{\\rm c}]\\times[0,\\pi\/2]\\times[0,2\\pi]$ with a mesh \nof $N_{\\rm r}=128\\times\\,N_{\\rm \\theta}=11\\times\\,N_{\\rm \\phi}=128$ grid zones, logarithmically expanding  \nalong the $r$-direction, as a cosine in the polar $\\theta$-direction, and uniformly spaced along the azimuthal $\\phi$-direction. \n\\textcolor{black}{As in~\\citet{2018arXiv181100574A}, we use a size of the sink cell of $20\\, \\rm au$, which is larger} than that of the \nfirst paper of this series~\\citep{meyer_mnras_464_2017} in order \nto reach longer integration times $t_{\\rm end}$, while avoiding very restrictive Courant-Friedrich-Levy conditions \non the time-step within the direct protostellar surroundings.  \nWe simulate the gravitational collapse of several pre-stellar cores with different initial masses Mc. \nEach core forms a central protostar and a massive circumstellar disc.\nThe accretion rate from the disc onto the protostar is computed as the rate of mass transport $\\dot{M}$ through the sink cell. \nThe pre-stellar core temperature is uniformly set to $T_{\\rm c}=10\\, \\rm K$ and its rotational-by-gravitational \nenergy ratio is set to a typical value of $\\beta=4\\, \\%$~\\citep{meyer_mnras_464_2017}. The models are run \nuntil the mass of the central star $M_{\\star}$ becomes equal to one third that of the initial mass core $M_{\\rm c}$. \nThe characteristics of our models are summarised in Tab.~\\ref{tab:models}. \n\n\n\nTo solve the evolution of the above described physical system, we numerically integrate the  \nequations of gravito-radiation-hydrodynamics with the {\\sc pluto} \ncode\\footnote{http:\/\/plutocode.ph.unito.it\/}~\\citep{mignone_apj_170_2007,migmone_apjs_198_2012}. \nOur method takes into account the direct irradiation of the protostar and radiation transport in \nthe accretion disc within the gray approximation using the scheme \nof~\\citet{kolb_aa_559_2013}\\footnote{http:\/\/www.tat.physik.uni-tuebingen.de\/~\\,pluto\/pluto\\_radiation\/} \nadapted following the prescriptions of~\\citet{meyer_mnras_473_2018}, see also equivalent radiation-hydrodynamics \nmethods implementations in e.g.~\\citet{commercon_aa_529_2011},~\\citet{flock_aa_560_2013} and~\\citet{bitsch_aa_564_2014}.\nThe photospheric photons are first ray-traced from the stellar atmosphere and propagate by flux-limited \ndiffusion into the circumstellar disc. Such method permits to treat the disc thermodynamics accurately, with \nits central heating together with the outer cooling of the discs, as predicted by~\\citet{vaidya_apj_742_2011}.  \nOpacities and calculation of the dust properties are as in~\\citet{meyer_mnras_473_2018}, i.e. estimated \nby assuming that disc silicate grains are in equilibrium with the total radiation field. \nThe gravitational force includes the gravity of the growing young massive star and the self-gravity of the gas, the latter computed using the prescriptions \nof~\\citet{black_apj_199_1975} and the implementation method inspired by~\\citet{hirano_sci_357_2017}\\footnote{https:\/\/shirano.as.utexas.edu\/SV.html} \nby time-dependently solving the Poisson equation with the help of the PETSC library\\footnote{https:\/\/www.mcs.anl.gov\/petsc\/}. \nWe assume that the angular momentum transport in the disc is essentially produced by the spiral arms in \nthe discs and we therefore do not include additional turbulent $\\alpha$-viscosity~\\citep{meyer_mnras_473_2018}. \n\n\n\n\\subsection{Stellar evolution calculations}\n\\label{sect:evol}\n\n\n\nWe used our accretion rate histories to compute the evolutionary tracks of \nour MYSOs in the Hertzsprung-Russel diagram. \nThe one-dimensional stellar evolution calculations were performed with the hydrostatic {\\sc Geneva} \ncode, the original version of which~\\citep{eggenberger_apss_316_2008} has recently been updated for disc accretion physics in \nthe context of pre-main-sequence massive protostars~\\citep{haemmerle_phd_2014}. The code was tested \nin the context of constant-accreting MYSOs~\\citep[see details relative to the numerical scheme and \nthe implementation method in][]{haemmerle_585_aa_2016} and showed full consistency \nwith the original results on high-constant-rate accreting MYSOs of~\\citet{hosokawa_apj_721_2010}. \nAccretion is treated within the so-called {\\it cold disc accretion} scenario~\\citep{palla_apj_392_1992}, which \nassumes that the inner disc region is geometrically thin when the accreted material reached the stellar surface. \nHence, we follow the mass growth of the hydrostatic core (i.e., the protostar), without considering a spherical \nenvelope during the accretion phase~\\citep{palla_apj_392_1992}. \nSubsequent theoretical and numerical works demonstrated that such assumption is fully \nreasonable~\\citep{vaidya_apj_742_2011,meyer_mnras_473_2018}. Any entropy excess is radiated away \nin direction perpendicular to the disc and it is channeled into the radiatively-driven outflow associated to young massive \nstars~\\citep{harries_2017}. The circumstellar material is advected inside the protostar assuming that its \nthermal properties are similar to those of the suface layer of the MYSOs. \n\n\nSince we assume that most of the energy produced by the accretion shock (not modelled in the \ncalculations) is radiated away before it reaches the protostellar surface, no additional entropy from the \nliberation of gravitational energy is added to the surface of the star. Such an assumption is the lower limit \non the entropy attained by the star during the accretion process, while the upper limit is the \nso-called spherical (or hot) accretion scenario, scenario, in which a fraction of accretion entropy is added \nto the star, see the\nsketch in fig.~1 of~\\citet{hosokawa_apj_721_2010}. We choose the cold scenario as it has recently been \nused in the context of accreting \\hii regions~\\citep{haemmerle_458_mnras_2016}. \nCalculations of the stellar structure are performed with the Henyey method within the Lagrangian formulation~\\citep{haemmerle_phd_2014}  \nat solar metallicity (Z=0.014) using the abundances of~\\citet{asplund_ASPC_2005} and~\\citet{cunha_apj_647_2006} \nand the deuterium mass fractions of~\\citet{norberg_aa_159_2000} and~\\citet{behrend_aa_373_2001}.  \nThe simulations make use of the Schwarzschild criterion for convection, overshooting is \nconsidered and they are initialised with fully convective stellar embryo because they are the most \ndifficult models to bloat~\\citep{haemmerle_585_aa_2016,haemmerle_458_mnras_2016}. \nHence, our method threats the stellar swelling most conservatively, avoiding any artificial effects \nthat can lead to excessive swelling. \nTo facilitate the stellar evolution calculations, we average the accretion rate histories \nover a time period of $10\\, \\rm yr$. This excludes the smallest variabilities. However, one can \njustify such an assumption as we know that high accretion rates ($\\ge 10^{-2}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$) responsible for \nprotostellar bloating are exclusively reached during the strongest and longest accretion \nbursts~\\citep{meyer_mnras_464_2017,meyer_mnras_473_2018}. \n\n\n\n\n\n\\section{Results}\n\\label{sect:results}\n\n\nThis section presents the accretion rate histories of our MYSOs and investigates the effects \nof the accretion of dense gaseous clumps on the structure and evolutionary path of high-mass \nprotostars in the Hertzsprung-Russell diagram. \n\n\n\\begin{figure}\n        \\centering\n        \\begin{minipage}[b]{ 0.49\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_structure.png}\n        \\end{minipage}        \n        \\caption{ \n        \t\t Kippenhahn diagram our of MYSOs generated with an initial $100\\, \\rm M_{\\odot}$ \n        \t\t pre-stellar core. It shows the evolution of the internal stellar structure as a  \n        \t\t function of time. The blue and orange regions denote the convective \n        \t\t and radiative parts of the protostar, respectively. \n                 }      \n        \\label{fig:structure}  \n\\end{figure}\n\n\n\\begin{figure*}\n        \\centering\n        \\begin{minipage}[b]{ 0.9\\textwidth}  \\centering\n                \\includegraphics[width=0.9\\textwidth]{.\/plot_lum_wave_1.pdf}\n        \\end{minipage}     \n        \\caption{ \n        \t\t Internal luminosity profiles (top panels) and gradient of luminosity profiles (bottom panels) \n        \t\t in our massive protostellar model Run-100-hydro. \n        \t\t The panels illustrate the development of the luminosity \n        \t\t wave (left panels) and the effect of a burst (right panels) onto the structure of a growing MYSO. \n                 }      \n        \\label{fig:lum_wave}  \n\\end{figure*}\n\n\n\\begin{figure*}\n        \\centering\n        \\begin{minipage}[b]{ 0.8\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_timescales_mdot.pdf}\n        \\end{minipage}        \n        \\caption{ \n        \t\t Characteristic timescales of our bursting protostars accreting at \n        \t\t variable accretion rates and generated from an initial $100\\, \\rm M_{\\odot}$ \n        \t\t (a,b) and $60\\, \\rm M_{\\odot}$ (c,d) pre-stellar core, respectively.\n        \t\t %\n                 Panels (a) and (c) plot the Kelvin-Helmholtz timescale $t_{\\rm KH}$ (thick solid \n                 blue line, in $\\rm kyr$), the accretion timescale $t_{\\rm acc}$ (thin dotted red \n                 line, in $\\rm kyr$) and the protostellar mass (thin solid black line, in $M_{\\odot}$).   \n                 %\n                 Panels (b) and (d) show the accretion rate onto the protostars (thick solid \n                 red line, in $M_{\\odot}\\, \\rm yr^{-1}$), the ratio $t_{\\rm KH}\/t_{\\rm acc}$ \n                 (thin solid black line) and the thin dotted black line corresponds to $t_{\\rm KH}\/t_{\\rm acc}=1$. \n                 }      \n        \\label{fig:timescale}  \n\\end{figure*}\n\n\n\\subsection{Accretion rate histories}\n\\label{sect:rates}\n\n\nFig.~\\ref{fig:rate} shows the accretion rate histories onto the MYSOs forming during the \ngravitational collapse of $100\\, \\rm M_{\\odot}$ (a) and $60\\, \\rm M_{\\odot}$ (b) pre-stellar cores, respectively. \nThe accretion rates (in $\\rm M_{\\odot}\\, \\rm yr^{-1}$) are plotted as a function of time (in $\\rm kyr$), from \nthe beginning of the collapse to the end of the simulation when $M_{\\star}=M_{\\rm c}\/3$. \nThe thin black vertical line marks the onset of disc formation when the free-collapsing material \nof the envelope begins to land on a centrifugally balanced circumstellar disc instead of keeping \non directly falling onto the protostellar surface. From this time instance on, the protostar \nstarts gaining mass via accretion from the disc. \nThe thick, solid lines represent the accretion rate onto the young high-mass stars while the dashed \nlines are the corresponding protostellar masses (in $\\rm M_{\\odot}$). \nFurthermore, we indicate by the orange circles the time instance when the MYSOs enter\nthe high-mass regime ($\\rm M_{\\star} > 8\\, \\rm M_{\\odot}$). \nWe run two distinct hydrodynamical simulation with $M_{\\rm c}=100\\, \\rm M_{\\odot}$ (model Run-100-hydro) \nand with $M_{\\rm c}=60\\, \\rm M_{\\odot}$ (model Run-60-hydro), and in order to particularly investigate the \neffects of the accretion spikes on the protostellar evolution, we construct additional accretion rate histories \nby modifying the accretion rate of our model Run-100-hydro (our Table~\\ref{tab:models}), by keeping constant \nthe final mass of $\\approx 33.3\\, \\rm M_{\\odot}$ and either (i) imposing a constant rate once the \ndisc has formed (model Run-100-constant, see black lines of Fig.~\\ref{fig:rate}a) or (ii) filtering out the \naccretion spikes (model Run-100-smoothed, see red lines of Fig.~\\ref{fig:rate}a) with a method similar to \nthat described in~\\citet{vorobyov_apj_805_2015}. \n\n\n\nThe free-fall collapse of the molecular pre-stellar core produces an initial infall of material through \nthe sink cell increasing the accretion rate during the first $\\approx 12$-$15\\, \\rm kyr$, up to reaching the \ncanonical value of $\\approx 10^{-3}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$ at which MYSOs are predicted \nto accrete~\\citep{hosokawa_apj_691_2009}. \nOnce the disc has formed (see vertical thin line of Fig.~\\ref{fig:rate}), variability \nimmediately develop in the accretion flow as a direct consequence of important anisotropies \nin the protostellar surroundings~\\citep{meyer_mnras_473_2018}. \nThe variations amplitude in the accretion rate continues increasing up to being interspersed by \nviolent accretion spikes becoming more numerous and more intense as a function of time. \nThey are regularly generated by the rapid migration of massive disc fragments forming in its \nouter region by gravitational fragmentation and falling onto the protostellar surface, provoking \nsudden increases of the accretion rate. Those dense gaseous clumps detached from \nspiral arms developing in the self-gravitational discs are responsible for violent accretion-driven \nluminosity bursts~\\citep{meyer_mnras_464_2017}. Such mechanism is connected \nto the formation of spectroscopic binary companions to the protostars, as long as the clumps sufficiently\ncontract in the core and heat up beyond the dissociation temperature of molecular \nhydrogen~\\citep{meyer_mnras_473_2018}. \nThe protostellar mass evolution reflects the accretion history from the disc in the sense that \nto each strong accretion events ($\\dot{M} \\ge 10^{-1}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$) corresponds \na step-like increase of the stellar mass~\\citep{meyer_mnras_464_2017}. \nThe integration time of model Run-60-hydro is longer ($\\approx 50\\, \\rm kyr$) than that of \nRun-100-hydro ($\\approx 65.2\\, \\rm kyr$) because we perform the simulations up to the time \ninstance when $M_{\\star}=M_{\\rm c}\/3$. This is longer in the case of the lower mass pre-stellar \ncore ($M_{\\rm c}=60\\, \\rm M_{\\odot}$) since the mass infall onto the disc is intrinsically smaller.  \n\n\n\n\\subsection{Stellar structure evolution}\n\\label{sect:structure}\n\n\nFig.~\\ref{fig:structure} plots the evolution of the stellar structure of our protostars as a function \nof the stellar age for our models with $M_{\\rm c}=100\\, \\rm M_{\\odot}$, distinguishing from different \naccretion histories. The figure indicates, in addition to the temporal variations in photospheric \nradius, the convective (blue) and radiative (orange) regions of the protostar, respectively.  \nThe protostars are initially fully convective (blue regions) with an internal temperature profile too \ncold to ignite Deuterium burning. As its center heats up by gravitational contraction, a radiative core \nforms and grows in mass while rapidly expanding towards the stellar surface once the Deuterium fuel is out. \nThe accreting stars further evolve once energetic photons are released out of the radiative core and propagate upwards to \nbe absorbed by the still cold and convective enveloppe. During the gradual diminishing of the convective envelope \nthickness (when the protostars of our Run-100-hydro is $\\approx 10\\, \\rm M_{\\odot}$, see Fig.~\\ref{fig:structure}a) \nconcludes the phase transition from radiative to convective stellar interior. \nAs the protostellar mass increases, the the so-called luminosity wave mechanism takes place~\\citep{larson_mnras_157_1972}. \nWe illustrate in Fig.~\\ref{fig:lum_wave} the development of the luminosity wave of the growing protostar in our model Run-100-hydro. \nA maximum in the luminosity profile forms (Fig.~\\ref{fig:lum_wave}a) and moves outwards until it reaches the stellar surface \nand adopts a strictly monotonically increasing shape, i.e. the luminosity gradient is positive everywhere (Fig.~\\ref{fig:lum_wave}c). \nThe energy equation of the stellar structure reads as, \n\\begin{equation}\n\t\\frac{ dL_\\star(r) }{ dM_\\star(r) } = -T_{\\rm eff} \\frac{ ds_\\star }{ dt }, \n\t\\label{eq:structure}\n\\end{equation}\nwhere $L_\\star(r)$, $M_\\star(r)$, $T_{\\rm eff}$ and $s_\\star$ are the internal luminosity, mass, temperature and specific entropy radial profiles. \nIt implies that interior to  the luminosity peak (at $r<R_{\\rm c}$) the entropy of the star decreases with time, i.e., \n\\begin{equation}\n\t\\frac{ dL_\\star(r) }{ dM_\\star(r)}>0 \\Rightarrow \\frac{ ds_\\star }{ dt }<0 \\mbox{ if } r<R_{\\rm c},\\\\\n\t\\label{}\n\\end{equation}\nwhile exterior to the luminosity peak (at $r>R_{\\rm c}$) the entropy of the star increases with time, i.e.,\t\n\\begin{equation}\n\t\\frac{ dL_\\star(r) }{ dM_\\star(r)}<0 \\Rightarrow \\frac{ ds_\\star }{ dt }>0 \\mbox{ if } r>R_{\\rm c}. \n\t\\label{}\n\\end{equation}\nTherefore, the entropy of the surface layers ($r>R_{\\rm c}$) \nincrease as they absorb energy triggering the stellar bloating, whereas when the $L(r)$ peak reaches the stellar surface at \n$r=R_{\\star}$ all the layers lose entropy and the protostar contracts. Each time a violent accretion event \nwith an accretion peak of $\\approx 10^{-2}$$-$$10^{-1}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$ deposits mass to the star, an \nother luminosity maximum sets in the interior (Fig.~\\ref{fig:lum_wave}b) and the luminosity wave forms again in \nthe upper layers of the protostar (Fig.~\\ref{fig:lum_wave}d), provoking a new swelling episod. \nOur model with $M_{\\rm c}=60\\, \\rm M_{\\odot}$ gives similar results. \n\n\n\n\n\n\n\n\n\\begin{figure}\n        \\centering\n        \\begin{minipage}[b]{ 0.475\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_stars.pdf}\n        \\end{minipage}        \n        \\caption{ \n        \t\t Evolution as a function of time (in $\\rm kyr$) of the stellar surface luminosity (a), \n        \t\t stellar radius (b) and effective temperature (c) of our MYSOs experiencing \n        \t\t variable disc accretion interspersed by bursts. The panels distinguish between the models assuming \n        \t\t an initial $60\\, \\rm M_{\\odot}$ (thick dotted red line) and $100\\, \\rm M_{\\odot}$ \n        \t\t (thin solid blue line) pre-stellar core, respectively. \n                 }      \n        \\label{fig:stellar_100}  \n\\end{figure}\n\n\n\\begin{figure*}\n        \\centering\n        \\begin{minipage}[b]{ 0.9\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_hdrs.pdf}\n        \\end{minipage}               \n        \\caption{ \n        \t\t Hertzsprung-Russell diagram of our proto-stellar models with a similar pre-stellar core mass of \n        \t\t $100\\, \\rm M_{\\odot}$, but experiencing different initial accretion rate histories (a) \n        \t\t and evolutionary tracks with pre-stellar cores of different initial masses \n        \t\t of $60$ and $100\\, \\rm M_{\\odot}$, respectively (b).\n        \t\t Grey dotted solid lines are isoradius and the thick black dashed line is the \n        \t\t zero-age-main-sequence (ZAMS) track of~\\citet{ekstroem_aa_537_2012}.  \n                 }      \n        \\label{fig:hdr}  \n\\end{figure*}\n\n\nFig.~\\ref{fig:timescale} presents two characteristic timescales, which are usually used to describe \nthe evolution of accreting protostars, and reports their time evolution next to the accretion rate \nhistory and the protostellar mass evolution. More specifically, the figure plots the so-called accretion timescale, \n\\begin{equation}\n\tt_{\\rm acc} = \\frac{ M_{\\star} }{ \\dot{M} }, \n\t\\label{eq:tacc}\n\\end{equation}\nrepresenting the characteristic timescale of mass growth of the accreting star (thin dashed red lines), and the Kelvin-Helmholtz timescale,\n\\begin{equation}\n\tt_{\\rm KH} = \\frac{G  M_{\\star}^{2} }{ R_{\\star} L_{\\star} }, \n\t\\label{eq:kh}\n\\end{equation}\nrelated to the entropy loss through radiation (thick solid blue lines). The protostar begins accreting at          \na variable rate once the disc forms (thick red lines), with $t_{\\rm acc}<t_{\\rm KH}$ at times when their evolution is \nonly governed by disc accretion instead of mass infall from the pre-stellar core. \nDuring the gravitational collapse and the early disc evolution, the protostars evolve by gaining \nentropy by advection, with negligible radiative loss ($t_{\\rm acc}<t_{\\rm KH}$, see Fig.~\\ref{fig:timescale}b,d). \nNote that our model Run-100-hydro is already a MYSOs when the first burst happens, whereas our model \nRun-60-hydro experiences its first swelling as an intermediate-mass star (thin solid line of Figs.~\\ref{fig:timescale}a,c). \nOnce the photospheric luminosity has grown enough, the energy loss governs the protostellar evolution \n($t_{\\rm acc}>t_{\\rm KH}$ because $t_{\\rm KH} \\propto 1\/R_{\\star}L_{\\star}$ throughout the bursts) \nand \\textcolor{black}{$L_{\\rm acc}<L_{\\star}$}, which means that $L_{\\rm acc}$ can be \nneglected (see Section~\\ref{sect:excursions}). \nInterestingly, the protostars experience episods with $t_{\\rm KH}\/t_{\\rm acc}>1$ at the same \ntime instance of the swelling. The MYSOs intermittently see their evolution ruled by mass \naccretion (the bloating) before recovering a more classical evolution \ngoverned by energy losses (the unswelling). This repeats each time an accretion burst happens. \nTherefore, each accretion-driven burst corresponds to a swelling episod of the MYSOs followed by a redistribution \nof the internal entropy restoring the internal thermal equilibrium. \n\n\n\n\\subsection{Evolution of the surface properties of the MYSOs}\n\\label{sect:episodic}\n\n\nIn Fig.~\\ref{fig:stellar_100}, we show the evolution of the protostellar iternal photospheric luminosity (a), \nradius (b), and effective temperature (c) as a function of the stellar age of our MYSOs. \nThe Figure distinguishes between the variable-accreting models with a \\textcolor{black}{$100\\, \\rm M_{\\odot}$} pre-stellar core (thin solid blue lines) \nand with a \\textcolor{black}{$60\\, \\rm M_{\\odot}$} pre-stellar core (thick dashed red line), respectively. \nWe assume that the MYSOs are black-bodies, therefore, the photospheric luminosity is estimated as,\n\\begin{equation}\n\tL_{\\star} = 4 \\pi R_{\\star}^{2} \\sigma T_{\\rm eff}^{4},\n\t\\label{eq:L}\n\\end{equation}\nwhere $\\sigma$ is the Stefan-Boltzman constant. \nThe stellar surface properties does not evolve much during the free-fall gravitational collapse onto the \nMYSOs and the early phase of the disc formation, at times $\\le 20\\, \\rm kyr$. Only a moderate and monotonical \nincrease of the stellar radius and corresponding photospheric luminosity occurs, as the deuterieum burning maintains the \ncentral temperature nearly constant (Fig.~\\ref{fig:stellar_100}a). \nWhen the variations of the accretion rate substantially increase in response to the growing strength of gravitational \ninstability and fragmentation in the circumstellar disc, the protostar grows faster, \nand, after the first luminosity wave, its surface becomes radiative so that any episodic deposit of \nmass on it generates an augmentation of the effective temperature and the surface luminosity. \nWith one important exception, the time evolution of stellar surface properties is similar to what was \nfound in the context of calculations carried out with a constant accretion rate -- the photospheric \nluminosity (Fig. 5a) and effective temperature (Fig. 5c) generally increase, while the stellar \nradius decreases (Fig. 5b) for as long as the star remains in the quiescent accretion phase.\nHowever, this monotonic behaviour is interspersed  with brief excursion events associated with violent accretion bursts, during \nwhich the MYSOs adopt opposite photospheric properties by becoming bigger and cooler. \nSuch a behaviour of $R_{\\star}$ and $T_{\\rm eff}$ is a direct consequence of the changes in the internal \nstructure of the MYSOs, which is sensitive to the accretion rate. In the next section, we demonstrate \nhow the variable accretion rate with episodic bursts can affect the evolutionary path of MYSOs in \nthe Hertzsprung-Russell diagram. \n\n\n\n\\subsection{Pre-main-sequence excursions in the Hertzsprung-Russell diagram}\n\\label{sect:excursions}\n\n\n\nFig.~\\ref{fig:hdr} shows the evolutionary tracks of our MYSOs in the Hertzsprung-Russell \ndiagram. More specifically, the evolution of the central star in the model with a $100\\, \\rm M_{\\odot}$ pre-stellar \ncore was calculated using a variable (thin blue \nline), constant (thick dashed red line) and smoothed (thick black line) accretion rates and are plotted in panel \n(a) together with the zero-age-main-sequence (ZAMS) track of~\\citet{ekstroem_aa_537_2012}. The tracks \nof the central stars in the models with different pre-stellar core masses of $60$ and $100\\, \\rm M_{\\odot}$ are shown in panel \n(b). In this case, only the variable accretion rate was considered. Grey solid lines are isoradii. \nFrom the definition of $t_{\\rm KH}$ and $t_{\\rm acc}$~\\citep{hosokawa_apj_691_2009,hosokawa_apj_721_2010} \nit can be shown that, \n\\begin{equation}\n\t\\frac{ t_{\\rm KH} }{ t_{\\rm acc} } \\propto \\frac{ L_{ \\rm acc } }{ L_{ \\star } },\n\t\\label{eq:rel_t_L}\n\\end{equation}\nwith the accretion luminosity $L_{ \\rm acc} \\propto G \\dot{M} M_{\\star} \/ R_{\\star}$, where $G$ is \nthe gravitational constant. \nThe initial stages of the stellar evolution calculations are similar as we assume identical accretion rates \nduring the free-fall collapse. Differences occur at the onset of the disc formation. \nIn Fig.