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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nA full century after the conception of general relativity (GR) the direct observation of gravitational waves (GWs) from merging black holes\nby the advanced LIGO-Virgo network of detectors~\\cite{Abbott:2016blz, TheLIGOScientific:2016pea, Abbott:2016nmj, Abbott:2017vtc, GW170814}\nhas finally opened the door to tests of relativistic gravity in the truly nonlinear strong field regime. Among the  prime objectives of present and\nnear-future GW astronomy is the materialisation of `black hole spectroscopy', that is, the observation of the so-called ringdown signal at the very\nend of the merger and the extraction\/identification of the final black hole's quasinormal modes (QNMs). The power of this method, much like its\natomic physics kin, lies in the fact that GR predicts a unique spectrum of complex QNM frequencies for a given black hole mass and spin and therefore\nthe simultaneous observation of more than two QNMs should, in principle, allow the Kerr hypothesis to be tested \\cite{Detweiler:1980gk,Dreyer:2003bv,Berti:2005ys}.\n\nAny programme aiming at probing the true nature of black holes should allow for deviations from GR's Kerr spacetime as well as for\ntheoretical input from alternative to GR theories of gravity. Perhaps the simplest `beyond-Kerr' strategy  is to use parametrised schemes (with the parameters controlling\nthe deformation away from GR) both for the black hole's spacetime metric~\\cite{Johannsen:2011dh, Johannsen2013PhRvD,Konoplya:2016jvv,Cardoso:2014rha}\nand the associated QNMs \\cite{Glampedakis:2017, Cardoso_etal2019,McManus_etal2019} without the need to commit to any particular theory\nof gravity.  The main drawback of this approach is that it may constitute nothing more than a null test of the Kerr metric, in the sense that the\ndeformations may not actually map onto any specific gravity theory. The alternative, more rigorous (and far more laborious) strategy is the\ntheoretical calculation of black hole spacetimes and their GW signature on a case-to-case basis within the zoo of modified gravity theories.\nNot surprisingly, this second approach is much more difficult to implement and as a result QNMs of non-GR black holes have been computed\nonly for a handful of cases, usually under the assumption of spherical\nsymmetry, e.g.~\\cite{molina2010, Kobayashi:2012kh, Kobayashi:2014wsa,Blazquez-Salcedo:2016enn, Blazquez-Salcedo:2016enn,Blazquez-Salcedo:2017txk, Brito:2018hjh, Tattersall_2018a, Tattersall_etal2018}\n(for a comprehensive review and further references see~\\cite{Bertietal2015}).\n\nIn this paper we study QNMs of black holes beyond GR by combining elements of the two aforementioned approaches.\nWe assume \\emph{spherically symmetric} black holes in which case, after separating out the angular dependence,  the wave dynamics\nand QNMs of the system are described by a set of radial wave equations for the perturbed spacetime metric (the tensorial field)\nand the other fields that are generically present in modified theories of gravity. In some instances one of the two symmetry sectors of the tensorial field\n(axial\/odd of polar\/even modes) couples to the other fields while the remaining one is described by the same Regge-Wheeler or Zerilli equation\nas in GR. Focusing, for obvious reasons, on the \\emph{coupled} case, we adopt a largely theory-agnostic approach and postulate\na pair of wave equations for the tensorial and the extra `scalar' field with parametrised potentials.\nAlthough far from representing the most general situation~\\cite{Tattersall_etal2018}, our parametrised model provides a useful benchmark\nfor describing perturbed non-GR black holes; in addition it has the merit of including as a special case at least a pair of modified theories of gravity,\nnamely, dynamical Chern-Simons gravity~\\cite{molina2010} and the sixth-order Proca theory~\\cite{Heisenberg:2017hwb,Tattersall_etal2018}.\n\n\nOur main main tool -- and second main simplification -- is the use of the \\emph{eikonal} (or geometric optics) approximation\nfor obtaining QNM solutions from the wave equations. The eikonal approximation  has a long and successful history in the study of\nSchwarzschild and Kerr black holes, dating back to the early 1970s~\\cite{Press:1971wr, Goebel:1972}\n(see~\\cite{Kokkotas:1999bd, Berti:2009kk} for reviews on the subject). According to the established eikonal picture, the fundamental QNM\ncan be visualised as a wavepacket localised in the radial direction at the peak of the wave potential;  in this approximation the peak itself coincides\nwith the location of the photon ring of null geodesics. The real part of the QNM frequency is found to be an integer multiple of the orbital angular frequency\nat the photon ring. Similarly, the decay rate (imaginary part) of the QNM is related to the Lyapunov exponent of the unstable null orbits at the photon ring\nradius~\\cite{Ferrari:1984zz, Mashhoon:1985cya,Cardoso:2008bp}. The same intuitive eikonal model has been used to establish a\nconnection between the $\\ell > |m|$ QNMs of Kerr black holes (where $\\ell, m$ are the usual spherical harmonic integers) and the nonequatorial\nspherical photon orbits~\\cite{Dolan:2010wr, Yang:2012he}. More recently, and inspired by the QNM-photon ring relation,\nan eikonal post-Kerr parametrised scheme was developed as a model for the fundamental $\\ell=|m|$ mode of non-GR black holes~\\cite{Glampedakis:2017}.\nAlthough this model can describe the astrophysically more interesting case of rotating black holes it also has the drawback that it does not\naccount for any extra field degrees of freedom. On the same topic of non-GR black holes it should be noted that some recent work has criticised\nthe validity of the connection between QNMs and photon geodesics~\\cite{Khanna:2016yow,Konoplya:2017wot}, although the models and arguments\nused in these papers are far from being conclusive.\n\nThe purpose of the eikonal calculation presented in this paper is therefore twofold. First, by including a coupling to an extra field,\nwe develop a parametrised eikonal QNM model that surpasses in rigorousness that of Ref.~\\cite{Glampedakis:2017}\n(although in doing so we restrict ourselves to the less physically interesting case of  spherical symmetry). Second, we examine to what\nextent these novel eikonal formulae preserve some connection between the fundamental QNM and photon geodesics.\nOur implementation of the eikonal approximation differs in one key aspect with respect to what is\nusually done in the context of GR black holes. In this latter case, the eikonal approximation consists in\ntaking the angular limit $\\ell \\gg 1$  in the Bohr-Sommerfeld formula originating from the WKB-approximated radial\nwave equations (for example, see \\cite{Berti:2014bla}). As no WKB approximation appears to exist (to the best of our knowledge)\nfor a system of coupled wave equations, we are forced to take the eikonal limit of the equations themselves both in the radial and angular\ndirections. In essence, our approach is very similar to that of Ref.~\\cite{Dolan:2010wr} for Kerr black holes, and this will become\napparent in the discussion of the following section.\n\nThe remainder of this paper is organised as follows. In Section~\\ref{sec:wave_eik}, and before embarking on\nour analysis of coupled wave equations, we study the eikonal limit of a single wave equation, albeit a generalised one that allows\nfor deviations from GR. Section~\\ref{sec:nonGR} contains the main calculation of this paper, namely, the derivation of eikonal formulae\nfor non-GR QNMs described by a pair of coupled wave equations in the context of modified theories of gravity with an extra scalar field degree of\nfreedom besides the standard tensorial one. A summary of these results can be found in Section~\\ref{sec:summary}.\nSection~\\ref{sec:CS} provides an application of our eikonal formulae for the case of QNMs\nof Schwarzschild black holes in dynamical Chern-Simons gravity. Our concluding remarks can be found in Section~\\ref{sec:conclusions}\nwhile the two appendices contain some additional material on photon geodesics and their correspondence to the eikonal limit.\n\nThroughout this paper we adopt geometric units $G=c=1$ and assume an $ e^{-i\\omega t}$ time dependence for the perturbed fields.\nWe use a prime to denote a $d\/dr$ radial derivative. For any function $f(r)$ we use abbreviations  $f_{\\rm ph} = f(r_{\\rm ph}), f_{\\rm max} = f(r_{\\rm max})$, etc.\n\n\n\\section{Generalised wave equation in a spherical black hole spacetime}\n\\label{sec:wave_eik}\n\n\\subsection{Eikonal approximation: leading order}\n\nIn order to set the stage for calculating QNMs of non-GR black holes in the eikonal approximation we first consider the case\nof a single (radial) wave equation describing perturbations of a field $\\psi$ in a background spherical spacetime.\nWe assume an equation,\n\\begin{equation}\n\\frac{d^2 \\psi}{dx^2} + \\left ( \\omega^2 - U  \\right ) \\psi = 0,\n\\label{gen_wave}\n\\end{equation}\nwhere $x(r)$ is a suitably defined tortoise coordinate which maps the black hole horizon (located at $r=r_{\\rm H}$) and infinity ($r=\\infty$)\nonto $x= -\\infty$ and $x=+\\infty$ respectively. The wave potential is assumed to be of the form\\footnote{Single-field wave equations of the\nform (\\ref{gen_wave})-(\\ref{UsinglePot}) can appear in extensions of GR like the higher-dimensional Einstein-Gauss-Bonnet\ntheory~\\cite{Konoplya:2017wot}.},\n\\begin{equation}\nU = f(r) \\left [\\, \\frac{\\ell(\\ell+1)}{r^2} \\alpha(r) - \\frac{6M}{r^3} \\zeta (r) \\, \\right ],\n\\label{UsinglePot}\n\\end{equation}\nwhere $\\ell$ is the familiar angular integer multipole and $f(r)$ a function with asymptotic behaviours $f(r\\to r_{\\rm H}) \\to 0$ and $f(r \\to \\infty)\\to 1$.\nThe functions $\\alpha (r), \\zeta (r)$ are assumed to carry no $\\ell$-dependence but are otherwise unspecified. Crucially, we require\n$U$ to be `black hole-like', that is, with a single peak and $U(x\\to \\pm \\infty) \\to 0 $.\n\nAccording to the eikonal\/geometric optics prescription  we look for wave solutions of (\\ref{gen_wave}) of the form,\n\\begin{equation}\n\\psi (x) = A(x) e^{iS(x)\/\\epsilon},\n\\end{equation}\nwhere $\\epsilon$ is the customary bookkeeping parameter. Then,\n\\begin{eqnarray}\n\\frac{d^2 \\psi}{dx} = e^{iS\/\\epsilon} \\left [\\, A_{,xx} + \\frac{i}{\\epsilon} \\left (\\,  2S_{,x} A_{,x} + S_{,xx} A \\,\\right ) - \\frac{(S_{,x})^2}{\\epsilon^2} A   \\, \\right ],\n\\nonumber \\\\\n\\end{eqnarray}\nand the wave equation becomes,\n\\begin{align}\n    A_{,xx} &+ \\frac{i}{\\epsilon} \\left (\\,  2S_{,x} A_{,x} + S_{,xx} A \\,\\right )\n + \\Big [ \\omega^2  - \\frac{(S_{,x})^2}{\\epsilon^2}\n \\nonumber \\\\\n & - f\\left (\\, \\frac{\\ell(\\ell+1)}{r^2} \\alpha - \\frac{6M}{r^3} \\zeta \\, \\right ) \\Big ] A = 0.\n \\label{full_eik}\n\\end{align}\nThe double limit $\\epsilon \\ll 1$ and $\\ell \\gg 1$ enforces the eikonal limit in the radial and angular directions\nand leads to the following leading-order equation,\n\\begin{equation}\n - \\frac{(S_{,x})^2}{\\epsilon^2} + \\omega^2 - \\ell^2 \\frac{f \\alpha}{r^2}  = 0 + {\\cal O}(\\ell, \\epsilon^{-1}).\n \\label{lead_eik}\n\\end{equation}\nThis expression, reminiscent of a radial Hamilton-Jacobi equation, only makes sense if the two expansion parameters balance,\n$\\epsilon \\ell = {\\cal O}(1)$. The physical picture behind this balance is that of wave packets equally localised in the radial and angular directions.\nThe same reasoning dictates a scaling $\\omega = {\\cal O} (\\ell)$.\n\nFor a QNM wave solution we need to impose the boundary conditions,\n\\begin{equation}\nS (x\\to  \\pm \\infty) = \\pm \\omega x.\n\\end{equation}\nFollowing a reasoning similar to the eikonal analysis of Ref. \\cite{Dolan:2010wr},\nwe expect the phase function $S$ to switch from outgoing ($x >0$) to ingoing ($x<0$) at the location of the potential peak (located at $x\\approx 0$);\nin other words $S$ should have a minimum at that radius.\n\n\nIf we define $\\tilde{U}$ as\n\\begin{equation}\nU(r) \\approx  \\ell^2   \\frac{f \\alpha}{r^2} \\equiv \\ell^2 \\tilde{U} (r), \\qquad (\\ell \\gg 1),\n\\end{equation}\nthe potential peak in the eikonal limit, $r=r_0$, should be\n\\begin{equation}\n\\tilde{U}^\\prime_0 = \\left (  \\frac{f \\alpha}{r^2} \\right )^\\prime_0 = 0,\n\\label{dU0eq}\n\\end{equation}\nwhere a prime stands for a derivative with respect to $r$.\nIn fact, the association of $S_{,x} = 0 $ with the potential peak can be enforced rather than being merely assumed\nby taking the $x$-derivative of (\\ref{lead_eik}):\n\\begin{equation}\n \\frac{2}{\\epsilon^2} S_{,x} S_{,xx} = -  \\ell^2 \\frac{dr}{dx} \\tilde{U}^\\prime ~\\Rightarrow ~  S_{,x} (r_0) = 0.\n\\end{equation}\nThe evaluation of (\\ref{lead_eik}) at the potential peak $r=r_0$ singles out the black hole's fundamental QNM frequency (at leading\neikonal order). We obtain\n\\begin{equation}\n\\omega^2 =  \\ell^2 \\tilde{U}_0 = \\ell^2 \\frac{f_0\\, \\alpha_0}{r^2_0}.\n\\label{omRgen}\n\\end{equation}\nNote that we could have arrived at the same result via the $\\ell \\gg 1$ limit of the standard WKB formula~\\cite{SchutzWill}\nwhich is still applicable for our wave equation with the assumed single-peak potential. In the GR limit of a Schwarzschild spacetime\nwe have $f = 1 -2M\/r, ~ \\alpha =1$, and our result reduces to the correct expression (see e.g.~\\cite{Berti:2009kk}).\n\nWriting the complex QNM frequency as\n\\begin{equation}\n\\omega = \\omega_R + i \\omega_I,\n\\end{equation}\nwe would normally expect $ \\omega_R \\gg |\\omega_I|$ in the eikonal limit as in, e.g. Schwarzschild black holes. In that case\n\\begin{equation}\n\\omega_R  \\approx  \\ell \\, \\tilde{U}_0^{1\/2}.\n\\label{omReik1}\n\\end{equation}\nIn principle, though, the leading-order result (\\ref{omRgen}) should apply equally well for a QNM with the opposite arrangement\n$|\\omega_I| \\gg \\omega_R$ of real and imaginary parts.\n\n\n\\subsection{Eikonal approximation: subleading order}\n\nMoving beyond the leading-order eikonal calculation, we expect  $\\omega_I$ to be determined by the ${\\cal O} (\\epsilon^{-1})$ terms\nof Eq.~(\\ref{full_eik}) evaluated at $r=r_0$.  At this order we should also account for the subleading piece of $\\omega_R$.  That is,\n\\begin{equation}\n\\omega_R = \\omega_R^{(0)} + \\omega_R^{(1)} + {\\cal O} (\\ell^{-1}),\n\\quad  \\omega_R^{(0)} = \\ell \\, \\tilde{U}_0^{1\/2},\n\\end{equation}\nwhere both $\\omega_I,~ \\omega_R^{(1)}$ are ${\\cal O} (1)$ quantities.\nThen,\n\\begin{equation}\n  \\frac{2 i}{\\epsilon} S_{,x} A_{,x} + \\left ( \\frac{i}{\\epsilon} S_{,xx} +2 i \\omega_R^{(0)} \\omega_I  + 2\\omega_R^{(0)} \\omega_R^{(1)}\n - \\ell \\tilde{U} \\right ) A = 0.\n \\end{equation}\n Setting $r=r_0$,\n \\begin{equation}\n\\frac{i}{\\epsilon} S_{,xx} (r_0) +2 i \\omega_R^{(0)} \\omega_I + 2\\omega_R^{(0)} \\omega_R^{(1)} - \\ell \\, \\tilde{U}_0 = 0.\n \\label{sublead_eik}\n\\end{equation}\nTo proceed, we must obtain $S_{,xx}$. By Taylor-expanding Eq.~(\\ref{lead_eik}) around $r_0$ and making use of Eqs.~(\\ref{dU0eq}) and (\\ref{omReik1})\nwe find\n\\begin{equation}\n \\frac{(S_{,x})^2}{\\epsilon^2}  = -\\frac{\\ell^2}{2}   \\tilde{U}^{\\prime\\prime}_0 (r-r_0)^2.\n\\end{equation}\nTaking the positive square root and then differentiating, we get\n \\begin{equation}\n \\frac{S_{,xx}}{\\epsilon}  =  \\frac{\\ell}{\\sqrt{2}}   \\frac{dr}{dx} | \\tilde{U}^{\\prime\\prime}_0 |^{1\/2}.\n \\label{SxxEq1}\n \\end{equation}\nThe imaginary part of (\\ref{sublead_eik}) gives\n\\begin{equation}\n\\omega_I =  -\\frac{1}{2} \\left (\\frac{dr}{dx}\\right )_0  \\sqrt{\\frac{| \\tilde{U}^{\\prime\\prime}_0 |}{2\\tilde{U}_0}}.\n\\label{omIeik1}\n\\end{equation}\nAs in the case of $\\omega_R^{(0)}$, this $\\omega_I$ result could have been derived by taking the $\\ell \\gg 1$\nlimit of a WKB formula.\n\nMoving on, we can see that the real part of (\\ref{sublead_eik}) furnishes the $\\omega_R^{(1)}$ correction,\n\\begin{equation}\n\\omega_R^{(1)} = \\frac{1}{2}  \\tilde{U}_0^{1\/2}.\n\\end{equation}\nThis means that we can write an expression for $\\omega_R$ featuring the famous `$\\ell +1\/2$' Langer factor,\n\\begin{equation}\n\\omega_R = \\left (\\ell + \\frac{1}{2} \\right )  \\tilde{U}_0^{1\/2} + {\\cal O} (\\ell^{-1}).\n\\label{omReik2}\n\\end{equation}\nTogether with Eq.~(\\ref{omIeik1}) this result completes our eikonal approximation analysis of the wave equation (\\ref{gen_wave}).\n\n\n\\subsection{An eikonal QNM-photon geodesics connection?}\n\\label{sec:eik_geod}\n\nAs already discussed in some detail, the eikonal analysis of Schwarzschild (or Kerr) QNMs establishes a direct\nconnection between the fundamental QNM associated with the peak of the wave potential and the geodesic photon\nring~\\cite{Ferrari:1984zz,Mashhoon:1985cya,Cardoso:2008bp, Dolan:2010wr, Berti:2009kk}. We can explore the validity\nof this connection within the general setup of the previous section by assuming a Schwarzschild background spacetime\nwith $f=1-2M\/r$.\n\nThe geodesic photon ring at $r_{\\rm ph}=3M$ is the solution of $ 2f= r f^\\prime$ (see Appendix~\\ref{sec:geodesics}\nfor a discussion of photon rings in a general spherical metric).\nOn the other hand, the peak $r_0$ of the eikonal wave potential solves the slightly different equation [see Eq.~(\\ref{dU0eq})]\n\\begin{equation}\nr ( f\\alpha)^\\prime = 2 f \\alpha,\n\\label{dU0eqx}\n\\end{equation}\nand therefore  $r_0 \\neq r_{\\rm ph}$, \\emph{unless} $\\alpha=\\mbox{const.}$ In this case\nwe inevitably face a breakdown of the eikonal QNM-photon ring correspondence [here we note that Ref.~\\cite{Konoplya:2017wot}\nconsiders higher-dimensional wave equations of the type (\\ref{gen_wave}) and arrives at a similar conclusion].\n\nAs it turns out, however, the eikonal $\\omega_R$ does `see' an \\emph{effective} spherical metric with,\n\\begin{equation}\n-g^{\\rm eff}_{tt} = \\frac{1}{ g_{rr}^{\\rm eff}} = f (r) \\alpha (r).\n\\label{geffsingle}\n\\end{equation}\nNull geodesics in this metric obey the radial equation (see Appendix~\\ref{sec:geodesics})\n\\begin{equation}\n(u^r)^2 = 1 - b^2 \\tilde{U} (r)  = V_r (r,b),\n\\end{equation}\nwhere $b$ is the impact parameter. Circular photon orbits are singled out by the two conditions  $V_r =  V^\\prime_r = 0$; these\nlead to the following equation for the `photon ring' radius [see Eq.~(\\ref{rpheqsph2})],\n\\begin{equation}\n r (g_{tt}^{\\rm eff} )^\\prime = 2 g_{tt}^{\\rm eff},\n\\label{rpheqsph2x}\n\\end{equation}\nwhich is identical to (\\ref{dU0eqx}); therefore, the photon ring coincides with the peak $r_0$.\nAt the same time, the photon ring's `angular frequency' $\\Omega_0$ is given by [this is Eq.~(\\ref{Omph}) evaluated at\n$r_{\\rm ph} = r_0$]\n\\begin{equation}\n\\Omega_0 = \\frac{\\sqrt{-g_{tt}^{\\rm eff}(r_0)}}{r_0} = \\frac{(f\\alpha)^{1\/2}_0}{r_0},\n\\label{Omphx}\n\\end{equation}\nand our earlier result (\\ref{omReik1}) becomes $\\omega_R \\approx \\ell \\, \\Omega_0$, i.e. the familiar relation between the\neikonal $\\omega_R$ and the photon ring's angular frequency~\\cite{Ferrari:1984zz,Mashhoon:1985cya,Cardoso:2008bp, Dolan:2010wr, Berti:2009kk}.