~\\ref{fig:hdr}a, we directly see that the final stellar mass at the end of the main accretion \nphase is not the key quantity that governs the evolutionary tracks of high-mass protostars \nin the Hertzsprung-Russell diagram. Despite all models are calculated up to reaching a similar \nstellar mass of $33.3\\, \\rm M_{\\odot}$, their track are qualitatively different. Furthermore, \nthe track of the Run-100-smoothed model with a smoothed accretion rate (but retaining small-amplitude \nvariations) is globally similar to the track of the Run-100-constant model, which assumes a constant \naccretion rate. The small-amplitude variations in the accretion rate only induce tiny \ndeviations from the constant accretion track. \nThe model with accretion spikes strongly deviates from the constant accretion track. Strong \naccretion bursts generate short but important changes in the pre-main-sequence \nevolutionary track of the MYSOs in the form of evolutionary loops to the red part of the Hertzsprung-Russell \ndiagram. These excursions repeat themselves as more and more accretion spikes occur. \nWe find similar behaviour for our model with a pre-stellar core mass of $60\\, \\rm M_{\\odot}$ (Fig.~\\ref{fig:hdr}b). \n\n\n\n\n\n\n\\section{Effect of the initial conditions on the stellar evolutionary tracks of MYSOs}\n\\label{sect:ic}\n\n\n\\textcolor{black}{\nThis section explores the effects of different initial conditions of the protostellar seed in \nthe evolution calculations on stellar structures, the various protostellar properties \nand on the pre-ZAMS evolutionnary track of the MYSOs in the Hertzsprung-Russell diagram. \n}\n\n\n\\begin{table*}\n\t\\centering\n\t\\caption{\n\t\\textcolor{black}{\n\tCharacteristics of the $2\\, \\rm M_{\\odot}$ protostellar seeds used in our comparison simulations of our models with $100\\, \\rm M_{\\odot}$ core and episodic accretion.   \n\t}\n\t}\n\t\\begin{tabular}{lcccr}\n\t\\hline\n\t${\\rm {Models}}$                          &  $R_{\\star}$ ($\\rm R_{\\odot}$)    &    $L_{\\star}$ ($\\rm L_{\\odot}$)   &   $T_{\\rm eff}$ ($\\rm K$)   & ${\\rm {Initial}\\, \\rm {conditions}}$  \\\\ \n\t\\hline    \n\t\\textcolor{black}{\\rm Run-100-hydro}       &  $20.4$                           &    $123$                           &  $4270$         &  \\textcolor{black}{ Convective embryo with large initial radius mimicing spherical, then disc accretion$^{\\star}$ }    \\\\  \n\t\\textcolor{black}{{\\rm Run-100-compact}}    &  $2.9$                            &    $9.7$                           &  $5960$         &  \\textcolor{black}{ Radiative embryo with small initial radius mimicing continuous disc accretion$^{\\star}$ }    \\\\ \n\t\\hline    \n\t\\end{tabular}\n\\label{tab:models_com}\\\\\n\\footnotesize{ ($\\star$) See Section 2.3 of~\\citet{haemmerle_458_mnras_2016} }\\\\\n\\end{table*}\n\n\n\n\\subsection{\\textcolor{black}{Stellar structures}}\n\\label{sect:ic_struc}\n\n\n\n\\textcolor{black}{\nThe seed core taken as a stellar embryo to initialise the stellar evolution calculations is a \nparameter which both depends on the properties of the pre-stellar core~\\citep{vaytet_aa_598_2017,bhandare_aa_618_2018} \nand influences the stellar evolution calculations~\\citep{haemmerle_458_mnras_2016}. \nIn order to investigate the effects of the bursts on the evolutionary tracks as a function of the \ninitial seeds, we carry out a comparison study between two cases using our accretion rate derived from the \nhydrodynamical simulation with a initial $100\\, \\rm M_{\\odot}$ pre-stellar core with variable \naccretion rates (Run-100-hydro and Run-100-compact), see our Table~\\ref{tab:models_com}. \nThe model Run-100-hydro is initialised with a convective core of $2\\, \\rm M_{\\odot}$ \nof radius $20.4\\, \\rm R_{\\odot}$, temperature $4270\\, \\rm K$ and luminosity $123\\, \\rm L_{\\odot}$, \nwhile the model Run-100-compact is started as a fully radiative star of $2\\, \\rm M_{\\odot}$ \nwith radius $2.9\\, \\rm R_{\\odot}$, temperature $5960\\, \\rm K$ and luminosity $9.7\\, \\rm L_{\\odot}$, \nrespectively. \nThey principally differ by the internal entropy profiles, i.e. the initial convective model \nRun-100-hydro exhibits a flat entropy profile towards the protostellar surface, whereas \nthe initial radiative model Run-100-compact has a positive entropy gradient. Therefore, Run-100-hydro  \nis adiabatically convective and is larger than Run-100-compact by about an order of magnitude. \nDetails about these protostellar seeds are given in~\\citet{haemmerle_458_mnras_2016}. \nParticularly, the authors stressed therein that the convective and radiative protostellar embryos have \nproperties similar to models of $M_{\\star}=2\\, \\rm M_{\\odot}$ built by hot and cold accretion, respectively, \n\\textcolor{black}{while the geometry of the accretion is cold for both cases throughout the stellar evolution calculations}. \nOur comparison models therefore investigate the effects of the early accretion geometry on the evolution of the MYSOs. \n}\n\n\n\\begin{figure}\n        \\centering\n        \\begin{minipage}[b]{ 0.49\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_ic_structure.pdf}\n        \\end{minipage}        \n        \\caption{ \n                 \\textcolor{black}{\n        \t\t As Fig.~\\ref{fig:structure} for our of MYSOs generated with an initial $100\\, \\rm M_{\\odot}$ pre-stellar \n        \t\t core, variable accretion rates, large (a) and compact (b) initial protostellar radius. \n                 }      \n                 }\n        \\label{fig:ic_structure}  \n\\end{figure}\n\n\n\n\\textcolor{black}{\nFig.~\\ref{fig:ic_structure} reports the Kippenhahn diagram two MYSOs calculated with the \nsame episodic accretion rate history but different initial conditions (Tab.~\\ref{tab:models_com}). \nDespite of differences in the evolution of the outer radius of the protostar, \nnotable swelling episods of the MYSO appear at the time instances corresponding \nto the accretion bursts. Although differences are visible, especially (i) during the \nclustered bursts at $15$$-$$20\\, \\rm M_{\\odot}$ in the model Run-100-hydro (a) which \nresults in a single inflation of $R_{\\star}$ in Run-100-compact (b), and (ii) in the \nburst at $\\sim 22.5\\, \\rm M_{\\odot}$ that is more pronounced in the initially \nconvective case (b), the major outbursts nevertheless generate similar effects \nregardless of the initial conditions of the calculations. \nOnce the protostar reaches $\\sim 10\\, \\rm M_{\\odot}$, both simulated MYSOs are \nstructured with a convective layer that inflates under the effects of the entropy \ndeposition by accretion of gaseous circumstellar clumps and a radiative interior \nwhich progressively develops a convective core once the protostar is heavy and hot enough \nfor H-burning at $\\sim 25\\, \\rm M_{\\odot}$. \nThe development of a luminosity wave in the outer layer of the MYSO each time a burst \nhappens is similar as above pictured in the context of an initially radiative protostellar \nembryo (Fig.~\\ref{fig:lum_wave}). \nNote that the radiative model is numerically more difficult to calculate and \nit has been simulated over a slightly smaller time interval. Both initial conditions \nproduce similar effects on the evolution of the radius of MYSOs as a response of the \naccretion of circumstellar gaseous clumps. \n}\n\n\n\n\\subsection{\\textcolor{black}{Surface properties and excursions in the Hertzsprung-Russel diagram}}\n\\label{sect:ic_surface}\n\n\n\n\\textcolor{black}{\nFig.~\\ref{fig:ic_prop} plots the evolution of the stellar surface luminosity $L_{\\star}$ \n(panel a, in $\\rm L_{\\odot}$), the stellar radius $R_{\\star}$ (panel b, in $\\rm R_{\\odot}$) and \n(the effective temperature $T_{\\rm eff}$ (panel c, in $\\rm K$) as \na function of time (in $\\rm kyr$) of our $\\rm M_{\\rm c}=100\\, \\rm M_{\\odot}$ collapsing \npre-stellar cores that are considered with large, convective (thin solid blue line) and compact, \nradiative (thick dotted red line) initial conditions, respectively. \nThe only difference in the calculations is the past evolution of the stellar embryo of \n$2\\, \\rm M_{\\odot}$ which is mimiced by the initial conditions of the models (Tab.~\\ref{tab:models_com}). \nAccretion onto the radiative embryo produces an immediate swelling ($>100\\, \\rm R_{\\odot}$) \nas a response of the deposition of entropy by cold accretion (Fig.~\\ref{fig:ic_prop}b), which also \ndimishes the surface temperature (Fig.~\\ref{fig:ic_prop}c) and increases the surface luminosity \n(Fig.~\\ref{fig:ic_prop}a). \nThese differences between the surface properties of the MYSOs persist up to $\\approx 20$$-$$25\\, \\rm kyr$, \nwhen the series of accretion-driven outbursts begins. As illustrated in Fig.~\\ref{fig:ic_structure}, \nthe effets of the accretion spikes onto the protostellar properties are qualitatively similar: the \nmost important luminosity rises and falls coincide with each other as a function of time and their respective \noffsets are due to slightly shifted values of $R_{\\star}$ and $T_{\\rm eff}$. \nNote that the initial compact model has a more pronounced swelling during the series of moderate \nbursts at $\\approx 20$$-$$25\\, \\rm kyr$ (Fig.~\\ref{fig:ic_prop}b). Once the strong accretion events \nonto the MYSO have started, the initial convective model has baseline values of $R_{\\star}$ \nand $T_{\\rm eff}$ more compact and hotter than that of the initial radiative calculation (see thick \nred dotted lines of Fig.~\\ref{fig:ic_prop}a-c). \n}\n\n\n\n\\begin{figure}\n        \\centering\n        \\begin{minipage}[b]{ 0.49\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_ic_stars.pdf}\n        \\end{minipage}        \n        \\caption{ \n                 \\textcolor{black}{\n        \t\t As Fig.~\\ref{fig:stellar_100} with the evolution as a function of time (in $\\rm kyr$) \n        \t\t of the stellar surface luminosity (a), stellar radius (b) and effective temperature (c) \n        \t\t of our $\\rm M_{\\rm c}=100\\, \\rm M_{\\odot}$ collapsing pre-stellar cores, considered with \n        \t\t different initial radii of the protostellar embryo in the stellar evolution calculations, \n        \t\t respectively. \n        \t\t }\n                 }      \n        \\label{fig:ic_prop}  \n\\end{figure}\n\n\n\n\\textcolor{black}{\nFig.~\\ref{fig:ic_hrd} shows the evolutionary tracks in the Hertzsprung-Russell diagram of the models \nRun-100-hydro (thin solid blue line) and Run-100-compact (thick dotted red line), respectively. The \ngrey dotted solid lines are isoradii and the thick black dashed line is the zero-age-main-sequence \n(ZAMS) track of~\\citet{ekstroem_aa_537_2012}. \nIt underlines the initial differences between the two models, especially the rapid \nswelling of the compact model by $\\sim 2$ orders of magnitude accompanied by a decrease of \nits temperature and an increase of its luminosity, respectively. It also further illustrates that \nthe early bursts are more pronounced in the compact case than it the convective case. \nThe protostellar radius of the initially radiative Run-100-compact (thick dotted red line) is \nlarger than that of the initially convective model Run-100-hydro in the quiescent phases \n(Fig.~\\ref{fig:ic_prop}b), therefore, its evolutionary track is closer to the $100\\, \\rm R_{\\odot}$ \nisoradius. This difference diminishes as a function of time and the two tracks overlap \neach other in the quiescent phase after the third excursion which almost reach the \n$1000\\, \\rm R_{\\odot}$ isoradius occurs. \nFinally, let notice that in both cases, the tracks cross the $1000\\, \\rm R_{\\odot}$ isoradius throughout \nthe strongest swelling episods and reach the Hayashi limit. In the initial convective case, when the peak \nof the excursions happens, the track of the MYSO slightly follows vertically the Hayashi track before it \nrecovers the values of $R_{\\star}$ and $L_{\\star}$ corresponding to the quiescent accretion phase. \nAlthough the properties of the first Larson core depend on the intrinsic pre-stellar core \ncharacteristics~\\citep{vaytet_aa_598_2017,bhandare_aa_618_2018}, our suite of comparison simulations shows that the \n\\textcolor{black}{mechanism triggering the excursions} of MYSOs in the Hertzsprung-Russell diagram \\textcolor{black}{is} \nindependent of their initial conditions.  \nThis shows that, under our assumptions, the accretion geometry induces little qualitative differences on the \nevolution of the surface properties of episodically-accreting MYSOs. \n}\n\n\n\n\\begin{figure}\n        \\centering\n        \\begin{minipage}[b]{ 0.48\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_ic_hdr.pdf}\n        \\end{minipage}     \n        \\caption{ \n                 \\textcolor{black}{\n        \t\t As Fig.~\\ref{fig:hdr} with the evolutionary tracks of our models \n        \t\t of $M_{\\rm c}=100\\, \\rm M_{\\odot}$ collapsing pre-stellar cores \n        \t\t and episodic accretion rates onto the protostar, considered with \n        \t\t different initial conditions of the stellar embryo, respectively. \n        \t\t }\n                 }      \n        \\label{fig:ic_hrd}  \n\\end{figure}\n\n\n\n\n\n\\section{Discussion}\n\\label{sect:discussion}\n\n\nIn this section, we present the limitation of our method, compare our outcomes to those of other \nstudies assuming burst-free disc accretion histories and extrapolate the findings of our study to \nthe formation of intermediate-mass stars, and we discuss the observable implications of our results. \n\\textcolor{black}{\nFinally, we review alternative explanations for FU-Orionis-like bursts from young stars. \n}\n\n\n\\begin{figure*}\n        \\centering\n        \\begin{minipage}[b]{ 0.695\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_hdr_mass.pdf}\n        \\end{minipage} \n        \\caption{ \n        \t\t\n        \t\t Comparison between our massive protostar with $M_{\\rm c}=100\\, \\rm M_{\\odot}$ experiencing variable accretion (thick coloured line) \n        \t\t and the evolutionary tracks for massive protostars accreting with constant rates (thin black lines) of~\\citet{haemmerle_585_aa_2016}. \n        \t\t The color coding of the track indicate the stellar mass (in $\\rm M_{\\odot}$) and the coloured dots are \n        \t\t isomasses, respectively.  \n        \t\t Grey dotted solid lines are isoradius and the thick black dashed line is the \n        \t\t zero-age-main-sequence (ZAMS) track of~\\citet{ekstroem_aa_537_2012}. \n                 }      \n        \\label{fig:hdr2}  \n\\end{figure*}\n\n\n\\subsection{Model caveats}\n\\label{sect:caveats}\n\n\n\nThe limitations of our method are two-fold. First, the numerical hydrodynamics simulations of disc \nformation are subject to caveats which merit future improvements, whereas the stellar calculations \nare also subject to assumptions potentially calling follow-up betterments. \nIn addition to the well-known limitation of disc fragmentation simulations given by the logarithmically-expanding \nradial grid~\\citep{meyer_mnras_473_2018}, the sink cell radius, which strongly influences the time-step of the simulations, \nis kept to a value making \nthe simulations affordable from the point of view of their numerical cost. Decreasing the inner hole would allow us to better follow  \nthe migration of the gaseous clumps responsible for the accretion-driven bursts, and therefore make our accretion histories more accurate. \nHowever, the value used in this paper is kept to a decent value ($r_{\\rm in}=20\\, \\rm au$) that is still smaller \nthan that of other studies on disc fragmentation, see e.g. the supermassive protostellar models of~\\citet{hosokawa_2015}.  \n\n\n\\textcolor{black}{For the sake of completeness,}\nfuture improvements should equivalently include the initial non-sphericity and differential \nrotation of the parent pre-stellar cores~\\citep[see, e.g.][]{banerjee_mnras_373_2006} and take into account \nstellar motion in response to the gravitational force of the disc, as massive disc substructures \ncan shift notably the center of mass of the system from coordinate center where the protostar resides~\\citep{regaly_aa_601_2017}. \n\\textcolor{black}{\nNevertheless, despite of the fact that the effects of the stellar inertia on the behaviour of accretion discs has been anaytically \nshown to play a role on the developement of asymetries in their structures~\\citep{adam_apj_347_1989}, recent numerical simulations \ndemonstrated that wobbling neither prevents disc fragmentation nor reduces the bursts intensities \\textcolor{black}{in the \nearly formation phase of young high-mass stars}~\\citep{2018arXiv181100574A}. \nA set of comparison simulations with and without wobbliofofng is shown in the Fig.~7 of their Section~4 and its illustrates that \nhigh-magnitudes accretion-driven outbursts develop similarly within the same timescale after the onset of the disc formation which \nfollowed the free-fall gravitational collapse. Only the time instance and perhaps the long-term occurence of the excursions \nof MYSOs in the Hertzsprung-Russell diagram would change if a moving sink-cell is used in the hydrodynamical simulations. \n}\nOur assumptions related to the hydrodynamical simulations are discussed in great detail in~\\citet{meyer_mnras_464_2017}. \n\n\n\n\nThe manner our stellar evolution calculations treat the accretion of circumstellar material can \nalso be improved. Although the so-called cold accretion scenario is a well-established method to include the accretion \nof disc material onto the stellar surface~\\citep[see][and references therein]{hosokawa_apj_691_2009}, it is the \nlower limit in terms of for the accretion of entropy~\\citep{hosokawa_apj_721_2010}. \nDespite of the fact that high-resolution observations recently demonstrated the disc-plus-jet structure surrounding \nMYSOs, the exact topology of the accretion flow within a few  tens of stellar radii from the protostellar \nsurface is unknown and may differ from accretion in the midplane, for example via the formation of accretion \ncolumns~\\citep{romanova_mnras_421_2012}. The deviations of the accretion geometry \nfrom the cold accretion scenario can be explored within the so-called hot accretion scenario \nor by hybridising the cold and hot accretion scenarios~\\citep{vorobyov_aa_605_2017}. \n\\citet{haemmerle_458_mnras_2016} showed how the stellar bloating can significantly change as a \nfunction of the early accretion geometry, regardless of the accretion rate. However, this concerned smaller \naccretion rates than ours and our calculations are performed with the initial convective model (\"CV\") \nof~\\citet{haemmerle_458_mnras_2016}, with a larger initial radius than that of the radiative models \n(\"RD\") for spherical accretion, for which the bloating of the radius is less pronounced. \n\n\n\nOur study make use of the methods developed in~\\citet{haemmerle_585_aa_2016} and~\\citet{haemmerle_458_mnras_2016}. These \nworks showed that (i) when a massive protostar accretes at very high constant rates ($\\ge 10^{-2}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$), its \ncorresponding evolutionary track derives towards the red part of the Hertzsprung-Russell diagram~\\citep{haemmerle_585_aa_2016}, and (ii) that variabilities in the accretion rate onto \n\\hii regions around massive protostar taken from large-scale hydrodycamical simulations~\\citep[see][]{peters_apj_711_2010} produce luminosity changes \nreflecting that of the fluctuating accretion rates~\\citep{haemmerle_458_mnras_2016}.\nSince we focus on the accretion in the vicinity of massive protostars, the accretion rates histories measured from small-scale \nhydrodynamical simulations~\\citep{meyer_mnras_464_2017,meyer_mnras_473_2018} are more realistic and we investigate how massive \nprotostellar structures reacts under the effect of such accretion variability. \nWe show that, high episodic accretion rates ($\\ge 10^{-2}$-$10^{-1}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$) produce repetitive excursions in \nthe Hertzsprung-Russell diagram. It was not the case in~\\citet{hosokawa_apj_691_2009} and~\\citet{hosokawa_apj_721_2010} \nbecause their accretion rates were constant and weaker than that ours (up to $10^{-3}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$). \nIndeed, the entropy accreted is higher than for cold accretion, hence, the radius will be larger according to \nthe homology relations, as confirmed for constant rates in~\\citet{hosokawa_apj_691_2009,hosokawa_apj_721_2010}, \nand, since the tracks already reached the Hayashi limit, substential differences \\textcolor{black}{in the maximal \nvalues of the prostellar radius should not be expected. \n}\n\n\n\\textcolor{black}{\nOne can nevertheless wonder whether the mean protostellar radius of the MYOS may be affected by \nhybrid\/hot accretion geometries, by making the protostars slightly colder during the phases of \nquiescent accretion and consequently diminishing the amplitudes of the excursions, as shown in \nthe context of low-mass star formation~\\citep{hosokawa_apj_738_2011}. \nHowever, the incidence of massive magnetic OB stars is small~\\citep{fossati_aa_582_2015,fossati_aa_592_2016} and \npre-main-sequence massive stars do have strong surface radiation field because of their high effective \ntemperatures~\\citep{hosokawa_apj_691_2009,hosokawa_apj_721_2010}. Therefore, quiescent accretion processes in most MYSOs should happen via boundary layer mechanisms at the equatorial plane, as investigated by radiation-hydrodynamics simulations~\\citep{kee_mnras_479_2018}. \n}\nThe assumption of cold accretion is therefore appropriate for this \\textcolor{black}{first} study on the effects of \n\\textcolor{black}{strong, episodic} accretion variability on the stellar evolutionnary tracks of pre-ZAMS massive protostars. \n\n\n\n\\begin{figure}\n        \\centering\n        \\begin{minipage}[b]{ 0.47\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_hdr_obs.pdf}\n        \\end{minipage}     \n        \\caption{ \n        \t\t Comparison between our evolutionary tracks with variable accretion rates \n        \t\t of $M_{\\rm c}=60\\, \\rm M_{\\odot}$ (thick dashed black line) and \n        \t\t $M_{\\rm c}=100\\, \\rm M_{\\odot}$ (thin solid black line) collapsing \n        \t\t pre-stellar cores, respectively, and observational data. \n        \t\t Grey dotted-dashed lines are isoradius and the thick black dashed line is \n        \t\t the zero-age-main-sequence (ZAMS) track of~\\citet{ekstroem_aa_537_2012}. \n                 }      \n        \\label{fig:hdr_data}  \n\\end{figure}\n\n\n\\subsection{Comparison with other works assuming constant disc accretion rates}\n\\label{sect:works}\n\n\nOur work extends previous studies on the modifications brought by disc mass accretion \nonto the evolutionary path of young massive stars in the Hertzsprung-Russell diagram. \nWe perform the most spatially-resolved numerical simulations of the inner accretion \ndisc around MYSOs that have revealed self-consistent fragmentation of the irradiated \ncircumstellar disc into spiral arms interspersed with dense gaseous clumps, which migration \nonto the protostar induces strong variability and bursts in the accretion rate histories. \nThese variabilities account for the specifics of the inner disc physics that was informations that were not \npassed to the protostars in previous works on the evolution of massive protostars. \nIndeed, constant accretion rate is considered in early studies~\\citep{palla_apj_392_1992,\npalla_1993,beech_apjs_95_1994,bernasconi_aa_307_1996,bernasconi_aa_120_1996} \nand in more recent works~\\citep{hosokawa_apj_691_2009,hosokawa_apj_721_2010,hosokawa_2015}. \nThe latter found that rapid strong accretion produces a strong swelling of the protostars, \nwhich bloat so that their radii reach $\\simeq 100\\, \\rm R_{\\odot}$. This bloating results in inducing a reduction \nof the effective temperature so that the \\hii region and reestablishes only when the star contracts \nagain and comes back to the bluer part of the Hertzsprung-Russell diagram~\\citet{haemmerle_458_mnras_2016}. \nVariabilities produced by the initial gravitational collapse of \nthe host pre-stellar core in which the star grows and by the effect of outflows launched \nperpendicularly to the disc have been explored with stellar calculations based on the 2.5D \naxisymmetric gravito-radiation-hydrodynamics models, leading to a single excursion \nto the redder region of the Hertzsprung-Russell diagram~\\citep{kuiper_apj_772_2013}. \n\n\n\\begin{figure}\n        \\centering\n        \\begin{minipage}[b]{ 0.495\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_photons.pdf}\n        \\end{minipage}     \n        \\caption{ \n        \t\t Evolution of the number of surface ionizing photons $S_{\\star}$ per second \n        \t\t produced by our young massive stars as a function of the protostellar age, \n        \t\t for our models with an initial $100\\, \\rm M_{\\odot}$ assuming different accretion \n        \t\t histories (a) and generated by different initial pre-stellar core masses but \n        \t\t experiencing variable accretion rate (b). \n                 }      \n        \\label{fig:hdr3}  \n\\end{figure}\n\n\nOur study investigates the effects of strong accretion-driven bursts predicted to happen \nin the context of massive protostars~\\citep{meyer_mnras_464_2017}. \nWe show that the swelling of the MYSOs, \\textcolor{black}{occasionally} going to the cold part of the \nHertzsprung-Russell diagram~\\citep{kuiper_apj_772_2013}, can be generalised in a  \nwider context, as a successive series of evolutionary loops to the red region of the \nHertzsprung-Russell diagram, before the protostars converge to the ZAMS. \nFig.~\\ref{fig:hdr2} compares our models with the tracks of~\\citet{haemmerle_585_aa_2016} \nby plotting our evolutionary tracks with a color coding informing on the stellar mass \n(in $\\rm M_{\\odot}$).  \nIn general, our MYSOs follow the track of a massive prototar accreting at a constant \nrate of $10^{-3}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$~\\citet{haemmerle_585_aa_2016}, but, in addition, our tracks exhibit \nexcursions to the red part of the Hertzsprung-Russell diagram caused by repetitive accretion bursts \nthat modify their surface properties. \nOur tracks are therefore consistent in terms of final mass and location on the ZAMS~\\citep{ekstroem_aa_537_2012}, \nmodulo the excursions which deviate from the constant-accretion solutions. Particularly, we recover \nthe non-monotoneous evolution of the radius and effective temperature of two-dimensional present-day models of disc-accreting \nMYSOs noted by~\\citet{kuiper_apj_772_2013}, but reveal their episodic nature together with the intermittency of their ionized \nflux, as it is known to exist in the context of primordial supermassive protostars gaining mass from a fragmented disc~\\citep{hosokawa_2015}. \n\n\n\\begin{figure*}\n        \\centering\n        \\begin{minipage}[b]{ 0.8\\textwidth}  \\centering\n                \\includegraphics[width=1.0\\textwidth]{.\/plot_ic_structure_ionisation.pdf}\n        \\end{minipage}     \n        \\caption{ \n                 \\textcolor{black}{\n                 %\n        \t\t Correlation between the evolution of the protostellar radius and the \n        \t\t number of surface UV ionizing photons $S_{\\star}$ in our model Run-100-hydro. \n        \t\t %\n        \t\t The protostellar radius and the emissivity is shown as a function of the mass \n        \t\t of the MYSO (a,b) and as a function of the stellar age for two of its main \n        \t\t bursts-producing excursions in the Herztsprung-Russel diagram (c,d). \n        \t\t %\n        \t\t The vertical thick dashed black line marks the beginning of a bloating episod \n        \t\t of the protostar. A similar figure illustrating the intermittency of the \\hii \n        \t\t region of supermassive protostars can be found in~\\citet{hosokawa_2015}.  \n        \t\t %\n        \t\t }\n                 }      \n        \\label{fig:yso_photons_flux}  \n\\end{figure*}\n\n\n\\subsection{Observational implications}\n\\label{sect:obs}\n\n\n\nFig.~\\ref{fig:hdr_data} compares the excursions of our protostars in the \nHertzsprung-Russell diagram with observational data of various high-mass stars. The figure \nclearly illustrates that the properties of young massive stars during the bursts may be similar \nto those of evolved, supergiant massive stars with high-luminosity, large radii and cold effective \ntemperature. One can note that the model with $M_{\\rm c}=60\\, \\rm M_{\\odot}$ \n(dashed black line) has surface characteristics at the peak of the burst \nthat are similar to the characteristics of a red supergiant~\\citep{levesque_apj_628_2005}. \nOn the other hand, the $M_{\\rm c}=60\\, \\rm M_{\\odot}$n model has surface characteristics \nsimilar to Be stars~\\citep{arcos_mnras_474_2018} when converging to the ZAMS in the post burst phase.\nAt the same time, the model with a more massive pre-stellar \ncore ($M_{\\rm c}=100\\, \\rm M_{\\odot}$) crosses the luminous blue variables (LBV) region~\\citep{groh_aa_558_2013} \nattains the characteristics a yellow supergiant (YSG) star at the peak of the burst~\\citep{jaeger_arv_8_1998}. \nTo avoid confusion with evolved massive stars, one should use unique spectral signature of young massive \nprotostars, such as infrared excess and line emission typically associated to accretion discs.\n\n\n\nMost important observational implication of our results is the intermittent character \nof the \\hii regions associated to massive protostars accreting from a fragmented \ncircumstellar disc. \nWe estimate its impact on the \\textcolor{black}{ionization feedback} with the stellar properties \nobtained from our stellar evolution calculations by computing the number of UV-ionizing photons \nper unit time $S_{\\star}$ as the integral of the blackbody spectrum above the ionizing \nenergy threshold~\\citep{haemmerle_458_mnras_2016}, i.e. \n\\begin{equation}\n\tS_{\\star} = 4 \\pi R_{\\star}^{2} \\int_{h\\nu>13.6\\, \\rm eV} \\frac{F_{\\nu}}{h\\nu} d \\nu,\n\t\\label{eq:photons}\n\\end{equation}\nwith, \n\\begin{equation}\n\tF_{\\nu} = \\frac{ 2 \\pi (h \\nu)^{3} }{ c^{2} h^{2}  } \\frac{ 1   }{ e^{ \\frac{ h\\nu}{k_{\\rm B}T_{\\rm eff}} } -1  },\n\t\\label{eq:flux}\n\\end{equation}\nwhere $h$, $\\nu$, $c$ and $k_{\\rm B}$ are the Planck constant, photon frequency, speed of light and Boltzman \nconstant, respectively. We report its evolution during the pre-main-sequence phase for our protostellar \nmodels in Fig.~\\ref{fig:hdr3}. \nThe number of photons gradually increases up to the time instance of the disc formation at \n$10$-$20\\, \\rm kyr$ (Fig.~\\ref{fig:hdr3}a) after the beginning of the gravitational collapse. \nThe accretion variability breaks the strict monotonic time-evolution of $S_{\\star}$ at \n$\\approx 25\\, \\rm kyr$ (Run-100-hydro with variable accretion, thick blue line of Fig.~\\ref{fig:hdr3}a) \nand at the time of each burst (equivalently each spectroscopic excursions), $S_{\\star}$ sharply decreases \nby up to $\\approx 8-9$ orders of magnitude, \\textcolor{black}{inducing dippering in the variability of the} \\hii regions of MYSOs. Such a phenomenon is \nonly a consequence of the bursts, as our models with constant and smoothed accretion histories do not \nexhibit sharp decreases in $S_{\\star}$ (thin solid black and thick dotted lines of Fig.~\\ref{fig:hdr3}a). A similar behaviour \nis found for our model Run-60-hydro (\\ref{fig:hdr3}b). \nThe \\hii regions become fainter, which makes them much more difficult to detect on timescales corresponding \nto those of the bursts. \n\\textcolor{black}{\nFig.~\\ref{fig:yso_photons_flux} further highlights the correlation between evolution of the protostellar \nradius and number of ionizing photons released by the protostellar surface $S_{\\star}$ as a function of the \ngrowing mass of the MYSOs. The time interval corresponding to the bloating phases is of the order of the \ncollisional recombinaison timescale in the plasma ($\\sim 100\\, \\rm yr$) derived for the intermittent \\hii \nregions around supermassive stars in~\\citet{hosokawa_2015}. \n}\n\n\n\n\n\\subsection{Implication for intermediate-mass star formation}\n\\label{sect:obs}\n\n\nBoth theoretical and observational works have recently highlighted a possible similarity  \nbetween the star forming processes in different mass regimes. First evidence came form \nthe direct observation of \\hii regions piercing opaque pre-stellar clouds in high-mass \nstar forming regions~\\citep{fuente_aa_366_2001,testi_2003,cesaroni_aa_509_2010}. Then, \nthe observation of accretion flow~\\citep{keto_apj_637_2006} and Keplerian disc structure \nsurrounding protostars of various mass~\\citep{johnston_apj_813_2015,ilee_mnras_462_2016,forgan_mnras_463_2016,2018arXiv180410622G} \nstrengthened that picture. \nThe numerical proof of disc fragmentation around MYSOs and triggering of FU-Orionis-like \nevents supported the scaled-up character of high-mass star formation with respect to that of low-mass \nand primordial stars~\\citep{vorobyov_apj_805_2015,hosokawa_2015,meyer_mnras_464_2017}. \nMoreover, the intermittent character of \\hii regions from young primordial stars has \nbeen demonstrated with the help of gravito-radiation-hydrodynamics models of the same \nkind as ours~\\cite{2016ApJ...824..119H}. By coupling the outcomes of the hydrodynamical \nresults to stellar evolution calculations, they derive the intermittent properties of the \nUV feedback that is channelled through the radiation-driven cavity perpendicular to the \naccretion disc of young supermassive stars. \n\n\n\nOur results extends such phenomenon to young massive stars, and we speculate that similar mechanisms may \nbe at work in the intermediate-mass systems as they equivalently \ngenerate ionizing photons~\\citep{haemmerle_458_mnras_2016} and form accretion disks and jets~\\citep{kessel_aa_337_1998,fuente_aa_366_2001,torrelles_mnras_442_2014,2017arXiv170604657R}. \nYoung intermediate-mass stars indeed have all necessary prerequisites, i.e. circumstellar disc and \\hii \nregions~\\citep{lumsen_mnras_424_2012,fontani_mnras_423_2012,menu_aa_581_2015,zakhozhay_mnras_477_2018}, \nto experience disc fragmentation and accretion variability. They may episodically produce FU-Orionis-type \nbursts which will subsequently make the UV feedback that fills their radiation-driven bubbles intermittent. \nSuch a prediction is supported by various observational results, including amongst others \nthe variability of the eruptive intermediate-mass stellar object IRAS 18507+0121~\\citep{nikoghosyan_aa_603_2017} \nand the ionised outflow of the intermediate-mass stellar object IRAS 05373+2349 VLA 2~\\citep{brown_mnras_463_2016}. \nAdditionally, we interpret the spectroscopic excursions of massive \nprotostars in the Hertzsprung-Russell diagram triggered by FU-Orionis bursts as a direct consequence \nof the inward migration of a gaseous clump in their fragmented accretion discs onto the stellar \nsurface, and, consequently, consider the intermittency of their \\hii regions as a possible signature \nof disc gravitational fragmentation. \n\n\n\n\\subsection{Alternative explanations for bursting young stellar objects}\n\\label{sect:alternative}\n\n\\textcolor{black}{\nALMA views of bursting objects such as protostars with EXor revealed that their accretion discs seem not to be Toomre-instable, \nwhich suggests that outbursts should be trigered by mechanisms different than gravitational instabilities followed \nby inward migration of clumps in the disc~\\citep{cieza_mnras_474_2018}. \nAlthough the massive accretion discs of MYSOs implies that the fragmentation scenario is likely to \nhappen~\\citep{kratter_mnras_373_2006,kratter_araa_54_2016}, various other mechanisms have indeed been proposed to \nexplain luminosity rises from young low-mass stellar objects, particularly in the low-mass regime of star formation. \nThe work of~\\citet{cieza_mnras_474_2018} lists the principal alternatives for the generation of bursts without the \nclassical picture of migrating disc gaseous clumps, i.e. (i) coupling between magneto-rotational and gravitational \ninstabilities, (ii) thermal-viscous instability, (iii) instabilities induced by planets or companions and (iv) the \ninfall clumpiness mechanism. \n}\n\n\n\\textcolor{black}{\nAlthough these mechanisms are very different, they all consist in providing a manner to suddenly\/episodically increase \nthe accretion rate onto young stars. \nFirst, a cyclic magnetohydrodynamical instability in the inner $\\sim 1\\, \\rm au$ of protoplanetary discs is proposed in~\\citet{armitage_mnras_324_2001}. \nIt is further demonstrated in~\\citet{zhu_apj_694_2009} that the magneto-rotational instability or the gravitational instability alone can \nnot be responsible for the radial mass transport over the overall disc and produce high fluctuations of the accretion rate \non young protostars that is required for FUor bursts. However, these two instabilities can couple together in the innermost au \nof the disc and induce accretion variabilities compatible with the observed infrared spectra of FU-Orionis objects~\\citep{zhu_apj_701_2009}. \nSecondly, a thermal ionisation instability at the inner region of accretion discs around low-mass protostars has been proposed \nto explain episodic increases of the disc-star mass transferts associated to luminous flashes from ionized gas. This model has \nsuccessfully been compared to the major observational characteristcs of FU-Orionis objects~\\citep{clarke_mnras_442_1990,bell_apj_427_1994,bell_apj_444_1995}. \nAdditionally, the thermal viscous ionisation instability scenario has been extended to a wider mechanism, which assumes that \nit naturally develops away from the disc edge by the presence of an embedded planet~\\citep{lodato_mnras_353_2004}. \nSimilarly, instable mass transferts in a binary system modelled with Lagrangian methods gave accretion rate and \nluminosity rises consistent with that of FU-Orionis protostars~\\citep{bonnell_apj_401_1992}. \nLast, the infall clumpiness scenario consists in assuming than the dense material which falls onto the \nyoung stars is directly formed by gas from converging, clumpy filamentary flows in the collapsing turbulent molecular \nclouds in which stellar clusters form~\\citep{padoan_apj_797_2014}. Such gravito-turbulent, large-scale models do not \nresolve small-scale structures as accretion discs and spot the forming stars with sink particles~\\citep{federrath_apj_713_2010}, \nhowever, it gives results consistent with the disc fragmentation scenario~\\citep{vorobyov_apj_633_2005,vorobyov_apj_805_2015}. \n}\n\n\n\\textcolor{black}{\nThese alterlative mechanisms can all explain the production of flares from young stellar objects, with or without the \npresence of an accretion disc around them, and they potentially can be applied to the high-mass regime of star formation. \nThe strong ionization feedback of MYSOs~\\citep{vaidya_apj_742_2011}, the efficient gravitational instability in their surrounding discs~\\citep{meyer_mnras_473_2018} and the \npossible therein magneto-rotational instability~\\citep{kratter_apj_681_2008} make the model of~\\citet{armitage_mnras_324_2001} and~\\citet{zhu_apj_694_2009} \napplicable in the context of massive protostars. \nThe thermal viscous ionisation instability scenario is conceivable in objects such as the high-mass proto-binary \nIRAS17216-3801~\\citep{kraus_apj_835_2017} and the infall clumpiness scenario applicable to regions such as the \nmassive collapsing and high-mass star-forming filament IRDC 18223~\\citep{beuther_aa_584_2015}. \n}\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{sect:cc}\n\n\n\nOur study explores for the first time the effects of a strongly variable protostellar accretion rate history, \nincluding including strong accretion-driven luminosity bursts~\\citep{meyer_mnras_464_2017}, on the \ninternal structure and evolutionary path of pre-main-sequence, massive young stellar objects (MYSOs). \nWe model growing massive protostars by performing three-dimensional gravito-radiation-hydrodynamics \nsimulations of the formation and evolution of their circumstellar discs, unstable to gravitational \ninstability, from which the protostars gain mass~\\citep{meyer_mnras_473_2018}. Direct stellar irradiation feedback and \nappropriate disc thermodynamics are taken account, in addition to a sub-au spatial resolution of the \ninner region of the self-gravitating circumstellar discs. \nGaseous clumps produced in the fragmented discs episodically migrate towards the protostar  \nand produce brief, but violent increases of the accretion rate onto the MYSOs, subsequently responsible \nfor luminous outbursts via the mechanism described in~\\citet{meyer_mnras_464_2017}. \nWe post-process our accretion rate histories by using them as inputs when feeding \na stellar evolutionary code including the physics of pre-main-sequence disc accretion at high rates \n($\\ge 10^{-3}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$) within the so-called {\\it cold accretion} \nformalism~\\citep{hosokawa_apj_691_2009,hosokawa_apj_721_2010,haemmerle_585_aa_2016,haemmerle_602_aap_2017}. \nThe internal and surface stellar properties are self-consistently calculated, together with the evolutionary \ntrack of the protostars in the Hertzsprung-Russell. \nOur models differ by the initial mass of the collapsing pre-stellar cores, taken to be $60$ and \n$100\\, \\rm M_{\\odot}$, respectively. \n\n\n\nThe protostars are initially fully convective and highly opaque, but soon develop a radiative core. \nAt the outer boundary of the radiative core the internal luminosity peaks and moves outwards to the \nsurface as a luminosity wave following the decrease of the internal opacity~\\citep{larson_mnras_157_1972}. \nAt the moment of the strong increase of the accretion rate ($\\ge 10^{-2}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$), \nthe MYSOs bloat as a consequence of the redistribution of entropy, thus reestablishing the internal thermal equilibrium. \nThe photospheric luminosity first sharply augments as a response of the increase of the accretion rate, \nbefore gradually decreasing up to recovering its quiescent, pre-burst value. In addition to the swelling of the stellar \nradius that accompanies the burst, the effective temperature decreases during the bursts. \nThis found decrease in the effective temperature is in stark contrast to what was found for \nlow-mass stars~\\citep{vorobyov_aa_605_2017}. In the latter case, the effective temperature increases \nand the protostars migrate to the left portion of the Hertzsprung-Russell diagram. \nOur models recover the principal feature of the protostars accreting at rates $\\ge 10^{-2}\\, \\rm M_{\\odot}\\, \\rm yr^{-1}$  \nextensively studied in~\\citet{hosokawa_apj_721_2010} and~\\citet{haemmerle_585_aa_2016}, i.e. their evolutionary track \nreach the forbidden Hayashi region. The decrease of the accretion rate which follows each accretion \nspike brings back the evolutionary track of our massive protostars to the standard ZAMS~\\citep{ekstroem_aa_537_2012}, \nas previously calculated in the models of~\\citet{kuiper_apj_772_2013}. \n\\textcolor{black}{\nImportantly, this phenomenon is \\textcolor{black}{qualitatively} independent of the choice of \nthe initial stellar radius considered in the stellar evolution calculations. \n}\nOur simulations with accretion histories derived from three-dimensional simulations show that this \nmechanism is repetitive. Consequently, while gradually gaining mass, the protostellar radius episodically \nswells and the MYSOs experience multiple pre-main-sequence spectroscopic excursions towards the colder regions of \nthe Hertzsprung-Russell diagram. Each additional evolutionary loop to the red corresponds to an ongoing \naccretion burst inducing intermittent changes in the surface entropy of the protostar. \n\n\n\nOur work demonstrates that the successive excursions of massive protostars in the Herztsprung-Russel are only \npossible during strong accretion bursts, and that mild disc-induced variability is insufficient to trigger them. \nAccretion bursts make MYSOs appear colder and much fainter than during their quiescent accretion phase, \neventually crossing the luminous-blue-variable and yellow supergiant sectors of the Hertzsprung-Russell diagram \nto reach the red supergiants region, particularly if the initial the pre-stellar core mass is sufficiently large \n($\\ge 100\\, \\rm M_{\\odot}$). \n\\textcolor{black}{\nTo each high-magnitude accretion bursts~\\citep{meyer_mnras_473_2018} corresponds an excursion assuming that none \nof them produce a close\/spectroscopic binary companion to the massive \nprotostar~\\citep{chini_424_mnras_2012,2013A&A...550A..27M}.\n}\nThe changes in the stellar structure cause a decrease the  \neffective temperature while increasing the stellar radius, which greatly affects number of ionizing UV photons \nreleased by the MYSOs and induces intermittencies in their surrounding \\hii region, as was previously noticed \nin the context of primordial stars~\\citep{2016ApJ...824..119H}. \n\\textcolor{black}{\nThe present study highlights the scaled-up character of star-forming processes, as models for the \nevolution of brown dwarfs and low-mass stars also revealed excursions in the Herztsprung-Rusell as a response \nto strong variable disc accretion~\\citep{baraffe_apj_702_2009,baraffe_aa_597_2017,vorobyov_aa_605_2017}. \n}\nFinally, we conjecture that such mechanism should equivalently affect star formation in the \nintermediate-mass regime and constitute a typical feature of hot, UV-feedback producing young stars. \n\n\n\n\n\n\\section*{Acknowledgements}\n\n\\textcolor{black}{\nThe authors thank the anonymous referee for useful advices \nand suggestions which greatly improved the manuscript. \n}\nD.~M.-A.~Meyer thanks N.~Castro-Ramirez for kindly sharing his knowledge on observational data  \n\\textcolor{black}{and B.~Stecklum for insightful remarks}. \nThis work was sponsored by the Swiss National Science Foundation (project number 200020-172505). \nE.~I.~Vorobyov acknowledges acknowledges support form the Russian Ministry of Education \nand Science grant 3.5602.2017.\n    \n\n\n\\bibliographystyle{mn2e}\n\n\n\\footnotesize{\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction. The main result}\nWe consider the system\n\\begin{equation}\\label{NSC}\n-\\,\\nabla \\cdot\n\\,\\big(\\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}\\,D\\,u\\,\\big) =\\,f \\\n\\mbox{ in } \\Omega\n\\end{equation}\nunder the Navier slip boundary condition \\eqref{bcns}. Here, and in\nthe sequel, $u$ is an $N$-dimensional vector field defined in a\nbounded, open, connected, subset  $\\,\\Omega\\,$ of ${\\mathbb R}^n\\,,$ locally\nsituated on one side of its boundary, a smooth manifold ${\\Gamma}$. We\ndenote by ${\\underline{n}}$ the outer unit normal to $\\partial\\Omega\\,$ and by\n$\\mu\\,\\geq \\,0$ a given parameter. The vector field $f$ is given.\nFor convenience, we will assume that $\\Omega\\,$ has not axis of\nsymmetry. The reason will be clear below. We are particularly\ninterested in the singular case\n${\\mu}=\\,0\\,.$\\par%\nBy\n$$\nD \\,u=\\,\\nabla u+\\,\\nabla^{T} u\n$$\nwe denote the symmetric gradient. So\n\\begin{equation}\nD_{i\\,j}(u)=\\,\\partial_i\\,u_j +\\,\\partial_j\\,u_i\\,,\n\\end{equation}\n where $i,\\,j=\\,1,2,...,n\\,$. Often we simply write $D$, provided that\nthe vector field under consideration follows from the context.\\par%\nEquation \\eqref{NSC} has been considered by mathematicians mostly\nwith $\\,D \\,u\\,$ replaced by $\\,\\nabla u\\,.$ It is worth noting that\n\\eqref{NSC} satisfies the Stokes Principle (see \\cite{stokes}, and\n\\cite{serrin} page 231), a significant physical requirement of\nisotropy,\nwhich does not hold if we replace $\\,D \\,u\\,$ by $\\,\\nabla u\\,.$\\par%\nIn the following we denote by  ${\\underline{t}}(u)\\,$ the Cauchy stress vector\n$$\n{\\underline{t}}(u)=\\,(\\,D\\,u)\\cdot\\,{\\underline{n}}\\,.\n$$\nSo,\n$$\nt_j=\\,(\\partial_i\\,u_j +\\,\\partial_j\\,u_i\\,)\\,n_i\\,,\n$$\nwhere (here and in the sequel) we use the summation convention on repeated indexes.\\par%\nThe homogeneous Navier slip type boundary condition, see\n\\cite{Navier}, says that the velocity is tangent to the boundary,\nand the tangential component of the stress vector ${\\underline{t}}(u)\\,$\nvanishes on the boundary. We write this condition in the following\nform\n\\begin{equation}\\label{bcns}\n\\left\\{\n\\begin{array}{l}\n{\\underline{u}}\\cdot {\\underline{n}}=\\,0,\\\\\n({\\underline{t}}(u))_\\tau=\\,0\\,,\n\\end{array}\n\\right.\n\\end{equation}\nwhere in general the subscript $\\tau$ denotes tangential component.\nFor a mathematical study of the above boundary condition see, for\ninstance, \\cite{bvadvances} and \\cite{so-sca}, where this boundary\ncondition is associated to the linear Stokes problem.\\par%\nBy $L^p(\\Omega)$ and $W^{m,p}(\\Omega)$, $\\,1 \\leq\\,p \\leq\\,\\infty\\,$,\n$\\,m\\,$ nonnegative integer, we denote the usual Lebesgue and\nSobolev spaces, with the standard norms $\\|\\cdot\\|_p\\,$ and\n$\\|\\,\\cdot\\,\\|_{m,p}\\,$, respectively.\\par%\nIn notation concerning  norms and functional spaces, we do not\ndistinguish between scalar and vector fields. For instance\n$L^p(\\Omega;{\\mathbb R}^N)= [L^p(\\Omega)]^N$, $N>1$, is denoted simply by\n$L^p(\\Omega)$. We define\n$$\nV_p=\\,V_p(\\Omega)=\\,\\{ v \\in\\,W^{1,\\,p}(\\Omega):\\, v\\cdot {\\underline{n}}=\\,0\n\\,\\textrm{\\,on\\,}\\, {\\Gamma}\\,\\}.\n$$\nThe linear space $\\,V_p(\\Omega)\\,$, endowed with one of the following\nnorms\n$$\n\\big(\\,\\|\\,v\\,\\|_p +\\,\\|\\,D\\,v\\,\\|_p\\,\\big)^\\frac1p \\,,\n\\quad\\big(\\,\\|\\,v\\,\\|_p +\\,\\|\\,{\\nabla}\\,v\\,\\|_p\\,\\big)^\\frac1p \\,,\\quad\n\\|\\,{\\nabla}\\,v\\,\\|_p\\,,\n$$\nis a Banach space. The above norms are equivalent in $\\,V_p(\\Omega)\\,.$\nFurther, since we assume that the domain $\\,\\Omega\\,$ has not axis of\nsymmetry, it follows that $\\,\\|\\,D\\,v\\,\\|_p\\,$ alone is a norm. For\na quite complete discussion on this point, we refer to\n\\cite{bvadvances}. Without this hypothesis, existence or uniqueness\nof the solution may fail, depending on the particular external force\n$\\,f\\,$. We believe that this should be not difficult to show, by\nappealing to counter examples. However, a complete study of the\npossible phenomena (due to nonlinearity) should be difficult but\nquite interesting. Note that, if we replace  $\\,D \\,u\\,$ by\n$\\,\\nabla u\\,,$ the above assumption on $\\,\\Omega\\,$\nis superfluous.\\par For the proof of Korn's inequality we refer,\nfor instance, to \\cite{so-sca} and \\cite{pares}.\\par%\nOur main result is the following.\n\\begin{theorem}\\label{teoremaq}\nAssume that $\\mu\\geq 0\\,,$ and let $f\\in L^q(\\Omega)\\,,$ where\n$\\,q>\\,n\\,.$ Let $C_q=\\,C(q,\\,\\Omega)\\,$ be the constant that appears in\nthe linear estimate \\eqref{tse} below. Assume that\n\\begin{equation}\\label{cqom}%\n(2-\\,p)\\,C_q <\\,1 \\,.\n\\end{equation}\nThen, the weak solution $\\,u\\,$ to the problem \\eqref{NSC},\n\\eqref{bcns} belongs to $\\,W^{2,q}(\\Omega)\\,$. Moreover, the following\nestimate holds\n\\begin{equation}\\label{dnq}\n \\|u\\|_{2,q}\\leq C\n \\,\\left(\\|f\\|_q+\\|f\\|_{q}^\\frac{1}{p-1}\\right)\\,.\n\\end{equation}\n\\end{theorem}\nThe proofs also apply, in a simpler way, to the Dirichlet boundary\nvalue problem. In section \\ref{ssei} we consider the boundary value\nproblem \\eqref{bcnos}. The singular case remains open. Finally, we\nrefer to section \\ref{ssetes} for an application to the\nfluid mechanics system \\eqref{NSCCC}. %\n\n\\vspace{0.2cm}\n\nRegularity of solutions for systems like \\eqref{NSC} has received\nsubstantial attention from many authors. We refer, for instance, to\nreferences \\cite{acerbi}, \\cite{Hamburger}, \\cite{Lieb88},\n\\cite{Liu}, \\cite{Tolk2}, \\cite{ura}. Other related results may be\nfound in \\cite{anton-sh-1}, \\cite{anton-sh-2}, \\cite{BDVCRIplap},\n\\cite{DB}, \\cite{DBM}, \\cite{FS}, \\cite{MP}, and references therein.\\par%\n\\vspace{0.2cm}\n\nThe plan of the paper is the following: In section \\ref{sdue} we\nrecall the existence and uniqueness result of the weak solution. In\nsection \\ref{stre} we introduce an auxiliary linear problem and\nstate (by appealing to well know classical results) the existence of\nsolutions to this linear problem in spaces $W^{2,\\,q}(\\Omega)\\,.