\n\nEncouraged by this result, the next objective is to examine whether the QNM's correspondence to an effective geodesic photon ring\nextends to the imaginary part $\\omega_I$ via a relation to the photon ring's  Lyapunov exponent.\nThe latter parameter is given by [see Ref.~\\cite{Cardoso:2008bp} and Eq.~(\\ref{Lyap1})]\n\\begin{equation}\n\\gamma^2_0 = -\\frac{1}{2} \\, r^2_0 \\, g_{tt}^{\\rm eff} (r_0)  \\left ( \\frac{g_{tt}^{\\rm eff}}{r^2} \\right )^{\\prime\\prime}_0.\n\\label{Lyap1x}\n\\end{equation}\nWe can then see that (\\ref{omIeik1}) becomes, after using the identification~\\eqref{geffsingle},\n\\begin{equation}\n\\omega_I = - \\frac{1}{2} \\left (\\frac{dr\/dx}{f \\alpha}  \\right )_0 |\\gamma_0 |.\n\\end{equation}\nThis reduces to the standard expression $\\omega_I = - |\\gamma_0 |\/2 $ provided\n\\begin{equation}\n\\frac{dr}{dx} = f \\alpha = -g_{tt}^{\\rm eff}.\n\\end{equation}\nIndeed, this is the same $x$ coordinate required for writing the scalar wave equation $ g_{\\mu\\nu}^{\\rm eff} \\nabla^\\mu \\nabla^\\nu \\Phi =0$\nin the form (\\ref{gen_wave}), see Appendix~\\ref{sec:qnmgeod} for details.\n\nWe can thus conclude that the eikonal QNM of the generalised wave equation (\\ref{gen_wave}), and in terms of the effective metric (\\ref{geffsingle}),\nretains the physical interpretation it enjoys in GR; i.e. it describes unstable null orbits in the photon ring of that metric.\n\nIn a slightly different scenario than the one considered here (see Appendix~\\ref{sec:qnmgeod}), one can show that the eikonal QNM\nof the usual scalar wave equation in a general spherical metric (and assuming that the present\nsection's model applies) is related to the properties of the metric's \\emph{true} photon ring via the same leading-order relations\n$\\omega_R = \\ell \\, \\Omega_{\\rm ph},~\\omega_I =  -|\\gamma_{\\rm ph} |\/2$. In the light of this section's calculation this\nresult should not come as a total surprise given that the eikonal potential of the scalar wave equation (\\ref{radscalar})\nis $-\\ell^2 \\, g_{tt}\/r^2$ in perfect analogy with this section's potential $U = -\\ell^2 \\, g_{tt}^{\\rm eff}\/r^2 $.\n\n\n\n\\section{Eikonal QNM of non-GR black holes}\n\\label{sec:nonGR}\n\n\\subsection{Coupled wave equations in a spherical black hole spacetime}\n\\label{sec:coupled}\n\nThe perturbation theory underpinning the calculation of QNMs of spherically symmetric black holes in modified theories of gravity\ntypically boils down to a coupled system of wave equations for the axial or polar components of the metric-tensor field ($\\psi$) and the\nadditional field degrees of freedom; see for example~\\cite{molina2010, Kobayashi:2012kh, Kobayashi:2014wsa, Blazquez-Salcedo:2016enn,Blazquez-Salcedo:2017txk,\nBertietal2015}. Our own discussion here is meant to be theory independent, but in order to keep this first analysis\nsimple, we assume a system comprising two perturbed fields $\\psi$ and $\\Theta$ (the scalar field, governed by a pair of coupled wave\nequations,\n\\begin{subequations}\n\\begin{align}\n&\\frac{d^2 \\psi}{dx^2} + \\left [\\, \\omega^2 - V_\\psi(r) \\, \\right ] \\psi = \\beta_\\psi (r) \\Theta,\n\\label{waveT}\n \\\\\n& \\frac{d^2 \\Theta}{dx^2} + g(r)  \\frac{d\\Theta}{dx} + \\left [\\, \\omega^2 - V_\\Theta (r) \\, \\right ] \\Theta = \\beta_\\Theta (r) \\psi,\n\\label{waveS}\n\\end{align}\n\\end{subequations}\nwhere $x$ is a common tortoise coordinate. The potential $V_\\psi$ is assumed to be identical to the Schwarzschild's spacetime\nRegge-Wheeler or Zerilli potential~\\cite{ChandraBook} while the scalar potential $V_\\Theta$ is allowed to deviate from GR,\n\\begin{equation}\nV_\\psi = \\{ V_{\\rm RW}, V_{\\rm Z} \\},\n\\quad V_\\Theta = f \\left [ \\frac{\\ell(\\ell+1)}{r^2}\\alpha + \\frac{2M}{r^3} \\zeta  \\right ],\n\\end{equation}\nwith $f=1-2M\/r$. The coupling functions $\\beta_\\psi(r)$,  $\\beta_{\\Theta} (r)$ and the potential functions\n$\\{g(r), \\alpha(r), \\zeta(r)\\}$ are left undetermined, but we expect them to scale with negative powers of $r$ so that\nasymptotic flatness is preserved\\footnote{This requirement implies a massless scalar field.}.\nWe also assume that the $\\ell (\\ell +1)$ factor represents the entire $\\ell$-dependence of the left-hand-side of (\\ref{waveS}).\nThis `reduced' system is modeled after the perturbation equations for the axial tensor-scalar\nperturbations of Schwarzschild black holes in Chern-Simons gravity~\\cite{molina2010}, and it includes them as\na special case.\n\nOur coupled system admits a QNM solution provided the appropriate boundary conditions are satisfied:\n\\begin{equation}\n\\psi (x\\to  \\pm \\infty) \\sim e^{\\pm i\\omega x}, \\quad \\Theta (x\\to  \\pm \\infty) \\sim e^{\\pm i\\omega x}.\n\\end{equation}\nFor this to be possible $\\{ V_\\psi$, $V_\\Theta\\}$ and $\\{\\beta_\\psi, \\beta_{\\Theta}, g\\}$ should vanish at $x = \\pm \\infty$.\nThe first set does indeed comply with this requirement provided $\\alpha$ and $\\zeta$ do not grow faster than $r^2$ and\n$r^3$, respectively. The second set of parameters follows suit if these parameters scale with $f$ and negative powers of $r$.\n\nWith regard to approximation methods for a system like  (\\ref{waveT})-(\\ref{waveS}), and as far as we are aware,\nno WKB formulae appear to exist in the literature. Fortunately, though, the eikonal\/geometric optics approximation is flexible\nenough to be applicable to a coupled system. To this end we assume the eikonal solution ansatz,\n\\begin{equation}\n\\psi (x) = A(x) e^{iS(x)\/\\epsilon}, \\quad \\Theta (x) = B(x) e^{i H(x)\/\\epsilon},\n\\end{equation}\nwhere we have allowed for different phase functions for the two fields.\n\n\nThe best strategy for dealing with the coupled system at hand is to first combine the two equations and subsequently\ntake the eikonal limit of the resulting expression. As in the previous case of the single wave equation, it quickly becomes\nevident that the two expansion parameters should balance, i.e.  $\\epsilon \\ell = {\\cal O}(1)$, and that $\\omega = {\\cal O} (\\ell)$.\nFollowing this recipe, we obtain the following expression after having eliminated the $B$ amplitude between the two wave equations:\n\\begin{align}\n& \\omega^4 -\\omega^2 \\left [\\,   \\ell^2  \\tilde{V}  (1+\\alpha)  + \\frac{1}{\\epsilon^2} \\left \\{ (S_{,x})^2 + (H_{,x})^2 \\right \\} \\, \\right ] -\\beta_{\\psi\\Theta}\n\\nonumber \\\\\n&+  \\ell^2  \\tilde{V}   \\left [  \\alpha \\left \\{ \\ell^2 \\tilde{V}  + \\frac{(S_{,x})^2}{\\epsilon^2} \\right \\}  + \\frac{(H_{,x})^2}{\\epsilon^2}   \\right ]\n+ \\frac{(S_{,x} H_{,x})^2}{\\epsilon^4}\n\\nonumber \\\\\n&   + \\frac{\\beta_\\psi}{A} e^{i(H-S)\/\\epsilon} \\left [ \\left (g + \\frac{2i}{\\epsilon} H_{,x} \\right ) B_{,x} + B_{,xx} \\right ] + 2 \\ell^3 \\tilde{V}^2  \\alpha\n\\nonumber \\\\\n& - \\frac{i}{\\epsilon^3}  \\left [\\, \\left ( g H_{,x}  +  H_{,xx} \\right )  (S_{,x} )^2  + \\left ( \\frac{2 A_{,x}}{A} S_{,x}  + S_{,xx} \\right ) (H_{,x})^2\\, \\right ]\n\\nonumber \\\\\n&- \\frac{i}{\\epsilon} \\ell^2 \\tilde{V} \\Big [\\, \\left ( \\frac{2A_{,x}}{A} S_{,x} +  S_{,xx} \\right ) \\alpha + g H_{,x}  + H_{,xx}   \\, \\Big ]\n\\nonumber \\\\\n&+  \\frac{\\ell \\tilde{V}}{\\epsilon^2} \\left [ \\alpha (S_{,x})^2 + (H_{,x})^2 \\right ]\n- \\omega^2 \\Big [\\, \\ell \\tilde{V} (1+\\alpha)  - \\frac{i}{\\epsilon} \\Big ( g H_{,x}\n\\nonumber \\\\\n& +  H_{,xx}  +  \\frac{2 A_{,x}}{A} S_{,x}  + S_{,xx} \\Big ) \\,\\Big ]  = 0  +  {\\cal O} \\left (\\epsilon^{-2} \\right),\n\\label{waveTSeik}\n\\end{align}\nwhere we have defined $\\beta_{\\psi \\Theta} \\equiv \\beta_\\psi \\beta_\\Theta$ and $\\tilde{V} (r) \\equiv f(r)\/r^2$. This expression includes all\nterms up to ${\\cal O}(\\epsilon^{-3})$ order; all terms beyond that order have been omitted as they will not play any role in the subsequent analysis.\nAt the same time we have retained all $\\beta$-coupling terms as a result of their unspecified eikonal order. From the above equation\nit is clear that $\\beta_{\\psi\\Theta} \\leq {\\cal O} (\\ell^4)$, which is the highest allowed eikonal order consistent with the rest of the terms.\nThe presence of the exponential term would in principle be a cause for concern; assuming a $\\beta_\\psi \\leq {\\cal O} (\\ell^2)$ pushes this\nterm into the group of ${\\cal O} (\\epsilon^{-3})$ subleading order terms where, as we shall see below, it makes no impact to the final results.\nThe same is true for the remaining amplitude-dependent terms.\n\nTaking the eikonal limit,  $\\epsilon \\ll 1$ and $\\ell \\gg 1$, we have at leading order [i.e. the $ {\\cal O} ( \\epsilon^{-4})$ terms of (\\ref{waveTSeik})]\n\\begin{align}\n& \\omega^4 -\\omega^2 \\left [\\, \\ell^2 \\tilde{V}  (1+\\alpha)  + \\frac{1}{\\epsilon^2} \\left \\{ (S_{,x})^2 + (H_{,x})^2 \\right \\} \\, \\right ]\n\\nonumber \\\\\n&+ \\ell^2  \\tilde{V}   \\left [  \\alpha \\left \\{ \\ell^2 \\tilde{V}  + \\frac{(S_{,x})^2}{\\epsilon^2} \\right \\}  + \\frac{(H_{,x})^2}{\\epsilon^2}   \\right ]\n+ \\frac{(S_{,x} H_{,x})^2}{\\epsilon^4} = \\beta_{\\psi\\Theta}.\n\\label{leadEq1}\n\\end{align}\n\n\nThe simplest situation is the one in which  $\\beta_{\\psi \\Theta} \\leq {\\cal O} (\\ell^3)$; in that\ncase the coupling term in (\\ref{leadEq1}) is subdominant with respect to the other terms\nand the system of the two equations effectively decouples. We discuss this case in more detail in the following section.\n\n\n\n\\subsection{The case of decoupled wave equations}\n\nHere we focus on the scenario of  `weak $\\ell$-coupling' where the term $\\beta_{\\psi\\Theta} $ does not appear\nin the leading-order eikonal equation (\\ref{leadEq1}). This obviously includes the trivial case of \\emph{no} coupling\n$\\beta_\\psi = \\beta_\\Theta = 0$. Under these circumstances (\\ref{leadEq1}) becomes the product of two factors,\n\\begin{equation}\n\\frac{(S_{,x})^2}{\\epsilon^2} = \\omega^2 - \\ell^2 \\tilde{V}, \\quad\n\\frac{(H_{,x})^2}{\\epsilon^2} = \\omega^2 -   \\ell^2 \\tilde{V}  \\alpha.\n\\end{equation}\nEach of these equations is a special case of the general wave equation of Sec.~\\ref{sec:wave_eik} and as a\nconsequence we can use the results obtained there.  For the $\\psi$-field we recover the standard Schwarzschild result,\n$\\omega_R \\approx \\ell \\, \\tilde{V}_0^{1\/2} =  \\ell \\, \\Omega_{\\rm ph}$ with $r_0 =r_{\\rm ph}=3M$.\nFor the $\\Theta$-field  we similarly obtain $\\omega_R\\approx \\ell  \\, ( \\tilde{V} \\alpha )_0^{1\/2}$ with a different peak\n$\\tilde{r}_0 \\neq 3M$ associated with the eikonal potential  $\\ell^2 \\tilde{V} \\alpha$ where $H_{,x} =0$.\n\nClearly, due to the presence of the $\\alpha(r)$ function the two eikonal frequencies do not coincide. This implies\nthat the two fields propagate independently, each one with its own QNM frequency.\n\n\n\n\\subsection{Leading-order analysis of the coupled system.}\n\\label{sec:fullcoupled}\n\nThe lesson from the preceding calculation is clear: the existence of a mixed scalar-tensor QNM wave\nwith a single frequency $\\omega$ in the eikonal limit \\emph{requires} the presence of a term\n$\\beta_{\\psi \\Theta} = {\\cal O}(\\ell^4)$ in the leading-order equation~(\\ref{leadEq1}).\n\nThe next step is to assume that both phase functions are simultaneously minimised, $S_{,x}  = H_{,x} = 0$,\nat the \\emph{same} radius $r_{\\rm m}$.  Provided $\\beta_{\\psi\\Theta}$ is $\\omega$-independent, the resulting expression\nis a  biquadratic equation  in $\\omega$:\n\\begin{equation}\n\\omega^4 - \\omega^2 \\ell^2 \\tilde{V}_{\\rm m} ( 1+ \\alpha_{\\rm m})\n+ \\ell^4  \\tilde{V}^2_{\\rm m} \\alpha_{\\rm m}   -  (\\beta_{\\psi \\Theta})_{\\rm m} = 0,\n\\label{coupled5}\n\\end{equation}\nadmitting the pair of roots\n\\begin{align}\n\\omega^2_{\\pm} = \\frac{\\ell^2}{2}  \\left [\\, \\tilde{V}_{\\rm m} (1+\\alpha_{\\rm m})\n\\pm \\sqrt{  \\tilde{V}^2_{\\rm m} (1-\\alpha_{\\rm m})^2 +  4 ( \\tilde{\\beta}_{\\psi \\Theta})_{\\rm m}} \\,  \\right ],\n\\nonumber \\\\\n\\label{coupled6}\n\\end{align}\nwhere $\\tilde{\\beta}_{\\psi \\Theta} = \\beta_{\\psi\\Theta}\/\\ell^4$ is $\\ell$-independent. Apart from its\nself-consistent scaling, $\\omega_{\\pm} = {\\cal O} (\\ell)$, this result also has the correct GR limit\n ($\\alpha \\to 1,~ \\beta_{\\psi \\Theta} \\to 0 $) provided $r_{\\rm m} \\to 3M$.\n\nAs pointed out earlier, the eikonal $\\omega_{\\pm}$ may not necessarily coincide with the QNM with frequency\n$\\omega_R \\gg |\\omega_I |$. In the spirit of our theory-agnostic framework, we should not rule out modes with\nthe opposite arrangement $| \\omega_I| \\gg \\omega_R $ or indeed ones with $| \\omega_I | \\sim \\omega_R$ in the eikonal limit\n[this last scenario requires at least one of the functions $\\alpha, \\beta_\\psi, \\beta_\\Theta$ to be complex-valued and\/or\n$( \\tilde{\\beta}_{\\psi \\Theta})_{\\rm m} < 0 $].\n\n\nThe radius $r_{\\rm m}$ can be calculated by taking the derivative of (\\ref{leadEq1}) and then evaluating it at\n$(r,\\omega)=(r_{\\rm m}, \\omega_\\pm)$,\n\\begin{equation}\n\\frac{\\omega^2_\\pm}{\\ell^2} \\left [  \\tilde{V} ( 1+ \\alpha)  \\right ]^\\prime_{\\rm m} - ( \\alpha  \\tilde{V}^2  )^\\prime_{\\rm m}\n+  (\\tilde{\\beta}_{\\psi \\Theta})^\\prime_{\\rm m} = 0.\n\\label{coupled8}\n\\end{equation}\nThis equation determines $r_{\\rm m}$; in the GR limit  it reduces to $ \\tilde{V}^\\prime =0$ and $ r_{\\rm m} = r_{\\rm ph}$.\n\n\nRemarkably, $r_{\\rm m} $ turns out to be a `peak' (in reality a local minimum or maximum)\nof an \\emph{effective} potential, albeit a frequency-dependent one. This can be easily verified by first defining\n\\begin{equation}\nV_{\\rm eff}(r,\\omega) \\equiv \\ell^2  \\tilde{V} \\left [   \\omega^2 ( 1+ \\alpha)  - \\ell^2   \\tilde{V} \\alpha \\right ] + \\ell^4  \\tilde{\\beta}_{\\psi \\Theta},\n\\label{Veff}\n\\end{equation}\nand then noticing that (\\ref{coupled8}) is equivalent to\n\\begin{equation}\nV_{\\rm eff}^\\prime (r,\\omega)=0, \\quad \\mbox{at}~ (r,\\omega)=(r_{\\rm m},\\omega_\\pm).\n\\label{dVeff}\n\\end{equation}\nWith the effective potential $V_{\\rm eff} (r, \\omega)$ defined in this way, Eq. (\\ref{coupled5}) becomes\n\\begin{equation}\n\\omega^4_\\pm = V_{\\rm eff} (r_{\\rm m}, \\omega_\\pm).\n\\label{omVeff}\n\\end{equation}\nInterestingly, one can define an alternative effective potential which is equivalent to (\\ref{Veff}) in the sense\nthat both potentials satisfy the condition (\\ref{dVeff}) and therefore have the same `peak' $r_{\\rm m}$.\nThis second potential can be produced from (\\ref{Veff}) after changing $\\omega \\to \\omega_\\pm(r)$, where\n$\\omega_\\pm (r)$ is the function of Eq.~(\\ref{coupled6}) with $r_{\\rm m} \\to r$.\nFor the same reason (\\ref{omVeff}) becomes $\\omega^4_\\pm (r) = V_{\\rm eff} (r,\\omega_\\pm (r))$. Then,\n\\begin{equation}\n\\frac{d}{dr}  V_{\\rm eff} (r,\\omega_\\pm (r)) = V_{\\rm eff}^\\prime + V_{\\rm eff,\\omega_\\pm}\\, \\omega^\\prime_\\pm,\n\\end{equation}\nand at the same time,\n\\begin{equation}\n\\\\\n4 \\omega^3_\\pm \\omega^\\prime_\\pm = \\frac{d}{dr}  V_{\\rm eff} (r,\\omega_\\pm (r)).\n\\end{equation}\nThe combination of these two equations implies $V_{\\rm eff}^\\prime =0 \\Leftrightarrow \\omega^\\prime_\\pm =0$,\nthus ensuring the equivalence of the two potentials.\n\n\n\\subsection{The coupled system at subleading eikonal order}\n\\label{sec:coupled_sub}\n\nGoing beyond the leading eikonal order, we assume a `canonical' QNM with $\\omega_R \\gg |\\omega_I | $\nand\n\\begin{equation}\n\\omega = \\omega_R^{(0)} + \\omega_R^{(1)} + i \\omega_I +  {\\cal O} (\\ell^{-1}), \\quad \\omega_R^{(0)} = \\omega_\\pm.\n\\end{equation}\nWe also expand the coupling parameter as\n\\begin{equation}\n\\beta_{\\psi\\Theta} =  \\beta_{\\psi\\Theta}^{(0)} +  \\beta_{\\psi\\Theta}^{(1)}+ {\\cal O}(\\ell^2), \\quad\n \\beta_{\\psi\\Theta}^{(0)} = \\ell^4 \\tilde{\\beta}_{\\psi\\Theta}.\n\\end{equation}\nThe subleading order eikonal equation consists of the ${\\cal O} (\\epsilon^{-3})$ terms of the general expression (\\ref{waveTSeik}).\nAfter setting $r=r_{\\rm m}$, all terms with $S_{,x}$ and $H_{,x}$ factors vanish (including terms with the `frictional' parameter $g$\nand exponentials) and we obtain\n\\begin{align}\n&   2 \\omega_\\pm \\omega_R^{(1)} \\left [\\,  2 \\omega^2_\\pm -  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m}) \\, \\right ]\n-\\ell \\tilde{V}_{\\rm m}\\Big [\\, \\omega_\\pm^2   (1+ \\alpha_{\\rm m} )\n\\nonumber \\\\\n&- 2 \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m}  \\Big ] - (\\beta_{\\psi\\Theta}^{(1)})_{\\rm m}\n+ i \\Big [ \\,  (H_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm -\\ell^2 \\tilde{V}_{\\rm m} \\right )\n \\nonumber \\\\\n&  + (S_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm -  \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m} \\right )\n+ 2 \\omega_\\pm \\omega_I \\Big \\{\\,  2 \\omega^2_\\pm\n\\nonumber \\\\\n& -  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m}) \\, \\Big \\}  \\, \\Big ]  = 0.\n\\end{align}\nWe decompose this into real and imaginary parts, assuming for simplicity an entirely real $\\beta_{\\psi\\Theta}$,\n\\begin{subequations}\n\\begin{align}\n&   2 \\omega_\\pm \\omega_R^{(1)} \\left [\\,  2 \\omega^2_\\pm -  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m}) \\, \\right ]\n-\\ell \\tilde{V}_{\\rm m}\\Big [\\, \\omega_\\pm^2   (1+ \\alpha_{\\rm m} )\n\\nonumber \\\\\n&- 2 \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m}  \\Big ] - (\\beta_{\\psi\\Theta}^{(1)})_{\\rm m} = 0,\n\\label{subleadEqR}\n\\\\\n\\nonumber \\\\\n& 2 \\omega_\\pm \\omega_I \\left [\\,  2 \\omega^2_\\pm  -  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m}) \\, \\right ]\n+ (H_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm -\\ell^2 \\tilde{V}_{\\rm m} \\right )\n\\nonumber \\\\\n&+ (S_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm -  \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m} \\right ) = 0.\n\\label{subleadEqIm}\n\\end{align}\n\\end{subequations}\nThis is a decoupled system of equations for the subleading frequency corrections $\\{\\omega_R^{(1)},\\omega_I \\}$\nwith roots\n\\begin{align}\n& \\omega_I =  \\frac{(H_{,xx})_{\\rm m} \\left ( \\omega^2_\\pm  -\\ell^2 \\tilde{V}_{\\rm m} \\right )\n+ (S_{,xx})_{\\rm m} \\left [ \\omega^2_\\pm - \\ell^2 \\tilde{V}_{\\rm m}  \\alpha_m  \\right ]}\n{2\\omega_\\pm \\left [\\,  \\ell^2 \\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m} ) -2 \\omega^2_\\pm \\,\\right ]},\n\\label{omIroot1}\n\\\\\n\\nonumber \\\\\n& \\omega_R^{(1)} = \\frac{ \\ell \\tilde{V}_{\\rm m} \\left [\\, \\omega^2_\\pm (1+\\alpha_{\\rm m})\n- 2\\ell^2 \\tilde{V}_{\\rm m}^2 \\alpha_{\\rm m}  \\, \\right ] + (\\beta^{(1)}_{\\psi\\Theta})_{\\rm m} }{2\\omega_\\pm \\left [\\, 2 \\omega^2_\\pm\n-  \\ell^2\\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m} ) \\,\\right ]}.\n\\label{omR1root1}\n\\end{align}\nIn order to calculate the second derivatives $\\{ S_{,xx}, H_{,xx}\\}$ we rewrite (\\ref{leadEq1}) in the equivalent form\n\\begin{align}\n& \\omega^4 -V_{\\rm eff} (r,\\omega)  - \\frac{(S_{,x})^2}{\\epsilon^2}  \\left ( \\omega^2 - \\ell^2 \\tilde{V} \\alpha \\right )\n \\nonumber \\\\\n& -\\frac{(H_{,x})^2}{\\epsilon^2} \\left (  \\omega^2 - \\ell^2 \\tilde{V} \\right ) + \\frac{(S_{,x} H_{,x})^2}{\\epsilon^4} = 0,\n\\end{align}\nand then expand around $r=r_{\\rm m}$. With the help of\n\\begin{align}\nS_{,x} (r) &\\approx \\left ( \\frac{dx}{dr} \\right )_{\\rm m} (S_{,xx})_{\\rm m} (r-r_{\\rm m}),\n\\\\\nH_{,x} (r) &\\approx  \\left ( \\frac{dx}{dr} \\right )_{\\rm m} (H_{,xx})_{\\rm m} (r-r_{\\rm m}),\n\\\\\nV_{\\rm eff} (r,\\omega_\\pm) & \\approx \\omega^4_\\pm  + \\frac{1}{2} V_{\\rm eff}^{\\prime\\prime}  (r_{\\rm m}, \\omega_\\pm) (r-r_{\\rm m} )^2,\n\\end{align}\nwe obtain at leading order in $r-r_{\\rm m}$,\n\\begin{align}\n& \\frac{(S_{,xx})^2_{\\rm m}}{\\epsilon^2} \\left ( \\omega^2_\\pm -\\ell^2 \\tilde{V}_{\\rm m}  \\alpha_{\\rm m} \\right )\n + \\frac{(H_{,xx})^2_{\\rm m}}{\\epsilon^2} \\left ( \\omega^2_\\pm - \\ell^2 \\tilde{V}_{\\rm m} \\right )\n\\nonumber \\\\\n& =  - \\frac{1}{2}   \\left ( \\frac{dr}{dx} \\right )_{\\rm m}^2 V_{\\rm eff}^{\\prime\\prime}  (r_{\\rm m}, \\omega_\\pm).\n\\label{coupled9}\n\\end{align}\nAs it stands, this expression cannot be manipulated further unless we assume a relation between the\nderivatives of the phase functions. The simplest choice is to set $ (S_{,xx})^2_{\\rm m} = (H_{,xx})^2_{\\rm m}$;\nthis would clearly be the case if the phase functions were equal (up to a constant)\\footnote{The equality between the phase functions\nwould imply a common wave propagation speed for the two fields.}, i.e. $S_{,x} =H_{,x}$. With this assumption (\\ref{coupled9}) becomes\n\\begin{align}\n& \\left [ 2 \\omega^2_\\pm  -\\ell^2 \\tilde{V}_{\\rm m} (1+  \\alpha_{\\rm m} )  \\right ] \\frac{(S_{,xx})^2_{\\rm m} }{\\epsilon^2}\n=  \\pm \\ell^2 \\Big [  \\tilde{V}_{\\rm m}^2 (1-  \\alpha_{\\rm m} )^2\n\\nonumber \\\\\n& +  4 ( \\tilde{\\beta}_{\\psi \\Theta} )_{\\rm m}   \\Big ]^{1\/2} \\frac{(S_{,xx})^2_{\\rm m} }{\\epsilon^2}\n= -\\frac{1}{2}  \\left ( \\frac{dr}{dx} \\right )_{\\rm m}^2 V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ).\n\\label{coupled10}\n\\end{align}\nAssuming a potential peak, $V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) < 0 $, it is always possible to choose the sign\nof the square root so that $(S_{,xx})_{\\rm m} >0$. We then have\n\\begin{align}\n\\frac{(S_{,xx})_{\\rm m}}{\\epsilon} &=   \\frac{1}{\\sqrt{2} \\ell} \\left (\\frac{dr}{dx} \\right )_{\\rm m}\n | V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) | ^{1\/2}\n \\nonumber \\\\\n&\\quad \\times \\left [ \\tilde{V}_{\\rm m}^2 (1-\\alpha_{\\rm m})^2 + 4 ( \\tilde{\\beta}_{\\psi\\Theta} )_{\\rm m}   \\right ]^{-1\/4},\n\\end{align}\nand solution (\\ref{omIroot1}) for $\\omega_I$ becomes\n\\begin{align}\n\\omega_I & = - \\frac{(S_{,xx})_{\\rm m}}{2 \\omega_\\pm}\n\\nonumber \\\\\n&= -\\frac{(dr\/dx)_{\\rm m}}{2 \\sqrt{2} \\omega_\\pm \\ell}  \\frac{  | V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) |^{1\/2}}\n{ \\left [\\,  \\tilde{V}_{\\rm m}^2 (1-\\alpha_{\\rm m})^2 + 4 ( \\tilde{\\beta}_{\\psi \\Theta} )_{\\rm m}  \\, \\right ]^{1\/4} }.\n\\label{omIroot2}\n\\end{align}\n\n\n\n\n\n\\subsection{Summary of eikonal formulae}\n\\label{sec:summary}\n\nHere we recap the eikonal results of this section for the complex QNM frequency $\\omega = \\omega_R + i \\omega_I$\nof the coupled system (\\ref{waveT}) and (\\ref{waveS}).\nWe have\n\\begin{equation}\n\\omega = \\omega_R^{(0)} + \\omega_R^{(1)} + i \\omega_I +  {\\cal O} (\\ell^{-1}),\n\\end{equation}\nwith\n\\begin{align}\n \\omega_R^{(0)}  &= \\omega_{\\pm} = \\frac{\\ell}{\\sqrt{2}}  \\Big [\\, \\tilde{V}_{\\rm m} (1+\\alpha_{\\rm m})\n \\nonumber \\\\\n&\\quad\\quad\\quad\\quad \\pm \\sqrt{  \\tilde{V}^2_{\\rm m} (1-\\alpha_{\\rm m})^2 +  4 ( \\tilde{\\beta}_{\\psi \\Theta})_{\\rm m}} \\,  \\Big ]^{1\/2},\n\\label{omR0_final}\n\\\\\n\\nonumber \\\\\n\\omega_R^{(1)} &= \\frac{ \\ell \\tilde{V}_{\\rm m} \\left [\\, \\omega^2_\\pm (1+\\alpha_{\\rm m})\n- 2\\ell^2 \\tilde{V}_{\\rm m}^2 \\alpha_{\\rm m}  \\, \\right ] + (\\beta^{(1)}_{\\psi\\Theta})_{\\rm m} }{2\\omega_\\pm \\left [\\, 2 \\omega^2_\\pm\n-  \\ell^2\\tilde{V}_{\\rm m} (1+ \\alpha_{\\rm m} ) \\,\\right ]},\n\\label{omR1_final2}\n\\end{align}\nfor the real part and\n\\begin{equation}\n\\omega_I = -\\frac{(dr\/dx)_{\\rm m}}{2 \\sqrt{2} \\omega_\\pm \\ell}  \\frac{  | V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) |^{1\/2} }\n{ \\left [\\,  \\tilde{V}_{\\rm m}^2 (1-\\alpha_{\\rm m})^2 + 4 ( \\tilde{\\beta}_{\\psi\\Theta} )_{\\rm m}  \\, \\right ]^{1\/4} },\n\\label{omI_final}\n\\end{equation}\nfor the imaginary part. For the coupling functions we have assumed $\\beta_\\psi \\leq {\\cal O} (\\ell^2)$\nand that $\\beta_{\\psi\\Theta} = \\beta_\\psi \\beta_\\Theta $ is real-valued, $\\omega$-independent and with an $\\ell \\gg 1$\nexpansion $ \\beta_{\\psi\\Theta} = \\ell^4   \\tilde{\\beta}_{\\psi\\Theta} + \\beta_{\\psi\\Theta}^{(1)} + {\\cal O}(\\ell^2) $.\nThe potentials appearing in these expressions are given by\n\\begin{align}\n    V_{\\rm eff} (r,\\omega) & =  \\ell^2  \\tilde{V} \\left[   \\omega^2 ( 1+ \\alpha)  - \\ell^2   \\tilde{V} \\alpha \\right] + \\ell^4  \\tilde{\\beta}_{\\psi \\Theta},\n\\label{Veff_final}\n\\\\\n\\tilde{V} (r) &= \\frac{f}{r^2},\n\\end{align}\nand the potential peak radius $r_{\\rm m}$ solves $V_{\\rm eff}^\\prime (r_{\\rm m},\\omega_{\\pm})=0$.\nIt is straightforward to verify that in the GR limit,\n\\begin{equation}\n V_{\\rm eff}^{\\prime\\prime} (r_{\\rm m}, \\omega_\\pm ) \\to 2 \\ell^2 \\left [\\, \\tilde{V}_{\\rm m}^{\\prime\\prime} (\\omega^2_\\pm -\\ell^2 \\tilde{V}_{\\rm m})\n -\\ell^2 (\\tilde{V}^\\prime_{\\rm m})^2 \\, \\right ],\n\\end{equation}\nand $ \\tilde{V}^\\prime_{\\rm m} \\to 0$. Then, the above results reduce to the familiar expressions\n\\begin{align}\n& \\omega_R^{(0)} \\to \\ell \\tilde{V}_{\\rm m}^{1\/2}, \\quad\n\\omega_R^{(1)}  \\to \\frac{1}{2} \\tilde{V}_{\\rm m}^{1\/2},\n\\\\\n&\\omega_I  \\to  -\\frac{1}{2} \\left (\\frac{dr}{dx} \\right )_{\\rm m} \\sqrt{ \\frac{|\\tilde{V}^{\\prime\\prime}_{\\rm m}|}{ 2\\tilde{V}_{\\rm m}} },\n\\end{align}\nwith $r_{\\rm m} \\to 3 M$.\n\n\n\n\\subsection{A `photon ring' for the coupled system}\n\\label{coupled_geod}\n\nThe emergence of $V_{\\rm eff}$ in the eikonal calculation of the preceding sections prompts us to explore the possibility of\na `geodesic' connection to an effective photon ring. The form of $V_{\\rm eff}$  and the correspondence\n$\\ell\/\\omega \\to b$ suggests an effective radial geodesic potential,\n\\begin{equation}\nV_r^{\\rm eff} (r,b) = 1 -  b^2  \\left [\\, \\tilde{V}(1 + \\alpha) - b^2 \\tilde{V}^2 \\alpha -  b^2 \\tilde{\\beta}_{\\psi\\Theta} \\,\\right ].\n\\label{Vreff}\n\\end{equation}\nThe clear dissimilarity to the radial potential (\\ref{pheom}) of null geodesics in a spherical metric suggests that we should\nnot be too optimistic about attaching a geodesic analogy to the eikonal QNM of the coupled system.\n\nA `photon ring' in the potential (\\ref{Vreff}) is defined by\n\\begin{equation}\nV_r^{\\rm eff} = 0, \\quad (V_{r}^{\\rm eff} )^\\prime = 0.\n\\end{equation}\nThe first condition leads backs to $b_\\pm^2 = \\ell^2\/\\omega_{\\rm \\pm}^2 $ and then the second one can be identified with Eq.~(\\ref{coupled8})\nfor $r_{\\rm m}$. In other words, $r_{\\rm m }$ is the photon ring radius of the `geodesic' potential $V^{\\rm eff}_r$.\n\nWe can rewrite the impact parameter result as,\n\\begin{equation}\n\\omega_\\pm  = \\ell \\, b_\\pm^{-1} \\equiv \\ell\\, \\Omega_\\pm,\n\\end{equation}\nin accordance with the definition (\\ref{Omph}) of angular frequency in spherical symmetry.\nThus defined, $\\Omega_\\pm$ represents the effective photon ring's angular frequency.\n\nThis is as far as the geodesic analogy can be pushed.  Although a Lyapunov exponent can be defined for the potential\n(\\ref{Vreff}), we find that it is not related to the $\\omega_I$ given by (\\ref{omI_final}) in the same way as in the\nsingle wave equation case.\nWith our eikonal analysis of the coupled system brought to an end we are now ready to study a specific example of\na modified theory of gravity.\n\n\n\\section{Application: Schwarzschild black holes in Chern-Simons gravity}\n\\label{sec:CS}\n\nChern-Simons gravity (dCS), in its dynamical version, represents an extension of GR achieved by adding a parity-violating term to\nthe standard Einstein-Hilbert action; see \\cite{jackiw2003,alexander2009, yunes2009} for further details. As a result of this modification\nthe gravitational field in this theory acquires a dynamical scalar field degree of freedom in addition to the standard tensorial one.\nAs far as black holes are concerned, the theory admits the Schwarzschild metric as a spherically symmetric vacuum solution with a\nvanishing background scalar field~\\cite{Yunes2008,Cardoso:2009pk}. The polar perturbations of these black holes are described by the\nfamiliar general relativistic Zerilli equation~\\cite{ChandraBook} while the axial sector ($\\psi$) of the perturbations couples to the perturbed\nscalar field ($\\Theta$). This coupling signals the breakdown of isospectrality between the polar and axial QNM sectors; the former modes\nremain the same as in GR while the latter are governed by a system of coupled wave equations of the form (\\ref{waveT}) and (\\ref{waveS})\nwith~\\cite{Cardoso:2009pk, molina2010}\n\\begin{subequations}\n\\begin{align}\n& V_\\psi = V_{\\rm RW} = f \\left [ \\frac{\\ell (\\ell+1)}{r^2} -\\frac{6M}{r^3}\\right ], \\quad g = 0,\n\\\\\n&  \\frac{dr}{dx} = f = 1 - \\frac{2M}{r}, \\quad \\alpha =1 + \\frac{576 \\pi M^2}{\\beta r^6}, \\quad \\zeta =1,\n\\\\\n& \\beta_\\psi = 96  \\pi M \\frac{f }{r^5}, \\quad \\beta_\\Theta =  \\frac{6 M}{ \\beta} \\frac{(\\ell+2)!}{(\\ell-2)!} \\frac{f}{r^5},\n \\end{align}\n\\end{subequations}\nwhere $\\beta$ is the theory's coupling constant (GR is formally recovered in the limit $ M^4 \\beta \\to \\infty$).\nWe can observe that the dependence of these functions on $\\{r,\\ell, f\\}$ is consistent with the constraints discussed\nin Secs.~\\ref{sec:coupled} and \\ref{sec:fullcoupled} in relation to the necessary QNM boundary conditions and\nthe $\\ell$-scaling of $\\beta_{\\psi\\Theta}$.\n\nThe aim of this section is to compare the numerically computed axial tensor-scalar QNMs of dCS black holes\nreported in Ref.~\\cite{molina2010} to the leading-order eikonal formulae (\\ref{omR0_final}) and (\\ref{omI_final}).\nWe limit our discussion to the more physically relevant case\\footnote{A $\\beta < 0$ leads to a negative kinetic energy term\nin the action and as a result the theory is infested with ghostlike instabilities.} $\\beta > 0$ \\cite{molina2010}.\n\nThe eikonal limit of the coupling functions,\n\\begin{equation}\n\\beta_{\\psi\\Theta} = \\frac{576 \\pi M^2}{\\beta} \\frac{f^2}{r^{10}} \\left [ \\ell^4 + 2 \\ell^3 -\\ell^2 + {\\cal O}(\\ell) \\right ],\n\\end{equation}\nallows for easy identification of $\\tilde{\\beta}_{\\psi \\Theta}$ and $\\beta_{\\psi\\Theta}^{(1)}$.\nFor the real part of the eikonal QNM, we find\n\\begin{align}\n\\omega^2_\\pm =  \\frac{\\ell^2 f_{\\rm m}}{r^2_{\\rm m}} \\left[  1 +   \\frac{288 \\pi M^2}{\\beta r_{\\rm m}^6}\n\\left (\\, 1 \\pm  \\sqrt{  1+ \\frac{\\beta r^6_{\\rm m} }{ 144 \\pi M^2} } \\, \\right ) \\right].\n\\nonumber \\\\\n\\label{omCS}\n\\end{align}\nThe expression for $\\omega_I$ is somewhat cumbersome and, as a consequence, is not shown here.\nEquation~(\\ref{coupled8}) for the potential peak radius $r_{\\rm m}$ takes the form\n\\begin{align}\n&  \\Big [ 288 \\pi M^2 \\left ( 4 r_{\\rm m}  -9 M  \\right ) +  \\beta r^6_{\\rm m}   (r_{\\rm m}  - 3 M )  \\Big ]\n\\nonumber \\\\\n& \\times \\Big [ 288 \\pi M^2   + \\beta r^6_{\\rm m}  \\pm 24M \\sqrt{\\pi(144 \\pi M^2  + \\beta r^6_{\\rm m}  )}  \\Big ]\n \\nonumber \\\\\n& - r^{12}_{\\rm m}   (r_{\\rm m} -3M) \\beta^2 = 0.\n\\label{rmCS}\n\\end{align}\nThe fact that the right-hand side of (\\ref{omCS}) is real  implies $\\omega_R \\gg | \\omega_I|$ or\n$| \\omega_I | \\gg \\omega_R$ for the eikonal modes. Considering the strong `anti-GR' coupling limit $ M^4 \\beta \\ll 1$,\nwe find the asymptotic expressions,\n\\begin{equation}\n\\omega_{+} \\approx \\frac{ 2048\\ell }{6561} \\sqrt{\\frac{\\pi}{\\beta}},\n\\quad r_{\\rm m} \\approx \\frac{9}{4} M \\left ( 1 + \\frac{19683}{524288} \\frac{\\beta}{\\pi} \\right ),\n\\end{equation}\nfor the `+ mode' and\n\\begin{equation}\n\\omega_{-} \\approx \\frac{\\ell f_{\\rm m}^{1\/2}}{r_{\\rm m}} \\left [ 1 -12 \\left (\\frac{\\pi M^2}{\\beta r_{\\rm m}^6} \\right )^{1\/2} \\right ],\n\\quad \\frac{\\beta r_{\\rm m}^6}{M^2} \\gg 1,\n\\end{equation}\nfor the `- mode'.  The divergence of $\\omega_+$ as $\\beta \\to 0$ suggests that this mode may not be physically relevant in the strong\ncoupling regime.\n\n\nTable~\\ref{tab:CSqnms} collects numerical QNM data from Ref.~\\cite{molina2010} and our eikonal frequencies\nfor the same values of $\\beta$ and $\\ell$. In all cases Eq.~(\\ref{rmCS}) leads to a unique real solution with $r_{\\rm m} > 2M$.\nStarting from the intermediate coupling regime, $M^4 \\beta \\sim 1$, we find good agreement between the two sets of results\nwith an overall precision ($\\sim 10\\%-20\\%$) which is typical of the eikonal approximation in the low-$\\ell$ regime.\nWe can observe that among the two eikonal modes, $\\omega_{-}$ is the least damped and the one that lies closer to a numerical QNM.\nThe precision of the $\\omega_I$ results can be seen to be far better than that of $\\omega_R$; in the examples\nshown here the agreement can extend to two or three significant digits!\n\n\nMoving towards the GR limit,  $M^4 \\beta \\gg 1$, the two eikonal modes approach each other, while the numerical modes converge to the usual\nQNM frequencies of decoupled gravitational and scalar perturbations in the Schwarzschild spacetime (while, at the same time, $r_{\\rm m} \\to 3M$).\nInterestingly, the eikonal $\\omega_R$ ($\\omega_I$) is seen to converge towards the corresponding component of the gravitational (scalar) QNM frequency.\n\n\\begin{figure*}[htb!]\n\\includegraphics[width=0.49\\textwidth]{PLOTS\/effective_potential_beta_5em1_minus.eps}\n\\includegraphics[width=0.49\\textwidth]{PLOTS\/effective_potential_beta_5em1_plus.eps}\n\\caption{\\emph{The effective potentials for axial tensor-scalar QNMs of Schwarzschild black holes in dCS gravity}.\nWe show the $M^4\\beta=0.5,~\\ell=2$ potentials $V_{\\rm eff} (r, \\omega_\\pm)$ and $V_{\\rm eff}(r, \\omega_\\pm (r))$ defined in Sec.~\\ref{sec:fullcoupled}.\nThe left (right) panel corresponds to the $-$ ($+$) mode solution. The corresponding $r_{\\rm m}$ solutions (cf. Table~\\ref{tab:CSqnms}) mark the common\nextrema of the two potentials. In the right panel, these extrema are both maxima, while in the left panel (see inset), it is a maximum for $V_{\\rm eff} (r, \\omega_{-}(r))$\nand a minimum for $V_{\\rm eff} (r, \\omega_{-})$.\n}\n\\label{fig:VeffCS}\n\\end{figure*}\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{c c c c c c c}\n\\hline\\hline\n$ M^4 \\beta$      & $\\ell$  & $+\/-$  & $ r_{\\rm m} \/M  $  & $ M \\omega_{\\rm eik}  $  & $ M \\omega_{\\rm num} $ & $\\Delta \\omega$ [\\%]  \\\\\n\\hline\n0.005             &  2     & $-$   &  9.96       &  0.133-0.0642$i$    & 0.186-0.0606$i$   & 39.8+5.6$i$ \\\\\n                  &  2     & $+$   &  2.25       &  15.7-0.148$i$     &                    &  \\\\\n                  & 10     & $-$   &  9.96       &  0.665-0.0642$i$    & 0.696-0.0636$i$   & 4.7+0.9$i$\\\\\n                  & 10     & $+$   &  2.25       &  78.3-0.14$i$      &                    & \\\\\n\\hline\n0.04              & 2      & $-$   &  7.24       &  0.178-0.0793$i$    & 0.220-0.0760$i$  & 23.6+4.1$i$  \\\\\n                  & 2      & $+$   &  2.25       &  5.55-0.148$i$     &                     \\\\\n                  & 7      & $-$   &  7.24       &  0.624-0.0793$i$    & 0.662-0.0687$i$ & 6.1+13.4$i$   \\\\\n                  & 7      & $+$   &  2.25       &  19.4-0.148$i$     &                     \\\\\n\\hline\n0.5               & 2      & $-$   &  5.07       &  0.244-0.0936$i$    & 0.276-0.0936$i$  & 13.1+0$i$ \\\\\n                  &        & $+$   &  2.26       &  1.620-0.144$i$     & 1.97-0.144$i$   & 21.6+0$i$ \\\\\n\\hline\n1                 & 2      & $-$   &  4.65       &  0.263-0.0959$i$    & 0.292-0.0971$i$  & 11.0+1.3$i$  \\\\\n                  &        & $+$   &  2.28       &  1.183-0.141$i$     & 1.43-0.142$i$   & 20.9+0.7$i$   \\\\\n\\hline\n100               & 2      & $-$   &  3.23       &  0.359-0.0968$i$   & 0.367-0.092$i$    & 2.2+4.9$i$ \\\\\n                  &        & $+$   &  2.77       &  0.421-0.0976$i$    & 0.501-0.0954$i$  & 19.0+2.3$i$   \\\\\n\\hline\n$10^4$            & 2      & $-$   &  3.02       &  0.382-0.0962$i$    & 0.374-0.0889$i$  & 2.1+7.6$i$  \\\\\n                  &        & $+$   &  2.98       &  0.388-0.0962$i$    & 0.484-0.0968$i$  & 24.7+0.6$i$  \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\emph{Axial tensor-scalar QNMs of a Schwarzschild black hole in dCS gravity}.  We show\neikonal results ($M \\omega_{\\rm eik}$) corresponding to the modes $\\omega_{\\pm}$\ncomputed with the help of Eqs.~(\\ref{omI_final}), (\\ref{omCS}) and (\\ref{rmCS}) against\nnumerical data ($M \\omega_{\\rm num}$)~\\cite{molina2010,molina_private} for a variety of coupling strengths $M^4 \\beta$ and multipoles $\\ell$.\nThe relative error $\\Delta \\omega = | [ (\\omega_{\\rm eik})_{R,I} -(\\omega_{\\rm num})_{R,I}]\/(\\omega_{\\rm eik})_{R,I}| $ is shown as a complex\nnumber in the last column. We also tabulate the radial location $r_{\\rm m}$ of the `peak' of the effective potential $V_{\\rm eff}$, as obtained\nfrom (\\ref{rmCS}).}\n\\label{tab:CSqnms}\n\\end{table}\nIn the opposite limit of strong coupling, $M^4 \\beta \\ll 1$, Ref.~\\cite{molina2010} finds that the black hole's ringdown is dominated\nby nonoscillatory QNMs. At first glance, this seems to be at odds with the eikonal formulae's prediction of oscillatory modes\n(no matter which of $\\omega_R, \\omega_I$ is the dominant component). As it turns out, however, these nonoscillatory modes do not\nrepresent the black hole's fundamental mode. Indeed, nonoscillatory modes also appear in the late part of the intermediate regime\n($M^4 \\beta \\sim 1$) time evolutions, immediately after the ringdown of the fundamental mode, see Fig. 2 in~\\cite{molina2010}.\nAs $\\beta$ is reduced,  the nonoscillatory mode emerges at an earlier stage thus dominating most of the time domain signal.\nA recent unpublished (and preliminary) QNM calculation~\\cite{molina_private} reveals the presence of oscillatory modes\nin the time domain signal of perturbed, strong coupling regime black holes; this are the data displayed in the top two rows of\nTable~\\ref{tab:CSqnms} for $M^4 \\beta = \\{0.005, 0.04\\}$.  