$ In\nsection \\ref{stre} we formulate the non-linear problem in a more\nexplicit, formally equivalent, form in which the non-linearities are\n(roughly speaking) concentrated in the right hand side (see equation\n\\eqref{formfina} below). Furthermore, we appeal to this formulation\nto define \"strong solution\". In section \\ref{squattro}, by assuming\n$\\,{\\mu}>\\,0\\,$ and by appealing to the result stated in section\n\\ref{stre} for the auxiliary linear problem, we show that the strong\nsolution introduced in section \\ref{squattro} exists and belongs to\n$\\,W^{2,\\,q}(\\Omega)\\,,$ for each $\\,{\\mu}>\\,0\\,.$ Moreover, the estimates\nobtained are independent of $\\,{\\mu}\\,.$ This last property allows us\nto extend, in section \\ref{scinque}, the regularity result to the\nsingular case $\\,{\\mu}=\\,0\\,$ by passing to the limit in the\nvariational formulation \\eqref{bufi} as $\\,{\\mu}\\,$ tends to zero. In\nsection \\ref{ssei} we consider the boundary value problem\n\\eqref{bcnos}. Finally, in  section \\ref{ssetes}, we appeal to a\nrecent result proved by Petr Kaplick\\'y and Jakub Tich\\'y, to show\nthat the result claimed in theorem \\ref{teoremaq} applies to\nsolutions to the system \\eqref{NSCCC}, in the particular case\n$\\,q=\\,{\\widehat{q}}\\,$, see \\eqref{errq2}.\n %\n\\section{Existence and uniqueness  of the weak solution}\\label{sdue}\nExistence and uniqueness of weak solutions follows from well know\nresults. Let us recall some basic points. Set\n\\begin{equation}\\label{buh}\nB(\\,D\\,u\\,)=\\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}\\,.\n\\end{equation}\nBy appealing to the identity  $\\,D_{ij}u \\,D_{ij}v=\\,2\\,D_{ij}u\n\\,\\partial_j v_i\\,,$ integration by parts shows that\n\\begin{equation}\\label{bufe} \\begin{array}{ll}\\vspace{1ex}\\displaystyle%\n\\frac12 \\,\\int_{\\Omega}\\ \\ B (D\\,u)\\, \\cdot D\\,u  \\cdot D\\, v \\,dx\n=-\\,\\int_{\\Omega}\\,{\\nabla} \\cdot\\,\\big(\\,B(D\\,u)\\,D(u)\\,\\big) \\cdot\\,v\n\\,dx\\,  \\\\\n\\displaystyle +\\,\\int_{{\\Gamma}} \\,B(D\\,u)\\,[\\,(D\\,u)\\,\\cdot\\,v \\,\\cdot\\,n\\,]\n\\,dS\\,.%\n\\end{array}%\n\\end{equation}\nHence,\n\\begin{equation}\\label{bufi} \\begin{array}{ll}\\vspace{1ex}\\displaystyle%\n\\frac12 \\,\\int_{\\Omega}\\ \\ B (D\\,u)\\, \\cdot D\\,u \\cdot D\\, v \\,dx\n=\\,-\\,\\int_{\\Omega}\\,{\\nabla} \\cdot\\,\\big(\\,B(D\\,u)\\,D(u)\\,\\big)\n\\cdot\\,v \\,dx\\,  \\\\\n\\displaystyle +\\,\\int_{{\\Gamma}} \\,B(D\\,u)\\,({\\underline{t}}(u))_{\\tau}\\cdot v \\, dS\\,,\n\\end{array}\\end{equation}%\nprovided that $\\,v\\cdot\\,n=\\,0\\,$ on $\\,{\\Gamma}\\,.$\\par%\nThis last identity justifies the following definition.\n\\begin{definition}\\label{noit}\nLet $f \\in V'_p(\\Omega)$. We say that $u$ is a {\\rm weak solution} of\nproblem \\eqref{NSC}, \\eqref{bcns} if $\\,u\\,\\in\\, V_p(\\Omega)$ satisfies\n\\begin{equation}\\label{bufi} \\begin{array}{ll}\\vspace{1ex}\\displaystyle%\n\\frac12 \\,\\int_{\\Omega}\\ \\ B (D\\,u)\\, \\cdot D\\,u \\cdot D\\, v \\,dx\n=\\,\\int_{\\Omega}\\, f\\, \\cdot\\, v \\,dx\\,,\n\\end{array}\\end{equation}%\nfor all $\\,v \\in \\,V_p(\\Omega)\\,$.\n\\end{definition}\nExistence and uniqueness of the above weak solution, for any fixed\n$\\,{\\mu} \\geq\\,0\\,,$ follows by appealing to the theory of monotone\noperators, see J.-L. Lions \\cite{lions}.%\n\\section{An auxiliary linear problem}\\label{stre}\nIn this section we consider the linear problem\n\\begin{equation}\\label{NSC2}\n-\\,\\nabla \\cdot \\,\\big(\\,D\\,u\\,\\big) =\\,F \\ \\mbox{ in } \\Omega\n\\end{equation}\nunder the boundary condition \\eqref{bcns}, and state an auxiliary\nresult to be used in the next sections. This particular result\nfollows from well known general results.\\par%\nNote that equation \\eqref{NSC2} may be also written in the\nequivalent form (not used in this section)\n\\begin{equation}\\label{forteF}%\n-\\,\\Delta\\,u -\\, \\nabla\\,(\\nabla \\cdot\\,u\\,)= F\\,.\n\\end{equation}%\n\\begin{definition}\\label{noit}\nLet $f \\in V'_2(\\Omega)$. We say that $u$ is a {\\rm{weak solution}} of\nproblem \\eqref{NSC2}, \\eqref{bcns} if $\\,u\\,\\in\\, V_2(\\Omega)$ satisfies\n\\begin{equation}\\label{basta}%\n\\frac12 \\,\\int_{\\Omega}\\ \\  D\\,u \\cdot D\\, v \\,dx =\\,\\int_{\\Omega}\\, F\\,\n\\cdot\\, v \\,dx\\,,\n\\end{equation}%\nfor all $\\,v \\in \\,V_2(\\Omega)\\,$.\n\\end{definition}\nCoerciveness of the bilinear form on the left hand side of\n\\eqref{basta} follows here by appealing to the fact that\n$\\,\\|\\,D\\,v\\,\\|\\,$ alone is a norm in $\\,V_2(\\Omega)\\,$, since we have\nassumed that $\\Omega$ has not axis of symmetry. Hence, existence,\nuniqueness, and the standard estimate holds for the above problem.\n\n\\vspace{0.2cm}\n\nNext we consider the regularity of the solutions to the above linear\nsystem \\eqref{NSC2} (or, equivalently, \\eqref{forteF}) under the\nboundary condition \\eqref{bcns}. The $\\,W^{2,\\,2}(\\Omega)\\,$ regularity\nmay be proved, for instance, by following \\cite{so-sca} and\n\\cite{bvadvances}. The reader may easily adapt the argument\ndeveloped in \\cite{so-sca}, section 4. Further, as claimed in\nreference \\cite{so-sca} section 4, by appealing to results proved in\nreference \\cite{so2} (see also \\cite{a-d-n}), the\n$\\,W^{2,\\,q}(\\Omega)\\,$ regularity, for arbitrarily large exponents $q$,\nfollows. Actually, under suitable, canonical, regularity assumptions\non $F$ and $\\Omega\\,,$ $\\,W^{m,\\,q}(\\Omega)\\,$ regularity for arbitrarily\nlarge values of $m\\geq\\,2\\,$ follows.\\par%\nAlternatively, we may follow \\cite{Sol71} to show that the system\n\\eqref{forteF}, \\eqref{bcns} is of Petrovks\\u\\i\\ type (a subclass of\nAgmon-Douglis-Nirenberg elliptic systems). See, in particular, the\nTheorem 5.1 in reference \\cite{Sol71}. This allows a simplified\nintegral representation formula for the solutions to the above\nlinear problem. Moreover, for Petrovks\\u\\i's systems, the\n$W^{2,\\,2}$-regularity yields full $W^{m,\\,q}$-regularity, provided\nthat the data are sufficiently smooth. In particular, there is a\nconstant $\\widetilde{C}_q \\,$ such that\n\\begin{equation}\\label{tse2}%\n\\|\\,u\\,\\|_{2,\\,q} \\leq\\,\\widetilde{C}_q \\,\\|\\,F\\,\\|_q \\,.\n\\end{equation}\nSummarizing, the following result holds.\n\\begin{theorem}\\label{seva}\nConsider the linear boundary value problem \\eqref{NSC2},\n\\eqref{bcns}. Assume that $\\,F \\in\\,L^q(\\Omega)\\,,$ for some $\\,q\n\\,\\geq\\,2\\,.$ Then, the solution $u$ to the above linear problem\nbelongs to $W^{2,\\,q}(\\Omega)\\,.$ Furthermore, there is a constant\n$\\,C_q =\\,C_q(q,\\,\\Omega)\\,,$ such that\n\\begin{equation}\\label{tse}%\n\\|\\,{\\nabla}\\,Du\\,\\|_q \\leq\\,C_q\\,\\|\\,F\\,\\|_q \\,.\n\\end{equation}\n\\end{theorem}\nClearly, $\\,C_q \\leq\\,\\widetilde{C}_q\\,.$ The pointwise estimate\n\\begin{equation}\\label{dirup}\n|\\nabla^2\\,u| \\leq\\,3\\,|\\nabla\\, D\\,u| \\leq\\,6\\,|\\nabla^2\\,u|\n\\end{equation}\nshows that $\\|\\,u\\,\\|_{2,\\,q}$ and $\\,\\|\\,{\\nabla}\\,Du\\,\\|_q \\,$ are\nequivalent norms in $\\,W^{2,\\,q}(\\Omega) \\cap\\,V_q(\\Omega)\\,.$\n\n\\vspace{0.2cm}\n\nUnder the homogeneous Dirichlet boundary value problem the constant\n$\\,C_q\\,$ is bounded from above by a constant $\\,K\\,$ times $\\,q\\,.$\nClearly, this nice behavior can not hold, with the same constant\n$\\,K\\,,$ for arbitrarily large values $q$. To each upper-bounded\ninterval of values $\\,q\\,$ it corresponds a distinct value $\\,K\\,.$\nSee \\cite{yud}. We do not know whether a similar (quite predictable)\nresult is known for the Navier boundary condition.\n\\section{The strong solution. Definition.}\nThe main lines followed in this section have their starting point in\nsome ideas already used, in a more complex context, in reference\n\\cite{bvlali} (see, for instance, equations (4.17), (4.25), (4.26),\nand (4.27) in this last reference). Since\n$$\n\\nabla\\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}=\\,\\frac{p-2}{2}\\,\n(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-4}{2}}\\,\\nabla\\,(\\,|D\\,u|^2)\\,,\n$$\nstraightforward calculations show that%\n\\begin{equation}\\label{formula}\\begin{array}{ll}\\vspace{1ex}\\displaystyle \\nabla \\cdot\n\\,\\big(\\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}\\,D\\,u\\,\\big)=\\,\n(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}\\,\n\\,\\nabla \\cdot (Du\\,) \\\\\n\\displaystyle +\\,(p-2)\\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-4}{2}}\\,I(u)%\n\\end{array}\\end{equation}%\nwhere, by definition,\n$$\nI(u)=\\,\\frac12 \\,\\nabla\\,(\\,|D\\,u|^2)\\cdot\\, D\\,u=\\,(\\,Du :\\,\\nabla\nDu\\,)\\cdot\\,Du\\,.\n$$\nThe $j$ component of the vector field $I(u)$ is given by\n$$\nI_j(u)=\\,\\sum_k \\,\\sum_{l,\\,m}\\, D_{l m} (\\partial_k\\,D_{l m}) \\,D_{k\nj}\\,.\n$$\nBy improving an argument already used in \\cite{bvlali}, we may prove\n(as in the proof of Lemma 3.4 in \\cite{nonhom}) the algebraic\nrelation\n$$\n|I\\cdot \\,\\xi| \\leq\\,|D|^2\\,|\\nabla \\,D\\,u|\\,|\\xi|\\,,\n$$\nfor each arbitrary vector field in $\\,\\xi\\in\\,{\\mathbb R}^N\\,$, where\n$$\n|\\nabla \\,D\\,u|^2=\\,\\sum_{m,l,k} \\,(\\partial_k\\,D_{m\\,l})^2\\,.\n$$\nConsequently, the pointwise estimate\n\\begin{equation}\\label{buf2}\n|I(u)| \\leq\\,|D|^2\\,|\\nabla \\,D\\,u|\n\\end{equation}\nholds.\\par%\nNext we introduce the notion of strong solution used in the next\nsection.\n\\begin{definition}\\label{noitidia}\nAssume that $\\,{\\mu} >\\,0\\,,$ and let $\\,f \\,\\in \\, L^q(\\Omega)\\,$ be\ngiven, $q>1\\,$. We say that $\\,u\\in\\,W^{2,\\,q}(\\Omega)\\,$ is a\n{\\rm{strong solution}} of problem \\eqref{NSC}, \\eqref{bcns} if\n$\\,u\\,$ satisfies \\eqref{bcns} in the trace sense and, moreover,\n the equation\n\\begin{equation}\\label{formfina}%\n-\\,\\nabla \\cdot\\,(D\\,u) =(p-\\,2)\\,G(u)+\\,\n(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}\\,f%\n\\end{equation}%\nholds almost everywhere in $\\Omega\\,,$ where\n$$\nG(v)=\\,(\\,{\\mu}+|\\,D\\,v|^2\\,)^{-1}\\,I(v)\\,.\n$$\n\\end{definition}\nNote that $ G(v) \\leq\\,|\\nabla \\,D\\,v|\\,,$ almost everywhere in\n$\\,\\Omega\\,,$ for all ${\\mu}\\,$. So,\n\\begin{equation}\\label{geve}\n\\|\\,G(v)\\,\\|_q \\leq\\,\\|\\nabla \\,D\\,v\\|q\\,.\n\\end{equation}\n\\section{Existence of the strong solution for ${\\mu}>\\,0\\,.$}\\label{squattro}\nFix $\\,{\\mu}>\\,0\\,,$ and let $\\,f \\in \\,L^q(\\Omega)\\,$. Following\n\\cite{BVCRI}, by appealing to a fixed point argument, one proves the\nexistence of a (unique) strong solution $\\,u\\in\\,W^{2,\\,q}(\\Omega)\\,$ of\nthe above problem. Let us sketch the proof.\n\n\\vspace{0.2cm}\n\nSince $q>n\\,,$ there is a constant $\\,{\\widehat{C}}(q,\\,\\Omega)\\,$ such that\n\\begin{equation}\\label{c7}%\n\\|\\,D\\,v\\|_{\\infty}\\leq \\,{\\widehat{C}}\\,\\|\\nabla\\,D\\,v\\,\\|_q\\,,%\n\\end{equation}%\nfor all $v\\,\\in\\,W^{2,\\,q}(\\Omega) \\cap\\, V_q(\\Omega)\\,.$  Hence,\n\\begin{equation}\\label{holdes}%\n\\|\\,|\\,D v |^{2-p}\\,f\\|_q\\leq \\displaystyle\\|\\,D v\\,\\|_{\\infty}^{2-p}\\,\n\\|\\,f\\|_q \\,\n\\leq\\,{\\widehat{C}}^{2-\\,p}\\,\\|\\nabla\\,D\\,v\\,\\|^{2-\\,p}_q\\,\\|\\,f\\|_q\\,.\n\\end{equation}%\nFurther, since $\\,(a+\\,b)^\\alpha \\leq \\,a^\\alpha + b^\\alpha\\,$\nfor nonnegative $a$ and $b$, and  $ 0 <\\alpha<1\\,,$ it follows that%\n\\begin{equation}\\label{cjen}\n({\\mu}+\\,|D\\,v|^2)^{\\frac{2-p}{2}}\\leq\n\\,\\mu^{\\frac{2-p}{2}}+\\,|D\\,v|^{2-p}\\,.%\n\\end{equation}%\nFrom \\eqref{holdes} and \\eqref{cjen} we show that\n\\begin{equation}\\label{lap1w}%\n\\|\\,({\\mu}+\\,|\\,D\\,v\\,|^2)^{\\frac{2-p}{2}} \\,f\\|_q \\leq\n\\,\\mu^{\\frac{2-p}{2}}\\,\\|\\,f\\,\\|_q\n+\\,{\\widehat{C}}^{2-\\,p}\\,\\,\\|\\,{\\nabla}\\,D v\\,\\|^{2-\\,p}_q\\,\\|\\,f\\|_q\\,.%\n\\end{equation}%\nNext we define the convex closed set\n\\begin{equation}\\label{krapa}%\n{\\mathbb K}=\\,{\\mathbb K}(R)=\\,\\{ v\\in\\,W^{2,\\,q}(\\Omega)\\cap\\,V_q(\\Omega)\\,:\\,\\| {\\nabla} D\\,v\\|_q\n\\leq\\,R\\,\\}\\,,\n\\end{equation}%\nand consider, for each $v \\in\\,{\\mathbb K}\\,,$ the solution $u=\\,T(v)\\,$ to\nthe problem\n\\begin{equation}\\label{formfina3}%\n-\\,\\nabla\\,\\cdot\\,D\\,u\\,=\\,F(v)\\equiv (p-\\,2)\\,G(v)+\\,\n\\,(\\,{\\mu}+|\\,D\\,v|^2\\,)^{\\frac{2-\\,p}{2}}\\,f\\,,%\n\\end{equation}%\nunder the boundary conditions \\eqref{bcns}\\,.\\par%\nBy appealing to equations \\eqref{tse}, \\eqref{geve}, and\n\\eqref{lap1w}, we obtain the estimate\n\\begin{equation}\\label{bingas}%\n\\|\\,{\\nabla}\\,Du\\,\\|_q \\leq\\,C_q\\,\\{\\,(2-\\,p)\\,\\|\\,{\\nabla}\\,Dv\\,\\|_q +\\,\n\\mu^{\\frac{2-p}{2}}\\,\\|\\,f\\,\\|_q\n+\\,{\\widehat{C}}^{2-\\,p}\\,\\,\\|\\,{\\nabla}\\,D v\\,\\|^{2-\\,p}_q\\,\\|\\,f\\|_q\\,\\}\\,.%\n\\end{equation}\nNext we show that if $\\|\\,{\\nabla}\\,Dv\\|_q\\leq R\\,$ then the\ncorresponding solution $u=\\,T(v)$ satisfies the same estimate,\nnamely $\\|\\,{\\nabla}\\,Du\\|_q\\leq R\\,$. This shows that $T({\\mathbb K}) \\subset \\,{\\mathbb K}\\,.$\\par%\nSince $\\,v \\in {\\mathbb K}$ it follows that%\n\\begin{equation}\\label{ssim}%\n\\|\\,{\\nabla}\\,Du\\,\\|_q \\leq \\,\\mu^{2-p}\\,C_q\\,\\|f\\|_q\\,+\\,(2-p)\\,\\,C_q\\,R\n\\,+\\,C_q\\, {\\widehat{C}}^{2-\\,p} \\,\\|\\,f\\|_q \\,R^{2-\\,p}\\,.\n\\end{equation}%\nBy assuming \\eqref{cqom}, we show that $\\,u \\in\\,{\\mathbb K}(R)\\,$ if\n$$\n[\\,1-\\,(2-\\,p)\\,C_q\\,]\\,R \\geq \\,\n\\,\\mu^{2-p}\\,C_q\\,\\|f\\|_q\\,+\\,C_q\\, {\\widehat{C}}^{2-\\,p} \\,\\|\\,f\\|_q\n\\,R^{2-\\,p}\\,.\n$$\nThis inequality is satisfied if, for instance, its left hand side is\nequal to two times the sum of the two terms on the right hand side.\nThis holds for\n\\begin{equation}\\label{condidois}\nR=\\,\\frac{2}{\\alpha}\\,\\mu^{2-p}\\,C_q\\,\\|f\\|_q +\\,\n(\\,\\frac{2\\,C_q\\,{\\widehat{C}}^{2-\\,p}}{\\alpha}\\,)^{\\frac{1}{p-\\,1}}\\,\\|\\,f\\|^{\\frac{1}{p-\\,1}}_q\\,,\n\\end{equation}\nwhere $\\,\\alpha=\\,1-\\,C_2(q)\\,(2-\\,p)\\,.$ Hence $\\|\\,{\\nabla}\\,Du\\,\\|_q\n\\leq\\,R\\,,$ and the inclusion $\\,T({\\mathbb K}) \\subset \\,{\\mathbb K}\\,$ follows. This\nis the main ingredient to prove the existence of a fixed point in\n$\\,{\\mathbb K}\\,.$ For the missing details we refer to\nthe argument developed in reference \\cite{BVCRI}.\\par%\nThe expression of $\\,R\\,$ shows that the uniform estimate\n\\eqref{dnq} follows. Actually, we have shown that,\nfor each positive ${\\mu}\\,,$ the estimate%\n\\begin{equation}\\label{dnq2}\n\\|u^{\\mu}\\|_{2,q}\\leq C \\,\\left(\\|f\\|_q+\\|f\\|_{q}^\\frac{1}{p-1}\\right)\n\\end{equation}\nholds, where $\\,u^{\\mu}\\,$ denotes the strong solution related to the\nparticular positive value $\\,{\\mu}\\,$.\n\\section{Existence of the strong solution for ${\\mu}=\\,0\\,.$}\\label{scinque}\nIn this section, since the estimate \\eqref{dnq2} is uniform with\nrespect to values $\\,{\\mu}\\,$ (assumed to be bounded from above), by\nappealing to a compactness argument, we pass to the limit, as ${\\mu}$\ntends to zero, in the weak formulation \\eqref{bufi} (which contains\nthe singular case ${\\mu}=0$) and prove that the weak solution $u$ to\nthe singular problem also belongs to $W^{2,q}(\\Omega)\\,,$ and satisfies\n\\eqref{dnq}.\\par%\nWe start by recalling the definition of weak solution $\\,u^{\\mu}\\,$ of\nproblem \\eqref{NSC}, for $\\,{\\mu} \\geq\\,\\,0\\,:$\n\\begin{equation}\\label{buf23}\n\\int_{\\Omega}\\\n\\left(\\,{\\mu}+|\\,D\\,u^{\\mu}|^2\\,\\right)^{\\frac{p-2}{2}}\\,D\\,u^{\\mu}\\, \\cdot\nD\\, v \\,dx\\ =\\,\\int_{\\Omega}\\, f\\, \\cdot\\, v \\,dx\\,,\n\\end{equation}\nfor all $\\,v \\in \\,V_p(\\Omega)\\,.$ This condition is satisfied by the\nstrong solutions $\\,u^{\\mu}\\,,$ for $\\,{\\mu} >\\,0\\,,$ constructed in the\nprevious section. Since these solutions are uniformly bounded in\n$W^{2,\\,q}(\\Omega)\\,,$ suitable sub-sequences, which we continue to\ndenote by $\\,u^\\mu\\,$, weakly converge in $W^{2,\\,q}(\\Omega)\\,$ to some\n$\\,u\\,$. The argument followed in \\cite{BVCRI} shows\nthat we may pass to the limit in \\eqref{buf23} to prove that%\n\\begin{equation}\\label{buf24} \\int_{\\Omega}\\\n\\,|\\,D\\,u|^{\\frac{p-2}{2}}\\,D\\,u\\, \\cdot D\\, v\n\\,dx\\ =\\,\\int_{\\Omega}\\, f\\, \\cdot\\,v \\,dx\\,,%\n\\end{equation}\nfor all $\\,v \\in \\,V_p(\\Omega)\\,.$ So, $\\,u \\in\\,W^{2,\\,q}(\\Omega)\\,$ is the\nsolution (known to be unique), corresponding to ${\\mu}=\\,0\\,.$ To prove\nthe above claim, we have to show that, for each fixed $\\,v \\in\n\\,V_p(\\Omega)$, the left hand side of equation \\eqref{buf23} converges\nto the left hand side of \\eqref{buf24}. Essentially, the proof\nfollowed in reference \\cite{BVCRI} section 4 applies here. For the\nreader's convenience, we repeat the main argument here.\\par%\nSince $\\,u^\\mu\\,\\rightharpoonup u\\,$ weakly in $W^{2,q}(\\Omega)\\,$, and\n$\\,q>\\,n\\,,$ strong convergence (of suitable subsequences) in\n$\\,W^{1,s}(\\Omega)\\,$, for any $\\,s\\,,$ follows. So, strong convergence in $W^{1,p}(\\Omega)$ holds.\\par%\nWe write the integral on the left-hand side of \\eqref{buf23} as\n\\begin{equation}\\label{aaa2} \\int_{\\Omega} \\,\\big[\\,\\left(\\,\\mu+|\\,D\nu^\\mu|^2\\,\\right)^{\\frac{p-2}{2}}\\,D u^\\mu\\, - \\,\\left(\\,\\mu+|\\,D\nu\\,|^2\\,\\right)^{\\frac{p-2}{2}}\\,D u\\,\\big] \\cdot D v \\,dx \\end{equation}\n$$\n +\\,\\int_{\\Omega} \\,\\left(\\,\\mu+|\\,D\nu|^2\\,\\right)^{\\frac{p-2}{2}}\\,D u\\, \\cdot \\,D v \\,dx\\,,\n$$\nand show that the first integral tends to zero, and the second\nintegral tends to the left hand side of \\eqref{buf24}. The\ninequality%\n\\begin{equation}\\label{tensorS1}|\\,(\\mu+|A|)^{\\frac{p-2}{2}}\nA-(\\mu+|B|)^{\\frac{p-2}{2}} B| \\leq\\,\nC\\,\\frac{|A-B|}{(\\mu+|A|+|B|)}{\\!\\atop ^{{2-p}}}\\,,\\end{equation}%\nwhere $C$ is independent of $\\mu$ (see \\cite{DER} equation (6.8)),\nshows that the absolute value of the first integral in equation\n\\eqref{aaa2} is bounded by%\n$$\nC\\,\\int_{\\Omega}\\! \\,\\left(\\,\\mu+|\\,D\\,u\\,|+|\\,D\\,\nu^\\mu|\\,\\right)^{p-2}\\,|\\,D\\,u-\\,D u^\\mu\\,|\\,|\\,D \\,v|\\, dx\\,.\n$$\nSince\n$$\n\\left(\\,\\mu+|D u\\,|+|D u^\\mu|\\,\\right)^{p-2}\\,|\\,D u-\\,D u^\\mu\\,|\n\\leq\\,\\left|\\,D u-\\,D u^\\mu\\,\\right|^{p-1}\\,,\n$$\nthe absolute value of the first integral in equation \\eqref{aaa2}\nis bounded by%\n$$\nC\\,\\|\\,D\\,\nu^\\mu\\!-\\,D\\,u\\,\\|_p^{p-1}\\,\\|\\,D\\,v\\,\\|_p\\,,%\n$$\nwhich tends to zero with $\\,\\mu\\,.$\\par%\nFinally,\n$$\n \\lim_{\\mu\\to 0^+}\\int_{\\Omega}\n\\,\\left(\\,\\mu+|\\,D\\,u\\,|^2\\,\\right)^{\\frac{p-2}{2}}\\,D\\,u\\, \\cdot\n\\,D\\,v  \\,dx= \\,\\int_{\\Omega} \\,|\\,D\\, u\\,|^{p-2}\\,\\,D\nu \\cdot \\,D\\,v \\, dx\\,,%\n$$\nby Lebesgue's dominated convergence theorem.\\par%\n\n\\section{On a related slip boundary condition}\\label{ssei}\nIn this section we consider the system \\eqref{NSC} under the slip\nboundary condition\n\\begin{equation}\\label{bcnos}\n\\left\\{\n\\begin{array}{l}\n{\\underline{u}}\\cdot {\\underline{n}}=\\,0,\\\\\n{\\underline{\\omega}}(u) \\times \\,{\\underline{n}}=\\,0\\,, \\; \\textrm{on} \\; {\\Gamma}\\,,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $\\,{\\underline{\\omega}}=\\,{\\underline{\\omega}}(u)=\\, curl \\,{\\underline{u}}\\,,$ and $\\,n=\\,N=\\,3\\,.$ The\nboundary condition \\eqref{bcnos}, introduced by C. Bardos in\nreference \\cite{bardos}, has been studied by a large number of\nauthors.\\par%\nIn the following preliminary approach to the above problem, we start\nby a sketch of the proof of the existence of the strong solution,\nunder the assumption $\\,{\\mu}>\\,0\\,$. Moreover, an uniform\n$\\,W^{2,\\,q}(\\Omega)\\,$ estimate is claimed. However, the case\n$\\,{\\mu}=\\,0\\,$ is not considered here, due to the lack of a suitable\ndefinition of weak solution. This point will be discussed below.\\par%\nConcerning the existence of a strong solution for each $\\,{\\mu}>\\,0\\,,$\nby taking into account definition \\ref{noitidia}, and by appealing\nto \\eqref{NSC} and \\eqref{formula}, we say that $u$ is a {\\rm{strong\nsolution}} of problem \\eqref{NSC}, \\eqref{bcnos} if\n$\\,u\\in\\,W^{2,\\,2}(\\Omega)\\,$ enjoys the boundary condition\n\\eqref{bcnos}, and satisfies equation \\eqref{formfina} almost\neverywhere in $\\Omega\\,$. We write here the\nequation \\eqref{formfina} in the equivalent form%\n\\begin{equation}\\label{formfina2}%\n\\begin{array}{ll}\\vspace{1ex}\\displaystyle%\n-\\,\\Delta\\,u -\\, \\nabla\\,(\\nabla \\cdot\\,u\\,)= \\\\\n\\displaystyle (p-\\,2)\\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{-1}\\,I(u)+\\,\n(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}\\,f\\,.%\n\\end{array}\\end{equation}%\n\n\\vspace{0.3cm}\n\nLet us prove the following result.\n\\begin{lemma}\nThe following identity holds.\n\n\\begin{equation}\\label{ospois} \\begin{array}{ll}\\vspace{1ex}\\displaystyle%\n\\int_{\\Omega} \\,\\nabla\\,(\\nabla \\cdot\\,u\\,)\\cdot \\,{\\Delta}\\,v\n\\,dx=\n\\\\\n\\displaystyle \\,\\int_{\\Omega} \\,{\\nabla}\\,(\\nabla \\cdot\\,u\\,)\\cdot \\,{\\nabla}\\,(\\nabla\n\\cdot\\,v\\,) \\,dx  -\\, \\int_{{\\Gamma}} \\,\\nabla\\,(\\nabla\n\\cdot\\,u\\,)\\cdot\\,(\\, {\\underline{n}} \\times \\,{\\underline{\\omega}}(v)\\,) \\,d{\\Gamma}\\,.%\n\\end{array}\\end{equation}%\nIn particular\n\\begin{equation}\\label{ochey3}\n\\int_{\\Omega} \\,\\nabla\\,(\\nabla \\cdot\\,u\\,)\\cdot\\,\\Delta\\,u \\,dx=\\,\n\\int_{\\Omega} \\,|\\nabla\\,(\\nabla \\cdot\\,u\\,)|^2 \\,dx  -\\, \\int_{{\\Gamma}}\n\\,\\nabla\\,(\\nabla \\cdot\\,u\\,)\\cdot\\,(\\, {\\underline{n}} \\times \\,{\\underline{\\omega}}(u)\\,)\n\\,d{\\Gamma}\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSince\n$$\n{\\Delta} \\,v =\\,\\nabla\\,(\\nabla \\cdot\\,v\\,)-\\, {\\nabla} \\times\\,{\\underline{\\omega}}(v)\\,,\n$$\nit follows that%\n\\begin{equation}\\label{antes}%\n\\int_{\\Omega} \\,\\nabla\\,(\\nabla \\cdot\\,u\\,)\\cdot \\,{\\Delta}\\,v\n\\,dx=\\,\\int_{\\Omega} \\,{\\nabla}\\,(\\nabla \\cdot\\,u\\,)\\cdot \\,{\\nabla}\\,(\\nabla\n\\cdot\\,v\\,) \\,dx -\\,\\int_{\\Omega}\n\\,\\nabla\\,(\\nabla \\cdot\\,u\\,)\\cdot\\, {\\nabla} \\times \\,{\\underline{\\omega}}(v)\\, dx\\,.%\n\\end{equation}%\nOn the other hand, by appealing to the identity\n$$\n\\int_{\\Omega} \\,f\\cdot\\,({\\nabla} \\times\\,g\\,) \\,dx =\\,\\int_{\\Omega} \\,({\\nabla}\n\\times\\,f\\,) \\cdot \\,g \\,dx +\\,\\int_{{\\Gamma}} \\,f\\cdot\\,({\\underline{n}} \\times\\,g\\,)\n\\,d{\\Gamma}\\,,\n$$\nwe get\n$$\n\\int_{\\Omega} \\,\\nabla\\,(\\nabla \\cdot\\,u\\,)\\cdot\\, {\\nabla} \\times \\,{\\underline{\\omega}}(v)\\,\ndx=\\, \\int_{{\\Gamma}} \\,\\nabla\\,(\\nabla \\cdot\\,u\\,)\\cdot\\,(\\, {\\underline{n}} \\times\n\\,{\\underline{\\omega}}(v)\\,) \\,d{\\Gamma}\\,.\n$$\n\\end{proof}\nNote that the boundary integral in equation \\eqref{ochey3} vanishes\nif $\\,u\\,$ satisfies the boundary condition $\\,({\\nabla} \\times\\,u)\n\\times \\,{\\underline{n}}=\\,0\\,$ on ${\\Gamma}$. So\n\\begin{equation}\\label{ochey}\n\\int_{\\Omega} \\,\\nabla\\,(\\nabla \\cdot\\,u\\,)\\cdot\\,\\Delta\\,u \\,dx=\\,\n\\int_{\\Omega} \\,|\\nabla\\,(\\nabla \\cdot\\,u\\,)|^2 \\,dx\\,.\n\\end{equation}\nMultiplication of both sides of \\eqref{formfina2} by\n$\\,\\Delta\\,u\\,,$ followed by integration in $\\,\\Omega\\,,$ together with\n\\eqref{ochey} and \\eqref{buf2}, leads to the following a priori\nestimate, uniform with respect to $\\,{\\mu}>\\,0\\,.$\n\n$$\n\\begin{array}{ll}\\vspace{1ex}\\displaystyle%\n\\|\\Delta\\,u\\|^2_2 +\\,\\|\\nabla\\,(\\nabla \\cdot\\,u\\,)\\|^2_2 \\leq\n\\\\\n\\displaystyle (2-\\,p)\\,\\int_{\\Omega} \\,|\\nabla \\,D\\,u|\\,\\cdot\\,\\Delta\\,u \\, dx \\,+%\n\\int_{\\Omega} \\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{2-\\,p}{2}} \\,|\\,f\\,|\n\\,|\\Delta\\,u| \\,dx\\,.%\n\\end{array}%\n$$\nThis shows that, for each $\\,{\\mu}> \\,0\\,,$ it should be not difficult\nto prove the existence of a strong solution $\\,u\\,$ in\n$\\,W^{2,\\,2}(\\Omega)\\,,$ under the smallness assumption on $\\,2-\\,p\\,.$\nBy appealing to Agmon-Douglis-Nirenberg results, this should lead to\nan estimate of $\\,u\\,$ in $\\,W^{2,\\,q}(\\Omega)\\,,$ uniform with respect\nto $\\,{\\mu}>\\,0\\,.$ Clearly, we have to assume a restriction (like\n\\eqref{cqom}) relying the exponents $\\,q\\,$ and $\\,p\\,.$ The\nestimates obtained are uniform with respect to $\\,{\\mu}>\\,0\\,$.\nHowever, a suitable definition of weak solution, for $\\,{\\mu}\n\\geq\\,0\\,,$ must be established, as a previous step to try to\n\"pass to the limit\" as $\\,{\\mu}\\,$ goes to zero. Let us discuss this point.\\par%\nWe start by some identities. Since $\\,(\\partial_i \\, u_k -\\,\\partial_k u_i\n)\\,n_i=\\,({\\underline{\\omega}} \\times\\,{\\underline{n}})_k\\,,\\,$ the identity\n\\begin{equation}\\label{frontum}%\n(D\\,u)\\,\\cdot\\,{\\underline{n}} \\,\\cdot\\,v \\equiv \\,(\\partial_k\\,u_i+\\,\\partial_i\\,u_k)\nn_i\\,v_k =\\,({\\underline{\\omega}} \\times\\,{\\underline{n}})\\cdot\\,v  +\\,2\\,(\\partial_k u_i ) v_k\\,n_i%\n\\end{equation}\nfollows. Further, from\n\\begin{equation}\\label{frontdoi}\n(\\partial_k u_i ) v_k\\,n_i =\\, {\\nabla} (u\\cdot\\,{\\underline{n}})\\,\\cdot v -\\,(\\partial_k n_i )\nv_k\\,u_i\\,,\n\\end{equation}\none gets\n\\begin{equation}\\label{frontre}\n(\\,D\\,u\\,)\\cdot\\,v\\cdot\\,n =\\,({\\underline{\\omega}} \\times\\,{\\underline{n}})\\cdot\\,v\\,+\\,2\\,{\\nabla}\n(u\\cdot\\,{\\underline{n}})\\,\\cdot v -\\,2\\,(\\partial_k n_i ) v_k\\,u_i\\,.\n\\end{equation}\nIt follows, in particular, that on flat portions of the boundary the\nconditions\n\\eqref{bcns} and \\eqref{bcnos} are equivalent.\\par%\nBy appealing to \\eqref{bufe} and \\eqref{frontre}, and by assuming\nthat $\\,u\\cdot\\,n=\\,v\\cdot\\,n=\\,0\\,$ on $\\,{\\Gamma}\\,,$ we show that\n(recall definition \\eqref{buh})\n\\begin{equation}\\label{bufia} \\begin{array}{ll}\\vspace{1ex}\\displaystyle%\n\\frac12 \\,\\int_{\\Omega}\\ \\ B (D\\,u)\\, \\cdot D\\,u \\cdot D\\, v \\,dx\n=\\,-\\,\\int_{\\Omega}\\,{\\nabla} \\cdot\\,\\big(\\,B(D\\,u)\\,D(u)\\,\\big)\n\\cdot\\,v \\,dx\\,  \\\\\n\\displaystyle +\\,\\int_{{\\Gamma}} \\,B(D\\,u)\\,({\\underline{\\omega}} \\times\\,{\\underline{n}})\\cdot\\,v\\, dS\n-\\,2\\,\\int_{{\\Gamma}} \\,B(D\\,u)\\,(\\partial_k n_i ) v_k\\,u_i\\, dS\\,.