The $\\ell=2$ eikonal frequency exhibits a deteriorated precision\n($\\sim 20\\% - 40 \\%$) with respect to the precision in the $M^4 \\beta \\gtrsim 1$ regime. On the other hand, the accuracy of the\neikonal damping rate is much better ($\\sim 5 \\%$). As expected, the accuracy of the eikonal results gets better for the\nhigher $\\ell$ multipoles. It is also likely that the overall precision of the eikonal approximation in the $M^4 \\beta \\ll 1$ regime\nmay get better when more accurate numerical QNM results become available.\n\n\nBesides the frequencies themselves, it is interesting to study the form of the effective potential $V_{\\rm eff}$ as defined in two\nequivalent ways at the end of Sec.~\\ref{sec:fullcoupled}. Figure~\\ref{fig:VeffCS} displays a typical example of this potential for the\n$M^4 \\beta=0.5$, $\\ell=2$ QNMs appearing in Table~\\ref{tab:CSqnms}. The shape of the potential depends on which definition we adopt, namely,\n$V_{\\rm eff} (r,\\omega_\\pm)$ or $V_{\\rm eff} (r,\\omega_\\pm (r))$. In this particular example, three out of the four potentials are found to be\nblack hole-like, with a single hump located at the same $r_{\\rm m}$ for a given mode. In contrast, the remaining fourth potential has a local\nminimum at $r_{\\rm m}$ and two maxima at different radii (see inset in Fig.~\\ref{fig:VeffCS}). In all cases $V_{\\rm eff} \\to 0$ as $r\\to \\{2M, \\infty\\}$\nin agreement with the QNM boundary conditions. Moreover, by taking the $M^4 \\beta \\to \\infty$ limit in (\\ref{Veff_final}),\nthe effective potential converges to $\\omega^2 \\ell^2 \\tilde{V}$.\n\n\n\n\n\\section{Concluding remarks}\n\\label{sec:conclusions}\n\nThe main results of this paper, Eqs.~(\\ref{omR0_final})--(\\ref{omI_final}), represent the leading-order\neikonal QNM complex frequency for black hole perturbations described by the general coupled system of\nwave equations (\\ref{waveT}) and (\\ref{waveS}). Furthermore, this eikonal mode (which is an approximation\nto the black hole's fundamental mode) can be associated with the peak of a single effective potential, Eq.~(\\ref{Veff_final}),\nin much the same way as it happens for the eikonal modes of Schwarzschild or Kerr black holes in GR~\\cite{Berti:2009kk}.\nAs a performance benchmark of our results, we have computed eikonal modes of Schwarzschild black holes in dCS gravity and found\nthem to be in good agreement with numerically computed data~\\cite{molina2010, molina_private}.\n\n\n\nThe strength of the eikonal method lies in the analytic form of its results and the present study is no exception to the rule.\nAlthough the QNM spectrum of spherical black holes in a given modified theory of gravity could, in principle, be obtained by means\nof direct numerical integration, our eikonal formulae have the key merit of describing the fundamental mode of black holes in a\nparametrised `post-GR' form and explicitly displaying the mode's dependence on the coupling parameters that generically\nappear in modified theories of gravity.\nIn addition, our framework is sufficiently general to cover background black hole solutions which are either\ngiven analytically or numerically.\n\nOur exploration of the connection between the eikonal QNMs and photon geodesics has been partially successful.\nWe have been able to identify an effective metric and photon ring that can be mapped on the QNM of a general class\nof single-field wave equations. The more complicated (and physically relevant) coupled system of equations has only allowed us to\nassociate an effective potential and its peak to the eikonal QNM. However, the connection of this potential to the true photon\ngeodesics of a given theory is an open question.\n\nSeveral gravity theories lead to more complex perturbation systems than the one considered here, featuring\nmore than two wave equations\/perturbed fields (including massive scalar fields) and\/or higher-order derivative terms (see for\nexample~\\cite{Blazquez-Salcedo:2016enn, Blazquez-Salcedo:2017txk, Brito:2018hjh, Tattersall_2018a, Tattersall_etal2018}).\nIn principle, the eikonal scheme of this paper should be adaptable to these more general scenarios at the cost of an increased\nalgebraic complexity. An equally important extension of this work -- and one that could lift our capability of testing the Kerr hypothesis\nwith GW observations beyond the level of null tests -- would be towards the study of rotating black holes, perhaps initially\nwithin a slow-rotation approximation. These are all important issues that need to be addressed in future work.\n\n\n\\acknowledgements\n\nWe are grateful to Carlos Molina for providing us with black hole QNM data in dCS gravity.\nWe also wish to thank Caio F. B. Macedo for helpful feedback on our paper.\nK.G. acknowledges networking support by the COST Actions GWverse CA16104 and PHAROS  CA16214.\nH.O.S. acknowledges financial support through NASA Grants No.~NNX16AB98G and No.~80NSSC17M0041.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this article, we describe a new mechanism for obtaining a scalar that is parametrically lighter than the UV cutoff of the \ntheory.  This mechanism involves a discrete $\\mathbb{Z}_N$ symmetry that is non-linearly realized on a scalar as a shift symmetry\nand manifests as an exchange symmetry on the $N$ copies of particles with which it interacts~\\footnote{Due to the presence of additional discrete symmetries, the $\\mathbb{Z}_N$ symmetry may be enhanced to a $\\mathbb{D}_N$ symmetry.}.  This approach allows one to obtain\na hierarchy between the mass of the scalar and the UV cutoff that is exponential in $N$, meaning that $N=3$ already solves most problems.\n\nWe first consider an explicit example of a Yukawa coupling between a scalar and a fermion.\nThe starting point is a periodic scalar $\\phi$ with period $2 \\pi f$.  $\\phi$ \nhas a spurion $\\epsilon$ which breaks the arbitrary shift symmetry $\\phi \\rightarrow \\phi + \\alpha$ down to \n$\\phi \\rightarrow \\phi + 2 \\pi f$.  Thus $\\phi$ only appears in the Lagrangian as\n\\begin{eqnarray}\n\\epsilon \\sin \\left ( \\frac{\\phi}{f} + \\theta \\right ).\n\\end{eqnarray}\nThis structure can result from a perturbative UV completion where $\\phi$ is the pseudo-goldstone\nof a complex scalar $\\Phi \\sim e^{i \\phi\/f}$.\n\nThe theory also has a discrete $\\mathbb{Z}_N$ symmetry under which $\\phi$ transforms as \n\\begin{eqnarray}\n\\phi \\rightarrow \\phi + \\frac{2 \\pi f}{N}.\n\\end{eqnarray}\nIt is crucial to the mechanism that the spurion of shift symmetry breaking is not invariant under this symmetry.\nNext, we introduce Weyl fermions $\\psi_k$ and $\\psi_k^c$ for $\\phi$ to couple to.  Under the $\\mathbb{Z}_N$ symmetry, the fermions transform as\n\\begin{eqnarray}\n\\psi_k , \\psi_k^c \\rightarrow \\psi_{k+1} ,  \\psi_{k+1}^c,\n\\end{eqnarray}  \nwhere we have identified $\\psi_0^{(c)}$ and $\\psi_N^{(c)}$.\n$\\phi$ and $\\psi$ are coupled in a $\\mathbb{Z}_N$-symmetric manner via the interaction\n\\begin{eqnarray}\n\\label{Eq: interact}\n\\mathcal{L} = \\sum_{k=1}^N \\left ( m_\\psi + \\epsilon \\sin \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k }{N} \\right ) \\right )  \\psi_k \\psi^c_k.\n\\end{eqnarray}\n\nThis Lagrangian describes a scalar $\\phi$ coupled to fermions with a Yukawa \ncoupling $\\sim \\epsilon\/f$.  Thus the natural \nexpectation for the mass of $\\phi$ is \n\\begin{eqnarray}\nm_\\phi \\gtrsim \\frac{\\epsilon \\Lambda}{f}.\n\\end{eqnarray}\nwhere the UV cutoff is $\\Lambda$.\nThis intuition turns out to be incorrect as the scalar described in Eq.~\\ref{Eq: interact} has its potential cancelled to\na very high degree of accuracy.  $N$ insertions of $\\epsilon$ will be needed until a non-trivial potential for $\\phi$ can be\nobtained.\n\nTo see the cancelations in action, we take $N > 2$ and consider the potential induced for $\\phi$ by closing the fermion loop with two insertions of $\\epsilon$:\n\\begin{eqnarray}\nV_{1-loop} \\supset \\sum_{k=1}^N \\epsilon^2 \\sin^2 \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k }{N} \\right ) \\Lambda^2 = \\frac{N}{2} \\epsilon^2 \\Lambda^2.\n\\end{eqnarray}\nThe quadratic divergence to the $\\phi$ mass coming from a fermion $\\psi_k$ has been cancelled by its $N-1$ partners.\nThere is a stronger version of the above cancelation, which is that for all integer $m$ with  $N > m \\geq 0$,\n\\begin{eqnarray}\n\\sum_{k=1}^N \\sin^m \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k }{N} \\right ) &=& \\begin{cases}\n  0 \\qquad \\qquad \\quad \\, \\, \\, m = \\text{odd} \\\\\n  \\frac{N}{2^m} \\frac{m!}{(m\/2)!^2} \\qquad m = \\text{even}\n\\end{cases}\n\\end{eqnarray}\nThus the leading-order UV contribution to the potential of $\\phi$ comes from the higher-dimensional operator with $N$ insertions of the spurion $\\epsilon$.\n\\begin{eqnarray}\nV(\\phi) &\\propto& \\frac{\\epsilon^N}{\\Lambda^{N-4}} \\sum_{k=1}^N \\sin^N \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k }{N} \\right )  \\\\\n&=& \n  (-1)^{ \\lfloor N\/2 \\rfloor} \\frac{N \\epsilon^N}{2^{N-1} \\Lambda^{N-4}} \\cos \\left ( \\frac{N \\phi}{f} -\\frac{\\pi}{2} (N \\% 2) \\right ) .\\nonumber\n\\end{eqnarray}\nThe UV contribution to the mass of $\\phi$ scales as\n\\begin{eqnarray}\nm_\\phi^2 \\sim \\frac{N^3 \\epsilon^N}{f^2 \\Lambda^{N-4}}\n\\end{eqnarray}\nand is exponentially suppressed in $N$.\n\nTo get an intuitive understanding of why the potential is so suppressed, we analyze the potential for $\\phi$ from a symmetry perspective.  Due to the $\\mathbb{Z}_N$ symmetry, the potential for $\\phi$ must be\n$2 \\pi f\/N$ periodic so that we can express it as\n\\begin{eqnarray}\nV(\\phi) = \\sum_k c_k \\sin \\left (\\frac{N k \\phi}{f} + \\theta_k \\right ) .\n\\end{eqnarray}\nWe will see below that under a broad set of assumptions,\n\\begin{eqnarray}\n\\label{Eq: ci}\nc_k \\sim \\epsilon^{N k}.\n\\end{eqnarray}\nso that by taking $\\epsilon$ small and $N$ large, we can parametrically separate the mass of the scalar from the UV cutoff.\n\nThe quick and dirty way to derive that $c_k \\sim \\epsilon^{N k}$ is to note that $e^{i N \\phi\/f} = ( e^{i \\phi\/f} )^N$ so that $N$ insertions of the symmetry-breaking spurion $\\epsilon e^{i \\phi\/f}$ are needed to generate a potential for $\\phi$.  Alternatively, the relation can be\nobtained by considering the potential in frequency space.\nThe interaction frequency of $\\phi$ is $\\omega = 1\/f$ ($ V \\sim \\cos(\\omega \\phi) \\psi \\psi^c$). \nThe frequency associated with the mass term and every other term in the potential for $\\phi$ is $N \\omega = N\/f$ due to the $\\mathbb{Z}_N$ symmetry.  \nIn order\nto construct the high-frequency mass term, $N$ contributions of the lower frequency $\\omega$ are needed.\nEach of these comes with its own factor of $\\epsilon$ giving the scaling shown in Eq.~\\ref{Eq: ci}.\n\nA critical assumption that was made implicitly in the previous discussion is that there are no phase transitions or massless particles as $\\phi$ varies in field space.  \nPhase transitions introduce discontinuities.  By cutting up a $2 \\pi f$ symmetric potential, a $2 \\pi f\/N$ symmetric potential can\nbe easily generated.  This point will be exploited later in the paper to obtain a light Higgs boson.\n\nSince $\\epsilon$ is a dimension-one number, we need to specify what the dimensionless expansion parameter is.\nWhen dealing with UV contributions to the potential, it is clear that the expansion parameter is\n$\\epsilon\/\\Lambda$.  In this case,\n\\begin{eqnarray}\nc_k \\sim \\Lambda^4 \\left ( \\frac{\\epsilon}{\\Lambda} \\right )^{N k} .\n\\end{eqnarray}\n\nIn addition to UV contributions to the potential, there will also be IR contributions.\nAs an example of how the IR contributions behave, consider the coupling to the fermions discussed before in Eq.~\\ref{Eq: interact}.  The effective potential \n will depend on the fermion mass $m_\\psi$ in some manner, e.g. $m_\\psi^4 \\log m_\\psi$.  From this,\n one sees that the expansion parameter for IR contributions to the potential is $\\epsilon\/m_\\psi$ and that\n\\begin{eqnarray}\nc_k \\sim \\Lambda_{IR}^4 \\left ( \\frac{\\epsilon}{m_\\psi} \\right )^{N k} .\n\\end{eqnarray}\nDepending on other IR parameters in the Lagrangian, the IR potential for $\\phi$ can be relatively unsuppressed.  Thus scalars of this type are sensitive to the IR but insensitive to the UV.\n\n\\section{Analytic bounds}\n\nIn the example given in the Introduction, we saw that a scalar could couple strongly to \nmatter yet remain light.  \nIn this section, we extend the previous result to more general situations. We describe\nunder what circumstances the mass of a scalar $\\phi$ is exponentially suppressed in $N$\nand under what circumstances it is only power-law suppressed in $N$.\n\nDue to the $\\mathbb{Z}_N$ symmetry, \nthe potential for $\\phi$ is of the form\n\\begin{eqnarray}\n\\label{Eq: assumption}\nV(\\phi) \\propto \\sum_{k=0}^{N-1} F(\\frac{\\phi}{f} + \\frac{2 \\pi k}{N} ).\n\\end{eqnarray}\nUnlike before, we do not assume the existence of a small coupling $\\epsilon$.  The results \npresented in this section are valid for any choice of the function $F$ and for any value of $N$ but are most useful in the case where $F$ does not depend explicitly on $N$.\n\nThe large-$N$ limit of Eq.~\\ref{Eq: assumption} is easy to understand as the sum is simply a Riemann sum that converges to an integral :  \n\\begin{eqnarray}\nV(\\phi) &\\propto& \\sum_{k=0}^{N-1} F(\\frac{\\phi}{f} + \\frac{2 \\pi k}{N} ) \\nonumber \\\\\n\\label{Eq: Riemann}\n&=& \\frac{N}{2 \\pi} \\int_0^{2\\pi} F(\\theta) d\\theta + \\mathcal{O}(N^0) .\n\\end{eqnarray}\nThe leading-order piece is completely $\\phi$ independent so that the mass of $\\phi$\nis subleading in the large-$N$ limit.  Let us denote the subleading piece that generates the\nmass of $\\phi$ as \n\\begin{eqnarray}\nE_N(F) = \\int_0^{2\\pi} F(\\theta) d\\theta - \\frac{2 \\pi}{N} \\sum_{k=0}^{N-1} F(\\frac{\\phi}{f} + \\frac{2 \\pi k}{N} ).\n\\end{eqnarray}\nThe scaling of the mass of $\\phi$ with $N$ is an issue of estimating the convergence rate of \nthe Riemann sum of $F''$.  Riemann sums of periodic functions are known to converge extremely \nquickly, and we present two useful theorems (the proofs can be found in Section 9.4 of Ref.~\\cite{book}).\n\nThe first theorem is a special case of the Euler-Maclaurin theorem.  If the function $F$ is $2 \\pi$\nperiodic and $2m + 1$ times differentiable, then\n\\begin{eqnarray}\n| E_N(F) | \\leq \\frac{2}{N^{2m+1}} \\left ( \\sum_{j=1}^\\infty \\frac{1}{j^{2m+1}} \\right ) \\int_{0}^{2 \\pi} | F^{(2 m + 1)} (\\theta) | d\\theta . \\nonumber\n\\end{eqnarray}\nPotentials where there are massless particles as $\\phi$ varies will have discontinuities and are not infinitely \ndifferentiable.  In these cases, the mass of $\\phi$ is only power-law suppressed by $N$.\n\n\nThe second theorem can be shown via the residue theorem.  Let $F$ be a $2 \\pi$-periodic function \nthat is also analytic.  Then there exists an open strip, which includes the real axis and the \ncomplex axis from $-i a$ to $i a$ with $a > 0$, upon which $F$ can be extended into a holomorphic, $2 \\pi$-periodic, bounded function with bound $M$.  \nIn this case,\n\\begin{eqnarray}\n| E_N(F) | \\leq \\frac{4 \\pi M}{e^{N a} - 1} .\n\\end{eqnarray}\nPotentials that are infinitely differentiable give\nan exponentially suppressed mass for $\\phi$ even if there is no small parameter in the problem.\n\nThese two theorems demonstrate how much the mass of $\\phi$ can be suppressed for any given\npotential.  In the case where there are massless particles as $\\phi$ varies, the mass of $\\phi$ is suppressed by\nhow differentiable the potential is.  In the case with no \ndiscontinuities, the mass is exponentially suppressed even if there is no small number\nin the problem.\n\n\n\\section{Examples}\n\nIn the example given in the Introduction, we showed how a light scalar (e.g. from a fifth force or dark matter) Yukawa coupled to the SM that naively looks tuned can actually arise naturally.  In this \nsection, we provide two more examples of how this new solution to the Hierarchy Problem can be applied to theories of interest.  The first example is that of a light axion.  \nThis example serves to highlight how different this solution is from other solutions such as supersymmetry, \nwhich cannot make the axion lighter than its QCD contributions.\nThe second example is a solution to the Little Hierarchy Problem.\n\n\\subsection{A light axion}\n\nUnlike the example in the Introduction, the axion~\\cite{Peccei:1977hh,Peccei:1977ur,Weinberg:1977ma,Wilczek:1977pj} does not have a small parameter $\\epsilon$ characterizing its couplings.  \nHowever, the axion potential is analytic so that by the\ntheorems presented before, its mass must be exponentially suppressed in the large-$N$ limit.\nAs before, we have a $\\mathbb{Z}_N$ symmetry with $N$ copies of the SM that are interchanged under the \nsymmetry, and an axion that non-linearly realizes the discrete symmetry.\nThe axion couples to the $N$ different sectors via the coupling\n\\begin{eqnarray}\n\\mathcal{L} = \\sum_k \\left ( \\frac{a}{f} + \\frac{2 \\pi k}{N} + \\theta \\right ) G_k \\tilde G_k .\n\\end{eqnarray}\n\n\nDue to confinement of the $N$ different QCD sectors, there will be a potential for the axion whose leading order contribution is\n\\begin{eqnarray}\nV(a) = &-& m_\\pi^2 f_\\pi^2 \\sum_k \\sqrt{1 - 4 \\frac{m_u m_d}{(m_u + m_d)^2} \\sin^2 \\left ( \\frac{a}{2f} + \\frac{\\pi k}{N} \\right ) } \\nonumber \\\\\n\\label{Eq: axion}\n&+& \\mathcal{O}(m_\\pi^4).\n\\end{eqnarray}\nAs the theta angles are all identical, we have shifted them away.\nWith a bit of algebra, one can show that $\\theta = 2 \\pi k\/N$ for integer $k$ is a minimum (maximum) for $N$ odd (even).\n\nTo use the results of the convergence theorems presented before, we define\n\\begin{eqnarray}\nF(z) = \\sqrt{1 - 4 \\frac{m_u m_d}{(m_u + m_d)^2} \\sin^2 \\left ( \\frac{z}{2} \\right ) } ,\n\\end{eqnarray}\nwhich is holomorphic until the square root of a negative number is taken.  We are thus considering\nthe open strip $(-i a , i a)$ where \n\\begin{eqnarray}\na = \\log ( c + \\sqrt{c^2 - 1} ), \\qquad c = \\frac{(m_u + m_d)^2}{2 m_u m_d} - 1 .\n\\end{eqnarray}\nInserting the measured values of the quark masses, we find that the mass of the axion is bounded to be $\\approx 1\/2^{N\/2}$ smaller than its natural value where we have neglected the subleading terms proportional to powers of $N$.\n\nEq.~\\ref{Eq: axion} contains subleading terms suppressed by more powers of the quark masses.  \nThese terms are also analytic and thus also give exponentially suppressed contributions to the axion mass.  \nThese contributions are also suppressed by at least $2^{N\/2}$.  To understand this scaling behavior,\nwe note that if $m_u = m_d$, there is a first-order phase transition at $\\theta = \\pi$~\\cite{Witten:1980sp}.  If the quark masses are equal, the axion\npotential is non-analytic and thus not exponentially suppressed.  