%\n\\end{array}\\end{equation}%\nThe identity \\eqref{bufia} would justify to call $\\,u\\,$ a weak\nsolution of problem \\eqref{NSC}, \\eqref{bcns} if $\\,u\\,\\in\\,\nV_p(\\Omega)$ satisfies\n\\begin{equation}\\label{bufies}\\begin{array}{ll}\\vspace{1ex}\\displaystyle%\n\\frac12 \\,\\int_{\\Omega}\\ \\ B (D\\,u)\\, \\cdot D\\,u \\cdot D\\, v \\,dx\n+\\,2\\,\\int_{{\\Gamma}} \\,B(D\\,u)\\,(\\partial_k n_i ) v_k\\,u_i\\, dS= \\\\\n\\displaystyle =\\,\\int_{\\Omega}\\, f\\, \\cdot\\, v \\,dx\\,,\n\\end{array}\\end{equation}%\nfor all $\\,v \\in \\,V_p(\\Omega)\\,$. Note that if the boundary integral in\nequation \\eqref{bufies} vanishes for all $\\,v\\,$ such that\n$\\,v\\cdot\\,n=\\,0\\,$, then $\\,B(D\\,u)\\,({\\underline{\\omega}} \\times\\,{\\underline{n}})=\\,0\\,$ on\n$\\,{\\Gamma}\\,.$ Since $\\,B(D\\,u)\\neq \\,0\\,$, the second boundary\ncondition \\eqref{bcns} follows. However, the boundary integral in\nequation \\eqref{bufies} is not well defined due to the term\n$B(D\\,u)\\,,$ except for $\\,p=\\,2\\,.$\\par%\nNote that, in equation \\eqref{bufies}, for $\\,v=\\,u\\,$  and  $\\Omega\\,$\nconvex, the integrand in the boundary integral is nonnegative. So,\nin a convex domain, we may obtain, at least, an a priori estimate in\n$\\,W^{1,\\,p}(\\Omega)\\,$.\n\\section{On the Fluid Mechanics system}\\label{ssetes}\nThe proof of theorem \\ref{teoremaq} may be immediately adapted to\nthe case $\\,q<\\,n\\,$, also considered in \\cite{BVCRI} and\n\\cite{nonhom}. In the case $\\,q<\\,n\\,$, see \\cite{BVCRI}, the\nassumption $\\,f\\in L^q(\\Omega)\\,$ does not imply\n$\\,u\\in\\,W^{2,q}(\\Omega)\\,.$ This last regularity result requires a\nstronger assumption on $\\,f\\,$, namely $\\,f\\in\\,L^{r(q)}(\\Omega)\\,,$\nwhere $\\,r(q)\\,$ is given by\n\\begin{equation}\\label{rqqq}%\nr(q)=\\,\\frac{nq}{n(p-1)+q(2-p)}\\,.%\n\\end{equation}%\nNote that $\\,r(q)>\\,q\\,,$ and $\\,r(n)=\\,n\\,$.\\par%\nSince, on the whole, regularity results are stronger for large\nvalues of $\\,q\\,$, in \\cite{BVCRI} the authors have assumed, for\nconvenience, that $\\,q\\geq\\,2\\,.$ This assumption excludes the\nsignificant case of square integrable external forces\n$\\,f\\in\\,L^2(\\Omega)\\,.$ In fact, by \\eqref{rqqq}, $\\,r(q)=\\,2\\,$ holds\nfor $\\,q=\\,{\\widehat{q}}\\,$ given by\n\\begin{equation}\\label{errq2}%\n{\\widehat{q}} =\\,\\frac{2\\,n\\,(\\,p-\\,1\\,)}{n-\\,2\\,(\\,2-\\-p\\,)}<\\,2\\,.\n\\end{equation}\nHowever, the proof shown in reference \\cite{BVCRI} also applies to\nvalues $\\,q<\\,2\\,$, in particular to $\\,{\\widehat{q}}\\,.$ The (really obvious)\nmodification required to adapt the proof to this case was shown in\nreference \\cite{BV-ARX}. In this last reference we were interested\nin the particular case $\\,r(q)=\\,2\\,,$ i.e. $\\,q=\\,{\\widehat{q}}\\,.$ In\nproposition 2.1 in \\cite{BV-ARX} we basically remark that if $ f\\in\nL^2(\\Omega)$ then $u$ belongs to $W^{2,\\,{\\widehat{q}}}(\\Omega)\\,.$ Moreover,\n\\begin{equation}\\label{dnqn}\n\\|u\\|_{2,{\\widehat{q}}}\\leq C\n\\,\\left(\\|f\\|_{{\\widehat{q}}}+\\|f\\|_{2}^\\frac{1}{p-1}\\right)\\,.\n\\end{equation}\nOn the other hand, during a recent meeting in Levico (December\n2012), Jakub Tich\\'y informed us about some new results obtained in\ncollaboration with Petr Kaplick\\'y, in reference \\cite{kapli}. The\nvery interesting results obtained by these authors concern the\ngeneralized Stokes problem\n\\begin{equation}\\label{NSCCC}\\left\\{\n\\begin{array}{ll}\\vspace{1ex}\n-\\,\\nabla \\cdot\n\\,\\big(\\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}\\,D\\,u\\,\\big) +\\,{\\nabla}\n\\,\\pi =\\,f\\,,\n\\\\%\n\\,{\\nabla} \\cdot\\,u=\\,0\\,,\n\\end{array}\\right.\n\\end{equation}\nunder the Navier boundary condition \\eqref{bcns}. Actually they\nconsider more general constitutive relations $\\,S(D(u))\\,.$ In\nparticular (see the theorem 1.8 in reference \\cite{kapli}), under\nnatural assumptions on the external force $\\,f\\,,$ the solution to\nproblem \\eqref{NSCCC}, under the Navier boundary condition,\nsatisfies\n\\begin{equation}\\label{naosei}\n-\\,\\nabla \\cdot\n\\,\\big(\\,(\\,{\\mu}+|\\,D\\,u|^2\\,)^{\\frac{p-2}{2}}\\,D\\,u\\,\\big)\n\\in\\,\\L^2(\\Omega)\\,.\n\\end{equation}\nSo, by appealing to our theorem \\ref{teoremaq} (adapted, as\ndescribed above, to the value $\\,q=\\,{\\widehat{q}}\\,$), it follows that the\nsolutions to problem \\eqref{NSCCC} belong to $W^{2,\\,{\\widehat{q}}}(\\Omega)\\,.$\nClearly, we have to assume that condition \\eqref{cqom} holds for the\nvalue $\\,q=\\,{\\widehat{q}}\\,.$\n\n\\vspace{0.2cm}\n\nWe take the occasion to announce that some new results concerning\nthe system \\eqref{NSCCC} in the torus will be shown in a forthcoming\npaper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFrustration in low dimensional strongly correlated magnetic systems leads to a plethora of fascinating  behaviors \\cite{Fazekas1999,Lauchli2006,Balents2010,Witczak-Krempa2014,Rau2016,Savary2017,Winter2017,Zhou2017}.\nAn unusual way of introducing magnetic frustrations is by strong spin-orbit couplings, which induce bond- and direction-dependent magnetic interactions \\cite{Jackeli2009,Chaloupka2010,Rau2014}. \nA famous example of frustrated magnetic system of this type is the two-dimensional (2D) Kitaev model on the honeycomb lattice \\cite{Kitaev2006}.\nThe model was proposed to host exotic fractionalized excitations including Majorana fermions and anyons \\cite{Kitaev2006},\nand  has triggered tremendous research interests in recent years \\cite{Singh2010,Reuther2011,Jiang2011,Price2012,Choi2012,Singh2012,Chaloupka2013,Modic2014,Plumb2014,Kim2015,Johnson2015,Sandilands2015,Sears2015,Banerjee2016,Yadav2016,Baek2017,Banerjee2017,Zheng2017,Ran2017,Wang2017,Janssen2017,Liu2018,Catuneanu2018,Gohlke2018,Jansa2018,Yu2018,Hentrich2018,Kasahara2018,Gordon2019,Motome2020}.\nThe 2D Kitaev model can be realized in Mott insulating A$_2$IrO$_4$ (A=Li, Na) compounds and $\\alpha$-RuCl$_3$ systems.\nIn real materials, additional symmetry allowed couplings also appear. \nThe generalized Kitaev model has been proposed to describe the real systems \\cite{Jackeli2009,Rau2014,Wang2017}, which includes Heisenberg and Gamma interactions in addition to the Kitaev coupling.\n\nSince quantum fluctuations are enhanced by reducing the spatial dimension,\nexotic behaviors are expected to emerge also in one-dimensional (1D)  strongly spin-orbit coupled quantum magnetic systems.\nA series of recent works have performed both analytical and numerical studies on the phase diagram of 1D spin-1\/2 generalized Kitaev models \\cite{Agrapidis2018,Yang2020a,Yang2020,Yang2020b}.\nThe two-leg ladder case has also been analyzed \\cite{Agrapidis2019,Catuneanu2019}, which already shows a similar phase diagram with the 2D case \\cite{Catuneanu2019}. \nIn particular, in Ref. \\onlinecite{Yang2020b}, the phase diagram of the 1D spin-1\/2 Kiteav-Heisenberg-Gamma chain has been studied in detail, which reveals a rich phase diagram with eleven distinct phases.\n\nIn this work, we perform a combination of  classical  and spin wave analysis on the  1D spin-$S$ Kitaev-Heisenberg-Gamma model with an antiferromagnetic (AFM) Kitaev coupling. \nThe phase diagram is shown in Fig. \\ref{fig:phase}.\nThe N\\'eel and ``$D_3$-breaking I, II\" phases for the spin-1\/2 case found in Ref. \\onlinecite{Yang2020b} are also confirmed for higher spins.\nOn the other hand, the classical analysis predicts an $O_h\\rightarrow D_3$ symmetry breaking for $J=0$, which is in contrast with the $O_h\\rightarrow D_4$ symmetry breaking for the spin-1\/2 case as discussed in Ref. \\onlinecite{Yang2020a}.\nOur DMRG numerics provide evidence for the $O_h\\rightarrow D_3$  symmetry breaking for $S=1$ and $3\/2$, based on which we conjecture that the spin-1\/2 case is the only exception where strong quantum fluctuations invalidate the classical analysis. \n\nWe have also constructed the spin wave theory which captures the small fluctuations around the classical configurations.\nThe lowest-lying spin wave mass $m_1$ is calculated perturbatively in the ``N\\'eel\", ``$O_h\\rightarrow D_3$\" and ``$D_3$-breaking I\" phases close to the hidden SU(2) symmetric ferromagnetic (FM) FM2 point in Fig. \\ref{fig:phase}. \nInterestingly, although $m_1\\propto (K-\\Gamma)^2$  in the ``$O_h\\rightarrow D_3$\" phase (where $J=0$)  and $m_1\\propto J^2$ in the ``$D_3$-breaking I\" phase for $K=\\Gamma$, the former requires a second order symplectic perturbation calculation, whereas to obtain the latter, one has to go to third order perturbation,\nwhere $K$, $\\Gamma$  and $J$ represent the Kitaev, Gamma and Heisenberg couplings, respectively.\nIn the ``$D_3$-breaking II\" phase, we encounter intrinsic difficulties in the perturbative calculation of the spin wave mass, and $m_1$ is studied numerically.\nThe origin of such difficulty is worth further explorations. \nFinally, we emphasize that the phase diagram in Fig. \\ref{fig:phase} possibly can only be trusted in a neighborhood of the FM2 point.\nWhen approaching the origin of Fig. \\ref{fig:phase} (i.e., the AFM Kitaev point), enhanced quantum fluctuations arising from frustrations may destroy the classical order.\n\n\n\n\n\\begin{figure}[h]\n\\includegraphics[width=8.5cm]{FM2_phase_diagram.pdf}\n\\caption{Classical phase diagram in the vicinity of the FM2 point.\nThe horizontal coordinate $\\varphi$ is defined through $K=\\cos(\\varphi)$, $\\Gamma=\\sin(\\varphi)$.\nThe $\\varphi$-coordinates of $K$, FM2 and $\\Gamma$ points when $J=0$ are $0$, $\\pi\/4$ and $\\pi\/2$, respectively.\nThe classical phase transition at $\\Gamma$ is shifted to $\\varphi_c$ by quantum fluctuations.\n} \\label{fig:phase}\n\\end{figure}\n\n\\section{Model Hamiltonian}\n\n\\subsection{The Hamiltonian}\n\nThe spin-$S$ Kitaev-Heisenberg-Gamma ($KH\\Gamma$) chain \\cite{Rau2014} is defined as\n\\begin{flalign}\n&H=\\sum_{<ij>\\in\\gamma\\,\\text{bond}}\\big[ KS_i^\\gamma S_j^\\gamma+ J\\vec{S}_i\\cdot \\vec{S}_j+\\Gamma (S_i^\\alpha S_j^\\beta+S_i^\\beta S_j^\\alpha)\\big],\n\\label{eq:Ham}\n\\end{flalign}\nin which $<ij>$ is used to denote that $i,j$ are nearest neighboring lattice sites;\n$\\gamma=x,y$ is the spin direction associated with the $\\gamma$ bond shown in Fig. \\ref{fig:bonds} (a); \n$\\alpha\\neq\\beta$ are the two remaining spin directions other than $\\gamma$; \n$K$, $J$, and $\\Gamma$ \nare the Kitaev, Heisenberg, and Gamma couplings, respectively;\nand the spin operators satisfy $\\sum_{\\alpha=x,y,z}(S_i^\\alpha)^2=S(S+1)$.\nSince $R(\\hat{z},\\pi)$ changes the sign of $\\Gamma$ but leaves $K$ and $J$ invariant, there is the equivalence \\cite{Yang2020b}\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n(K,J,-\\Gamma)\\simeq (K,J,\\Gamma),\n\\label{eq:equivalence}\n\\eea\nwhere the notation $R(\\hat{n},\\alpha)$ is used to represent a global spin rotation around the $\\hat{n}$-direction by an angle $\\alpha$.\nParametrizing $K$ and $\\Gamma$ as \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nK=\\cos(\\varphi), ~\\Gamma=\\sin(\\varphi),\n\\eea\nit is enough to consider $\\varphi\\in[0,\\pi]$ due to the equivalence in Eq. (\\ref{eq:equivalence}).\nOccasionally, we also use the following parametrization\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nK=\\sin(\\theta)\\cos(\\varphi), ~\\Gamma=\\sin(\\theta)\\sin(\\varphi),~J=\\cos(\\theta).\n\\label{eq:parametize_theta_phi}\n\\eea\nIn this work, we will be interested in the region with an antiferromagnetic Kitaev coupling, i.e., $\\varphi\\in[0,\\pi\/2]$.\nIn particular, we mainly study the region in the vicinity of the FM2 point in Fig. \\ref{fig:phase} where the coordinates of FM2 are $\\varphi=\\pi\/4$, $J=0$ (i.e., $\\theta=\\pi\/2$).\nHere we note that the notation ``FM2\" is chosen in accordance with Ref. \\onlinecite{Yang2020b}.\n\n\\begin{figure}\n\\includegraphics[width=0.48\\textwidth]{bonds.pdf}\n\\caption{Bond structures (a) before and (b) after the six-sublattice rotation.\nThe rectangular boxes denote unit cells.\n}\n\\label{fig:bonds}\n\\end{figure}\n\nA particularly useful six-sublattice rotation $U_6$ is defined as \n\\cite{Stavropoulos2018,Yang2020a}\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\text{Sublattice $1$}: & (x,y,z) & \\rightarrow (x^{\\prime},y^{\\prime},z^{\\prime}),\\nonumber\\\\ \n\\text{Sublattice $2$}: & (x,y,z) & \\rightarrow (-x^{\\prime},-z^{\\prime},-y^{\\prime}),\\nonumber\\\\\n\\text{Sublattice $3$}: & (x,y,z) & \\rightarrow (y^{\\prime},z^{\\prime},x^{\\prime}),\\nonumber\\\\\n\\text{Sublattice $4$}: & (x,y,z) & \\rightarrow (-y^{\\prime},-x^{\\prime},-z^{\\prime}),\\nonumber\\\\\n\\text{Sublattice $5$}: & (x,y,z) & \\rightarrow (z^{\\prime},x^{\\prime},y^{\\prime}),\\nonumber\\\\\n\\text{Sublattice $6$}: & (x,y,z) & \\rightarrow (-z^{\\prime},-y^{\\prime},-x^{\\prime}),\n\\label{eq:6rotation}\n\\eea\nin which \"Sublattice $i$\" ($1\\leq i \\leq 6$) represents the collection of the sites $\\{ i+6n\\}_{n\\in \\mathbb{Z}}$, and $S^\\alpha$ ($S^{\\prime \\alpha}$) is abbreviated as $\\alpha$ ($\\alpha^\\prime$) for short, where $\\alpha=x,y,z$.\nThe transformed Hamiltonian $H^\\prime=U_6 H U_6^{-1}$ acquires the form\n\\begin{eqnarray}\nH^\\prime&=\\sum_{<ij>\\in \\gamma\\,\\text{bond}}\\big[ -KS_i^\\gamma S_j^\\gamma-\\Gamma (S_i^\\alpha S_j^\\alpha+S_i^\\beta S_j^\\beta) \\nonumber\\\\\n& -J(S_i^\\gamma S_j^\\gamma+S_i^\\alpha S_j^\\beta+S_i^\\beta S_j^\\alpha)\\big],\n\\label{eq:6rotated}\n\\end{eqnarray}\nin which the bond $\\gamma=x,z,y$ is periodic under translation by three sites as shown in Fig. \\ref{fig:bonds} (b),\nand the  prime has been dropped in $\\vec{S}_i^\\prime$ for simplicity.\nThe explicit form of $H^\\prime$ is included in Appendix \\ref{app:Ham}.\nIt is clear from Eq. (\\ref{eq:6rotated}) that the FM2 point is SU(2) invariant in the six-sublattice rotated frame with an FM coupling. \n\nIn the remaining parts of the paper, we will stick to the six-sublattice rotated frame from here on  unless otherwise stated.\n\n\\subsection{Review of the  symmetries}\n\\label{sec:review_Oh}\n\nIn this section, we give a quick review of the symmetries of the model within the six-sublattice rotated frame.\n\n\\begin{figure}[h]\n\\includegraphics[width=7.3cm]{Oh_ops.pdf}\n\\caption{Actions of the elements in $G\/\\mathopen{<}T_{3a}\\mathclose{>}$ in  spin space as symmetry operations of a cube.\n} \\label{fig:Oh_ops}\n\\end{figure}\n\nWe first consider the $J=0$ case, i.e., the Kitaev-Gamma chain.\nThe symmetry group has been discussed in detail in Ref. \\onlinecite{Yang2020a}.\nThe symmetry transformations include:\n\\begin{eqnarray}\n1.&T &:  (S_i^x,S_i^y,S_i^z)\\rightarrow (-S_{i}^x,-S_{i}^y,-S_{i}^z)\\nonumber\\\\\n2.& R_aT_a&:  (S_i^x,S_i^y,S_i^z)\\rightarrow (S_{i+1}^z,S_{i+1}^x,S_{i+1}^y)\\nonumber\\\\\n3.&R_I I&: (S_i^x,S_i^y,S_i^z)\\rightarrow (-S_{10-i}^z,-S_{10-i}^y,-S_{10-i}^x)\\nonumber\\\\\n4.&R(\\hat{x},\\pi)&: (S_i^x,S_i^y,S_i^z)\\rightarrow (S_{i}^x,-S_{i}^y,-S_{i}^z)\\nonumber\\\\\n5.&R(\\hat{y},\\pi)&: (S_i^x,S_i^y,S_i^z)\\rightarrow (-S_{i}^x,S_{i}^y,-S_{i}^z)\\nonumber\\\\\n6.&R(\\hat{z},\\pi)&: (S_i^x,S_i^y,S_i^z)\\rightarrow (-S_{i}^x,-S_{i}^y,S_{i}^z),\n\\label{eq:syms_KG}\n\\end{eqnarray}\nin which $T$ is time reversal; $T_a$ is translation by one lattice site;\n$I$ is the spatial inversion around the point $C$ in Fig. \\ref{fig:bonds} (b); \nand $R_a=R(\\hat{n}_a,-2\\pi\/3)$, $R_I=R(\\hat{n}_I,\\pi)$ where \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\hat{n}_a=\\frac{1}{\\sqrt{3}}(1,1,1)^T,~\\hat{n}_I=\\frac{1}{\\sqrt{2}}(1,0,-1)^T.\n\\label{eq:na_nI}\n\\eea\nWe note that the inversion center $C$ can be chosen modulo three.\nThe symmetry group $G$ is generated by the above transformations as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nG=\\mathopen{<}  T,R_aT_a,R_I I, R(\\hat{x},\\pi),R(\\hat{y},\\pi),R(\\hat{z},\\pi) \\mathclose{>}.\n\\eea\n\nSince $T_{3a}=(R_aT_a)^3$ is an abelian normal subgroup of $G$, we can consider the quotient group $G\/\\mathopen{<}T_{3a}\\mathclose{>}$.\nIt has been worked out in Ref. \\onlinecite{Yang2020a} that the quotient group is isomorphic to $O_h$, \nwhere $O_h$ is the full octahedral group which is the symmetry group of a cube.\nThere is an intuitive understanding of this isomorphism.\nNeglecting the spatial components in the operations,\nthe actions in spin space are all symmetries of a spin cube as shown in Fig. \\ref{fig:Oh_ops}.\nIt is proved in Ref. \\onlinecite{Yang2020a} that the isomorphism still holds even if the spatial components are also included.\nHence we conclude that\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nG\\cong O_h \\ltimes 3\\mathbb{Z},\n\\eea\nwhere $3\\mathbb{Z}=\\mathopen{<}T_{3a}\\mathclose{>}$ and $\\ltimes$ is the semi-direct product.\n\nNext we consider the $J\\neq 0$ case, i.e., a general Kitaev-Heisenberg-Gamma chain.\nIn this case, the system is no longer invariant under the operations $R(\\hat{\\alpha},\\pi)$ ($\\alpha=x,y,z$).\nThus the symmetry group $G_1$ is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nG_1=\\mathopen{<}  T,R_aT_a,R_I I \\mathclose{>}.\n\\eea\nIt has been shown in Ref. \\onlinecite{Yang2020} that the group structure of $G_1$ is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nG_1\\cong D_{3d}\\ltimes 3\\mathbb{Z},\n\\eea\nin which $\\mathopen{<}T,R_aT_a,R_II\\mathclose{>}\/\\mathopen{<}T_{3a}\\mathclose{>}\\cong D_{3d}$ is used. \n\n\\subsection{Summary of the classical phase diagram}\n\\label{sec:cl_phase}\n\n\\begin{figure}\n\\includegraphics[width=7cm]{spin_orders.pdf}\n\\caption{``Center of mass\" directions of the three spins within a unit cell \nin the six-sublattice rotated frame as represented by: \nthe eight solid blue circles  for the eight degenerate ground states in the ``$O_h\\rightarrow D_3$\" phase;\nthe two solid light blue circles (along the $\\pm\\hat{n}_a$-directions) for the two degenerate ground states in the N\\'eel phase; \nthe six solid red circles  for the six degenerate ground states in the ``$D_3$-breaking I\" phase;\nthe six solid dark blue circles (removing the two light blue ones among the eight) for the six degenerate ground states in the ``$D_3$-breaking II\" phase.\n In the ``$D_3$-breaking II\" phase, the plots are for $J\\rightarrow 0$ according to the classical analysis.\nIn this paper, the convention  of the coordinates is taken such that  the eight vertices of the cube are  located at $(\\pm 1,\\pm1,\\pm1)$.\n} \n\\label{fig:spin_orders}\n\\end{figure}\n\nHere we give a brief summary  on the classical phase diagram as shown in Fig. \\ref{fig:phase}.\n\nThe system has a long-range N\\'eel order for $J>0$ where N\\'eel refers to the original frame \\cite{Yang2020b}. \nIn the six-sublattice rotated frame, the ``center of mass\" directions of the three spins in a unit cell are along $\\pm\\hat{n}_a$-directions as shown by the two solid light blue circles in Fig. \\ref{fig:spin_orders},\nwhere $\\hat{n}_a$ is defined in Eq. (\\ref{eq:na_nI}).\nFor $|\\Delta|, |J|\\ll1$, the lowest-lying spin wave mass is calculated to be $(\\frac{4}{81}\\Gamma \\Delta^2+\\frac{2}{3}J)S$,\nwhere $\\Delta=(K-\\Gamma)\/\\Gamma$.\n\nWhen $J=0$, the ground states are eight-fold degenerate with an $O_h\\rightarrow D_3$ symmetry breaking. \nOur DMRG numerics provide evidence for the $O_h\\rightarrow D_3$ symmetry breaking for $S=1$ and $3\/2$,\nthough the spin-1\/2 case is different which has  an $O_h\\rightarrow D_4$ symmetry breaking as discussed in Ref. \\onlinecite{Yang2020a}. \nThe ``center of mass\" spin directions of a unit cell in the eight degenerate $O_h\\rightarrow D_3$ ground states are shown by the eight solid blue circles in Fig. \\ref{fig:spin_orders}.\nThe classical phase transition point for $J=0$ is located at the $\\Gamma$-point (i.e., $\\varphi=\\pi\/2$), which is shifted to a different point $\\varphi_c$ due to quantum fluctuations.\nFor $|\\Delta|\\ll1$, the lowest-lying spin wave mass is calculated to be $\\frac{4}{81}S \\Gamma  \\Delta^2$.\n\nWhen $J<0$, there are two phases, namely ``$D_3$-breaking I, II\", both having six-fold degenerate ground states.\nThe symmetry breaking patterns of the two phases are $D_{3d}\\rightarrow \\mathbb{Z}_2^{\\text{(I)}}$ and $D_{3d}\\rightarrow \\mathbb{Z}_2^{\\text{(II)}}$, respectively, \nwhere $\\mathbb{Z}_2^{\\text{(I)}}$ and $\\mathbb{Z}_2^{\\text{(II)}}$ are two different symmetry groups albeit both isomorphic to $\\mathbb{Z}_2$.\nIn the ``$D_3$-breaking I\" phase, the ``center of mass\" spin directions of a unit cell \nin the six degenerate ground states\nwithin the six-sublattice rotated frame are plotted as the six solid red circles in Fig. \\ref{fig:spin_orders}. \nWe have calculated the lowest-lying spin wave mass $m_1$ for $\\Delta=0$, $|J|\\ll 1$\nand the result is $S J^2\/\\Gamma $.\nAlthough $m_1$ is proportional to $J^2$, it requires a third order symplectic perturbation calculation as discussed in Sec. \\ref{sec:sw_D3I}.\nIn the ``$D_3$-breaking II\" phase, the ``center of mass\" spin directions\nin the six degenerate ground states\n in the limit $J\\rightarrow 0$ are plotted as the six solid dark blue circles in Fig. \\ref{fig:spin_orders}.\nFor larger $|J|$, the ``center of mass\" directions are distorted away from the vertices of the cube.\nDue to intrinsic difficulties in doing perturbation in the ``$D_3$-breaking II\" phase, we are not able to obtain a perturbative expression for the spin wave mass.\nOn the other hand, the lowest-lying spin wave mass has been studied numerically as shown in Fig. \\ref{fig:sw_mass_D3}.\nWe note that our DMRG numerics provide evidence for  the spin ordering patterns in both ``$D_3$-breaking I, II\" phases for $S=1,3\/2$. \n\nFinally we make a comment on the numerical methods that we employ in this work.\nThe DMRG method\\cite{White1992} was used on chains with length of $L=18$ sites and periodic boundary conditions within the six-sublattice rotated frame. The calculation of the first ten eigenstates was performed using standard DMRG multi-targeting approaches\\cite{White1993}.\nEven though it is known that DMRG convergence is hard for periodic boundary conditions, we have checked that for the system size considered our results are converged using up to m= 1000 states with a truncation error below $10^{-6}$ as in previous investigations\\cite{Yang2020,Yang2020a,Yang2020b}.\n\n\n\\section{The ``$O_h\\rightarrow D_3$\" phase for $J=0$}\n\nIn this section, we perform a combination of classical and spin wave analysis for $J=0$ in the vicinity of the FM2 point in Fig. \\ref{fig:phase}.\nIn Sec. \\ref{sec:classical_equator}, the trial classical solution is demonstrated to be a minimum of the classical free energy by showing that the eigenvalues of the Hessian matrix are all positive. \nIn Sec. \\ref{sec:sym_break_equator}, the symmetry breaking pattern of the classical solution is shown to be $O_h\\rightarrow D_3$, exhibiting an eight-fold degeneracy.\nThen in Sec. \\ref{sec:SW_equator}, we derive the spin wave theory by quantizing the Gaussian fluctuations around the classical minima in the long wavelength limit.\nThe smallest spin wave mass is shown to be $\\frac{4}{81}S\\Gamma \\Delta^2$ up to the leading nonvanishing order in $\\Delta$.\nFinally in Sec. \\ref{sec:numerics_OhD3}, we provide numerical evidence for the ``$O_h\\rightarrow D_3$\" symmetry breaking for $S=1$  and $S=3\/2$.\nWe work in the six-sublattice rotated frame throughout this section unless otherwise stated.\n\n\n\\subsection{The classical solutions}\n\\label{sec:classical_equator}\n\nThe classical analysis is the saddle point approximation in the spin path integral formalism which is valid in the large-$S$ limit. \nIn what follows, we neglect  quantum fluctuations of the spins and approximate them as classical three-vectors, i.e., \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\vec{S}_i = S\\hat{n}_i,\n\\eea\nin which $S$ is the spin magnitude, and $\\hat{n}_i=(x_i,y_i,z_i)^T$ is a unit vector.\nThe classical free energy of a general $KH\\Gamma$ chain is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nf&=&\\sum_n (f_{1+3n}+f_{2+3n}+f_{3+3n}),\n\\label{eq:f}\n\\eea\nin which\n\\begin{flalign}\n&f_{1+3n}=-(K+J)S^2 x_{1+3n}x_{2+3n}-\\Gamma S^2[y_{1+3n}y_{2+3n}\\nonumber\\\\\n&~~+z_{1+3n}z_{2+3n}]-JS^2[y_{1+3n}z_{2+3n}+z_{1+3n}y_{2+3n}],\\nonumber\\\\\n&f_{2+3n}=-(K+J)S^2 z_{2+3n}z_{3+3n}-\\Gamma S^2[x_{2+3n}x_{3+3n}\\nonumber\\\\\n&~~+y_{2+3n}y_{3+3n}]-JS^2[x_{2+3n}y_{3+3n}+y_{2+3n}x_{3+3n}],\\nonumber\\\\\n&f_{3+3n}=-(K+J)S^2 y_{3+3n}y_{4+3n}-\\Gamma S^2[z_{3+3n}z_{4+3n}\\nonumber\\\\\n&~~+x_{3+3n}x_{4+3n}]-JS^2[z_{3+3n}x_{4+3n}+x_{3+3n}z_{4+3n}].\n\\end{flalign}\nThe constraints\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nx_j^2+y_j^2+z_j^2=1\n\\label{eq:constraints}\n\\eea \ncan be introduced via Lagrange multipliers $\\{\\lambda_j\\}_{j\\in \\mathbb{Z}}$\nso that the free energy becomes\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nf^\\prime=f-\\frac{1}{2}\\sum_j \\lambda_j(x_j^2+y_j^2+z_j^2-1).\n\\label{eq:fprime}\n\\eea\nWe will first write down the saddle point equations for a general $J$, and later take $J=0$ in this section.\n\nSeeking classical minima that are invariant under $T_{3a}$, i.e.,\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nx_{i+3n}\\equiv x_i,y_{i+3n}\\equiv y_i, z_{i+3n}\\equiv z_i,~ (1\\leq i\\leq 3),\n\\eea\nthe energy per unit cell $F=3f^\\prime\/L$ becomes\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nF&=& -(K^\\prime+J^\\prime) x_1x_2-\\Gamma^\\prime(y_1y_2+z_1z_2)-J^\\prime(y_1z_2+z_1y_2)\\nonumber\\\\\n&&-(K^\\prime+J^\\prime) z_2z_3-\\Gamma^\\prime(x_2x_3+y_2y_3)-J^\\prime(x_2y_3+y_2x_3)\\nonumber\\\\\n&&-(K^\\prime+J^\\prime) y_3y_1-\\Gamma^\\prime(z_3z_1+x_3x_1)-J^\\prime(z_3x_1+x_3z_1)\\nonumber\\\\\n&&-\\sum_{i=1}^3 \\frac{1}{2}\\lambda_i (x_i^2+y_i^2+z_i^2-1),\n\\label{eq:energy_KHG}\n\\eea\nin which $\\Gamma^\\prime,K^\\prime,J^\\prime$ are defined as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\Gamma^\\prime=\\Gamma S^2, ~K^\\prime=K S^2, ~J^\\prime=J S^2.\n\\eea\nFrom Eq. (\\ref{eq:energy_KHG}), the saddle point equations  can be derived as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\frac{\\partial F}{\\partial x_1}&=&-(K^\\prime+J^\\prime)x_2-\\Gamma^\\prime x_3-J^\\prime z_3-\\lambda_1 x_1=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial y_1}&=&-\\Gamma^\\prime y_2-(K^\\prime+J^\\prime)y_3-J^\\prime z_2-\\lambda_1 y_1=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial z_1}&=&-\\Gamma^\\prime z_2-\\Gamma^\\prime z_3-J^\\prime x_3-J^\\prime y_2-\\lambda_1 z_1=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial \\lambda_1}&=&-(x_1^2+y_1^2+z_1^2-1)=0\n\\label{eq:partial_F1}\n\\eea\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\frac{\\partial F}{\\partial x_2}&=&-(K^\\prime+J^\\prime)x_1-\\Gamma^\\prime x_3-J^\\prime y_3-\\lambda_2 x_2=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial y_2}&=&-\\Gamma^\\prime y_1-\\Gamma^\\prime y_3-J^\\prime z_1-J^\\prime x_3-\\lambda_2 y_2=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial z_2}&=&-\\Gamma^\\prime z_1-(K^\\prime+J^\\prime) z_3-J^\\prime y_1-\\lambda_2 z_2=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial \\lambda_2}&=&-(x_2^2+y_2^2+z_2^2-1)=0\n\\label{eq:partial_F2}\n\\eea\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\frac{\\partial F}{\\partial x_3}&=&-\\Gamma^\\prime x_2-\\Gamma^\\prime x_1-J^\\prime y_2-J^\\prime z_1-\\lambda_3 x_3=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial y_3}&=&-\\Gamma^\\prime y_2-(K^\\prime+J^\\prime) y_1-J^\\prime x_2-\\lambda_3 y_3=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial z_3}&=&-(K^\\prime+J^\\prime) z_2-\\Gamma^\\prime z_1-J^\\prime x_1-\\lambda_3 z_3=0\\nonumber\\\\\n\\frac{\\partial F}{\\partial \\lambda_3}&=&-(x_3^2+y_3^2+z_3^2-1)=0.\n\\label{eq:partial_F3}\n\\eea\n\nFor the purpose of discussing the Kitaev-Gamma chain in this section, $J$ should be taken as zero.\nTaking $J=0$, and plugging the following trial solutions \n\\begin{flalign}\n&\\hat{n}^{(0)}_1=(x_1,y_1,z_1)^T=(a,a,b)^T,\\nonumber\\\\\n&\\hat{n}^{(0)}_2=(x_2,y_2,z_2)^T=(a,b,a)^T,\\nonumber\\\\\n&\\hat{n}^{(0)}_3=(x_3,y_3,z_3)^T=(b,a,a)^T,\\nonumber\\\\\n&\\lambda_1=\\lambda_2=\\lambda_3=\\lambda,\n\\label{eq:spins_OhD4}\n\\end{flalign}\ninto Eqs. (\\ref{eq:partial_F1},\\ref{eq:partial_F2},\\ref{eq:partial_F3}), \nwhere the superscript ``$(0)$\" is used to indicate that these are saddle point solutions,\nwe find that Eqs. (\\ref{eq:partial_F1},\\ref{eq:partial_F2},\\ref{eq:partial_F3}) are reduced to\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n-(\\lambda+K^\\prime)a -\\Gamma^\\prime b&=&0 \\nonumber\\\\\n-2\\Gamma^\\prime a -\\lambda b&=& 0\\nonumber\\\\\n2a^2+b^2-1&=&0.\n\\label{eq:partial_red_equator}\n\\eea\nSince there are three variables $a,b,\\lambda$ and three equations,\nthe solution of Eq. (\\ref{eq:partial_red_equator}) exists.\nIn particular, $\\lambda$ can be determined from the secular equation\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\det \\left(\\begin{array}{cc}\n-(\\lambda+K^\\prime) & -\\Gamma^\\prime \\\\\n-2\\Gamma^\\prime & -\\lambda\n\\end{array}\\right)=0.\n\\label{eq:lambda_eq_equator}\n\\eea\n\nWhen $K=\\Gamma$, there are two solutions of $\\lambda$ solved from Eq. (\\ref{eq:lambda_eq_equator}), i.e., $\\lambda^{(1)}=\\Gamma^\\prime$ and $\\lambda^{(2)}=-2\\Gamma^\\prime$.\nThe solution $\\lambda^{(2)}$ should be kept, since the free energy $F$ in Eq. (\\ref{eq:energy_KHG}) acquires a larger value for $\\lambda^{(1)}$ than for $\\lambda^{(2)}$.\nWhen $K\\neq \\Gamma$, Eq. (\\ref{eq:partial_red_equator}) can be solved perturbatively in an expansion over $\\Delta$, where the parameter $\\Delta$ is defined as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\Delta=(K-\\Gamma)\/\\Gamma.\n\\label{eq:define_Delta}\n\\eea \nThe results up to $O(\\Delta^2)$ are\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\lambda(\\Delta)&=& (-2-\\frac{2}{3}\\Delta-\\frac{2}{27} \\Delta^2)\\Gamma^\\prime+O(\\Delta^3),\\nonumber\\\\\na(\\Delta)&=& \\frac{1}{\\sqrt{3}} (1+\\frac{1}{9} \\Delta-\\frac{2}{81} \\Delta^2)+O(\\Delta^3),\\nonumber\\\\\nb(\\Delta)&=& \\frac{1}{\\sqrt{3}} (1-\\frac{2}{9} \\Delta+\\frac{1}{81} \\Delta^2)+O(\\Delta^3).\n\\label{eq:saddle_Oh_D4}\n\\eea\nWe note that among the two solutions of $\\lambda$, the one which reduces to $-2\\Gamma^\\prime$ for $\\Delta=0$ is kept in Eq. (\\ref{eq:saddle_Oh_D4}).\n\nOn the other hand, Eq. (\\ref{eq:saddle_Oh_D4}) only represents a saddle point solution, not necessarily a global minimum of the free energy.\nNext we perturbatively show that the eigenvalues of the Hessian matrix of the free energy $F$ are all positive at least for $|\\Delta|\\ll 1$,\nthereby confirming that Eq. (\\ref{eq:saddle_Oh_D4}) constitutes a minimal solution.\nNumerics of the classical analysis provide evidence for Eq. (\\ref{eq:saddle_Oh_D4}) to be a  global  minimum of the free energy as discussed in Appendix \\ref{app:numerics_classical}.\n\nBecause of the constraints in Eq. (\\ref{eq:constraints}), the $T_{3a}$-invariant spin configurations form a six-dimensional manifold in the nine-dimensional Euclidean space spanned by the nine coordinates $\\{x_i,y_i,z_i\\}_{1\\leq i\\leq 3}$.\nSince the $\\lambda_i$ terms in Eq. (\\ref{eq:energy_KHG}) vanish as a consequence of the constraints in Eq. (\\ref{eq:constraints}), $f^\\prime$ in Eq. (\\ref{eq:fprime}) acquires the same value as $f$ in Eq. (\\ref{eq:f}) on the six-dimensional manifold, where $L\\in 3\\mathbb{Z}$ is the number of lattice sites.  \nTherefore, we will equivalently consider $F=3f^\\prime\/L$ instead of $3f\/L$ in what follows to calculate the Hessian matrix.\nThe advantage of using $F$ is that its gradient vanishes at the saddle point,\n unlike the case of $3f\/L$, where the gradient is perpendicular to the tangent space at the saddle point.\n\nConsider the six eigenvalues of the Hessian matrix of the free energy $F$ restricted to the six-dimensional manifold.\nRight at the FM2 point, two of the eigenvalues are zero, \nwhich is reasonable since there are two gapless spin waves for an FM Heisenberg chain.\nBased on this, we expect that for $|\\Delta|\\ll 1$, \nthe Hessian matrix contains two low-lying eigenvalues. \nSince the other four high-lying eigenvalues remain to be gapped with a small correction  dependent on $\\Delta$,\nit is enough to check that the two-lying eigenvalues are positive.\nIn what follows, we demonstrate this by perturbatively calculating the two smallest eigenvalues of the Hessian matrix in an expansion in $\\Delta$.\n\nBefore proceeding on, we first set up some notations.\nDenote $R(\\Delta)=(\\hat{n}^{(0),T}_1(\\Delta),\\hat{n}^{(0),T}_2(\\Delta),\\hat{n}^{(0),T}_3(\\Delta))^T$  to be the saddle point solution for a fixed value of $\\Delta$ within the nine-dimensional space where $\\{\\hat{n}^{(0)}_i(\\Delta)\\}_{1\\leq i\\leq 3}$ are given by Eqs. (\\ref{eq:spins_OhD4},\\ref{eq:saddle_Oh_D4}).\nIn what follows, we will ignore the transpose operation on the superscripts,\nbearing in mind that we are  always considering a nine-component column vector.\nDenote $T_R(\\Delta)$ to be the tangent space of the six-dimensional manifold at the point $R(\\Delta)$,\nand $P(\\Delta)$ to be the projection to the tangent space $T_R(\\Delta)$.\nExplicitly, the expression of $P(\\Delta)$ is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nP=\\mathbbm{1}_{9\\times 9}-r_1r_1^T-r_2r_2^T-r_3r_3^T,\n\\label{eq:projection_classical}\n\\eea\nin which $r_i$ is $r_i(\\Delta)$ for short, \nwhere \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nr_1&=&(\\hat{n}^{(0)}_1,\\vec{0},\\vec{0}),\\nonumber\\\\\nr_2&=&(\\vec{0},\\hat{n}^{(0)}_2,\\vec{0}),\\nonumber\\\\\nr_3&=&(\\vec{0},\\vec{0},\\hat{n}^{(0)}_3).\n\\label{eq:define_ri}\n\\eea \nNow let $H_F(\\Delta)$ be the $9\\times 9$ Hessian matrix of $F$,\nin which the derivatives are taken with respect to the unconstrained coordinates $\\{x_i,y_i,z_i\\}_{1\\leq i\\leq3}$,\ni.e.,\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n(H_F)_{\\alpha_i,\\beta_j}(\\Delta)=\\frac{\\partial^2 F}{\\partial \\alpha_i \\partial \\beta_j}(\\Delta), \n\\label{eq:HF_equator}\n\\eea\nwhere $1 \\leq i,j\\leq 3$ are the site indices in a unit cell\nand $\\alpha,\\beta=x,y,z$.\nNotice that if $P(\\Delta)H_F(\\Delta)P(\\Delta)$ is viewed as a $9\\times 9$ matrix, \nthen there are always three zero eigenvalues, and the three  corresponding null vectors are given by Eq. (\\ref{eq:define_ri}), \nsince $r_i(\\Delta)$ ($1\\leq i \\leq 3$) are always annihilated by $P(\\Delta)$.\nDenote $v_1(\\Delta),v_2(\\Delta)$ to be the eigenvectors of the two low-lying eigenvalues,\nand $w_1(\\Delta),w_2(\\Delta),w_3(\\Delta),w_4(\\Delta)$ the other four eigenvectors of the high-lying eigenvalues.\nWe will be only interested in $v_1(\\Delta),v_2(\\Delta)$.\n\nConsider an FM configuration with all spins aligning along $\\hat{n}$-direction.\nLet $\\hat{e}_{\\theta},\\hat{e}_\\phi$ be the two unit vectors perpendicular to $\\hat{n}$\nwhich are along tangent directions of the $\\theta$ and $\\phi$ coodinates, respectively,\nwhere $\\theta,\\phi$ are the polar and azimuthal angles of a unit sphere.\nWhen $\\Delta=0$, $v_1=v_1(\\Delta=0)$ and  $v_2=v_2(\\Delta=0)$ are the two acoustic eigenvectors given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nv_1&=& (\\frac{1}{\\sqrt{3}} \\hat{e}_\\theta, \\frac{1}{\\sqrt{3}} \\hat{e}_\\theta,\\frac{1}{\\sqrt{3}} \\hat{e}_\\theta),\\nonumber\\\\\nv_2&=& (\\frac{1}{\\sqrt{3}} \\hat{e}_\\phi, \\frac{1}{\\sqrt{3}} \\hat{e}_\\phi,\\frac{1}{\\sqrt{3}} \\hat{e}_\\phi),\n\\label{eq:v1v2}\n\\eea\nwhereas $\\{w_i=w_i(\\Delta=0)\\}_{1\\leq i\\leq4} $ are the optical ones:\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nw_1&=&(\\frac{1}{\\sqrt{2}}\\hat{e}_\\theta,-\\frac{1}{\\sqrt{2}}\\hat{e}_\\theta,\\vec{0}),\\nonumber\\\\\nw_2&=&(\\frac{1}{\\sqrt{2}}\\hat{e}_\\phi,-\\frac{1}{\\sqrt{2}}\\hat{e}_\\phi,\\vec{0}),\\nonumber\\\\\nw_3&=&(\\frac{1}{\\sqrt{2}}\\hat{e}_\\theta,\\frac{1}{\\sqrt{6}}\\hat{e}_\\theta,-\\sqrt{\\frac{2}{3}}\\hat{e}_\\theta),\\nonumber\\\\\nw_4&=&(\\frac{1}{\\sqrt{6}}\\hat{e}_\\phi,\\frac{1}{\\sqrt{6}}\\hat{e}_\\phi,-\\sqrt{\\frac{2}{3}}\\hat{e}_\\phi).\n\\label{eq:wi_s}\n\\eea\nWe note that the eigenvalues of the Hessian matrix for $\\Delta=0$ corresponding to $v_i$-eigenvectors ($i=1,2$) are both $0$, and those corresponding to $w_i$'s ($i=1,2,3,4$) are all $3$.\nSince when $\\Delta\\rightarrow 0$, the solution reduces to $a=b$ as can be seen from Eq. (\\ref{eq:saddle_Oh_D4}),\n$\\hat{n}$ should be chosen as $\\hat{n}_a=\\frac{1}{3}(1,1,1)^T$ to determine the zeroth order terms in $v_1(\\Delta)$ and $v_2(\\Delta)$ in a perturbative expansion over $\\Delta$.\nAs a result, $\\hat{e}_\\theta$ and $\\hat{e}_\\phi$ in Eqs. (\\ref{eq:v1v2},\\ref{eq:wi_s}) are given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\hat{e}_\\theta&=&(-\\frac{1}{\\sqrt{6}},-\\frac{1}{\\sqrt{6}},\\sqrt{\\frac{2}{3}})^T,\\nonumber\\\\\n\\hat{e}_\\phi&=&(-\\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}},0)^T.\n\\eea\n\nThe projected Hessian matrix\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\mathcal{H}_F(\\Delta)=P(\\Delta) H_F(\\Delta) P(\\Delta)\n\\label{eq:proj_HF_equator}\n\\eea\ncan be expanded in a power series of $\\Delta$\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\mathcal{H}_F(\\Delta)=\\mathcal{H}_F^{(0)}+ \\mathcal{H}_F^{(1)}+\\mathcal{H}_F^{(2)} +....,\n\\eea\nin which $\\mathcal{H}_F^{(n)}$ is proportional to $\\Delta^n$.\nSince both $v_1$ and $v_2$ have zero eigenvalues of $\\mathcal{H}_F^{(0)}$,\na degenerate first order perturbation theory should be considered,\nand the first order perturbation Hamiltonian is \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{(1)}=\\left(\\begin{array}{cc}\nv_1^T \\mathcal{H}_F^{(1)} v_1& v_1^T \\mathcal{H}_F^{(1)} v_2\\\\\nv_2^T \\mathcal{H}_F^{(1)} v_1 & v_2^T \\mathcal{H}_F^{(1)} v_2\n\\end{array}\n\\right).\n\\label{eq:h1}\n\\eea\nHowever, straightforward calculation shows that $h^{(1)}$ vanishes\nand we have to go to second order perturbation.\n\nThe second order perturbation Hamiltonian can be obtained as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{(2)}=\\left(\\begin{array}{c}\nv_1^T\\\\\nv_2^T\n\\end{array}\n\\right)\n\\big(\\mathcal{H}^{(2)}_F+\\mathcal{H}^{(1)}_F\\sum_{i=1}^4 \\frac{w_iw_i^T}{E_0-E_i} \\mathcal{H}^{(1)}_F\\big)(v_1\\,v_2),\n\\eea\nin which $E_0=0,E_i=3$ are the eigenvalues of $\\mathcal{H}_F^{(0)}$\ncorresponding to the acoustic and optical eigenvectors, respectively. \nCalculations show that \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{(2)}=\\frac{4}{27}\\Gamma^\\prime \\Delta^2 \\sigma_0,\n\\label{eq:h2_equator}\n\\eea\nwhere $\\sigma_0$ is the $2\\times 2$ identity matrix.\nSince the eigenvalue $\\frac{4}{27}\\Gamma S^2  \\Delta^2$ is positive,\nwe arrive at the conclusion  that the solution in Eq. (\\ref{eq:spins_OhD4})\nis indeed a minimum of the classical free energy $F$ regardless of the sign of $\\Delta$\nat least when $|\\Delta|$ is small.\n\nNotice that up to $O(\\Delta^2)$, $v_1(\\Delta)$ and $v_2(\\Delta)$ are degenerate according to Eq. (\\ref{eq:h2_equator}).\nIn fact, this degeneracy holds to all orders in $\\Delta$.\nThis is explained in detail in Appendix \\ref{app:proof_degeneracy} based on a group theory analysis.\n\n\\subsection{The symmetry breaking pattern}\n\\label{sec:sym_break_equator}\n\nTo identify the symmetry breaking pattern, we work out the unbroken symmetry group of the spin alignments in Eq. (\\ref{eq:spins_OhD4}) in the six-sublattice rotated frame.\n\nIt is straightforward to verify that the spin orientations in Eq. (\\ref{eq:spins_OhD4}) \nare invariant under the symmetry operations $R_aT_a$ and $TR_I I$.\nTherefore the unbroken symmetry group $N$ is \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nN=\\mathopen{<}R_aT_a,TR_I I \\mathclose{>}.\n\\eea\nSince $T_{3a}$ is unbroken, in what follows within this subsection,\nwe will consider the quotient group $N\/\\mathopen{<}T_{3a}\\mathclose{>}$. \nAs proved in Ref. \\onlinecite{Yang2020}, $N\/\\mathopen{<}T_{3a}\\mathclose{>}$ is isomorphic to $D_3$.\nHere we give a quick demonstration of this isomorphism.\nThe group $D_n$ (i.e, the dihedral group of order $2n$)  has the following generator-relation representation\n\\begin{eqnarray}\nD_n=\\mathopen{<} \\alpha,\\beta| \\alpha^n=\\beta^2=(\\alpha\\beta)^2=e \\mathclose{>}.\n\\label{eq:generator_Dn}\n\\end{eqnarray}\nDefine $\\alpha=R_aT_a$, and $\\beta=TR_I I$.\nIt is straightforward to verify that the relations in Eq. (\\ref{eq:generator_Dn}) are satisfied for $\\alpha,\\beta$ modulo $T_{3a}$.\nFurthermore, it can be checked that $N\/\\mathopen{<}T_{3a}\\mathclose{>}$ contains at least six elements.\nSince $|D_3|=6$, we conclude that  $N\/\\mathopen{<}T_{3a}\\mathclose{>}\\cong D_3$.\nThis analysis shows that the symmetry breaking pattern predicted by the classical theory is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nO_h\\rightarrow D_3.\n\\eea\nWe note that the classical prediction is  different from the symmetry breaking pattern for the spin-1\/2 case \\cite{Yang2020a} which is numerically identified as $O_h\\rightarrow D_4$.\nThis  indicates strong quantum fluctuations in the spin-1\/2 case.\nOn the other hand, numerical calculations provide evidence for the $O_h\\rightarrow D_3$ symmetry breaking for $S=1$  and $S=3\/2$ as will be discussed in Sec. \\ref{sec:numerics_OhD3}.\nBased on this, we conjecture that  spin-1\/2 is the only exception and all other spins exhibit an $O_h\\rightarrow D_3$ symmetry breaking as predicted by the classical analysis.\n\n\nThe classical solutions are degenerate, and Eq. (\\ref{eq:spins_OhD4}) only gives one of the possibilities.\nSince $|O_h|=48$ and $|D_3|=6$,\nthe number of degenerate classical minima is $|O_h\/D_3|=8$.\nThe other minima are related to Eq. (\\ref{eq:spins_OhD4})\nby $O_h$ operations. \nNote that only operations in different equivalent classes of $O_h\/D_3$\ngive distinct classical spin configurations. \nIn fact, the eight degenerate spin orientations of $\\vec{S}_1$\nare $(\\pm a,\\pm a, \\pm b)$,\nand the orientations for $\\vec{S}_2$ and $\\vec{S}_3$\ncan be obtained by permuting $a$ and $b$\nin accordance with Eq. (\\ref{eq:spins_OhD4}).\nFor a pictorial illustration, the ``center of mass\" directions of the three spins within a unit cell\ncorresponding to the eight classical minima \nare represented as solid blue circles  located at the vertices of a cube as shown in Fig. \\ref{fig:spin_orders}.\n\n\n\n\\subsection{Spin wave theory}\n\\label{sec:SW_equator}\n\nIn this section, we derive the spin wave theory in the path integral formalism which characterizes the small fluctuations around the classical spin configurations.\nWe focus on the $|\\Delta|\\ll 1$ region, and only the lowest-lying spin wave will be considered.\n\n\\subsubsection{The spin wave Lagrangian}\n\\label{sec:sw_lag_equator}\n\nThe Lagrangin of the spin coherent state path integral is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nL=S\\sum_j \\vec{A}_j\\cdot \\partial_t\\hat{n}_j-f^\\prime[\\{\\hat{n}_j\\}],\n\\label{eq:spin_path_integral}\n\\eea\nin which the first term is the Berry phase term; the Berry connection $\\vec{A}_j(\\theta_j,\\phi_j)$ can be chosen as $\\frac{1-\\cos \\theta_j}{\\sin \\theta_j}\\hat{e}_{j\\phi}$ where  $\\theta_j$ and $\\phi_j$ are the polar and azimuthal angles of $\\hat{n}_j$, respectively,\nand $\\hat{e}_{j\\phi}$ is the unit vector along the azimuthal direction at $\\hat{n}_j$;\nthe functional $f^\\prime$ is given by Eq. (\\ref{eq:fprime}).\nNotice that again by virtue of the constraints in Eq. (\\ref{eq:constraints}), \nthere is no difference between $f^\\prime$ and $f$ in Eq. (\\ref{eq:f}).\nTherefore, it would be legitimate to write $f^\\prime$ instead of $f$ in Eq. (\\ref{eq:spin_path_integral}).\n\nNext we expand the Lagrangian around the classical solution in Eq. (\\ref{eq:spins_OhD4}).\nIn the spin wave approximation, only the Gaussian fluctuations will be kept.\nFor small fluctuations, $\\hat{n}_j$ moves in the tangent space of the unit sphere at the point $\\hat{n}_j^{(0)}$, in which $\\hat{n}_j^{(0)}=\\hat{n}_{[j]}^{(0)}$, $[j]\\equiv j\\mod 3$ and $1\\leq [j]\\leq 3$,\nwhere $\\hat{n}_{[j]}^{(0)}$ is given by Eqs. (\\ref{eq:spins_OhD4},\\ref{eq:saddle_Oh_D4}).\nThe local coordinate frame of the tangent space at site $j$ can be set up as $\\{\\hat{e}^{(0)}_\\theta(j),\\hat{e}^{(0)}_\\phi(j)\\}$, where $\\hat{e}^{(0)}_\\theta(j)$  and $\\hat{e}^{(0)}_\\phi(j)$ are the unit vectors along the polar and azimuthal directions at $\\hat{n}^{(0)}_{[j]}$, respectively.\nThen the deviations away from the equilibrium position are characterized by $\\{\\chi_\\theta(j), \\chi_\\phi(j)\\}$ which are the displacements along the $\\hat{e}^{(0)}_\\theta(j)$ and  $\\hat{e}^{(0)}_\\phi(j)$ directions.\n\nWith the above setup, the Berry phase term becomes\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\frac{1}{2}S\\sum_j [\\chi_\\theta(j)\\partial_t\\chi_\\phi(j)-\\chi_\\phi(j)\\partial_t\\chi_\\theta(j)].\n\\label{eq:Berry_chi}\n\\eea\nAs can be easily checked, the integration of Eq. (\\ref{eq:Berry_chi}) over  time gives the area swept out by the trajectory of $\\hat{n}_j$ within the tangent space, which coincides with the geometric meaning of the Berry phase term. \nWe note that $\\chi_\\theta(j)$ and $\\chi_\\phi(j)$ form a pair of canonical conjugates which can be clearly seen from Eq. (\\ref{eq:Berry_chi}).\nAlternatively, choosing the quantization axis along $\\hat{n}_j^{(0)}$, the angular momentum commutation relation becomes\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n[S\\chi_\\theta(j),S\\chi_\\phi(j)]=i\\hat{n}^{(0)}_j\\cdot \\vec{S}_j.\n\\label{eq:commutation}\n\\eea\nReplacing $\\hat{n}^{(0)}_j\\cdot \\vec{S}_j$ with its classical value $S$,\nEq. (\\ref{eq:commutation}) becomes\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n[\\chi_\\theta(j),\\chi_\\phi(j)]=i\\frac{1}{S},\n\\eea\nwhich is the canonical commutation relation where $1\/S$ plays the role of $\\hbar$.\nThis also indicates that the classical and spin wave analysis only applies in the large-$S$ (i.e., small $\\hbar$) limit.\n\nFor later convenience, we rewrite Eq. (\\ref{eq:Berry_chi}) in the Cartesian coordinates $\\{x_i,y_i,z_i\\}$ in the spin space.\nThe expression under the summation in Eq. (\\ref{eq:Berry_chi}) can be written as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\hat{n}_j^T [\\hat{e}^{(0)}_\\theta(j)\\hat{e}^{(0),T}_\\phi(j)-\\hat{e}^{(0)}_\\phi(j)\\hat{e}^{(0),T}_\\theta(j)] \\partial_t \\hat{n}_j.\n\\label{eq:Berry_chi2}\n\\eea\nNotice that the matrix kernel in Eq. (\\ref{eq:Berry_chi2}) is simply the $\\pi\/2$-rotation matrix around the $\\hat{n}^{(0)}_j$-direction.\nSince such rotation can be implemented by a cross product with $\\hat{n}^{(0)}_j$,\nthe matrix kernel in Eq. (\\ref{eq:Berry_chi2}) is  equal to\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nM_j=\\left(\\begin{array}{ccc}\n0& -n^{(0)}_{jz}&n^{(0)}_{jy}\\\\\nn^{(0)}_{jz}&0&-n^{(0)}_{jx}\\\\\n-n^{(0)}_{jy} &n^{(0)}_{jx}&0\n\\end{array}\\right),\n\\label{eq:Mj}\n\\eea\nin which $n^{(0)}_{j\\alpha}$ is the $\\alpha$-component of $n^{(0)}_{j}$, where $\\alpha=x,y,z$ and $j=1,2,3$.\nTherefore, for small fluctuations, Eq. (\\ref{eq:spin_path_integral}) becomes\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nL=S\\sum_j\\frac{1}{2} \\delta\\hat{n}_j^T M_j \\partial_t \\delta\\hat{n}_j -f^\\prime[\\{\\hat{n}_j\\}],\n\\label{eq:spin_path_integral3}\n\\eea\nin which $\\delta \\hat{n}_j=\\hat{n}_j-\\hat{n}_j^{(0)}$, and $M_j$ is given by Eq. (\\ref{eq:Mj}).\n\nTo discuss the spin wave dispersion, it is convenient to transform into the Fourier space. \nIn what follows, the Fourier transform of the Cartesian coordinates  $\\alpha_{i+3n}$ ($\\alpha=x,y,z$; $i=1,2,3$; $n\\in \\mathbb{Z}$) will be defined as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\alpha_{i+3n}=\\frac{1}{\\sqrt{L\/3}} \\sum_n e^{i kn}\\alpha_i(k),\n\\label{eq:Fourier_sw}\n\\eea\nin which $L$ is the system size.\nPlugging Eq. (\\ref{eq:Fourier_sw}) into Eq. (\\ref{eq:spin_path_integral3}) (setting $J=0$), we obtain\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nL&=&-f_0+S\\sum_k\\frac{1}{2} N^T(k) M \\partial_t N(-k)\\nonumber\\\\\n&& -\\frac{1}{2}\\sum_k N^T(k)[\\mathcal{H}_F+\\delta H (k)] N(-k),\n\\label{eq:f_fourier}\n\\eea\nin which $f_0$ is the classical free energy at the saddle points given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nf_0=-L\\Gamma S^2 (1+\\frac{1}{3}\\Delta+\\frac{1}{27}\\Delta^2)+O(\\Delta^3);\n\\eea\n$M$ is a $9\\times 9$ matrix\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nM=\\left(\\begin{array}{ccc}\nM_1&0&0\\\\\n0&M_2&0\\\\\n0&0&M_3\n\\end{array}\n\\right),\n\\label{eq:M}\n\\eea\nwhere $M_j$ ($j=1,2,3$) is defined in Eq. (\\ref{eq:Mj});\n $N^T(k)$ defined as\n \\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n N^T(k)=(\\hat{n}^T_1(k),\\hat{n}^T_2(k),\\hat{n}^T_3(k))\n \\label{eq:Nk}\n \\eea \n is a nine-component row vector where $\\hat{n}^T_i(k)=(x_i(k),y_i(k),z_i(k))$ ($i=1,2,3$); $\\mathcal{H}_F$ is given by Eq. (\\ref{eq:proj_HF_equator}); and the $9\\times 9$ matrix $\\delta H(k)$ can be derived as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\delta H^{(1,3)}(k)&=&\\Gamma S^2 (1-e^{-ik}) \\text{diag}(1,1+\\Delta,1),\\nonumber\\\\\n\\delta H^{(3,1)}(k)&=&[\\delta H^{(1,3)}(k)]^\\dagger,\\nonumber\\\\\n\\delta H^{(\\alpha,\\beta)}(k)&=&0, ~\\text{for}~\\{\\alpha,\\beta\\}\\neq\\{1,3\\},\n\\label{eq:deltaHk}\n\\eea\nwhere $\\text{diag}(\\cdot\\cdot\\cdot)$ denotes the diagonal matrix, and $\\delta H^{(\\alpha,\\beta)}(k)$ is the $3\\times 3$  matrix at the $(1,3)$-block of $\\delta H(k)$.\n\n\\subsubsection{Zero wavevector}\n\\label{sec:sw_equator_zero_momentum}\n\nLet's first consider the zero wavevector spin waves. \nThe Hamiltonian in Eq. (\\ref{eq:f_fourier}) for $k=0$ is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\frac{1}{2} N^T(k=0)H_F N(k=0).\n\\label{eq:Ham_sw_0}\n\\eea\nTo get the spin wave masses, the matrix kernel in Eq. (\\ref{eq:Ham_sw_0}) needs to be diagonalized. \nNaively, the matrix $\\mathcal{H}_F$ has already been diagonalized in Sec. \\ref{sec:classical_equator}.\nHowever, in Eq. (\\ref{eq:Ham_sw_0}), $\\mathcal{H}_F$ must be diagonalized by symplectic transformation which leaves the symplectic form $M$ in Eq. (\\ref{eq:M}) invariant,\nunlike the case in Sec. \\ref{sec:classical_equator} where $\\mathcal{H}_F$ is  diagonalized by an orthogonal transformation.\nRecall that we have already proved $\\mathcal{H}_F$ to be positive definite which is the restriction of $H_F$ in Eq. (\\ref{eq:HF_equator})  to  the six-dimensional tangent space.\nThen by the symplectic theory \\cite{DelaCruz2016}, $\\mathcal{H}_F$ (which is viewed as a $6\\times 6$ matrix) can be diagonalized by a symplectic transformation $U$, i.e., \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\mathcal{H}_F=U^T \\Lambda U;\n\\eea\nin which $U$ satisfies\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nU^T MU=M\n\\eea\nand the diagonal matrix $\\Lambda$ is of the form\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\Lambda=\\left(\\begin{array}{ccc}\nm_1\\sigma_0 & 0&0\\\\\n0&m_2\\sigma_0&0\\\\\n0&0&m_3\\sigma_0\n\\end{array}\n\\right)\n\\eea\nwhere $m_3>m_2>m_1>0$ ($i=1,2,3$); $\\sigma_0$ is the $2\\times 2$ identity matrix;\nand $M$ is viewed as a $6\\times 6$ matrix acting in  the six-dimensional tangent space.\nWe will be interested in $m_1$ which is related to the smallest spin wave mass.\nNotice that in general, $m_i$'s do not coincide with the eigenvalues of $\\mathcal{H}_F$. \nIn what follows,  $m_1$ will be calcualted to the leading nonvanishing order in $\\Delta$ by perturbation theory.\nThe result happens to be the same as the two lowest eigenvalues of $\\mathcal{H}_F$ (i.e., $\\frac{4}{27}\\Gamma S^2  \\Delta^2$) derived in Sec. \\ref{sec:classical_equator}.