Any non-zero mass difference\nresults in an analytic potential, and the parameter governing the exponential suppression is necessarily $m_u\/m_d \\sim 1\/2$.\nAs the exponential suppression of the mass is due to how far away the potential is from discontinuities, a.k.a. phase transitions, the axion mass\nis necessarily suppressed by at least $1\/2^{N\/2}$.\n\nAside from analytically bounding the mass of the axion in the large-$N$ limit, we also numerically fit the exponential dependence of the axion mass on $N$ and find that\n\\begin{eqnarray}\n\\frac{m_a(N)}{m_a(N=1)} \\sim \\frac{4}{2^{N\/2}}\n\\end{eqnarray}\nThere is good agreement between the analytically-derived limit on the mass and actual mass.\n\nThe price of obtaining an exponentially lighter axion is the linear problem associated with the fact that only one of the $N$ copies of the SM has $\\theta = 0$.  The rest have $\\theta = 2 \\pi k\/N$.  Thus one has traded an exponential fine tuning in the mass for a linear tuning of why we are in the sector with $\\theta = 0$.  This secondary problem may be solvable via other mechanisms.\n\n\n\\subsection{A naturally light Higgs boson} \\label{Sec: Higgs}\n\nSolving the Hierarchy Problem is fundamentally about finding a reason that a Higgs mass of zero is special.  \nThe scalars discussed in this article are sensitive to phase transitions.  As a phase transition occurs when the Higgs mass crosses zero,\nthese scalars should be able to favor a small Higgs mass.\n\nFollowing this train of thought, in this subsection we develop a theory for the modulus of the Higgs mass.  \nA modulus coupling to $N=3$ or $4$ copies of the Higgs bosons can result in one of them\nbeing lighter than what naturalness would otherwise imply by a factor of 10, thereby solving the Little Hierarchy Problem.\nIf the other $N-1$ copies of the Higgs boson had positive masses, then there would only need to be N copies of the Higgs boson with the rest of the SM transforming trivially under the $\\mathbb{Z}_N$.\nIn the model presented below, the other Higgs bosons obtain a negative mass squared so that the entire SM needs to be copied.\n\n\\subsubsection{$N=3$\/$N=4$ case}\n\nConsider a $\\mathbb{Z}_N$ symmetry under which there are $N$ copies of the SM\nand a scalar $\\phi$, which is the modulus of the Higgs mass~\\footnote{In principle, $SU(3)_c \\times U(1)_Y$ could transform trivially under this exchange symmetry,\nbut the resulting light colored and charged particles have been excluded by experiment.}.  For the rest of this section,\n$N$ will be 3 or 4.  \nWe couple $\\phi$ to the Higgs with a shift symmetry-breaking parameter $\\epsilon^2$:\n\\begin{eqnarray}\n\\label{Eq: Higgs}\nV &=& \\sum_k m_{H,k}^2(\\phi) H_k H^\\dagger_k + \\lambda (H_k H^\\dagger_k)^2 ,\\\\\n\\nonumber m_{H,k}^2(\\phi) &=& -m_H^2 + \\epsilon^2 \\cos \\left ( \\frac{\\phi}{f} + \\frac{2 \\pi k}{N} \\right ) .\n\\end{eqnarray}\nFor simplicity, we will take all cross-quartic couplings between the Higgses to be zero, but our results will not depend\non this assumption.  We will take $m_H^2 > 0$ and \n$\\Lambda^2 > \\epsilon^2 \\gtrsim m_H^2 = 3 y_t^2 \\Lambda^2\/8 \\pi^2$.  As discussed before, the UV contribution to the $\\phi$ potential\nwill be suppressed by $\\epsilon^{2N}\/\\Lambda^{2N-4}$.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{3negative.pdf}\n  \\caption{ The $N=3$ tree-level potential when $\\epsilon^2 = 1.3 \\, m_H^2$.  The solid line is the potential with no UV contribution, while the dotted (dashed) line includes the UV generated potential with $\\theta = 0$ ($\\theta = \\pi$) in Eq.~\\ref{Eq: potential}.  If the UV-generated potential has a minimum where the Higgs masses are all negative, then all Higgs masses are at their large natural value.  If the UV-generated potential has a minimum where one of the Higgs masses is positive, then there is a light Higgs.\n    \\label{fig}}\n\\end{figure}\n\nThe Higgs naturally has a phase transition when its mass changes sign.  Thus the potential for $\\phi$ will be sensitive to changes in\nthe sign of the Higgs mass.  To see this effect explicitly, we\nintegrate out the Higgs classically.  Only if the total Higgs mass is negative will the Higgs induce a non-zero tree-level potential for \n$\\phi$.  The tree-level potential for $\\phi$ is\n\\begin{eqnarray}\nV = &-& \\frac{\\epsilon^{2N}}{\\Lambda^{2N-4}} \\cos \\left( N \\frac{\\phi}{f} + \\theta \\right )  \\nonumber \\\\\n\\label{Eq: potential}\n&-& \\sum_k \\frac{m_{H,k}^4(\\phi)}{4 \\lambda} \\quad \\Theta \\left ( -m_{H,k}^2(\\phi) \\right ) .\n\\end{eqnarray}\nBy previous arguments, if all three Higgs masses are negative~\\footnote{Requiring that all three Higgs masses are negative for certain values of $\\phi$ and that one of them becomes positive for other values of $\\phi$ corresponds to the choice that\n$2 m_H^2 > \\epsilon^2 > m_H^2$.},\n then the contribution to the potential from the Higgs is $\\phi$ independent.  However, as soon as some of the Higgs masses become positive, a phase transition occurs and there is a potential for $\\phi$.  \n\n\n\nAn example $N=3$ potential is shown in Fig.~\\ref{fig} for some specific choices of parameters.\nThe preference for small Higgs masses can be seen by considering the Higgs's contribution to the\npotential of $\\phi$.  \nOver some of parameter space, all three Higgs vevs are negative and $\\phi$ does not acquire a potential \nfrom the Higgses.  However, whenever one of the Higgs masses becomes positive, there is no longer \na cancelation and the potential quickly increases.  Thus this contribution to the potential has a \nminimum whenever all of the Higgs masses are negative.\nThis preference for negative Higgs masses is balanced against the $\\epsilon$-suppressed UV contribution to the potential.  Choosing the phase of the UV contribution to favor positive Higgs masses gives a theory where at the minimum of the potential, one of the sectors has a Higgs with a small positive mass.\n\nWhen $N \\gtrsim 3$, the UV contribution becomes subdominant to the 1-loop potential for $\\phi$.\nThe 1-loop Coleman-Weinberg potential gives a potential for $\\phi$ that is of the form\n\\begin{eqnarray}\nV_\\text{1-loop} = \\frac{\\beta}{16 \\pi^2} \\sum_k \\left ( H_k H_k^{\\dagger} \\right )^2 \\log H_k H_k^{\\dagger} \/m_H^2.\n\\end{eqnarray}\nThe sign and value of $\\beta$ is determined by the beta functions at the natural scale of the Higgs masses.\nThe $N=4$ potential including the 1-loop potential ($\\beta = 0.2$) is shown in Fig.~\\ref{fig2}.  \n\n\nThe previous two examples of $N=3$ and $N=4$ gave a small positive Higgs mass, as opposed to the observed small negative Higgs mass.  There are two simple ways of obtaining a small negative Higgs mass.  The first is to introduce a small amplitude but high-frequency sine wave\npotential for $\\phi$,\n\\begin{eqnarray}\nV = \\alpha \\cos \\left ( \\frac{M \\phi}{f} \\right ) .\n\\end{eqnarray}\nThis will introduce additional minima, but can result in a small negative Higgs\nmass  in the {\\it absolute} minimum.  For example, including the 1-loop potential for $N=4$, the\nvalues $\\alpha = -0.01 \\, m_H^4$, $M=36$ and $\\beta = 0.1$ give a negative Higgs mass at the absolute minimum.\nFor this parameter set, the UV cutoff is 10 TeV.\n\nAnother way of obtaining a small negative Higgs mass is to introduce an additional $\\mathbb{Z}_2$ symmetry under which the $N$ copies of the SM are taken to another $N$ copies.  In the limit\nof an exact $\\mathbb{Z}_2$ symmetry, there are two light Higgses with identical small positive masses.  \nThe $\\mathbb{Z}_2$ symmetry is softly broken by giving the two sectors slightly different Higgs masses, $m_H^2$ and $m_H^2 + \\delta^2$.  A small negative Higgs mass can result if $\\delta \\gtrsim 125$ GeV.\nFor example, in the $N=4$ case above, a small $\\delta^2 \\sim 0.02 m_H^2$ and $\\beta = 0.05$ results in two light Higgses each a factor of 10 lighter than the bare mass.  One of the two has a positive mass squared while the other has a negative mass squared.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{4negative.pdf}\n  \\caption{ The $N=4$ potential including 1-loop effects for $\\epsilon^2 = 1.3 \\, m_H^2$ and $\\beta = 0.2$.  At the minimum, one of the four Higgs masses is $\\sim 0.1 \\, m_H$ and is positive.  As the Higgs is a factor of $\\sim 10$ lighter than its natural value, it solves the Little Hierarchy Problem modulo getting the sign of the Higgs mass wrong.\n    \\label{fig2}}\n\\end{figure}\n\nTaking the large-$N$ limit of the above solution to the Little Hierarchy Problem does not produce any parametric\nenhancements, though there are small numerical benefits going beyond $N=3$ or $4$.  The main issue is that requiring that there is only one Higgs mass that is positive\nat a time means that $m_H^2\/\\cos \\left ( \\pi\/N \\right ) > \\epsilon^2 > m_H^2$.  In the large-$N$ limit, this amounts to a $1\/N^2$ tuning.\nEven accepting this tuning, the other major issue is that the 1-loop Coleman-Weinberg potential is not parametrically suppressed.  By explicit calculation, the Coleman-Weinberg potential is only $1\/N^2$ suppressed, which does not result in a small enough Higgs mass.\n\nMany intriguing possibilities arise if the Coleman-Weinberg potential were highly suppressed due to very small beta functions.  Taking $N \\sim 10$ would suppress the UV contributions to a point where this would be a solution to the full Hierarchy problem.  Additionally, the minimum with a negative Higgs mass near 100 GeV could be generated by features in the beta function, e.g. a small increase and decrease in the quartic around 100 GeV.  This offers a unique twist on how conformal dynamics might result in a light Higgs mass.\n\n\\subsubsection{Phenomenology}\n\nThe phenomenology of this solution to the Little Hierarchy Problem will be discussed in detail in future \nwork.  Here we summarize the salient features.\n\nPreferential reheating of the sector with a light negative Higgs mass is built into the model as $\\phi$ couples to the SM \nHiggses through scalar mixing and is exactly the scalar reheating model described in \nNNaturalness~\\cite{Arkani-Hamed:2016rle}.  Since $\\phi$ is naturally lighter than the lightest Higgs, its decays\npreferentially reheat the sector with the lightest Higgs, evading all current cosmological \nconstraints.  Thus if $\\phi$ mediates reheating to the SM, then all cosmological problems are \nnaturally avoided.\n\nAnother feature of this solution is that the only particle required to interact with the Higgs is $\\phi$.  Much like the axion solution to the \nstrong CP problem, this mechanism works for any value of $f$ and a large value of $f$ results in $\\phi$ being very difficult to detect.  \nIn this limit, $\\phi$ shares all of the same\nbenefits and problems as the axion, e.g. care is needed so as not to overclose the universe, but on the flip side, $\\phi$ can provide a dark matter candidate.\n\nThe details will depend on the particular models, but parametrically $m_\\phi \\sim \\text{TeV}^2\/f$ and the mixing with the Higgs scales as $\\sim 10$ TeV\/$f$.\n$\\phi$ with masses down to $\\sim$ 0.1 GeV ($f \\lesssim 10^7$ GeV) are excluded by meson decays and beam dumps (see Ref.~\\cite{Alekhin:2015byh,Flacke:2016szy} for a compendium of constraints).\nHorizontal branch star cooling constrains scalar couplings to electrons and excludes $\\phi$ in the range $10^{13}$ GeV $\\gtrsim f \\gtrsim 10^{10}$ GeV, while fifth-force experiments exclude $f \\gtrsim 10^{17}$ GeV (see Ref.~\\cite{Raffelt:2012sp,Hardy:2016kme} for a compendium of constraints).  These estimates are very rough and detailed constraints will be model dependent.\n\nFinally, if there is a non-zero cross-quartic coupling between the Higgses, then the Higgs can mix with\nthe other Higgses.  Due to the requirement of vacuum stability and the relatively small value of the SM quartic coupling,\nnegative cross-quartics larger than a few percent are excluded.  Positive cross-quartics larger than a few percent are\nalso excluded, as large cross-quartics generally push the theory out of the parameter space where all three Higgses\ncan obtain vevs.  As a result, the mixing between the multiple Higgses is suppressed by $\\approx$ few $\\times \\, 10^{-3}$.\nThe resulting exotic collider signatures are very similar to Twin Higgs models (see e.g. Refs.~\\cite{Burdman:2014zta,Craig:2015pha}) only with much smaller production rates.\nAnother difference is the absence of the $v\/f$ tuning needed to make electroweak symmetry breaking\nwork in Twin Higgs models.\n\n\n\\section{Conclusion}\n\nTo conclude, we briefly compare our new solution to similar solutions to the Hierarchy Problem,\nTwin Higgs~\\cite{Chacko:2005pe} and Little Higgs~\\cite{ArkaniHamed:2001nc}.  In unitary gauge, both Little Higgs and Twin\nHiggs solve the gauge and Yukawa divergences by coupling the Higgs as $\\sin v\/f$ to our sector and as $\\cos v\/f$\nto our partners.  The cancelations are then just a $\\mathbb{Z}_4$ version of the previous arguments, where two copies\nhave been removed because gauge invariance cancels the odd powers of $v$ so that the extra two copies are not needed\nfor the cancelation.  This discrete symmetry solution utilizes a generalization of these sin\/cos identities though in a completely different manner.\n\nThe approach most similar to the one presented in this paper are dimensional deconstructions of an extra dimension where the scalars \nonly pick up a mass non-locally~\\cite{ArkaniHamed:2001ca,ArkaniHamed:2001nc}.  The dimensional deconstruction based theories result in effective field theories that are a \nsubset of our more general approach.  In that language, the scalar is light due to 5D gauge invariance and locality.\nIn contrast, as can be seen from our discrete symmetry based approach, the $N$ sectors do not need to be Higgsed down to the \ndiagonal subgroup (no fifth dimension is required) and no sense of locality is needed in\ntheory space.  Any interaction is allowed as long as the interactions satisfy the $\\mathbb{Z}_N$ symmetry.\n\n\nThe most obvious extension for this approach is to use it on the Higgs boson directly.  There are two challenges for this approach.  The first is that our solution to the Hierarchy Problem suppresses the Higgs quartic.\nThe second is that while gauge charged scalars can be made compact, they cannot be made \nperiodic.  If there were a non-abelian equivalent  of frequency, then there might be a way to solve the quadratic divergences coming from the Yukawa \ncouplings in this manner.\n\nThere is still much to be explored with this new solution to the Hierarchy Problem.  We briefly list a few below:\n\\begin{itemize}\n\\item The $\\mathbb{Z}_N$ solutions to the Little Hierarchy Problem need to be \nexplored both theoretically and phenomenologically.  Ideally there exists a model where the only particles transforming under the $\\mathbb{Z}_N$ symmetry are the Higgs boson and the modulus.  In this case, the top quark would be its own partner.\n\\item UV completions of these theories would also be interesting, since moduli frequently appear in String Theory and very often have discrete symmetries associated with them.\n\\item Many theories of flavor involve discrete symmetries, hence a scalar that realizes the discrete symmetry non-linearly \nmay allow for interesting flavor physics and possibly even explain why the universe has three generations.\n\\item This new solutions allows for fifth forces and scalar dark matter that would otherwise appear tuned.\n\\item The moduli can be dark matter, leading to new and interesting signatures in the early universe and at experiments.\n\\item This mechanism allows for the inflaton to have large couplings and not ruin its flat potential.\n\\end{itemize}\n\nVery optimistically, the sensitivity of this scalar to phase transitions\nleads one to hope that this approach could help solve the cosmological constant problem as well.\nAt the very least, the recent spate of solutions to the Hierarchy Problem~\\cite{Graham:2015cka,Arkani-Hamed:2016rle} demonstrates that there is still much to be learned about naturalness.\n\n\n\n\\section*{Acknowledgements}\n\nA.H. is supported by NSF Grant PHY-1620074, DOE Grant DE- SC0012012 and by the Maryland Center for Fundamental Physics (MCFP).  A.H. thanks Savas Dimopolous, Junwu Huang, and Raman Sundrum for useful discussions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction } \n\nKibble-Zurek mechanism (KZM) evolved from the scenario for defect creation in cosmological symmetry-breaking phase transitions \\cite{K-a, *K-b, *K-c}. As the post-Big-Bang Universe cools, causally disconnected regions must choose broken symmetry vacuum independently. Such random choices lead to topologically nontrivial configurations that survive phase ordering as topological defects. In the cosmological setting average size of the causally connected regions (hence, the average density of defects) is set by the Hubble radius at the time of the transition. This early Universe scenario relies on the speed of light and does not apply to the laboratory phase transitions. However, it was the point of departure for the dynamical theory \\cite{Z-a, *Z-b, *Z-c, Z-d} that employs critical exponents of the transition and the quench time to predict the scaling of the resulting density of defects. KZM was successfully tested using numerical simulations \\cite{KZnum-a,KZnum-b,KZnum-c,KZnum-d,KZnum-e,KZnum-f,KZnum-g,KZnum-h,KZnum-i,KZnum-j,KZnum-k,KZnum-l,KZnum-m} and laboratory experiments in condensed matter systems \\cite{KZexp-a,KZexp-b,KZexp-c,KZexp-d,KZexp-e,KZexp-f,KZexp-g,KZexp-gg,KZexp-h,KZexp-i,KZexp-j,KZexp-k,KZexp-l,KZexp-m,KZexp-n,KZexp-o,KZexp-p,KZexp-q,KZexp-r,KZexp-s,KZexp-t,KZexp-u,KZexp-v,KZexp-w,KZexp-x}. More recently, KZM was adapted to quantum phase transitions \\cite{Polkovnikov2005,QKZ1,QKZ2,d2005,d2010-a, d2010-b}. Theoretical developments \\cite{QKZteor-a,QKZteor-b,QKZteor-c,QKZteor-d,QKZteor-e,QKZteor-f,QKZteor-g,QKZteor-h,QKZteor-i,QKZteor-j,QKZteor-k,QKZteor-l,QKZteor-m,QKZteor-n,QKZteor-o,QKZteor-p,QKZteor-q,QKZteor-r,QKZteor-s,QKZteor-t} and experimental tests \\cite{QKZexp-a, QKZexp-b, QKZexp-c, QKZexp-d, QKZexp-e, QKZexp-f, QKZexp-g,deMarco2,Lukin18} of quantum KZM (QKZM) followed. The recent experiment \\cite{Lukin18}, where a quantum Ising chain in the transverse field is emulated using Rydberg atoms, is fully consistent with the predicted scaling \\cite{QKZ2,d2005}. \n\nThe KZM is often presented in its cartoon version where -- due to the critical slowing down \/ closing of the energy gap -- the dynamics of the system literally freezes-out in the neighborhood of the critical point. Today, as the experiments are able not only to count the final number of defects but can also monitor and probe the state of the system during the transition, it is timely to re-investigate the causally limited spreading of correlations during the putative ``freeze-out'' stage of the evolution. \n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.9\\columnwidth,clip=true]{Fig1.pdf}\n\\vspace{-0.1cm}\n\\caption{  \n{\\bf Adiabatic-impulse-adiabatic view of KZM.} \nLinear ramp crosses the critical point at time $t=0$. \nThe instantaneous transition rate, $\\left|\\dot{\\epsilon}\/\\epsilon\\right| = 1\/|t|$, diverges at the critical point and the relevant energy gap closes like $|\\epsilon|^{z\\nu}$. Consequently, while before $-\\hat t$ the state follows the adiabatic ground state, near the critical point (between $-\\hat t$ and $\\hat t$) its evolution is non-adiabatic.\nThe freeze-out assumes that the state is ``frozen'' at $-\\hat t$ \n-- size of the domains of the nascent phase does not change until $+\\hat t$, where the state starts to ``catch up'' with the Hamiltonian. \nThis version of KZM ignores propagation of the new phase front in the time interval $(- \\hat t, +\\hat t)$. It yields correct scalings, but it does not capture what happens -- for example -- in the paramagnetic-ferromagnetic quantum phase transition in the quantum Ising chain in transverse field \\cite{KZscaling1,Francuzetal}. Nevertheless, it may well be relevant in phase transitions where the conserved order parameter or other causes (localization) impede propagation of phase fronts of the broken symmetry phase.\n}\n\\label{fig:KZcartoon}\n\\end{figure}\n\nIn QKZM a system initially prepared in its ground state is smoothly ramped across a critical point to the other side of the quantum phase transition. A distance from the critical point, measured by a dimensionless parameter $\\epsilon$ controlling a Hamiltonian, can be linearized close to the critical point as\n\\begin{equation}\n\\epsilon(t)=\\frac{t}{\\tau_Q}.  \n\\label{epsilont}\n\\end{equation}\nHere $\\tau_Q$ is a quench time. Initially, far from the critical point, the evolution is adiabatic, and the system follows its adiabatic ground state, see Fig.~\\ref{fig:KZcartoon}. The adiabaticity fails at $-\\hat t$ when the reaction time of the system given by the inverse of the gap becomes slower than the timescale $|\\epsilon\/\\dot \\epsilon| = |t|$ on which the transition is being imposed.\nThe gap closes like $\\Delta\\simeq|\\epsilon|^{z\\nu}$, where $z$ and $\\nu$ are the dynamical and correlation length exponents, respectively. From the equation $|t|\\simeq |t\/\\tau_Q|^{-z\\nu}$ we obtain $\\hat t\\simeq \\tau_Q^{z\\nu\/(1+z\\nu)}$ and the corresponding $\\hat\\epsilon=\\hat t\/\\tau_Q\\simeq\\tau_Q^{1\/(1+z\\nu)}$. In the naive ``freeze-out'' version of the impulse approximation the ground state at $-\\hat\\epsilon$, with a corresponding correlation length\n\\begin{equation}\n\\hat\\xi \\simeq \\tau_Q^{\\nu\/(1+z\\nu)},  \n\\label{hatxi}\n\\end{equation}\nis expected to characterize the state of the system until $+\\hat t$, when the evolution can restart. In this way, $\\hat\\xi$ becomes imprinted on the initial state for the final adiabatic stage of the evolution after $+\\hat t$. Simplistic as it is, the adiabatic-impulse-adiabatic approximation correctly predicts the scaling of the characteristic lengthscale $\\hat\\xi$ and the timescale \n\\begin{equation} \n\\hat t\\simeq \\hat\\xi^z,\n\\label{hatt}\n\\end{equation} \nwith the critical exponents and $\\tau_Q$.\nThey both diverge in the adiabatic limit, $\\tau_Q\\to\\infty$, where they become the unique relevant scales in the KZ scaling ansatz~\\cite{KZscaling1,KZscaling2,Francuzetal}. For instance, a two-point correlation function $C_R(t)$, between two sites separated by a distance $R$, should satisfy\n\\begin{equation} \n\\hat\\xi^{\\Delta} C_R(t) = F\\left(t\/\\hat\\xi^z,R\/\\hat\\xi\\right).\n\\label{CRscaling}\n\\end{equation}\nHere $\\Delta$ is a scaling dimension and $F$ a non-universal scaling function. Eq.~\\eqref{CRscaling} is expected to be accurate in the long-wavelength and low-frequency limit.\nIt is worth to observe here, that the crude adiabatic-impulse-adiabatic approximation is consistent with the scaling hypothesis \\eqref{CRscaling}. However, it implies a particular  (time independent) form of the scaling function $F$.\n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.9\\columnwidth,clip=true]{Fig2.pdf}\n\\vspace{-0.1cm}\n\\caption{  \n{\\bf Sonic horizon view of KZM.} Initially, the correlation length $\\xi$ follows adiabatically the equilibrium healing length that -- in the adiabatic ground state (black) -- diverges at the critical point. Critical slowing down means that the size of the correlation length will begin to lag behind the values dictated by the ground state of the Hamiltonian at about $- \\hat t$. Pre-transition fluctuations reach size $\\hat \\xi$ at that instant and seed subsequent evolution of the system. The new broken symmetry phase is therefore selected by fluctuations in domains if size $\\hat \\xi$ at $-\\hat t$. Broken symmetry spreads within the impulse time interval of $2 \\hat t$ with the velocity $2\\hat v$ in every direction, enlarging the resulting ``sound cone'' to roughly $ 5 \\hat \\xi$ by $\\hat t$. In the freeze-out approximation (blue), after $-\\hat t$ the correlation length freezes, and remains close to the adiabatic correlation length at $-\\hat t$. Both the freeze-out and the sonic horizon views lead to the same scalings, but they result in different estimates of the pre-factors for domain sizes and defect densities.\n}\n\\label{fig:KZreal}\n\\end{figure}\n\n\nAs emphasized already in the early papers, see Ref.~\\onlinecite{Z-a, Z-b, Z-c, Z-d}, the freeze-out is not the complete story, and often not even a good approximation. A simple ``sonic horizon'' argument appealing to causality that goes beyond the impulse approximation is often more accurate. It is illustrated schematically in Fig.~\\ref{fig:KZreal}. As long as the evolution is adiabatic, the rate of growth of the diverging adiabatic correlation length, $\\xi\\simeq|\\epsilon|^{-\\nu}$, is  \n\\begin{equation} \n\\frac{d\\xi}{dt}=\n\\frac{d\\epsilon}{dt}\\frac{d\\xi}{d\\epsilon}=\n\\frac{1}{\\tau_Q}\n\\frac{\\nu}{|\\epsilon|^{\\nu+1}}. \n\\end{equation} \nThis rate diverges at the critical point. Hence there must be time $-\\hat t$ when it exceeds the speed limit set by twice\n\\begin{equation} \n\\hat v\\simeq\\frac{\\hat\\xi}{\\hat t}\\simeq\\tau_Q^{-\\nu(z-1)\/(1+z\\nu)}.\n\\label{hatv}\n\\end{equation} \nThe scaling of $-\\hat t$ obtained in this way is the same as in Eq.~\\eqref{hatt}. \n\nCausality and the KZ velocity $\\hat v$ are also central for the short-cuts to adiabaticity via inhomogeneous KZM. Therein, the external driving field has a smooth position dependence, gradually taking the system across the critical point---one part after another. Velocity of the driven critical front below $\\hat v$ (which in general depends on the shape of the above position dependence) is expected to pave the way to adiabatic dynamics, both for classical\\cite{ inhomo_classical-a, KZnum-c, inhomo_classical-c, inhomo_classical-d, inhomo_classical-e, inhomo_classical-f} and for quantum\\cite{inhomo_quantum-a, *inhomo_quantum-aa, inhomo_quantum-b, *inhomo_quantum-c, inhomo_quantum-d, inhomo_quantum-e, inhomo_quantum-f, inhomo_quantum-g} systems.\n\nIn the QKZM the speed limit is central to the causal argument. It originates from the dispersion of quasiparticles at the critical point: $\\omega\\simeq k^z$. Their speed for a quasimomentum $k$ is $v=d\\omega\/dk\\simeq k^{z-1}$. Between $-\\hat t$ and $\\hat t$ the quench excites quasiparticles with the magnitude of $k$ up to $\\hat k\\simeq \\hat\\xi^{-1}$. The speed of quasiparticles with the largest excited $k$ is therefore $\\hat v\\simeq\\hat k^{z-1}\\simeq\\hat\\xi^{1-z}\\simeq\\hat\\xi\/\\hat t$. When $z\\geq1$, $\\hat v$ is an upper bound on the velocity of quasiparticles. A quench in a translationally invariant system excites entangled pairs of quasiparticles with opposite quasimomenta: $k$ and $-k$. When moving apart, they are spreading correlations across the system. For $z\\geq1$ the rate of correlation spreading is limited by twice the speed $\\hat v$ of the fastest quasiparticles. \n\nIn the crudest version, neglecting in particular dependence of $\\hat v$ on the distance from the critical point, the argument implies that after $-\\hat t$ the correlation range (i.e., the ``sonic cone'') continues to grow at the rate $2\\hat v$ until $+\\hat t$. By this time, the range increases from the initial $\\hat\\xi$ at $-\\hat t$ to a final $\\hat\\xi+2\\hat v \\times 2\\hat t\\approx 5\\hat\\xi$. The growth is roughly five-fold. Even if the $5$ is just a very rough estimate, it shows how much the actual evolution can differ from the impulse approximation. Nevertheless, this argument confirms the role of $\\hat\\xi$ as the key relevant scale of length. With or without the prefactor of $5$, the final correlation length is proportional to $\\hat\\xi$, i.e., it scales with $\\tau_Q$ in the same way as given by Eq.~\\eqref{hatxi}. \n\nFrom another perspective, in the impulse approximation, the adiabatic ground state $|\\psi\\rangle$ corresponding to $-\\hat t$ freezes out as the state of the system. After $-\\hat t$ the adiabatic ground state departs from the frozen state. The frozen state becomes a superposition over adiabatic eigenbasis $|n\\rangle$: $\\sum_n c_n(t) |n\\rangle$, where $c_n(t)=\\langle n|\\psi\\rangle$. As a first step beyond the impulse approximation, we can include approximate dynamical phases: $\\sum_n c_n(t) e^{-i\\omega_n t}|n\\rangle$, where $\\omega_n$ is the adiabatic eigenfrequency at the critical point. In a (non-interacting) translationally invariant system, the eigenstates consist of pairs of excited quasiparticles, $|k,-k\\rangle$, and the eigenfrequencies are sums of $2\\omega_k$. The dynamical phase factors become scrambled -- and the phases begin to appear random -- when the largest of them, $2\\omega_{\\hat k}t\\propto\\hat k^z t$, becomes comparable to $1$ near $\\hat t$. The dephasing begins when the evolution crosses over from the non-adiabatic KZ stage to the post-KZ adiabatic stage. That is when the phases definitely can no longer be ignored, but even before the cross-over the phase factors $e^{-2i\\omega_k t}$ make the quasiparticle phase fronts propagate and let the quasiparticles spread the correlations across the system.\n\nFor $z=1$, when the dispersion is linear in $k$ and the quasiparticles have a definite speed of sound. This effect was termed the quasiparticle event horizon \\cite{EventHorizon}. In the QKZM context, it was considered in Refs. \\onlinecite{KZscaling1,Francuzetal}---see Fig.~\\ref{fig:Ising} for an example of the\nprototypical 1D quantum Ising model.\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\columnwidth]{fig2a.pdf}\n\\vspace{-0.7cm}\n\\caption{ {\\bf Quantum Ising chain.} \nThe correlation length during the quench in the 1D quantum Ising model, $H=-\\sum_n(1-\\epsilon) \\sigma_n^z + \\sigma^x_n \\sigma^x_{n+1} $, where the dynamical critical exponent $z=1$ and the excited quasiparticles posses definite speed of sound. Compare with Fig.~\\ref{fig:KZreal}.\nData from Ref.~\\onlinecite{Francuzetal}.\n}\n\\label{fig:Ising}\n\\end{figure}\n\n\nIn this paper we go beyond $z=1$ and present two examples with $z>1$: the classical Ising model with Glauber dynamics in Section \\ref{sec:Onsager} and the generalized quantum XY chain in Section \\ref{sec:XY}. They both exhibit an effective event horizon with a speed limit that depends on the quench time $\\tau_Q$.\nThe generic scenario is delimited by two examples where the sonic horizon effect cease to manifest because one of its underlying assumptions is not satisfied. The first one is the random Ising model in Section \\ref{sec:Random}, where localization of excited quasiparticles prevents the spreading of correlations, thus yielding in effect a ``freeze-out''. The other is the extended Ising model with long-range interactions in Section \\ref{sec:LR}, where the dynamical exponent $z$ is less than $1$. The excited quasiparticles with $k\\to0$ have infinite velocity, and the speed $\\hat v$ at the maximal excited $\\hat k$ is not an upper but a lower speed limit. Consequently, there is no sonic horizon effect, and the correlations have a long-range power-law tail that can evolve in time. After $-\\hat t$ the tail begins to lag behind its adiabatic evolution. However, instead of completely freezing out, it continues to grow at a finite rate.\n\n\\begin{figure}[b]\n\\vspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig3.pdf}\n\\vspace{-0.7cm}\n\\caption{ {\\bf Classical Ising in 2D on a square lattice.} \nIn (a),\nenergy per site, $E$, as a function of $\\beta(t)$ for different quench times $\\tau_Q$.\nFor reference, \nthe black line is the equilibrium internal energy $U(\\beta)$. \nWith increasing quench time, the quench curves converge to the equilibrium one.\nIn (b),\nexcitation energy per site, $\\Delta E=E-U$, as a function of scaled time for different quench times $\\tau_Q$.\nHere, both $\\Delta E$ and $t$ are rescaled according to the KZM predictions.\nFor large $\\tau_Q\\gg1$ the scaling makes them collapse in the KZ regime between \n$t\\approx-\\hat t$ and $t\\approx\\hat t$, corresponding to the rescaled times $-1$ and $+1$, respectively.\nAt later times the phase ordering kinetics steps in, which goes beyond the KZ physics.\n}\n\\label{fig:onsagerenergy}\n\\end{figure}\n\\section{ Classical Ising model: ${\\mathbf z=2}$ } \n\\label{sec:Onsager}\n\nWe begin with the classic example of the classical Ising model on a periodic square lattice of size $L\\times L$:\n\\begin{equation}\nH~=~-\\sum_{\\langle j,j'\\rangle} \\sigma^z_j \\sigma^z_{j'} ~.\n\\label{Honsager}\n\\end{equation}\nOn an infinite lattice, the critical inverse temperature would be $\\beta_c=\\ln(1+\\sqrt2)\/2\\simeq 0.4407$.\nThe relevant equilibrium exponents are $\\nu=1$ and $\\eta=1\/4$ \\cite{Schultz1964}.\nWe model relaxation to an external heat bath with the Glauber dynamics: \nMonte Carlo update thermalizes one spin (chosen at random) at a time.\nThe time needed for $L^2$ such one-spin updates sets unit of time.   \nFor such simple relaxation, the dynamical exponent is $z=2$, belonging to the universality class of model-A dynamics~\\cite{hohenberg1977theory}. \nWe performed all our numerical simulations on a $4096\\times 4096$ lattice. This lattice size is $100$ times longer than the longest correlation range encountered in the simulations, hence any finite size effects are eliminated with a wide safety margin. All results were averaged over $50$ repetitions of the quench, each of them starting from a different initial random spin configuration at infinite temperature.\n\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth]{fig4.pdf}\n\\vspace{-0.5cm}\n\\caption{  \n{\\bf Classical Ising model in 2D.} The scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$\nfor scaled times $t\/\\hat t=-1,0,1$ (left to right) and different quench times $\\tau_Q$.\nFor each scaled time,\nwhen $\\tau_Q\\gg1$ the plots with different $\\tau_Q$ collapse to a single scaling function $F(t\/\\hat t,R\/\\hat\\xi)$\ndemonstrating the KZ scaling (\\ref{CRscaling}) hypothesis for slow enough quenches.\n}\n\\label{fig:corrcollapse}\n\\end{figure*}\n\n\nThe inverse temperature of the heat bath is ramped linearly in time,\n\\begin{equation} \n\\beta(t)=\\beta_c\\left(1+\\frac{t}{\\tau_Q}\\right),\n\\end{equation}\nstarting with random spin configuration at $\\beta(-\\tau_Q)=0$. Figure~\\ref{fig:onsagerenergy}(a) shows the energy $E$ during the ramp as a function of $\\beta$ for different quench times $\\tau_Q$. Generally, the system is more ordered for slower quenches. For slow enough quenches, the KZ picture emerges. The system evolves adiabatically until it begins to go out of equilibrium around $-\\hat t$, where $\\hat t\\simeq\\tau_Q^{2\/3}$ is the KZ timescale.\n\nIn order to see how the excitation energy should depend on the quench time, let us consider the equilibrium internal energy. Near the critical point it is\n\\begin{equation} \nU(\\beta) = - \\left[ 1 + A (\\beta-\\beta_c) \\ln\\left|\\beta-\\beta_c\\right| \\right]\/\\tanh\\beta_c,\n\\end{equation}\nwhere $A\\simeq 1$ is a constant. In the adiabatic-impulse approximation the state becomes effectively frozen near $-\\hat t$ when $\\beta_c-\\hat\\beta\\simeq\\tau_Q^{-1\/3}$. At $\\beta_c$ the energies of the frozen state (i.e. the instantaneous state at $-\\hat t$) and the equilibrium one differ by\n\\bea\n\\Delta E(\\beta_c) &=&\nU(\\hat\\beta)-U(\\beta_c) \\nonumber\\\\\n&=&\nA (\\beta_c-\\hat\\beta) \\ln\\left|\\hat\\beta-\\beta_c\\right|\/\\tanh\\beta_c \\nonumber\\\\\n&\\simeq &\n\\tau_Q^{-1\/3}\\ln\\left(\\tau_Q\/\\tau_0\\right).\n\\end{eqnarray} \nHere $\\tau_0\\simeq1$ is a constant. We can see that, up to a subleading logarithmic correction, the KZ scale of energy is $\\simeq\\tau_Q^{-1\/3}$. Accordingly, in Figure~\\ref{fig:onsagerenergy}(b) we show scaled excitation energy $\\tau_Q^{1\/3}(E-U)$ in function of scaled time $t\/\\hat t$. For $\\tau_Q\\gg1$ the plots for different $\\tau_Q$ collapse in the KZ regime: $-1<t\/\\hat t<1$. There is no collapse at later times when phase ordering kinetics \\cite{POK} brings in new physics beyond the KZ mechanism.\nHaving seen how the excitations build up, we can have a closer look at the ferromagnetic correlator\n\\begin{equation} \nC_R(t) =\n\\langle \\sigma^z_i \\sigma^z_{j} \\rangle -\n\\langle \\sigma^z_i \\rangle \\langle \\sigma^z_{j} \\rangle.\n\\end{equation}\n\n\\begin{figure}[b]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\columnwidth]{fig5.pdf}\n\\vspace{-0.3cm}\n\\caption{  {\\bf Classical Ising model in 2D.}\nIn (a),\nthe scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$ for $\\tau_Q=2^{15}$ at different scaled times $t\/\\hat t\\in[-1,1]$. The dashed line marks a cut-off $c$ (here $c=0.1$) whose crossing point is a working definition of a scaled correlation range $\\xi_c\/\\hat\\xi$.\nIn (b),\nthe scaled correlation range $\\xi_c\/\\hat\\xi$ as a function of the scaled time $t\/\\hat t$ for the cut-off $c=0.1$.\nThe best linear fit yields a slope $0.72$ as an estimate of the scaled velocity.\nIn (c),\nthe scaled velocity (slope) as a function of the cut-off. A linear extrapolation to zero cut-off yields $0.93$. This number is an estimate for (scaled) velocity at which the furthest correlation tail is spreading.\n}\n\\label{fig:spread}\n\\end{figure}\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth]{fig6.pdf}\n\\vspace{-0.7cm}\n\\end{center}\n\\caption{{\\bf Extended quantum XY chain.} Scaled ferromagnetic correlations $C^{xx}_R$ as a function of scaled distance $R\/\\hat{\\xi}$ for the extended XY model in (\\ref{XYHsigma}) plotted for different quench times $\\tau_Q$ and at different scaled times $t\/\\hat{t}=-1,0,1$. For large enough $\\tau_Q$ plots collapse towards a universal scaling function $F(t\/\\hat{t},R\/\\hat{\\xi})$ demonstrating the KZ scaling hypothesis (\\ref{CRscaling}).\n}\n\\label{fig:XYcollapse}\n\\end{figure*}\n\n\n\nHere, the sites $i$ and $j$ are separated by a distance $R$ along one of the axes and the average is taken in the state at time $t$. Figure~\\ref{fig:corrcollapse} shows the scaled correlator $\\hat\\xi^\\eta C_R$ in function of the scaled distance $R\/\\hat\\xi$ for three scaled times $t\/\\hat t=-1,0,1$. Here $\\hat\\xi=\\tau_Q^{1\/3}$ is the scale of length in Eq.