\n\nThe calculations of $m_i$'s can be converted into an eigenvalue problem by considering the matrix $M\\mathcal{H}_F$ \\cite{DelaCruz2016}.\nIn fact, according to the symplectic linear algebra, \nthe eigenvalues of $M\\mathcal{H}_F$ are $\\pm i m_j$ ($j=1,2,3$).\nThe basics of the symplectic transformations for our purpose are collected in Appendix \\ref{app:symplectic}.\nIn what follows, we will view both $M$ and $\\mathcal{H}_F$ as $9\\times 9$ matrices.\nSince $M_j$ is defined as a cross product operation in Eq. (\\ref{eq:Mj}),\n$\\hat{n}_j^{(0)}$ must be a null vector of $M_j$.\nAs a result, $r_i(\\Delta)$ in Eq. (\\ref{eq:define_ri}) are always annihilated by $M$.\nHence the $9\\times 9$ matrix $M\\mathcal{H}_F$ always has three zero eigenvalues, and we will be interested in the other six eigenvalues.\nWhen $\\Delta=0$, among the other six eigenvalues, $M(\\Delta=0)\\mathcal{H}_F(\\Delta=0)$ contains two zero eigenvalues with eigenvectors given by \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nv_\\pm=v_1\\pm iv_2,\n\\label{eq:v_pm}\n\\eea\n where $v_i=v_i(\\Delta=0)$ ($i=1,2$) are given by Eq. (\\ref{eq:v1v2}).\nFor $\\Delta\\neq 0$, $v_\\pm$ evolve into $v_\\pm(\\Delta)$ which have eigenvalues $\\pm im_1(\\Delta)$.\n\nLet's first consider $\\pm im_1(\\Delta)$ to the linear order of $\\Delta$.\nDefine $ \\mathcal{H}_F^{M,(n)}$ in terms of the power expansion as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nM(\\Delta)\\mathcal{H}_F(\\Delta)\n&=& \\mathcal{H}_F^{M,(0)}+\\mathcal{H}_F^{M,(1)}+ \\mathcal{H}_F^{M,(2)}+...,\n\\eea\nwhere $ \\mathcal{H}_F^{M,(n)}$ is proportional to $\\Delta^n$.\nDefine $P_1(\\Delta)$ to be projection to the subspace spanned by $v_\\pm(\\Delta)$.\nAt a nonzero $\\Delta$, the first order degenerate perturbation theory is  given by \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(1)}=P_1^{(0)} \\mathcal{H}_F^{M,(1)} P_1^{(0)},\n\\eea\nwhere $P_1^{(0)}=P_1(\\Delta=0)$.\nCalculations show that $h^{M,(1)}$ vanishes,\nhence  second order degenerate perturbation has to be considered.\nWe note that there is a quick way to see $h^{M,(1)}=0$.\nIn fact, there is the relation\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(1)}=M^{(0)}P_1^{(0)}h^{(1)}P_1^{(0)},\n\\label{eq:hM1_equator_relation}\n\\eea\nin which $h^{(1)}$ is defined in Eq. (\\ref{eq:h1}).\nA proof of Eq. (\\ref{eq:hM1_equator_relation}) is given in Appendix \\ref{app:proof_symplectic_first}.\nSince $P_1^{(0)}h^{(1)}P_1^{(0)}$ vanishes according to the discussion below Eq. (\\ref{eq:h1}), $h^{M,(1)}$ has to vanish as a result.\n\n\n\nNext we proceed to second order perturbation.\nDefine $u_i$ ($i=1,2,3,4$) as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nu_1&=&w_1+iw_2,\\nonumber\\\\\nu_2&=&w_1-iw_2,\\nonumber\\\\\nu_3&=&w_3+iw_4,\\nonumber\\\\\nu_4&=&w_3-iw_4,\n\\label{eq:Def_us}\n\\eea\nin which $w_i$ are given in Eq. (\\ref{eq:wi_s}).\nThen $u_i$ are eigenvectors of $\\mathcal{H}^{M,(0)}_F$ with eigenvalues equal to $\\epsilon_i$, where \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\epsilon_1=3i,~ \\epsilon_2=-3i,~ \\epsilon_3=3i, ~\\epsilon_4=-3i.\n\\eea\nThe second order degenerate perturbation theory is captured by the following matrix,\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(2)}&=&\n\\left(\\begin{array}{c}\nv_+^\\dagger\\\\\nv_-^\\dagger\n\\end{array}\n\\right)\n\\big[\\mathcal{H}^{M,(2)}_F+\\mathcal{H}^{M,(1)}_F\\sum_{i=1}^4 \\frac{u_iu_i^\\dagger}{E_0-\\epsilon_i} \\mathcal{H}^{M,(1)}_F\\big]\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times(v_+\\,v_-),\n\\eea\nin which $E_0=0$ are the eigenvalues of $\\mathcal{H}^{M,(0)}_F$ for $v_{\\pm}$.\nCalculations show that \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(2)}=i\\frac{4}{27}\\Gamma^\\prime \\Delta^2 \\sigma_3,\n\\eea\nin which $\\sigma_3$ is the third Pauli matrix.\nThis shows that to the leading nonvanishing order,\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nm_1=\\frac{4}{27}\\Gamma^\\prime \\Delta^2.\n\\label{eq:m1_perturbative}\n\\eea\nWe have numerically calculated the eigenvalues of $M(\\Delta)\\mathcal{H}_F(\\Delta)$ and the result for $m_1$ is displayed in Fig. \\ref{fig:m1_Delta}.\nAs can be seen from Fig. \\ref{fig:m1_Delta}, the numerical results agree well with Eq. (\\ref{eq:m1_perturbative}).\n\n\n\\begin{figure}[h]\n\\includegraphics[width=8.0cm]{m1_Delta.pdf}\n\\caption{$m_1$ vs. $\\Delta$ for $J=0$ represented by the hollow circles as obtained by numerical diagonalization of $M(\\Delta)\\mathcal{H}_F(\\Delta)$,\nwhere $\\Delta$ is defined in Eq. (\\ref{eq:define_Delta}).\nThe solid line represents $\\frac{4}{27}\\Delta^2$.\nThe vertical axis is in unit of $\\Gamma^\\prime=\\Gamma S^2$.\n} \\label{fig:m1_Delta}\n\\end{figure}\n\n\\subsubsection{Nonzero wavevectors and the spin wave dispersions}\n\nNext, we consider nonzero wavevectors and diagonalize the matrix kernel $H_F+\\delta H(k)$ in Eq. (\\ref{eq:f_fourier}).\nWe will consider the long wavelength limit $k\\ll 1$ where the lattice constant has been taken as $1$.\nAs can be seen from Eq. (\\ref{eq:deltaHk}), the matrix elements of $\\delta H(k)$ are very small in the long wavelength limit,\nhence $\\delta H(k)$ can be treated as a perturbation of $\\mathcal{H}_F$.\n\nMultiplying with the symplectic matrix, the first order degenerate perturbation is  implemented by the following $2\\times 2$ matrix\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\delta h^{(1)}(k)=P_1^{(0)}M^{(0)} \\delta H(\\Delta=0,k)P_1^{(0)},\n\\eea\nin which we have taken $\\Delta=0$ since we are only interested in the leading nonvanishing order terms in $\\Delta$.\nStraightforward calculations show that the eigenvalues of $\\delta h^{(1)}(k)$ are $\\pm i\\frac{1}{6} \\Gamma S^2 k^2$.\nThus, by keeping only the lowest-lying spin wave, the spin wave Lagrangian in Eq. (\\ref{eq:f_fourier}) becomes\n\\begin{flalign}\n&L=-f_0+S\\sum_k\\frac{1}{2} (\\xi_\\theta(k) \\partial_t \\xi_\\phi(-k) -\\xi_\\phi(k) \\partial_t \\xi_\\theta(-k) )\\nonumber\\\\\n& -\\frac{1}{2}\\Gamma S^2 \\sum_k (\\frac{4}{27} \\Delta^2+\\frac{1}{3}k^2)[\\xi_\\theta(k)\\xi_\\theta(-k)+\\xi_\\phi(k)\\xi_\\phi(-k)],\n\\label{eq:f_fourier_low}\n\\end{flalign}\nin which \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\xi_\\theta(k)=N^T(k)v_1,~\\xi_\\phi(k)=N^T(k)v_2,\n\\eea\nwhere $N(k)$ is defined in Eq. (\\ref{eq:Nk}).\n\nFinally, we rewrite the spin wave Hamiltonian (i.e., the second line in Eq. (\\ref{eq:f_fourier_low})) in real space in the continuum limit.\nThe summation over $k$ can be converted to $\\sum_n=\\frac{1}{3}\\int dx$ where $x$ is the real space coordinate in the continuum limit.\nThe momentum $k$ can be converted to $i\\partial_n=3i\\partial_x$.\nUsing these, we see that the spin wave Hamiltonian $H_{sw}$ in the real space is\n\\begin{flalign}\nH_{sw}=\\Gamma S^2 \\int dx[ \\frac{1}{2} (\\partial_x \\xi_\\theta)^2+\\frac{1}{2} (\\partial_x \\xi_\\phi)^2+ \\frac{2}{81}\\Delta^2 (\\xi_\\theta^2+\\xi_\\phi^2)],\n\\label{eq:Hsw_equator}\n\\end{flalign}\nin which $\\xi_\\theta(j),\\xi_\\phi(j)$ is a pair of canonical conjugates satisfying $[\\xi_\\theta(j),\\xi_\\phi(j^\\prime)]=i\\delta_{jj^\\prime}\\frac{1}{S}$.\nFrom Eq. (\\ref{eq:Hsw_equator}) and the fact that $\\hbar=1\/S$,\nthe dispersion of the spin wave can be obtained as\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nE(k)=\\Gamma S (k^2+\\frac{4}{81}\\Delta^2).\n\\eea\nSince the spin wave mass $\\frac{4}{81}\\Gamma S \\Delta^2$ is very small, it would be very difficult to determine numerically (for example, in DMRG numerics). \nWe note that the path integral calculations to derive the spin wave Hamiltonian in Eq. (\\ref{eq:Hsw_equator}) is equivalent with the Bogoliubov transformation based on the Holstein-Primakoff transformation as explained in detail in Appendix \\ref{app:HP_equiv}.\n\n\n\\subsection{DMRG numerics}\n\\label{sec:numerics_OhD3}\n\n\\begin{table}\n                \\begin{tabular}[t]{|c|c|c|c|c|}\n\t\t  \\multicolumn{5}{c}{}\\\\ \\hline\n$E(S=1)$ & No field & $h_{\\hat{x}}=10^{-4}$ & $h_{\\hat{n}_{a}}=10^{-4}$ & $h_{\\hat{n}_{I}}=10^{-4}$ \\\\ \\hline\n$E_1$ &\\tikzmark{top left 1a} -12.01911&\\tikzmark{top left 1b} -12.02010&\\tikzmark{top left 1c}  -12.02067\\tikzmark{bottom right 1c}&\\tikzmark{top left 1d} -12.02106\\\\\n$E_2-E_1$ & $1.57\\cdot 10^{-4}$ & $1.47\\cdot 10^{-4}$ & $1.17\\cdot 10^{-3}$ & $2.3\\cdot 10^{-4}$\\tikzmark{bottom right 1d}\\\\\n$E_3-E_1$ & $1.57\\cdot 10^{-4}$ & $2.21\\cdot 10^{-4}$ & $1.17\\cdot 10^{-3}$ & $2.06\\cdot 10^{-3}$\\\\\n$E_4-E_1$ & $1.57\\cdot 10^{-4}$ & $4.07\\cdot 10^{-4}$\\tikzmark{bottom right 1b}& $1.21\\cdot 10^{-3}$ & $2.06\\cdot 10^{-3}$\\\\\n$E_5-E_1$ & $3.38\\cdot 10^{-4}$ & $2.09\\cdot 10^{-3}$ & $2.36\\cdot 10^{-3}$ & $2.22\\cdot 10^{-3}$\\\\\n$E_6-E_1$ & $3.38\\cdot 10^{-4}$ & $2.21\\cdot 10^{-3}$ & $2.40\\cdot 10^{-3}$ & $2.25\\cdot 10^{-3}$\\\\\n$E_7-E_1$ & $3.38\\cdot 10^{-4}$ & $2.28\\cdot 10^{-3}$ & $2.40\\cdot 10^{-3}$ & $4.16\\cdot 10^{-3}$\\\\\n$E_8-E_1$ & $5.55\\cdot 10^{-4}$\\tikzmark{bottom right 1a}& $2.41\\cdot 10^{-3}$ & $3.60\\cdot 10^{-3}$ & $4.30\\cdot 10^{-3}$\\\\\n$E_9-E_1$ & $4.33\\cdot 10^{-3}$ & $4.34\\cdot 10^{-3}$ & $4.60\\cdot 10^{-3}$ & $4.50\\cdot 10^{-3}$\\\\\n$E_{10}-E_1$ & $4.33\\cdot 10^{-3}$ & $4.48\\cdot 10^{-3}$ & $5.56\\cdot 10^{-3}$ & $5.32\\cdot 10^{-3}$\\\\ \\hline\n                \\end{tabular}\n\t\t\t \t\\DrawBox[ultra thick, red]{top left 1a}{bottom right 1a}\n                \\DrawBox[ultra thick, blue]{top left 1b}{bottom right 1b}\n                \\DrawBox[ultra thick, green]{top left 1c}{bottom right 1c}\n                \\DrawBox[ultra thick, orange]{top left 1d}{bottom right 1d}\n                \\hfill\n                \\begin{tabular}[t]{|c|c|c|c|c|}\n\t\t  \\multicolumn{5}{c}{}\\\\ \\hline\n$E(S=3\/2)$ & No field & $h_{\\hat{x}}=10^{-4}$ & $h_{\\hat{n}_{a}}=10^{-4}$ & $h_{\\hat{n}_{I}}=10^{-4}$ \\\\ \\hline\n$E_1$ &\\tikzmark{top left 2a} -26.99084&\\tikzmark{top left 2b} -26.99237&\\tikzmark{top left 2c} -26.99342 \\tikzmark{bottom right 2c}&\\tikzmark{top left 2d} -26.99389\\\\\n$E_2-E_1$ & $6.6\\cdot 10^{-5}$ & $6.3\\cdot 10^{-5}$ & $1.78\\cdot 10^{-3}$ & $8.2\\cdot 10^{-5}$\\tikzmark{bottom right 2d}\\\\\n$E_3-E_1$ & $6.6\\cdot 10^{-5}$ & $8.2\\cdot 10^{-5}$ & $1.78\\cdot 10^{-3}$ & $3.09\\cdot 10^{-3}$\\\\\n$E_4-E_1$ & $6.6\\cdot 10^{-5}$ & $1.48\\cdot 10^{-4}$\\tikzmark{bottom right 2b}& $1.78\\cdot 10^{-3}$& $3.09\\cdot 10^{-3}$\\\\\n$E_5-E_1$ & $1.34\\cdot 10^{-4}$ & $3.11\\cdot 10^{-3}$ & $3.57\\cdot 10^{-3}$ & $3.15\\cdot 10^{-3}$\\\\\n$E_6-E_1$ & $1.34\\cdot 10^{-4}$ & $3.16\\cdot 10^{-3}$ & $3.57\\cdot 10^{-3}$ & $3.15\\cdot 10^{-3}$\\\\\n$E_7-E_1$ & $1.34\\cdot 10^{-4}$ & $3.18\\cdot 10^{-3}$ & $3.57\\cdot 10^{-3}$ & $6.21\\cdot 10^{-3}$\\\\\n$E_8-E_1$ & $2.04\\cdot 10^{-4}$\\tikzmark{bottom right 2a}& $3.23\\cdot 10^{-3}$ & $1.27\\cdot 10^{-3}$ & $6.26\\cdot 10^{-3}$\\\\\n$E_9-E_1$ & $1.00\\cdot 10^{-2}$ & $1.01\\cdot 10^{-2}$ & $1.21\\cdot 10^{-2}$ & $1.02\\cdot 10^{-2}$\\\\\n$E_{10}-E_1$ & $1.00\\cdot 10^{-2}$ & $1.01\\cdot 10^{-2}$ & $5.37\\cdot 10^{-2}$ & $1.10\\cdot 10^{-2}$\\\\ \\hline\n                \\end{tabular}\n\t\t\t \t\\DrawBox[ultra thick, red]{top left 2a}{bottom right 2a}\n                \\DrawBox[ultra thick, blue]{top left 2b}{bottom right 2b}\n                \\DrawBox[ultra thick, green]{top left 2c}{bottom right 2c}\n                \\DrawBox[ultra thick, orange]{top left 2d}{bottom right 2d}\t\t\t\t\\hfill\n\\caption{Energies of the first ten lowest lying states computed with DMRG. The data refer to $L=18$ sites, $J=0$, and $\\phi=0.2\\pi$.\nThe energies enclosed by the colored squares are approximately degenerate.\n}\n\\label{table:OhD3}\n\\end{table}\n\n\\begin{figure*}[htbp]\n\\includegraphics[width=12.0cm]{OhD3_111.pdf}\n\\caption{Spin expectation values $\\left<S_i^\\alpha\\right>$ ($\\alpha=x,y,z$) under a small field $h_{\\hat{n}_a}=10^{-4}$ along $(1,1,1)$-direction \nat a representative point $(\\theta=\\pi\/2,\\phi=0.2\\pi)$ in the $O_h\\rightarrow D_3$ phase\n for (a) $S=1$, and (b) $S=3\/2$.\n The parametrization $(\\theta,\\phi)$ is defined in Eq. (\\ref{eq:parametize_theta_phi}).\nDMRG simulations are performed on a system of  $L=18$ sites.\n} \\label{fig:OhD3_spin_aligns_111}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\includegraphics[width=12.0cm]{OhD3_11m1.pdf}\n\\caption{Spin expectation values $\\left<S_i^\\alpha\\right>$ ($\\alpha=x,y,z$) under a small field $h_{\\hat{n}_b}=10^{-4}$ along $(1,1,-1)$-direction \nat a representative point $(\\theta=\\pi\/2,\\phi=0.2\\pi)$ in the $O_h\\rightarrow D_3$ phase\n for (a) $S=1$, and (b) $S=3\/2$.\n The parametrization $(\\theta,\\phi)$ is defined in Eq. (\\ref{eq:parametize_theta_phi}).\nDMRG simulations are performed on a system of  $L=18$ sites.\n} \\label{fig:OhD3_spin_aligns_11m1}\n\\end{figure*}\n\n\nIn this section, we present DMRG numerical results for $S=1,3\/2$, which provide evidence for the revealed $O_h\\rightarrow D_3$ symmetry breaking based on a classical analysis.\n\nTable \\ref{table:OhD3} displays the results for the energies of the ten lowest eigenstates under different magnetic fields at a representative point $(J=0,\\phi=0.2\\pi)$ in the $O_h\\rightarrow D_3$ phase,\nin which the first and second tables are for $S=1$ and $S=3\/2$, respectively.\nDMRG is performed on a system of $L=18$ sites in obtaining the data. \nAs can be clearly seen from Table \\ref{table:OhD3}, the system is approximately eight-fold degenerate at zero field,\nwith a ground state energy splitting (characterized by $E_8-E_1$) about one order of magnitude smaller than the excitation gap $E_9-E_1$,\nwhich is consistent with the eight-fold degeneracy predicted by the $O_h\\rightarrow D_3$ symmetry breaking as discussed in Sec. \\ref{sec:sym_break_equator}.\n\nTo test the pattern of the spin alignments as shown in Fig. \\ref{fig:spin_orders}, we apply  small magnetic fields $h_{\\hat{x}}$, $h_{\\hat{n}_a}$ and $h_{\\hat{n}_I}$  along  $\\hat{x}$, $\\hat{n}_a$, and $\\hat{n}_I$-directions (within the six-sublattice rotated frame), respectively, where $\\hat{n}_a$ and $\\hat{n}_I$ are defined in Eq. (\\ref{eq:na_nI}).\nThe magnitude of the field $h=10^{-4}$ is chosen to satisfy\n$\\Delta E\\ll  S|h|L \\ll E_g$, in which $L$ is the system size, $E_g$ is the excitation gap, and $\\Delta E$ is the finite size splitting of the ground state octet at zero field.  \nSuch choice of $h$ ensures a degenerate perturbation within the eight-dimensional ground state subspace, and at the same time, no mixing between the ground states and the excited states is induced. \nHence, it is a thermodynamically small field which only perturbs the ground state subspace. \n\nAs can be read from Fig. \\ref{fig:spin_orders}, \nthe field $h_{\\hat{x}}$ is predicted to lower the energies of the four states located at vertices $(1,\\pm1,\\pm1)$;\n$h_{\\hat{n}_a}$ lowers the energy of state at $(1,1,1)$;\nand $h_{\\hat{n}_I}$ lowers the energies of the two states at $(1,\\pm1,-1)$.\nIndeed, as can be seen from Table \\ref{table:OhD3}, \nthe ground state degeneracy becomes $4$-, $1$- and $2$-fold under the fields $h_{\\hat{x}}$, $h_{\\hat{n}_a}$, and $h_{\\hat{n}_I}$, respectively, \nwhich are consistent with the above analysis.\nThis provides further evidence for the predicted $O_h\\rightarrow D_3$ symmetry breaking.\n\nIn addition, we have also directly measured the expectation values of the spin operators under the fields $h_{\\hat{n}_a}$ and $h_{\\hat{n}_b}$ where $\\hat{n}_b$ is along the $(1,1,-1)$-direction.\nThe results are displayed in Figs. (\\ref{fig:OhD3_spin_aligns_111}, \\ref{fig:OhD3_spin_aligns_11m1}). \nAccording to the discussions in Secs. (\\ref{sec:classical_equator}, \\ref{sec:sym_break_equator}), since the vertices located at $(1,1,1)$ and $(1,1,-1)$ are picked out by $h_{\\hat{n}_a}$ and $h_{\\hat{n}_b}$, respectively,\nthe spin alignments are predicted to be:\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\left<\\vec{S}_{1+3n}\\right>&=&(a,a,b)^T,\\nonumber\\\\\n\\left<\\vec{S}_{3+3n}\\right>&=&(a,b,a)^T,\\nonumber\\\\\n\\left<\\vec{S}_{3+3n}\\right>&=&(b,a,a)^T,\n\\label{eq:aab}\n\\eea\nfor $h_{\\hat{n}_a}$;\nand \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\left<\\vec{S}_{1+3n}\\right>&=&(a,a,-b)^T,\\nonumber\\\\\n\\left<\\vec{S}_{2+3n}\\right>&=&(a,b,-a)^T,\\nonumber\\\\\n\\left<\\vec{S}_{3+3n}\\right>&=&(b,a,-a)^T,\n\\label{eq:aamb}\n\\eea\nfor $h_{\\hat{n}_b}$.\nIndeed, Fig. \\ref{fig:OhD3_spin_aligns_111} (Fig. \\ref{fig:OhD3_spin_aligns_11m1}) is consistent with the pattern in Eq. (\\ref{eq:aab}) (Eq. (\\ref{eq:aamb})).\n\n\n\n\n\\section{The N\\'eel phase for $J>0$}\n\nIn this section, we perform a combination of classical and spin wave analysis for $J>0$ in the vicinity of the FM2 point in Fig. \\ref{fig:phase}.\nSince the spin alignments exhibit an antiferromagnetic pattern in the original frame,\nthe region corresponds to a N\\'eel phase.\nThe mass of the lowest spin wave is calculated to the leading nonvanishing order in an expansion over $J$ and $\\Delta$.\nThroughout this section, we work in the six-sublattice rotated frame unless otherwise stated.\n\n\\subsection{Classical analysis and spin wave theory}\n\nThe saddle point equations have been derived in Eqs. (\\ref{eq:partial_F1},\\ref{eq:partial_F2},\\ref{eq:partial_F3}).\nAssuming the same pattern of spin alignments and relations between $\\lambda_i$'s ($1\\leq i\\leq 3$) as those in Eq. (\\ref{eq:spins_OhD4}), the saddle point equations reduce to\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n-(\\lambda+K^\\prime+2J^\\prime) a-\\Gamma^\\prime b&=&0,\\nonumber\\\\\n-2\\Gamma^\\prime a-(\\lambda+2J^\\prime)b&=&0,\\nonumber\\\\\n2a^2+b^2-1&=&0.\n\\label{eq:Saddle_Neel}\n\\eea\nSince there are three variables and three equations, a solution in general exists.\nOn the other hand, to confirm that this is a minimum of the free energy,\nwe still need to show that the eigenvalues of the Hessian matrix are all positive.\nWe will do a perturbative analysis and demonstrate that this is true at least in the vicinity of the FM2 point.\n\nFor simplicity, let's first take $\\Delta=0$ and turn on a small $J>0$.\nThe solution of Eq. (\\ref{eq:Saddle_Neel}) is given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\lambda=-2\\Gamma^\\prime-2J^\\prime,~\na=b=\\frac{1}{\\sqrt{3}}.\n\\label{eq:classical_Neel}\n\\eea\nFollowing the same logic in Sec. \\ref{sec:classical_equator}, \nwe define the matrix\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\mathcal{H}_F(J)=P(J) H_F(J) P(J),\n\\label{eq:HFJ}\n\\eea\nin which $H_F(J)$ is the $9\\times 9$ Hessian matrix of the free energy in Eq. (\\ref{eq:energy_KHG}),\nand $P(J)\\equiv P(J=0)$ is given by Eq. (\\ref{eq:projection_classical})\nwhere $\\vec{r}_i=\\hat{n}_a$ ($i=1,2,3$) with $\\hat{n}_a=\\frac{1}{\\sqrt{3}}(1,1,1)^T$.\nTaking the two gapless acoustic eigenvectors $v_1$ and $v_2$ (defined in Eq. (\\ref{eq:v1v2})) as the zeroth order  vectors,\nthe first order degenerate perturbation  matrix is given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{(1)}(J)=\\left(\\begin{array}{c}\nv_1^T\\\\\nv_2^T\n\\end{array}\\right)\n\\Delta \\mathcal{H}_F(J)\n(v_1~v_2),\n\\eea\nin which \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\Delta \\mathcal{H}_F(J)=\\mathcal{H}_F(\\Delta=0,J)-\\mathcal{H}_F(\\Delta=0,J=0).\n\\eea\nStraightforward calculations show that \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{(1)}(J)=2JS^2 \\sigma_0,\n\\eea\nwhere $\\sigma_0$ is the $2\\times 2$ identity matrix. \nThus the eigenvalues of the Hessian matrix are positive when $J>0$,\nthereby confirming the solution in  Eq. (\\ref{eq:classical_Neel})\nto be at least a local minimum.\nIn fact, numerical minimization of the free energy shows that it is also a global minimum\nas discussed in Appendix \\ref{app:numerics_classical}. \n\nWe note that the above  analysis can be extended to the case where both $J$ and $\\Delta$ are nonzero but small (i.e.,  $J,|\\Delta|\\ll 1$).\nTo the lowest nonvanishing order in perturbation, the wavefunction is unchanged. \nHence the eigenvalues are additive for $J$ and $\\Delta$.\nTherefore, two lowest eigenvalues of the Hessian matrix are both $2JS^2+\\frac{4}{27}\\Delta^2\\Gamma S^2$.\n\nWe also briefly discuss the symmetry breaking in the N\\'eel phase.\nThe unbroken symmetry group is the same as the $O_h\\rightarrow D_3$ phase, since the spins have the same pattern of alignments.\nAs discussed in Sec. \\ref{sec:review_Oh}, the full symmetry group for a nonzero $J$ is $D_{3d}$ (modulo $T_{3a}$),\ntherefore, the symmetry breaking in the N\\'eel phase is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nD_{3d}\\rightarrow D_3.\n\\label{eq:D3d_D3}\n\\eea\nSince $|D_{3d}\/D_3|=2$, there are two degenerate ground states.\nThe ``center of mass\" directions for the three spins within a unit cell in the two degenerate states are plotted as the two solid light blue circles in Fig. \\ref{fig:spin_orders}.\n\nWe make a comment on the spin ordering in the original frame.\nRotating the spin orientations in Eq. (\\ref{eq:spins_OhD4}) back to the original frame using Eq. (\\ref{eq:6rotation}),\nit is straightforward to verify that the spins align in a N\\'eel pattern with a two-site periodicity, i.e.,\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\vec{S}_{1+2n}=S(a,a,b)^T,~\\vec{S}_{2+2n}=S(-a,-a,-b)^T.\n\\label{eq:Neel_orig}\n\\eea\nThus this phase is termed as ``N\\'eel\" in the phase diagram in Fig. \\ref{fig:phase}.\n\nFinally we build up a spin wave theory for the small fluctuations around the classical configurations.\nTo obtain the spin wave mass, we need to calculate the eigenvalues of the matrix $M(\\Delta,J)\\mathcal{H}_F(\\Delta,J)$.\nThe contribution from the $\\Delta$-part is the same as Sec. \\ref{sec:SW_equator}.\nFor the $J$-part, within first order perturbation theory, the contribution is the same as the eigenvalues of $\\mathcal{H}_F$ as can be seen from Eq. (\\ref{eq:hM1_equator_relation}).\nTherefore, the spin wave Hamiltonian for the lowest spin wave is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nH_{sw}&=& \\frac{1}{2}\\Gamma S^2\\int dx[ (\\partial_x \\xi_\\theta)^2+(\\partial_x \\xi_\\phi)^2]\\nonumber\\\\\n&&+(\\frac{2}{81}\\Gamma \\Delta^2+\\frac{1}{3}J)S^2\\int dx (\\xi_\\theta^2+\\xi_\\phi^2),\n\\label{eq:Hsw_Neel}\n\\eea\nin which $\\xi_\\theta(j),\\xi_\\phi(j)$ is a pair of canonical conjugates satisfying $[\\xi_\\theta(j),\\xi_\\phi(j^\\prime)]=i\\delta_{jj^\\prime}\\frac{1}{S}$.\n\n\\subsection{DMRG numerics}\n\\label{sec:Neel_numerics}\n\n\\begin{table}\n                \\begin{tabular}[t]{|c|c|c|}\n\t\t  \\multicolumn{3}{c}{}\\\\ \\hline\n$E(S=1)$ & No field & $h_{\\hat{n}_{a}}=10^{-4}$\\\\ \\hline\n$E_1$ & \\tikzmark{top left 1a}-17.96054 &\\tikzmark{top left 1b} -17.96348\\tikzmark{bottom right 1b}\\\\\n$E_2-E_1$ & $4.57\\cdot 10^{-6}$\\tikzmark{bottom right 1a} & $5.887\\cdot 10^{-3}$\\\\\n$E_3-E_1$ & $6.054\\cdot 10^{-2}$ & $6.056\\cdot 10^{-2}$\\\\\n$E_4-E_1$ & $6.054\\cdot 10^{-2}$ & $6.110\\cdot 10^{-2}$\\\\\n$E_5-E_1$ & $6.515\\cdot 10^{-2}$ & $6.518\\cdot 10^{-2}$\\\\\n\\hline\n                \\end{tabular}\n                \\DrawBox[ultra thick, red]{top left 1a}{bottom right 1a}\n                \\DrawBox[ultra thick, blue]{top left 1b}{bottom right 1b}\n                \\hfill\n                \\begin{tabular}[t]{|c|c|c|}\n\t\t  \\multicolumn{3}{c}{}\\\\ \\hline\n$E(S=3\/2)$ & No field & $h_{\\hat{n}_{a}}=10^{-4}$\\\\ \\hline\n$E_1$ & \\tikzmark{top left 1a}-39.50646& \\tikzmark{top left 1b}-39.51099\\tikzmark{bottom right 1b}\\\\\n$E_2-E_1$ & $8.9\\cdot 10^{-12}$\\tikzmark{bottom right 1a} & $9.060\\cdot 10^{-3}$\\\\\n$E_3-E_1$ & $8.908\\cdot 10^{-2}$ & $8.910\\cdot 10^{-2}$\\\\\n$E_4-E_1$ & $8.908\\cdot 10^{-2}$ & $8.997\\cdot 10^{-2}$\\\\\n$E_5-E_1$ & $9.487\\cdot 10^{-2}$ & $9.489\\cdot 10^{-2}$\\\\\n\\hline\n                \\end{tabular}\n                \\DrawBox[ultra thick, red]{top left 1a}{bottom right 1a}\n                \\DrawBox[ultra thick, blue]{top left 1b}{bottom right 1b}\n\\caption{Energies of the five lowest lying states computed with\nDMRG simulations. The data refer to $L=18$ sites, $\\theta=0.4\\pi$, and $\\phi=0.2\\pi$.\nThe energies enclosed by the colored squares are approximately degenerate.\n\\label{table:energies_Neel}\n}\n\\end{table}\n\n\\begin{figure*}[htbp]\n\\includegraphics[width=12.0cm]{Neel_spin_align.pdf}\n\\caption{Spin expectation values $\\left<S_i^\\alpha\\right>$ ($\\alpha=x,y,z$) under a small field $h_{\\hat{n}_a}=10^{-4}$ along $(1,1,1)$-direction \nat a representative point $(\\theta=0.4\\pi,\\phi=0.2\\pi)$ in the N\\'eel phase\n for (a) $S=1$, and (b) $S=3\/2$.\n The parametrization $(\\theta,\\phi)$ is defined in Eq. (\\ref{eq:parametize_theta_phi}).\nDMRG is performed on a system of  $L=18$ sites.\n} \\label{Neel_spin_align}\n\\end{figure*}\n\nIn this section, we present DMRG numerical results for $S=1,3\/2$, which provide evidence for the revealed $D_{3d}\\rightarrow D_3$ symmetry breaking based on a classical analysis.\nWe proceed similarly as Sec. \\ref{sec:numerics_OhD3}.\n\nTable \\ref{table:energies_Neel} displays the results for the energies of the five lowest eigenstates under different magnetic fields at a representative point $(\\theta=0.4\\pi,\\phi=0.2\\pi)$ in the N\\'eel phase,\nin which the first and second tables are for $S=1$ and $S=3\/2$, respectively,\nand $\\theta,\\phi$ are defined in Eq. (\\ref{eq:parametize_theta_phi}).\nDMRG is performed for a system of $L=18$ sites in obtaining the data.\nOn a $L=12$ size system, we have checked that the DMRG results are in agreement with Lanczos Exact Diagonalization.\nAs can be clearly seen from Table \\ref{table:energies_Neel}, the system is approximately two-fold degenerate at zero field,\nwith a ground state energy splitting (characterized by $E_2-E_1$)  orders of magnitude smaller than the excitation gap $E_3-E_1$,\nwhich is consistent with the two-fold degeneracy predicted by the $D_{3d}\\rightarrow D_3$ symmetry breaking as discussed in Eq. (\\ref{eq:D3d_D3}).