~(\\ref{hatxi}) and $\\hat t=\\hat\\xi^z=\\tau_Q^{1\/3}$ is the scale of time in Eq.~(\\ref{hatt}). In accordance with the scaling hypothesis (\\ref{CRscaling}), for each scaled time the plots with different $\\tau_Q$ collapse to a common scaling function when $\\tau_Q$ is large enough. The plot for the longest $\\tau_Q=2^{15}$ is practically equal to the scaling function $F(t\/\\hat t,R\/\\hat\\xi)$.\n\n\nComparing the three panels in Figure~\\ref{fig:corrcollapse} we can see that (at odds with the impulse approximation) the range of correlations increases several times between $-\\hat t$ and $\\hat t$. This observation is further corroborated by Fig.~\\ref{fig:spread}(a) where we collect scaled plots for $\\tau_Q=2^{15}$ at different scaled times. Using $\\hat\\xi$ as a natural unit of length, this figure shows how the correlation spreads in the KZ regime between $-\\hat t$ and $+\\hat t$. \n\nFigure~\\ref{fig:spread}(b) shows how a scaled range of the correlator in Fig.~\\ref{fig:spread}(a) depends on the scaled time $s=t\/\\hat t$. Here, the range is defined as the distance at which the scaled correlator falls below a threshold---set here at $0.1$. The nearly linear time dependence is fitted with a line whose slope estimates the scaled velocity of correlation spreading. We repeated the same procedure for thresholds down to $0.02$ (below $0.02$ the tail of the correlator becomes too noisy for an unambiguous estimation of the correlation range). The estimated slopes\/velocities are collected in Fig.~\\ref{fig:spread}(c). The velocity increases rather slowly with a decreasing threshold. In the limit of zero threshold -- probing the longest correlation tail -- it extrapolates to $0.93$. We can conclude that the speed limit in this model is $\\hat v$, which allows the size of the sonic horizon to expand with\n\\begin{equation} \n2\\hat v = \n0.93~ \\frac{\\hat\\xi}{\\hat t} = \n0.93~ \\tau_Q^{-1\/2}.\n\\end{equation}\nIn natural units, in accordance with KZ prediction in Eq.~\\eqref{hatv}, the speed limit becomes slower for slower quenches.\n\\section{ Extended quantum XY chain: ${\\mathbf z=3}$ } \n\\label{sec:XY}\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{fig7.pdf}\n\\caption{{\\bf Extended quantum XY chain.}\nIn (a), \nfunction (\\ref{fitting}) is fitted (dashed lines) to the scaled ferromagnetic correlators $\\hat{\\xi}^{1\/4}C^{xx}_R$ for different scaled times during the non-adiabatic stage of the evolution: $t\/\\hat{t}\\in\\left[-2,2\\right]$.\nHere $\\tau_Q=2^{20}$.\nIn (b),\nthe fits provide the scaled correlation length $\\xi\/\\hat{\\xi}$ plotted against the scaled time $t\/\\hat{t}$. The slope of this plot is an estimate of the scaled velocity $v\/\\hat{v}=0.84$ at which the correlations are spreading. \nIn (c),\na more accurate approximation of the velocity is obtained by taking a derivative of the plot (b) with respect to the scaled time, which yields the scaled velocity $v\/\\hat{v}$ as a function of the scaled time.\nMaximal velocity is $v_{\\rm max}\/\\hat v=0.84$ for $\\tau_Q=2^{20}$. \nIn (d),\nthe maximal velocity is shown as a function of $\\tau_Q$.\nThe red line is a fit, which, for large $\\tau_Q$, saturates at $v_{max}\/\\hat v= 0.868$. \n}\n\\label{fig:XYspeed}\n\\end{figure*}\n\n\nWe step into the quantum realm \nwith the extended quantum XY periodic spin chain,\n\\bea\nH&=&-\\sum_{n=1}^N\n\\left[\n(1-\\epsilon) \\sigma^z_n +\nJ^{xx}\\sigma^x_n\\sigma^x_{n+1}+ \nJ^{yy}\\sigma^y_n\\sigma^y_{n+1}+\n\\right. \\nonumber \\\\\n&&\n\\left.\nJ^{xzx}\\sigma^x_n \\sigma^z_{n+1} \\sigma^x_{n+2}+ \nJ^{yzy}\\sigma^y_n \\sigma^z_{n+1} \\sigma^y_{n+2}\n\\right],\n\\label{XYHsigma}\n\\end{eqnarray} \na generalization of the quantum XY chain. It combines the desired $z>1$ with exact solvability.\nHere we consider an anisotropic model:\n\\bea\n&&\\left(J^{xx},J^{yy},J^{xzx},J^{yzy}\\right)=\\nonumber\\\\\n&&\n\\left(a\\frac{1+\\gamma}{2},a\\frac{1-\\gamma}{2},\n      b\\frac{1+\\delta}{2},b\\frac{1-\\delta}{2}\\right)\n\\end{eqnarray}\nwith the external magnetic field parametrized by $\\epsilon\\in[-1,1]$. The parameter is driven linearly (\\ref{epsilont}) from an initial $\\epsilon=-1$ in the paramagnetic phase, across the critical point at $\\epsilon=0$, to a final $\\epsilon=1$ in the ferromagnetic phase. After the Jordan-Wigner and Fourier transformations (\\ref{JordanWigner},\\ref{Fourier}), the Hamiltonian becomes:\n\\bea\nH &=\\sum_{k>0} &\nA_k(\\epsilon) \\left(c_k^\\dag c_k+c_{-k}^\\dag c_{-k}\\right) \\nonumber \\\\ \n&&\n+B_k \\left(c_kc_{-k}+c_{-k}^\\dag c_k^\\dag\\right),\n\\label{XYHk}\n\\end{eqnarray}\nwhere,\n\\bea \nA_k(\\epsilon) &=& 1-\\epsilon-a\\cos k-b\\cos 2k,  \\nonumber\\\\  \nB_k           &=& a\\gamma\\sin k+b\\delta\\sin 2k.\n\\end{eqnarray}\nIt is diagonalized by eigenmodes of the stationary Bogoliubov-de Gennes equations:\n\\bea\n\\omega_k\n\\left(\n\\begin{array}{c}\nU_k \\\\\nV_k\n\\end{array}\n\\right)&=&\n2\\left[\\sigma^z A_k(\\epsilon)+\\sigma^x B_k \\right] \n\\left(\n\\begin{array}{c}\nU_k \\\\\nV_k\n\\end{array}\n\\right),\n\\label{XYBdGk}\n\\end{eqnarray}\nwith eigenfrequencies\n$\n\\omega_k=2\\sqrt{A_k^2(\\epsilon)+B_k^2}.\n$\nHere, we set $a = 4\/3, b=-1\/3, \\gamma =1\/2, \\delta=1$. For the critical $\\epsilon=0$, we obtain a cubic dispersion relation\n\\begin{equation} \n\\omega_k\\simeq |k|^3,\n\\label{XYomegakz}\n\\end{equation} \nexpanding $\\omega_k$ around $k=0$, and the dynamical exponent $z=3$. On the other hand, setting $k=0$ and expanding in small $\\epsilon$, we obtain the gap opening as $\\omega_0\\simeq|\\epsilon|^1$. Hence $z\\nu=1$ and $\\nu=1\/3$. The dominant static ferromagnetic correlations have oscillatory tails:\n\\bea\nC_R     &=&\\langle\\sigma_i^x\\sigma_{i+R}^x\\rangle-\n           \\langle\\sigma_i^x\\rangle\\langle\\sigma_{i+R}^x\\rangle \\nonumber \\\\\n        &\\sim & R^{-\\eta} e^{-R\/\\xi}\\cos(b~R\/\\xi+c).\n\\end{eqnarray} \nHere $\\xi\\sim|\\epsilon|^{-\\nu}$ is a static correlation length and $\\eta=1\/4$. Given the exponents $z$ and $\\nu$, we can define the dynamical length (\\ref{hatxi}) and time (\\ref{hatt}) scales: $\\hat\\xi=\\tau_Q^{1\/6}$ and $\\hat t=\\tau_Q^{1\/2}$, respectively. The fastest excited quasiparticles have velocity $\\hat v\\simeq\\tau_Q^{-1\/3}$, see Eq.~\\eqref{hatv}. \n\nThe time-dependent quench is solved in appendix \\ref{XYtime} in a standard way \\cite{d2005} by mapping to the Landau-Zener problem, see appendix \\ref{LZ}. The KZ scaling hypothesis (\\ref{CRscaling}) is demonstrated in Fig.~\\ref{fig:XYcollapse} by collapse of the plots for different $\\tau_Q$. A perfect collapse requires very large $\\tau_Q\\simeq2^{20}$, as explained by Eq.~\\eqref{rescuv}. The oscillatory behaviour of $C_R$ in the original paramagnetic phase survives through the transition. In order to estimate the range of the scaled correlators, in Fig.~\\ref{fig:XYcollapse} we fit their tails with oscillatory functions of the form:\n\\begin{equation} \n\\hat{\\xi}^{1\/4} C_R(t)=\na~(R\/\\hat{\\xi})^{-1\/4}~\\mathrm{e}^{-\\frac{R\/\\hat{\\xi}}{\\xi\/\\hat{\\xi}}}\\cos(b~R\/\\hat{\\xi}+c),\n\\label{fitting}\n\\end{equation}\nwhere $a,b,c$ and $\\xi\/\\hat\\xi$ are the fitting parameters. We are interested how fast the scaled correlation length $\\xi\/\\hat\\xi$ grows with the scaled time $t\/\\hat t$. In order to reveal the universal behavior undisturbed by any short-range effects, we perform the fit in the range of scaled distances $R\/\\hat\\xi>2.5$. Ferromagnetic correlations for different scaled times $t\/\\hat{t}$, together with the fits, are shown in Fig.~\\ref{fig:XYspeed}(a) (for $\\tau_Q=2^{20}$). The correlation length $\\xi\/\\hat{\\xi}$ as a function of scaled time is shown in Fig.~\\ref{fig:XYspeed}(b). Its slope, equal to $0.838$, provides an estimate of the scaled velocity at which the correlations are spreading. For completeness, in Fig.~\\ref{fig:XYspeed}(c) we plot the derivative of the plot in Fig. \\ref{fig:XYspeed}(b) with respect to the scaled time. Its maximal value is shown in Fig.~\\ref{fig:XYspeed}(d) as a function of $\\tau_Q\\in [2^{10},2^{20}]$. For large $\\tau_Q$ it extrapolates to\n\\begin{equation} \n2 \\hat v = 0.868~ \\frac{\\hat{\\xi}}{\\hat{t}} = 0.868~ \\tau_Q^{-1\/3}.\n\\end{equation}\nAs in the classical 2D Ising model, where also $z>1$, the correlation spreading becomes slower for slower quenches.\n\n\\section{ Random Ising model } \n\\label{sec:Random}\n\nIn order to break the translational invariance and impede the propagation of excited quasiparticles that could spread correlations, next we consider the random Ising model (RIM) in one dimension defined by the Hamiltonian\n\\bea\nH=\n-\\sum_{n=1}^N\n\\left[ (h_{n} - \\epsilon)\\sigma^z_n \\ +\n J_{n}\\sigma^x_n\\sigma^x_{n+1}\n\\right].\n\\label{Hamil_RIM}\n\\end{eqnarray} \nHere both transverse fields $h_n$'s and nearest-neighbor couplings $J_n$'s are randomly selected from a uniform distribution between $0$ and $1$. The quantum critical point at $\\epsilon=0$ is separating the paramagnetic phase ($\\epsilon<0$) from the ferromagnetic one ($\\epsilon>0$). The critical point in this model is surrounded by Griffiths region \\cite{Griffiths-a, Griffiths-b, Griffiths-c}, where the presence of the so-called rare regions primarily manifests in two features: the activated dynamical scaling (the dynamical exponent $z\\rightarrow\\infty$) at the quantum critical point and the existence of singular regions where the linear susceptibility diverges even away from the critical point. The locally ordered rare regions act as giant spins that flip as a whole and are responsible for the exponentially slow dynamics near the critical point. These features are encapsulated in a dynamical exponent that diverges at the critical point as $z \\sim|\\epsilon|^{-1}$, see Ref. \\onlinecite{Fisher1-a, Fisher1-b} which also shows that the correlation length exponent is $\\nu = 2$. The average correlation function at criticality is a power law\\cite{McCoyWu}:\n\\begin{equation} \nC^{xx}_R=\\langle \\sigma_{i}^{x}\\sigma_{i+R}^{x}\\rangle \\sim R^{-\\eta},\n\\end{equation}\nwhere $\\eta = \\frac{3-\\sqrt{5}}{2}\\approx 0.38$. \n\nWe consider ramping the parameter $\\epsilon$ linearly as a function of time (\\ref{epsilont}) driving the Hamiltonian from the initial paramagnetic ground state, across the critical point at $t=0$, into the ordered phase. Ref. \\onlinecite{dziarmagaRIM-a, dziarmagaRIM-b, dziarmagaRIM-c} showed that the KZ correlation length $\\hat \\xi$ of the model varies logarithmically with the quench rate $\\tau_Q$:\n\\bea\n\\centering\n\\hat{\\xi} = \\ln^2{(\\tau_Q\/a)},\n\\label{hatxiRIM}\n\\end{eqnarray}\nwhen $\\ln{(\\tau_Q\/a)}\\gg 1$. Here $a\\simeq1$ is a non-universal constant. We can see, that the dependence on $\\tau_Q$ is very weak compared to any of the usual KZ power-law scalings (\\ref{hatxi}). Taking into account that the correlation-length exponent near the critical point is $\\nu = 2$, we get an estimate of the characteristic timescale,\n\\bea\n\\centering\n\\hat{t} = \\frac{\\tau_{Q}}{\\ln{(\\tau_{Q}\/a)}},\n\\label{hattRIM}\n\\end{eqnarray}\n for $\\ln{(\\tau_Q\/a)} \\gg 1$. \n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{fig8.pdf}\n\\caption{{\\bf Random Ising model.} Average scaled density of excitations $\\hat\\xi d_{\\rm ex}$ during the quench as a function of scaled time $t\/\\hat t$ for different quench times $\\tau_Q$. Here, we use Eqs.~(\\ref{hatxiRIM},\\ref{hattRIM}) with $a = 0.118$. This $a$ was tuned to obtain the best possible collapse at $t\/\\hat t=0$ but the plots collapse well during the whole quench. The KZ diabatic stage (shaded in green) extends roughly from $-0.25t\/\\hat t$ to $0.25t\/\\hat t$. This is where the excitation grows before it saturates in the last adiabatic stage. Averaging was done over 30 random realizations for a lattice of size $N = 128$.\n}\n\\label{FigDoERIM}\n\\end{figure}\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth,clip=true]{fig9.pdf}\n\\vspace{-0.1cm}\n\\caption{ {\\bf Random Ising model.} Average scaled ferromagnetic correlations $\\hat\\xi^\\eta C^{xx}_R$ as a function of scaled distance $R\/\\hat{\\xi}$ for the random Ising model (\\ref{Hamil_RIM}) plotted for different quench times $\\tau_Q$ and at different scaled times $t\/\\hat{t}=-0.25,0,0.25$. For large enough $\\tau_Q$ plots collapse towards a universal scaling function $F(t\/\\hat{t},R\/\\hat{\\xi})$ demonstrating the KZ scaling hypothesis\n(\\ref{CRscaling}).\n}\n\\label{fig:RIMcollapse}\n\\end{figure*}\n\nIn order to solve the dynamics, we map the Hamiltonian by the Jordan-Wigner transformation (\\ref{JordanWigner}) to a quadratic spinless free-fermionic model\n\\bea\nH = \\sum_{n}(h_{n}-\\epsilon)c_{n}^\\dag c_{n} -J_{n}c_{n}^\\dag c_{n+1} - J_{n}c_{n+1} c_{n} + {\\rm h.c}.\n\\label{JW_RIM}\n\\end{eqnarray}\nFollowing the convention of this article, we confine ourselves to the subspace of even parity of $c$-quasiparticles (anti-periodic boundary conditions, i.e., $c_{N+1} = -c_1$) and in this subspace, we diagonalize the Hamiltonian by a Bogoliubov transformation:\n\\bea\n\\centering\nc_{n} = \\sum\\limits_{m=1}^{N}(U_{nm}\\gamma_{m} + V_{nm}^* \\gamma_{m}^\\dag)\n\\label{bogoliubov_RIM}\n\\end{eqnarray}\nThe index $m$ labels the (Bogoliubov) eigenmodes of the stationary Bogoliubov-de Gennes equation\n\\bea\n\\centering\n\\omega_{m}U_{n,m}^{\\pm} = 2h_{n}U_{n,m}^{\\mp} - 2j_{n-1}U_{n-1,m}^{\\mp},\n\\label{stationary_B-DeG}\n\\end{eqnarray}\nwhere $\\omega_{m}>0$, $U_{nm}^{\\pm} = U_{nm}\\pm V_{nm}$ and anti-periodic boundary conditions $(U_{N+1,m}^{\\pm} = -U_{1,m}^{\\pm},U_{0,m}^{\\pm}=-U_{N,m}^{\\pm})$ is implemented. The eigenstates ($U_{nm},V_{nm}$), with positive energy, $\\omega_{m}>0$ normalized so that $\\sum_{n}(|U_{nm}|^2 + |V_{nm}|^2) = 1$, define quasiparticle operators $\\gamma_{m}=U_{nm}^{*}c_{n}+V_{nm}c_{n}^{\\dag}$. We have corresponding negative energy components of the eigenstates labelled ($U_{nm}^{neg},V_{nm}^{neg}$) with energy $-\\omega_{m}$, which defines a quasiparticle operator $\\gamma_{m}^{neg}=(U_{nm}^{neg})^{*}c_{n}+V_{nm}^{neg}c_{n}^{\\dag}$. The Bogoliubov transformation renders the Hamiltonian to be $H = \\sum_{m = 1}^{N}\\omega_{m}(\\gamma_{m}^{\\dag}\\gamma_{m} - \\frac{1}{2})$. In the even parity subspace only states with even number of quasi-particles belong to the spectrum of $H$.\n\nIn order to find whether the RIM fits into our narrative of the uniform scaled velocity of the correlation spreading by quasiparticles,\nwe proceed with the numerical simulation of the quench. \nWe prepare the initial state of the system deep in the paramagnetic phase in the ground state, i.e., in the Bogoliubov vacuum state for quasiparticles at an initial time $t_{0}$ where $\\epsilon(t_{0}) = 5$. As we tune the parameter $\\epsilon(t)$ towards $0$, see Eq.~\\eqref{epsilont}, the state of the systems departs from its adiabatic ground state and gets excited due to closing of the energy gap near the critical point. We work in the Heisenberg picture where we assume that the excited state is a Bogoliubov vacuum while the time-dependence is ascribed to a set of time-dependent quasi-particle operators\n\\bea\n\\centering\n\\gamma_{m}(t)=u_{nm}^{*}(t)c_{n}+v_{nm}^{*}(t)c_{n}^{\\dag}\n\\end{eqnarray}\nThe Bogoliubov modes $u_{nm}$ and $v_{nm}$ solve the time-dependent Bogoliubov-de Gennes  equations:\n\\bea\n\\centering\ni\\frac{du_{nm}^{\\pm}}{dt} = 2(h_{n}-\\epsilon(t))u_{nm}^{\\mp}-2J_{n}u_{n\\mp1,m}^{\\mp}\n\\label{t_B-deG}\n\\end{eqnarray}\nWe integrate equation (\\ref{t_B-deG}) numerically using the 2nd order Suzuki-Trotter method, see Appendix \\ref{tB-deG_RIM}. The density of excited Bogoliubov quasiparticles $d_{ex}(t)$ can be calculated at each time step by projecting the time-dependent Bogoliubov modes ($u_{nm}(t),v_{nm}(t)$) onto the corresponding instantaneous static negative Bogoliubov modes ($U_{nm}^{\\text{neg}},V_{nm}^{\\text{neg}}$):\n\\bea\n\\centering\nd_{ex}(t) = \\frac{1}{N} \\sum_{s}^N\\sum_{p}^N \\left|\\langle U_{p}^{\\text{neg}},V_{p}^{\\text{neg}}|u_{s}(t),v_{s}(t)\\rangle\\right|^2.\n\\label{doe_RIM}\n\\end{eqnarray}\n\nWhile in the initial adiabatic stage, the system remains in the instantaneous ground state and hence the density of quasiparticle excitations is $0$. On tuning the field $\\epsilon(t)$ towards its critical value at $t\/\\hat{t} = 0$, the system gets excited and consequently $d_{ex}$ starts to grow. As we keep increasing the field $\\epsilon(t)$, $d_{ex}$ saturates as the system ends its non-adiabatic journey across the critical point. In Fig.~\\ref{FigDoERIM} we see that the scaled plots (averaged over disorder) for different $\\tau_Q$ collapse in accordance with the dynamical scaling hypothesis (\\ref{CRscaling}). \n\n\\begin{figure}[b]\n\\includegraphics[width=\\columnwidth]{fig10.pdf}\n\\caption{   {\\bf Random Ising model.}\nIn (a),\nthe scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$ for $\\tau_Q=2^{11}$\nat different scaled times $t\/\\hat t$ in $[-0.25,0.25]$.\nThe dashed line marks a cut-off (here $1$) whose crossing point \nis a working definition of a scaled correlation range $\\xi_c\/\\hat\\xi$.\nIn (b),\nthe scaled correlation range $\\xi_c\/\\hat\\xi$ as a function of the scaled time $t\/\\hat{t}$ for the cut-off $c=0.5$. The best linear fit yields a slope of $0.113$ as an estimate of the scaled velocity.\nIn (c),\nthe scaled velocity (slope) as a function of the cut-off, with the error bars indicating fitting errors.\n}\n\\label{fig:RIMspread}\n\\end{figure}\n\nThe hypothesis for correlations in Eq.~\\eqref{CRscaling} is verified by a similar collapse of the plots in Fig.~\\ref{fig:RIMcollapse}. We consider large $\\tau_Q$ as the logarithmic KZ scaling laws are only valid in that regime. The correlation function is averaged over $100$ instances of disorder and lattice translations on a lattice of $N=256$. This lattice is a few times longer than the longest correlation range. The details of calculating the correlation functions are given in Appendix \\ref{appcorrRIM}.\n\nA close look at Fig.~\\ref{fig:RIMcollapse} reveals that the scaled dynamical correlation function grows with scaled time. In order to appreciate the growth, we collect the scaled correlators for different scaled times in Fig.~\\ref{fig:RIMspread}(a). In order to estimate the scaled correlation range $\\xi_c\/\\hat{\\xi}$ as a function of scaled time, we choose a cut-off $c$ for the scaled correlation and calculate scaled distances corresponding to the cut-off by spline interpolation. Fig.~\\ref{fig:RIMspread}(b) shows the scaled range in function of the scaled time for the cut-off (selected equal to $0.5$) in Fig.~\\ref{fig:RIMspread}(a).\nNear the critical point, we approximate it with linear dependence where the slope is $0.113\\hat\\xi\/\\hat t$. We repeat the same procedure for other cut-offs. We present the enumerated slopes for various cut-offs in Fig.