\nWe have also applied a small magnetic field along the $\\hat{n}_a$-direction,\nwhich should be able to pick out the state located at the $(111)$-vertex as shown in Fig. \\ref{fig:spin_orders}.\nIndeed, as can be seen from Table \\ref{table:energies_Neel},\nthe system becomes nondegenerate when $h_{\\hat{n}_a}$ is applied.\n\nIn addition, we have also directly measured the expectation values of the spin operators under the fields $h_{\\hat{n}_a}$.\nThe results are displayed in Fig. \\ref{Neel_spin_align} (a) for $S=1$ and (b) for $3\/2$.\nClearly, the spin alignments revealed in Fig. \\ref{Neel_spin_align}\nare consistent with the pattern in Eq. (\\ref{eq:aab}).\n\n\\section{The ``$D_3$-breaking I, II\" phases for $J<0$}\n\nIn this section, we discuss the ``$D_3$-breaking I, II\" phases in the negative $J$ region.\nWe work within the six-sublattice rotated frame unless otherwise stated.\n\n\\subsection{Classical phase diagram}\n\nWe first briefly describe the classical phase diagram in the negative $J$ region as shown in Fig.  \\ref{fig:phase},\nwith calculations included in the next two subsections. \nThere are two phases denoted as ``$D_3$ breaking I\" and ``$D_3$ breaking II\".\nBoth phases break the $D_3$ symmetry albeit in different ways,\nhence the ground states are six-fold degenerate.\nHowever, the symmetry breaking patterns are not the same.\n\nTo clarify this point, recall that the symmetry group is $G_1\\simeq D_{3d}\\ltimes 3\\mathbb{Z}$  as discussed in Sec. \\ref{sec:review_Oh}.\nSince $T_{3a}$ is not broken, we consider $G_1^\\prime=G_1\/\\mathopen{<}T_{3a}\\mathclose{>}\\simeq D_{3d}$ and the spins within a unit cell in what follows.\nIn the ``$D_3$ breaking I\" phase,\nthe spin orientations in one of the six degenerate ground states are\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\vec{S}_1=S\\left(\\begin{array}{c}\nx\\\\\ny\\\\\nz\n\\end{array}\\right),~\n\\vec{S}_2=S\\left(\\begin{array}{c}\n-\\frac{1}{\\sqrt{2}}\\\\\n0\\\\\n\\frac{1}{\\sqrt{2}}\n\\end{array}\\right),~\n\\vec{S}_3=S\\left(\\begin{array}{c}\n-z\\\\\n-y\\\\\n-x\n\\end{array}\\right),\n\\label{eq:order_D6I}\n\\eea\nin which $x^2+y^2+z^2=1$.\nAs can be checked, the little group of Eq. (\\ref{eq:order_D6I}) is generated by $R_I I$.\nHence the symmetry breaking is $D_{3d}\\rightarrow \\mathopen{<} R_I I \\mathclose{>} $.\nOn the other hand, in the ``$D_3$ breaking II\" phase,\nthe spin orientations in one of the six degenerate ground states are\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\vec{S}_1=S\\left(\\begin{array}{c}\nx\\\\\ny\\\\\nz\n\\end{array}\\right),~\n\\vec{S}_2=S\\left(\\begin{array}{c}\nm\\\\\nn\\\\\nm\n\\end{array}\\right),~\n\\vec{S}_3=S\\left(\\begin{array}{c}\nz\\\\\ny\\\\\nx\n\\end{array}\\right),\n\\label{eq:order_D6II}\n\\eea\nin which $x^2+y^2+z^2=2m^2+n^2=1$.\nThe little group of Eq. (\\ref{eq:order_D6II}) is generated by $TR_I I$,\nand the symmetry breaking is $D_{3d}\\rightarrow \\mathopen{<} TR_I I \\mathclose{>} $.\nThus we see that although the symmetry breaking in the two phases are both \n$D_{3d}\\rightarrow \\mathbb{Z}_2$, the group $\\mathbb{Z}_2$ represents different little groups.\nWe also note that since $D_{3d}\/\\mathbb{Z}_2 \\simeq D_3$,\nthe two phases both exhibit $D_3$-breaking\nwhich is the origin of the names of the two phases.\nThe ``center of mass\" directions of the three spins within a unit cell in the six degenerate ground states are shown in Fig. \\ref{fig:spin_orders},\nwhere the red (dark blue) solid circles correspond to the ``$D_3$-breaking I (II)\" phases.\n\n\n\\subsection{The ``$D_3$ breaking I\" phase}\n\n\\subsubsection{The classical solution}\n\\label{sec:D3I_classical}\n\nWe perform a classical analysis in the ``$D_3$ breaking I\" phase.\nFor simplicity, we consider the $\\Delta=0$ case with a small negative $J$.\nWe will use the normalized parameter $\\bar{J}=J\/\\Gamma$.\n\nWe take the trial solution given by Eq. (\\ref{eq:order_D6I}) and  assume $\\lambda_3=\\lambda_1$.\nSetting $K=\\Gamma$ and plugging the trial solution into Eq. (\\ref{eq:partial_F1},\\ref{eq:partial_F2},\\ref{eq:partial_F3}), the saddle point equations reduce to\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n(-\\lambda_1+J^\\prime) x+\\Gamma^\\prime z+\\frac{1}{\\sqrt{2}} (K^\\prime+J^\\prime)&=&0\\nonumber\\\\\n(\\Gamma^\\prime+J^\\prime-\\lambda_1)y-\\frac{1}{\\sqrt{2}} J^\\prime&=&0\\nonumber\\\\\n\\Gamma^\\prime x+(J^\\prime -\\lambda_1 )z-\\frac{1}{\\sqrt{2}} \\Gamma^\\prime&=&0\\nonumber\\\\\n-(K^\\prime+J^\\prime)x+J^\\prime y+\\Gamma^\\prime z+\\frac{1}{\\sqrt{2}}\\lambda_2&=&0\\nonumber\\\\\nx^2+y^2+z^2-1&=&0.\n\\label{eq:Saddle_eq_D6I}\n\\eea\nSince there are five variables $x,y,z,\\lambda_1,\\lambda_2$ and five equations,\ngenerically a solution exists.\nEq. (\\ref{eq:Saddle_eq_D6I}) can be solved perturbatively in an expansion over $J$.\nThe results up to $O(J^3)$ are\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nx&=& -\\frac{1}{\\sqrt{2}}-\\frac{1}{6\\sqrt{2}}\\bar{J}+\\frac{5}{72\\sqrt{2}} \\bar{J}^2-\\frac{7}{432\\sqrt{2}} \\bar{J}^3+O(\\bar{J}^4),\\nonumber\\\\\ny&=& \\frac{1}{3\\sqrt{2}}\\bar{J}-\\frac{1}{18\\sqrt{2}} \\bar{J}^2 +\\frac{1}{216\\sqrt{2}} \\bar{J}^3+O(\\bar{J}^4),\\nonumber\\\\\nz&=&\\frac{1}{\\sqrt{2}}-\\frac{1}{6\\sqrt{2}}\\bar{J}-\\frac{1}{72\\sqrt{2}} \\bar{J}^2+\\frac{5}{432\\sqrt{2}} \\bar{J}^3+O(\\bar{J}^4),\\nonumber\\\\\n\\label{eq:D3I_3rd_order_A}\n\\eea\nand\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\lambda_1&=&\\Gamma^\\prime (-2+\\frac{1}{2}\\bar{J}-\\frac{1}{24\\sqrt{2}} \\bar{J}^2-\\frac{1}{144} \\bar{J}^3)+O(\\bar{J}^4),\\nonumber\\\\\n\\lambda_2&=&\\Gamma^\\prime(-2-\\bar{J}-\\frac{5}{12} \\bar{J}^2+\\frac{7}{72} \\bar{J}^3)+O(\\bar{J}^4).\n\\label{eq:D3I_3rd_order_B}\n\\eea\nDetailed derivations of Eqs. (\\ref{eq:D3I_3rd_order_A},\\ref{eq:D3I_3rd_order_B}) are included in Appendix \\ref{app:D3I}.\n\n\nConsider the projected Hessian matrix  defined in Eq. (\\ref{eq:HFJ}) for $J<0$.\nThe perturbation Hamiltonian is\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\Delta\\mathcal{H}_F(J)&=&P(J) H_F(J) P(J)\\nonumber\\\\\n&&- P(J=0) H_F(J=0) P(J=0).\n\\eea\nThen the first order degenerate perturbation Hamiltonian is given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{(1)}(J)=\\left(\\begin{array}{cc}\nv_1^T \\Delta\\mathcal{H}_F(J) v_1& v_1^T \\Delta\\mathcal{H}_F(J) v_2\\\\\nv_2^T \\Delta\\mathcal{H}_F(J) v_1 & v_2^T\\Delta \\mathcal{H}_F(J) v_2\n\\end{array}\n\\right),\n\\eea\nin which $v_1,v_2$ are given by Eq. (\\ref{eq:v1v2}) where\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\hat{e}_\\theta=(-\\frac{1}{\\sqrt{2}},0,-\\frac{1}{\\sqrt{2}})^T,~\n\\hat{e}_\\phi=(0,-1,0)^T.\n\\label{eq:e_thetaphi_D3I}\n\\eea\nCalculations show that \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{(1)}(J)=-JS^2\\left(\\begin{array}{cc}\n\\frac{4}{3}&\\frac{2\\sqrt{2}}{3}\\\\\n\\frac{2\\sqrt{2}}{3}&\\frac{2}{3}\n\\end{array}\\right).\n\\label{eq:h1J_D3I}\n\\eea\nThe two eigenvalues of $h^{(1)}(J)$ are $0$ and $-2JS^2$.\nThus we see that although the first order perturbation already breaks the degeneracy,\none eigenvalue remains zero up to $O(J)$ and higher order perturbation is needed to obtain a nonzero value.\nIn fact, calculations show that the first nonvanishing term for this eigenvalue appears at $O(J^3)$.\nHere we only mention that the result is $-\\frac{1}{2}\\Gamma S^2\\bar{J}^3$,\nand detailed derivations are given in Appendix \\ref{app:D3I}.\n\nIn summary, the two low-lying eigenvalues are\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n-\\Gamma S^2\\bar{J},~-\\frac{1}{2}\\Gamma S^2\\bar{J}^3,\n\\label{eq:eigenvalues_D6I}\n\\eea\nwhich are both positive when $J<0$.\nThis shows that Eqs. (\\ref{eq:D3I_3rd_order_A},\\ref{eq:D3I_3rd_order_B}) represent a minimum of the free energy.\nNumerical calculations provide evidence for Eqs. (\\ref{eq:D3I_3rd_order_A},\\ref{eq:D3I_3rd_order_B}) to be a global minimum as discussed in Appendix \\ref{app:numerics_classical}. \n\n\n\n\\subsubsection{Spin wave theory}\n\\label{sec:sw_D3I}\n\n\n\\begin{figure*}[htbp]\n\\includegraphics[width=5.0cm]{0p21pi.pdf}\n\\includegraphics[width=6.2cm]{0pi.pdf}\n\\includegraphics[width=6cm]{0p30pi.pdf}\n\\caption{Numerically obtained smallest spin wave mass $m_1$ vs. $J$ represented by the hollow circles for (a) $\\varphi=0.21\\pi$, (b) $\\varphi=0.25\\pi$,\nand (c) $\\varphi=0.30\\pi$.\nIn all figures, $m_1$ is in units of $\\Gamma S^2$, where $\\Gamma=\\sin(\\varphi)$.\nIn (b), the solid line represents $\\Gamma^\\prime \\bar{J}^2$ where $\\Gamma^\\prime=\\frac{1}{\\sqrt{2}}$.\n} \n\\label{fig:sw_mass_D3}\n\\end{figure*}\n\nIn this subsection, we  calculate the lowest-lying spin wave mass for the $\\Delta=0$ case with a small negative $J$.\nLet's first consider the case of a zero wavevector.\nAgain, we need to diagonalize the Hessian matrix $\\mathcal{H}_F(J)$ using symplectic transformations.\nAs discussed in Sec. \\ref{sec:sw_equator_zero_momentum}, the spin wave masses are given by the eigenvalues of $M(J)\\mathcal{H}_F(J)$.\nWe will calculate the smallest spin wave mass up to the leading nonvanishing order of $J$.  \n\nBefore proceeding on, notice that the definitions of $v_i$ ($i=1,2$), $w_j$ ($j=1,2,3,4$) are the same as Eq. (\\ref{eq:v1v2}) and Eq. (\\ref{eq:wi_s}), where $\\hat{e}_\\theta$ and $\\hat{e}_\\phi$ should be taken as Eq. (\\ref{eq:e_thetaphi_D3I}).\nWe emphasize that we will use the same notations as Sec. \\ref{sec:sw_equator_zero_momentum} for simplicity.\nHowever, the expressions of the quantities are different from those in Eq. (\\ref{sec:sw_equator_zero_momentum}),\nwhich are determined by the form of the Hamiltonian and the saddle point solutions.\nLet $P_1^{(0)}$ be the projection operation to the subspace spanned by $\\{v_1,v_2\\}$.\nLet $\\mathcal{H}^{M,(n)} (J)$ be the order $J^n$ term in the expansion of $M(J)\\mathcal{H}_F(J)$ over $J$.\nThen the first order degenerate perturbation is given by the following $2\\times 2$ matrix,\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(1)} (J)=P^{(0)}_1 \\mathcal{H}^{M,(1)}_F (J) P^{(0)}_1.\n\\label{eq:first_D3I_mass}\n\\eea\nAccording to Eq. (\\ref{eq:hM1_equator_relation}), this is simply\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(1)} (J)=i\\sigma_2 h^{(1)}(J),\n\\eea\nin which $h^{(1)}(J)$ is given by Eq. (\\ref{eq:h1J_D3I}), and $i\\sigma_2$ is the projection of $M^{(0)}$ to the subspace spanned by $\\{v_1,v_2\\}$ where $\\sigma_\\alpha$ ($\\alpha=1,2,3$) are the Pauli matrices.\nAs can be readily checked, since one of the two eigenvalues of $h^{(1)}(J)$  vanishes, the two eigenvalues of $h^{M,(1)} (J)$ are both zero.\nHence, we need to go to second order perturbation.\n\nThe second order perturbation is given by the following matrix\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(2)}(J)&=&\n\\left(\\begin{array}{c}\nv_1^T\\\\\nv_2^T\n\\end{array}\n\\right)\n\\big[\\mathcal{H}^{M,(2)}_F+\\mathcal{H}^{M,(1)}_F\\sum_{i=1}^4 \\frac{u_iu_i^\\dagger}{E_0-\\epsilon_i} \\mathcal{H}^{M,(1)}_F\\big]\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\\times(v_1\\,v_2),\n\\label{eq:second_D3I_mass}\n\\eea\nin which $v_\\pm$ and $u_i$ ($i=1,2,3,4$) are defined in the same way as \nEq. (\\ref{eq:v_pm}) and Eq. (\\ref{eq:Def_us});\nand the eigenvalues $\\epsilon_i$'s ($i=1,2,3,4$) are given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\epsilon_1=-3i,~ \\epsilon_2=3i,~ \\epsilon_3=-3i, ~\\epsilon_4=3i.\n\\eea\nCalculations show that \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(2)}(J)=0.\n\\eea\nHence, the second order perturbation also vanishes, which means that we have to go to the third order perturbation theory.\n\nThe third order perturbation matrix $h^{M,(3)}(J)$ is given by\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nh^{M,(3)}(J)=\\bar{J}^3 \\left(\\begin{array}{cc}\n-\\frac{19}{54\\sqrt{2}}& \\frac{35}{108}\\\\\n-\\frac{4}{27}& \\frac{19}{54\\sqrt{2}}\n\\end{array}\\right).\n\\label{eq:hM3J}\n\\eea\nDetailed derivation of Eq. (\\ref{eq:hM3J}) is included in Appendix \\ref{app:D3I_third_order}.\n\nThe eigenvalues of $h^{M,(1)}+h^{M,(2)}+h^{M,(3)}$ are $\\pm i\\Gamma S^2 \\bar{J}^2$, which gives\n$m_1(\\Delta=0,J)=\\Gamma S^2 \\bar{J}^2$.\nIn Fig. \\ref{fig:sw_mass_D3} (b),\n the hollow circles represent the numerical results for $m_1$ by numerically solving the eigenvalues of $M(J)\\mathcal{H}_F(J)$,   \n and the solid line  represents $\\Gamma^\\prime \\bar{J}^2$.\n As can be clearly seen, the numerical results agree well with the obtained perturbative results up to $O(J^2)$.\n \n  \n\\begin{figure*}[htbp]\n\\includegraphics[width=15.0cm]{D3_break_Seq1.pdf}\n\\caption{(a,b,c) $\\langle S_j^x\\rangle$, (d,e,f) $\\langle S_j^y\\rangle$, and (g,h,i) $\\langle S_j^z\\rangle$ vs $j$ under $h_{\\text{I}}$ (black squares) and $h_{\\text{II}}$ (red dots) fields for $S=1$ at several different points.\n(a,d,g) are for $(\\theta=0.52\\pi,\\phi=0.15\\pi)$;\n(b,e,h)  for $(\\theta=0.52\\pi,\\phi=0.25\\pi)$;\nand (c,f,i)  for $(\\theta=0.52\\pi,\\phi=0.30\\pi)$.\nDMRG numerics are performed on $L=18$ sites with periodic boundary conditions. \nBoth $h_{\\text{I}}$ and $h_{\\text{II}}$ fields are taken to be $10^{-4}$.\n} \\label{fig:D3_break_Seq1}\n\\end{figure*}\n\nBased on the above discussions, we are able to obtain the spin wave Hamiltonian for $\\Delta=0,|\\bar{J}|\\ll 1$ as \n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nH_{sw}&=& \\frac{1}{2}\\Gamma S^2\\int dx[ (\\partial_x \\xi_\\theta)^2+(\\partial_x \\xi_\\phi)^2]\\nonumber\\\\\n&&+\\frac{1}{6}\\Gamma S^2\\bar{J}^2\\int dx (\\xi_\\theta^2+\\xi_\\phi^2).\n\\label{eq:Hsw_D3I}\n\\eea\n\n\\subsection{The ``$D_3$ breaking II\" phase}\n\nIn this subsection, we discuss the ``$D_3$ breaking II\" phase.\nTo obtain an intuitive understanding, let's start with the case of $\\Delta\\neq 0, J=0$, and then turn on a small negative $J$.\nAt $J=0$, the symmetry breaking is $O_h\\rightarrow D_3$ and there are eight degenerate ground states.\nIf $J\\neq 0$, since $J$ is planar-like, the two states along $\\pm(111)$-direction  in Fig. \\ref{fig:spin_orders} will have higher energies than the other six states.\nAs a result, the two solid light blue circles at the $\\pm(111)$ vertices should be removed compared with the $J=0$ case as shown  in Fig. \\ref{fig:spin_orders}.\nThus, the ground states now are six-fold degenerate and the spin orientations are slightly distorted away from the $J=0$ case.\nThis is different from the ``$D_3$ breaking I\" phase where the ``center of mass\" direction is perpendicular to the $(111)$-direction.\nOn the other hand, when $J$ is large enough, the ``center of mass\" spin orientations will eventually be bent to the plane perpendicular to the $(111)$-direction.\nThus we expect a ``$D_3$ breaking II\" to ``$D_3$ breaking I\" phase transition classically,\nthis is indeed the case as shown in Fig. \\ref{fig:phase}.\n\n\nWe take the trial solution in Eq. (\\ref{eq:order_D6II}) and assume $\\lambda_3=\\lambda_1$.\nUnder these assumptions, Eqs. (\\ref{eq:partial_F1},\\ref{eq:partial_F2},\\ref{eq:partial_F3}) reduce to\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n-(J^\\prime+\\lambda_1)x-\\Gamma^\\prime z-(K^\\prime+J^\\prime) m&=&0\\nonumber\\\\\n-(K^\\prime+J^\\prime+\\lambda_1)y-J^\\prime m-\\Gamma^\\prime n&=&0\\nonumber\\\\\n-\\Gamma^\\prime x-(J^\\prime+\\lambda_1)z-\\Gamma^\\prime m-J^\\prime n&=&0\\nonumber\\\\\n-(K^\\prime+J^\\prime) x-J^\\prime y-\\Gamma^\\prime z-\\lambda_2 m&=&0\\nonumber\\\\\n-2\\Gamma^\\prime y-2J^\\prime z-\\lambda_2 n &=&0\\nonumber\\\\\nx^2+y^2+z^2-1&=&0\\nonumber\\\\\n2m^2+n^2-1&=&0.\n\\label{eq:eq_D6II}\n\\eea\nSince there are seven variables and seven equations, \ngenerically a solution exists.\n\nNext we try to solve Eq. (\\ref{eq:eq_D6II}) in a perturbative expansion over $J$.\nHowever, we find difficulty in carrying out a perturbative expansion. \nThe $J=0$ case has been already solved in Sec. \\ref{sec:classical_equator},\nwhich is taken as the zeroth order solution.\nWhen $J\\neq 0$, up to $O(J)$, the solution is (see Appendix \\ref{app:D3II} for details)\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nx&=&\\frac{1}{\\sqrt{3}}\\big[x_0+(\\frac{3}{\\Delta^2}+\\frac{7}{6\\Delta})\\bar{J}\\big]+O(\\bar{J}^2)\\nonumber\\\\\ny&=&\\frac{1}{\\sqrt{3}}\\big[-x_0+(\\frac{6}{\\Delta^2}+\\frac{1}{3\\Delta})\\bar{J}\\big]+O(\\bar{J}^2)\\nonumber\\\\\nz&=&\\frac{1}{\\sqrt{3}}\\big[z_0+(\\frac{3}{\\Delta^2}+\\frac{1}{6\\Delta})\\bar{J}\\big]+O(\\bar{J}^2),\n\\label{eq:sol_D6II_A}\n\\eea\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\nm&=&\\frac{1}{\\sqrt{3}}\\big[x_0+(\\frac{3}{\\Delta^2}-\\frac{5}{6\\Delta})\\bar{J}\\big]+O(\\bar{J}^2)\\nonumber\\\\\nn&=&\\frac{1}{\\sqrt{3}}\\big[-z_0+(\\frac{6}{\\Delta^2}+\\frac{1}{3\\Delta})\\bar{J}+O(\\bar{J}^2)\\big],\n\\label{eq:sol_D6II_B}\n\\eea\nand\n\\begin{eqnarray}} \\def\\eea{\\end{eqnarray}\n\\lambda_1&=&\\lambda_0+\\frac{2}{\\Delta}\\bar{J}+O(\\bar{J}^2)\\nonumber\\\\\n\\lambda_2&=&\\lambda_0-\\frac{4}{\\Delta}\\bar{J}+O(\\bar{J}^2),\n\\label{eq:sol_D6II_C}\n\\eea\nin which the results are obtained up to $O(J)$.\nIn particular, since the $J$-dependent terms contain negative powers of $\\Delta$,\nthe perturbation is valid only when $|\\bar{J}|\/\\Delta^2\\ll 1$.\n\nAs usual, the eigenvalues of the Hessian matrix should be calculated to verify that the saddle point solution in Eqs. (\\ref{eq:sol_D6II_A},\\ref{eq:sol_D6II_B},\\ref{eq:sol_D6II_C}) corresponds to a minimum of the free energy. \nHowever,  the nonanalyticity in $\\Delta$ in Eqs. (\\ref{eq:sol_D6II_A},\\ref{eq:sol_D6II_B},\\ref{eq:sol_D6II_C}) complicates the calculation.\nAs discussed in detail in Appendix \\ref{app:D3II},\n one possibly has to go up to at least fifth order perturbation in $\\Delta$.\nWe will not perform such a difficult  fifth order perturbation, \nand in fact, we suspect if a good perturbation exists because of the nonanalytical dependence of the saddle point solution on $\\Delta$. \nThe smallest eigenvalue is studied by numerics as discussed in Appendix \\ref{app:D3II}.\n\nDue to the above mentioned difficulty, the spin wave mass will not be perturbatively calculated.\nInstead, we study the spin wave mass numerically by computing the eigenvalues of the matrix $M(\\Delta,J)\\mathcal{H}_F(\\Delta,J)$.\nThe dependence of $m_1$ on $J$ at three representatively values of $\\varphi=0.21\\pi$ and $0.30\\pi$ are shown in  Fig. \\ref{fig:sw_mass_D3} (a) and (c), respectively.\nThe value of $J=J_c(\\varphi)$  where $m_1$ vanishes is the transition point between the ``$D_3$-breaking I\" and the ``$D_3$-breaking II\" phases.\nNotice that for fixed value of $\\varphi$,  the ``$D_3$-breaking I (II)\" phase occupies the region $J>J_c(\\varphi)$ ($J<J_c(\\varphi)$).\n\n\\subsection{DMRG numerics}\n\nIn this subsection, we present the DMRG numerical results which provide numerical evidence for the predicted ``$D_3$-breaking I, II\" phases for both $S=1$ and $3\/2$.\n\n\n\n\\begin{figure*}[htbp]\n\\includegraphics[width=12.0cm]{energy_vs_field.pdf}\n\\caption{$\\Delta E$ ($=E(h_{\\hat{n}_a})-E(h_{\\hat{n}_a}=0)$) vs $h_{\\hat{n}_a}$ for (a,c) $S=1$, and (b,d) $S=3\/2$ at fixed values of $\\theta$ and $\\varphi$.\nThe magnetic field $h_{\\hat{n}_a}$ is taken along the $(111)$-direction with a magnitude $h_{\\hat{n}_a}=5\\times 10^{-4}$.\nED numerics are performed on $L=18$ sites with periodic boundary conditions. \n} \\label{fig:energy_vs_field}\n\\end{figure*}\n\nBefore proceeding on, we mention a subtlety in numerical calculations,\nwhich has already been discussed in detail in Ref. \\onlinecite{Yang2020b}.\nIn either the ``$D_3$-breaking I\" or ``$D_3$-breaking II\" phases, the six symmetry breaking ground states only become exactly degenerate in the thermodynamic limit. \nIn a finite size system, the ground state can be some arbitrary linear combination of the six states,\nand the coefficients depend on the system size and numerical details.\nBecause of this, random cancellations occur if the correlation functions $\\langle S_i^\\alpha S_{i+r}^\\beta\\rangle$ or the expectation values of the spin operators $\\langle\\vec{S}_i\\rangle$ are directly computed.\nTo circumvent such difficulty, a small magnetic field has to be applied such that the system is polarized into one of the six degenerate ground states.\n\nFor our purpose, we choose the field to be $h_{\\text{I}}$ along the $(-1,0,1)$-direction in the ``$D_3$-breaking I\" phase, and $h_{\\text{II}}$ along the $(1,-1,1)$-direction in the ``$D_3$-breaking II\" phase.\nAccording to Fig. \\ref{fig:spin_orders},\nwe expect that the red solid circle located at $(-1,0,1)$ is picked out in the the ``$D_3$-breaking I\" phase,\nand the solid dark blue circle located at $(1,-1,1)$ is picked out in the the ``$D_3$-breaking II\" phase.\nThen with the application of such fields, the spins should align according to the pattern given in Eq. (\\ref{eq:order_D6I}) (Eq. (\\ref{eq:order_D6II})) in the `$D_3$-breaking I (II)\" phase. \nHowever, as discussed in Ref. \\onlinecite{Yang2020b},\nthe ``$D_3$-breaking I (II)\" phase responds to the $h_{\\text{II}}$- ($h_{\\text{(I)}}$-) field as does the ``$D_3$-breaking II (I)\" phase.\nTherefore, this method is not able to distinguish the two $D_3$-breaking phases.\nHowever, the method is still useful since it can test the existence of either ``$D_3$-breaking I\" or ``$D_3$-breaking II\" orders.\n\nWe have calculated the spin expectation values $\\left<S_j^\\alpha\\right>$ ($\\alpha=x,y,z$) at three representative points $(\\theta=0.52\\pi, \\phi=0.15\\pi)$, $(\\theta=0.52\\pi, \\phi=0.25\\pi)$ and $(\\theta=0.52\\pi, \\phi=0.30\\pi)$ under the $h_{\\text{I}}$ and $h_{\\text{II}}$ fields.\nThe results for $S=1$ are displayed in Fig. \\ref{fig:D3_break_Seq1}.\nDMRG numerics are performed on a system of  $L=18$ sites with periodic boundary conditions,\nand both $h_{\\text{I}}$ and $h_{\\text{II}}$ fields are taken to be $10^{-4}$.\nAs can be clearly seen from Fig. \\ref{fig:D3_break_Seq1},\nthe spin alignments are consistent with the patterns given in Eqs. (\\ref{eq:order_D6I},\\ref{eq:order_D6II}),\nthereby confirming the existence of the ``$D_3$-breaking I, II\" phases.\nWe have also studied the $S=3\/2$ case, and the results are included in Appendix \\ref{app:more_numerics}.\n\nAs discussed in Ref. \\onlinecite{Yang2020b},\nthe two $D_3$-breaking phases can be distinguished by studying the response of the system to a small field $h_{\\hat{n}_a}$ along the $(111)$-direction,\nsince the ``$D_3$-breaking I\" phase does not respond to $h_{\\hat{n}_a}$,\nwhereas the ``$D_3$-breaking II\" phase does have an response.\n\n\\begin{figure*}[htbp]\n\\includegraphics[width=13.0cm]{D3_break_111field.pdf}\n\\caption{$\\left< S_j^x \\right>$ vs $j$ for (a) $S=1$, and (b) $S=3\/2$ at fixed values of $\\theta$, $\\varphi$.\nThe magnetic field is taken along the $(111)$-direction with a magnitude $h_{\\hat{n}_a}=5\\times 10^{-4}$. \nDMRG numerics are performed on $L=18$ sites with periodic boundary conditions. \n} \\label{fig:D3_break_111field}\n\\end{figure*}\n\nFig. \\ref{fig:energy_vs_field} shows the energy change $\\Delta E= E(h_{\\hat{n}_a})-E(h_{\\hat{n}_a}=0)$ as a function of $h_{\\hat{n}_a}$ at several representative points in the negative $J$ region for both $S=1$ and $S=3\/2$.\nClearly, while the system  has a huge response  at some  points, \nthe response nearly vanishes at others. \nBased on the results in Fig. \\ref{fig:energy_vs_field}, we arrive at the conclusion that the points \n$(\\theta=0.52\\pi,\\phi=0.2\\pi,0.24\\pi)$ and $(\\theta=0.55\\pi,\\phi=0.15\\pi,0.2\\pi,0.25\\pi)$ are within the ``$D_3$-breaking I\" phase,\nwhereas the points\n$(\\theta=0.52\\pi,\\phi=0.15\\pi,0.3\\pi)$ and $(\\theta=0.55\\pi,\\phi=0.3\\pi)$\nare in the ``$D_3$-breaking II\" phase.\nIn particular, as can be seen from Fig. \\ref{fig:energy_vs_field}, the range of the ``$D_3$-breaking I\" phase expands by increasing $\\theta$,\nwhich is consistent with the classical phase diagram as shown in Fig. \\ref{fig:phase}.\n\nFig. \\ref{fig:D3_break_111field} displays the response of $\\left<S_j^x\\right>$ to $h_{\\hat{n}_a}=5\\times 10^{-4}$ at several different points for both $S=1$ and $S=3\/2$.\nAs can be  seen from Fig. \\ref{fig:D3_break_111field},\nthe response at  the point $(\\theta=0.52\\pi,\\phi=0.24\\pi)$ is very small,\nhence this point should locate within the ``$D_3$-breaking I\" phase.\nOn the other hand, the response at the points $(\\theta=0.52\\pi, \\phi=0.15\\pi,0.3\\pi)$\nare significant, and they should be within the ``$D_3$-breaking II\" phase.\n\n\n\\section{Conclusions}\n\nIn conclusion, we have studied the classical phase diagram of the one-dimensional spin-$S$ Kitaev-Heisenberg-Gamma model in the region of an antiferromagnetic Kitaev coupling,\n based on a combination of classical and spin wave analysis.\nThe  revealed ``N\\'eel\" and ``$D_3$-breaking I, II\" phases are in accordance with the spin-1\/2 case as discussed in Ref. \\onlinecite{Yang2020b}.\nOn the other hand, the ``$O_h\\rightarrow D_3$\" phase in the absence of the Heisenberg term is not the same as the ``$O_h\\rightarrow D_4$\" phase in the spin-1\/2 case.\nDMRG numerics provide evidence for the ``$O_h\\rightarrow D_3$\" symmetry breaking for higher spins including $S=1$ and $3\/2$, which are consistent with the classical results.\nWe have also obtained analytic expressions of the lowest-lying spin wave mass perturbatively in the vicinity of the hidden  SU(2) symmetric ferromagnetic point. \n\n\n\n{\\it Acknowledgments}\nWe thank H.-Y. Kee for interesting remarks and helpful discussions.\nWY and IA acknowledge support from NSERC Discovery Grant 04033-2016.\nAN acknowledges computational resources and services provided by Compute Canada and\nAdvanced Research Computing at the University of British Columbia.\nAN is supported by the Canada First Research Excellence Fund.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}