~\\ref{fig:RIMspread}(c), with the error bars indicating the fitting error. Finally, we estimate the sonic horizon expansion speed in this model with the maximal observed value\n\\bea\n\\centering\n2\\hat{v} = 0.12 \\frac{\\hat{\\xi}}{\\hat{t}} = 0.12 \\frac{\\ln^3(\\tau_Q\/a)}{\\tau_Q}\n\\end{eqnarray}\nCompared to the previous models, with the prefactor $0.12$, the speed limit is significantly below expectations in this case. The entangled Bogoliubov quasiparticles get excited but, due to their localization by disorder, they do not propagate to spread correlations appreciably. In the spin language, the locally ordered regions are excited, but they essentially stay where they are. One may even argue that in this case, the adiabatic-impulse-adiabatic freeze-out approximation captures the essential physics.\n\n\n\\section{Long-range extended Ising model: $0<z<1$ } \n\\label{sec:LR}\n\n\nThe long-range extended Ising model is a further generalization of the extended XY model that we have discussed in Sec. \\ref{sec:XY}. Now the Hamiltonian of the system includes not only the nearest-neighbor and next-nearest-neighbor interaction, but all possible long-range cluster terms: \n\\bea\nH=\n-\\sum_{n=1}^N\n\\left[\n (1-\\epsilon) \\sigma^z_n +\n\\sum_{r=1}^{N-1} \nJ_r\\sigma^x_n \\sigma^x_{n+r} \\prod_{i=n+1}^{n+r-1} \\sigma^z_i\n\\right].\n\\label{Hamil_LR}\n\\end{eqnarray} \nHere \n\\begin{equation}\nJ_r ~=~ \\frac{1}{\\zeta(\\alpha)} ~ \\frac{1}{r^\\alpha},\n\\end{equation}\nand the normalization by the Riemann zeta function, possible for $\\alpha>1$, is such that $\\sum_r J_r=1$. The case of $\\alpha>2$ is not the most interesting here because the model behaves effectively like the short-range one \\cite{LRKitaev1,LRKitaev2,LR1,LR2,LRBosonic}. Similarly, when $0\\le\\alpha\\le1$ we would need to restrict ourselves to a finite system because the thermodynamic limit does not exist in this case, and the model behaves effectively like the Lipkin-Meshkov-Glick model \\cite{LMG} with infinite-range interactions. Therefore, we focus here on the intermediate $1<\\alpha<2$, where a cross-over between the short and infinite-ranges happens. In this regime, we consider the linear quench (\\ref{epsilont}) driving the system from the initial paramagnetic phase at $t=-\\infty$, across the critical point at $t=0$, to the final ferromagnetic phase. \n\n\\begin{figure}[b]\n\\includegraphics[width=\\columnwidth]{fig11.pdf}\n\\caption{  {\\bf Long-range extended Ising model.}\nIn (a),\nthe static correlation function $C_R$ at the critical $\\epsilon=0$ for $\\alpha=3\/2$. \nThe solid line is a linear fit with a slope $-1.002\\approx -1 $.\nThe same panel shows the same function far from the critical point at $\\epsilon=-2$. \nThe solid line is a linear fit with a slope $-1.45 \\approx -(3 -\\alpha) $.  \nIn (b),\nthe scaled static correlation function $\\xi C_R$ as a function of scaled distance $R\/\\xi$.\nIn accordance with the static scaling hypothesis (\\ref{staticSH}),\nthe plots for different $\\epsilon$ collapse to a unique scaling function.\nWe can see a crossover from the critical $(R\/\\xi)^{-1}$ to the off-critical $(R\/\\xi)^{-3\/2}$\nnear $R\/\\xi=1$. The straight lines are linear fits with slopes $-1.085\\approx-1$ and $-1.42\\approx -(3-\\alpha)$ for smaller and larger $R\/\\xi$, respectively. \n}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{fig12.pdf}\n\\caption{ {\\bf Long-range extended Ising model.}\nThe scaled dynamical correlation function $\\hat\\xi C_R$ for different quench times $\\tau_Q$ at different scaled times, $s={t}\/{\\hat{t}}$, during the quench.\nHere $\\alpha=3\/2$.\nFrom left to right: $s=-1,0,0.35,0.4$. \nIn (a) at $s=-1$,\nthe evolution is still adiabatic and the plots for different $\\tau_Q$ collapse according to the static scaling hypothesis which is equivalent to the KZ hypothesis in this early adiabatic regime.\nIn (b) at $s=0$,\nwhen crossing the critical point the evolution is diabatic. Different $\\tau_Q$ collapse to a common scaling function $F(s,R\/\\hat\\xi)$ in accordance with the KZ scaling hypothesis (\\ref{CRscaling}).\nIn (c) at $s=0.35$,\nthe correlation function begins to bend down at distances between $R\/\\hat\\xi=10$ and $100$.\nDifferent $\\tau_Q$ collapse, as the evolution is still in the KZ regime.\nFinally in (d) at $s=0.4$,\nthe collapse fails due to the phase ordering after the system left the KZ regime and entered the adiabatic one. At the distances between $R\/\\hat\\xi=10$ and $100$ the correlations become negative for longer $\\tau_Q$.\n}\n\\label{fig:corrp0m}\n\\end{figure*}\n\n\nThanks to the string operator in the long-range interaction terms, the model can be mapped to a quadratic free fermion model and solved analytically. After the Jordan-Wigner transformation (\\ref{JordanWigner}) the Hamiltonian (\\ref{Hamil_LR}) becomes:\n\\bea\nH &=& -\\sum_n \\left(1-\\epsilon\\right)\\left(c_nc_n^\\dag-c_n^\\dag c_n\\right) \\nonumber \\\\\n&&- \\sum_{n,r} J_r\\left( c_n^\\dag c_{n+r} + c_n^\\dag c_{n+r}^\\dag + {\\rm h.c.}  \\right),\n\\label{HcLR}\n\\end{eqnarray}\nfor the anti-periodic boundary conditions. This representation is also known as the long-range Kitaev model \\cite{LRKitaev1, LRKitaev2}, where the hopping and pairing terms are of equal strength. After Fourier transformation (\\ref{Fourier}),\n\\bea\nH &=&\n-2\\sum_{k>0} \n(1-\\epsilon -  \\Re(\\tilde{J_k})) \\left(c_k^\\dag c_k+c_{-k}^\\dag c_{-k}\\right) \\nonumber \\\\\n&&+  ~\\Im(\\tilde{J_k})\\left(c_{k}^\\dag c_{-k}^\\dag + c_{-k}c_{k}\\right),\n\\label{HkLR}\n\\end{eqnarray}\nwhere  $\\Re(\\tilde{J_k})$ and  $\\Im(\\tilde{J_k})$ are, respectively, real and imaginary parts of the Fourier transform $\\tilde{J_k}=\\sum_r J_r e^{ikr}$. We have $\\tilde{J_k}=\\frac{{\\rm Li}_{\\alpha}(e^{ik})}{\\zeta(\\alpha)}$, where ${\\rm Li}$ is the polylogarithm function: ${\\rm Li}_\\alpha(x)=\\sum_{n=1}^\\infty\\frac{x^n}{n^\\alpha}$. \n\n\n\nWe can now find the stationary Bogoliubov-de Gennes equations (\\ref{XYBdGk}) with\n$\nA_k(\\epsilon)=1-\\epsilon-\\Re(\\tilde{J_k}),~~ B_k=\\Im(\\tilde{J_k})\n$\nand eigenfrequencies\n$\n\\omega_k=2\\sqrt{A_k^2(\\epsilon)+B_k^2}.\n$\nWe have a critical point at $\\epsilon=0$ where the gap closes for $k=0$. Another critical point, \nnot to be considered here, is at $\\epsilon=2(1-2^{-\\alpha})$ and $k=\\pi$. The dispersion relation at the critical $\\epsilon=0$ is\n\\bea\n\\omega_k \\simeq |k|^{\\alpha-1},\n\\end{eqnarray}\nhence the dynamical exponent is $z=\\alpha-1\\in(0,1)$. On the other hand, for small $\\epsilon$, the gap at $k=0$ closes as\n$ \n\\omega_0 = 2|\\epsilon|,\n$\nhence $z\\nu=1$. \n\n\n\nIn a short range model for large $R$ the correlation function decays exponentially with $R$ when the system is away from the critical point but this does not need to hold for long-range interactions. Indeed, our system has a power law scaling even far away from the critical point \\cite{LRKitaev1,LRKitaev2}:\n\\bea \nC_R &=&\n\\langle \\sigma_i^x \\sigma_{i+1}^z \\ldots \\sigma_{i+R-1}^z \\sigma_{i+R}^x\\rangle\n\\sim 1\/R^{3-\\alpha},\n\\label{Corr}\n\\end{eqnarray}\ncompare Fig.~\\ref{fig2}(a). On the other hand, as discussed in more detail in Ref.~\\onlinecite{natun}, at the critical point we expect a critical power law $C_R \\sim 1\/r^{\\eta}$ with $\\eta<\\alpha$. Indeed, in Fig.~\\ref{fig2}(a) we find that $\\eta=1$ for $\\alpha=3\/2$. Similarly as for short-range interactions, for a small $\\epsilon$ we expect a cross-over between the two power laws when $R$ is close to $\\xi\\sim\\epsilon^{-\\nu}$. Note that here $\\xi$ is not the usual exponential correlation length, even though it scales with $\\epsilon$ in the characteristic way. The crossover can be verified with a static scaling hypothesis:\n\\bea \n\\xi^\\eta ~C_R = F_{\\rm st}\\left( R\/\\xi \\right).\n\\label{staticSH}\n\\end{eqnarray} \nWith $\\xi=\\epsilon^{-\\nu}$ the plots of the scaled correlator $\\xi^\\eta C_R$ as a function of the scaled distance $R\/\\xi$ for different $\\epsilon$ should collapse to a common static scaling function $F_{\\rm st}(x)$. We expect $F_{\\rm st}(x)$ to cross-over around $x=1$ between the critical tail $x^{-1}$ for small $x$ and $x^{3-\\alpha}$ for large $x$. Indeed, this is what we see in Fig.~\\ref{fig2}(b). The unscaled leading static tail is, therefore, \n\\begin{equation} \nC_R\\simeq \\xi^{1\/2} R^{-3\/2}\n\\label{lrtailstatic}\n\\end{equation} \nfor $R\\gg\\xi$.\n\n\n\\begin{figure}[b]\n\\vspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig13.pdf}\n\\vspace{-0.5cm}\n\\caption{ {\\bf Long-range extended Ising model.}\nIn (a), \nscaled correlator $\\hat\\xi C_R$ for $\\alpha=3\/2$ as a function of scaled distance for different scaled times in the log-log scale.\nIn (b),\nin the inset the coefficients $A,B,C$ in Eq.~\\eqref{ABC(s)fit} fitted to the tail of the scaled correlator are shown as a function of scaled time $s=t\/\\hat t$. The main picture shows $A(t\/\\hat t)$ together with the adiabatic $A_{\\rm ad}(t\/\\hat t)$.\nIn the initial adiabatic stage $A$ follows $A_{\\rm ad}$. \nIn the KZ stage $A_{\\rm ad}$ diverges to infinity at the critical point forcing $A$ to lag behind.\n}\n\\label{fig:fig2a}\n\\end{figure}\n\n\nHaving verified the static hypothesis, we are encouraged to propose its dynamical version in the same form (\\ref{CRscaling}) as for local interactions. The dynamics is described by the time-dependent Bogoliubov-de Gennes equations (\\ref{tBdGk}) whose solution is presented in Appendix \\ref{AppLR}. The KZ mechanism has recently been verified in a similar model~\\cite{amitdutta}, where defect density was found to scale as $\\tau_Q^{-1\/2(\\alpha-1)}$ in agreement with the KZ prediction $\\tau_Q^{-\\nu\/(1+z\\nu)}$, see also Ref.~\\cite{KZLR2}. Further developments, like scaling of excitation energy (consistent with KZ) or possible experimental implementation, were considered in Refs.~\\onlinecite{MorigiLR,LRPlenio}.\nHere, we are interested in the build-up of correlations during the quench. In Fig.~\\ref{fig:corrp0m}(a) we show scaled correlation functions at different scaled times. For definiteness, in this figure we set $\\hat t=\\tau_Q^{1\/2}$ and $\\hat\\xi=\\tau_Q^{1\/2(\\alpha-1)}$ with both numerical prefactors equal to $1$. We can see that up to $t=0.35\\hat t$ the scaled correlators for different $\\tau_Q$ collapse demonstrating the KZ scaling hypothesis in Eq.~\\eqref{CRscaling}. At $t\/\\hat t\\approx 0.4$ they already fail to collapse, signalling the end of the KZ regime and the beginning of phase ordering in the second adiabatic stage. \n\nIn Fig.~\\ref{fig:fig2a}(a), we collect scaled correlators for different scaled times, $s=t\/\\hat t$, in the KZ regime. In order to gain better insight, in Fig.~\\ref{fig:fig2a}(b) we fit their tails with a function\n\\begin{equation} \n\\hat \\xi C_R(s)=A(s)(R\/\\hat\\xi)^{-3\/2}+B(s)(R\/\\hat\\xi)^{-2}+C(s)(R\/\\hat\\xi)^{-5\/2}.\n\\label{ABC(s)fit}\n\\end{equation} \nIts form is motivated by correlation tails after a sudden quench considered in appendices \\ref{SuddenAppInfinite} and \\ref{SuddenAppFinite}. Here, the coefficients $A,B,C$ are functions of the scaled time shown in Fig.~\\ref{fig:fig2a}(c). On top of the leading $A(s)$ tail, sub-leading dynamical correlations build up in time, with $B(s)$ growing negative and $C(s)$ positive. Their behavior is similar to what is observed after a sudden quench to the critical $\\epsilon=0$ either from $\\epsilon=-\\infty$ or $\\epsilon=-1$, see appendices \\ref{SuddenAppInfinite} and \\ref{SuddenAppFinite}, respectively. The main Fig.~\\ref{fig:fig2a}(b) shows $A(s)$, i.e., the coefficient of the leading long-range tail $\\propto R^{-3\/2}$. \n\n\nThe same figure shows $A_{\\rm ad}(s)$, i.e., the same coefficient but in case the evolution were adiabatic. $A_{\\rm ad}(s)$ was obtained by a fit to the far tail of the static correlation function at $\\epsilon(s)=s\/(\\tau_Q\/\\hat t)$. The same $A_{\\rm ad}(s)$ can be obtained by equating the static tail (\\ref{lrtailstatic}) with the adiabatic tail $C_R=A_{\\rm ad}(s)\\hat\\xi^{1\/2}\/R^{3\/2}$, compare (\\ref{ABC(s)fit}):\n\\begin{equation} \nA_{\\rm ad}(s) \\simeq\n\\left[\\frac{\\xi(s)}{\\hat\\xi}\\right]^{1\/2} \\simeq\n\\left[\\frac{\\epsilon(s)^{-\\nu}}{\\hat\\xi}\\right]^{1\/2} \\sim\ns^{-\\nu\/2}.\n\\end{equation} \n$A_{\\rm ad}(s)\\sim s^{-1}$ for $\\alpha=3\/2$ is consistent with the divergence we observe in Fig.~\\ref{fig:fig2a}(b). We can also see that, as expected, $A(s)$ follows $A_{\\rm ad}(s)$ in the initial adiabatic stage up to $s\\simeq -1$. After that, in the diabatic KZ stage, it lags behind as it cannot catch up with the diverging $A_{\\rm ad}(s)$. If it did not, and the evolution were adiabatic, then the correlation range -- defined by $R$ where $C_R$ falls below a fixed small cutoff -- would diverge near the critical point as $|t|^{-2\/3}$. \n\n\n\nAfter $A(s)$ begins to lag behind the adiabatic evolution, it continues to grow at a finite rate until the critical point. This is the stage where the long-range model is the most reminescent of the standard causal KZ picture. After the critical point $A(s)$ begins to dip down around $s\\approx 0.2$ but this is where the KZ scaling hypothesis (\\ref{CRscaling}) begins to be violated, see Fig.~\\ref{fig:corrp0m}(d).\n\n\\section{ Conclusion } \n\\label{SectionConclusion}\n\nWe recall the causal\/sonic horizon version of the Kibble-Zurek mechanism (KZM).\nIn the initial adiabatic stage of the evolution, \nthe correlation range follows its adiabatic counterpart.\nAt $-\\hat t$ it begins to lag behind the diverging adiabatic range and continues to grow at a finite rate set by the speed limit $\\simeq \\hat\\xi\/\\hat t$.\nThis way, between $-\\hat t$ and $+\\hat t$, the correlation range can increase several times at odds with the impulse approximation where it remains frozen. \n\nThere are notable exceptions, like the random quantum Ising chain, where the increase is small due to localization of excited quasiparticles that prevents them from spreading entanglement. There are interesting generalizations, like the long-range extended Ising model, where it is the power law correlation tail, rather than the exponential correlation length, that lags behind its diverging adiabatic evolution.\n\n\n\\acknowledgements\nAS would like to thank Titas Chanda for useful discussions.\nWe acknowledge funding by National Science Centre (NCN), \nPoland under projects No.~2016\/23\/B\/ST3\/00830 (JS,AF) \nand No. 2016\/23\/D\/ST3\/00384 (MMR), \nNCN together with European Union through QuantERA ERA NET \nprogram No.~2017\/25\/Z\/ST2\/03028 (AS,DS,JD),\nand Department of Energy under the Los Alamos National Laboratory \nLDRD Program (WHZ).\nWHZ was also supported by the U.S. Department of Energy, Office of Science, Basic Energy\nSciences, Materials Sciences and Engineering Division, Condensed Matter Theory Program.\nAF acknowledges financial support by Polish Ministry of Science and Education, project No. DI2015 021345, from the budget funds for science in 2016-2020 under the \"Diamond Grant\" program.\nThis research was carried out with the equipment purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (contract No. POIG.02.01.00-12-023\/08). \n\n\n\n\\section{ Introduction } \n\nKibble-Zurek mechanism (KZM) evolved from the scenario for defect creation in cosmological symmetry-breaking phase transitions \\cite{K-a, *K-b, *K-c}. As the post-Big-Bang Universe cools, causally disconnected regions must choose broken symmetry vacuum independently. Such random choices lead to topologically nontrivial configurations that survive phase ordering as topological defects. In the cosmological setting average size of the causally connected regions (hence, the average density of defects) is set by the Hubble radius at the time of the transition. This early Universe scenario relies on the speed of light and does not apply to the laboratory phase transitions. However, it was the point of departure for the dynamical theory \\cite{Z-a, *Z-b, *Z-c, Z-d} that employs critical exponents of the transition and the quench time to predict the scaling of the resulting density of defects. KZM was successfully tested using numerical simulations \\cite{KZnum-a,KZnum-b,KZnum-c,KZnum-d,KZnum-e,KZnum-f,KZnum-g,KZnum-h,KZnum-i,KZnum-j,KZnum-k,KZnum-l,KZnum-m} and laboratory experiments in condensed matter systems \\cite{KZexp-a,KZexp-b,KZexp-c,KZexp-d,KZexp-e,KZexp-f,KZexp-g,KZexp-gg,KZexp-h,KZexp-i,KZexp-j,KZexp-k,KZexp-l,KZexp-m,KZexp-n,KZexp-o,KZexp-p,KZexp-q,KZexp-r,KZexp-s,KZexp-t,KZexp-u,KZexp-v,KZexp-w,KZexp-x}. More recently, KZM was adapted to quantum phase transitions \\cite{Polkovnikov2005,QKZ1,QKZ2,d2005,d2010-a, d2010-b}. Theoretical developments \\cite{QKZteor-a,QKZteor-b,QKZteor-c,QKZteor-d,QKZteor-e,QKZteor-f,QKZteor-g,QKZteor-h,QKZteor-i,QKZteor-j,QKZteor-k,QKZteor-l,QKZteor-m,QKZteor-n,QKZteor-o,QKZteor-p,QKZteor-q,QKZteor-r,QKZteor-s,QKZteor-t} and experimental tests \\cite{QKZexp-a, QKZexp-b, QKZexp-c, QKZexp-d, QKZexp-e, QKZexp-f, QKZexp-g,deMarco2,Lukin18} of quantum KZM (QKZM) followed. The recent experiment \\cite{Lukin18}, where a quantum Ising chain in the transverse field is emulated using Rydberg atoms, is fully consistent with the predicted scaling \\cite{QKZ2,d2005}. \n\nThe KZM is often presented in its cartoon version where -- due to the critical slowing down \/ closing of the energy gap -- the dynamics of the system literally freezes-out in the neighborhood of the critical point. Today, as the experiments are able not only to count the final number of defects but can also monitor and probe the state of the system during the transition, it is timely to re-investigate the causally limited spreading of correlations during the putative ``freeze-out'' stage of the evolution. \n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.9\\columnwidth,clip=true]{Fig1.pdf}\n\\vspace{-0.1cm}\n\\caption{  \n{\\bf Adiabatic-impulse-adiabatic view of KZM.} \nLinear ramp crosses the critical point at time $t=0$. \nThe instantaneous transition rate, $\\left|\\dot{\\epsilon}\/\\epsilon\\right| = 1\/|t|$, diverges at the critical point and the relevant energy gap closes like $|\\epsilon|^{z\\nu}$. Consequently, while before $-\\hat t$ the state follows the adiabatic ground state, near the critical point (between $-\\hat t$ and $\\hat t$) its evolution is non-adiabatic.\nThe freeze-out assumes that the state is ``frozen'' at $-\\hat t$ \n-- size of the domains of the nascent phase does not change until $+\\hat t$, where the state starts to ``catch up'' with the Hamiltonian. \nThis version of KZM ignores propagation of the new phase front in the time interval $(- \\hat t, +\\hat t)$. It yields correct scalings, but it does not capture what happens -- for example -- in the paramagnetic-ferromagnetic quantum phase transition in the quantum Ising chain in transverse field \\cite{KZscaling1,Francuzetal}. Nevertheless, it may well be relevant in phase transitions where the conserved order parameter or other causes (localization) impede propagation of phase fronts of the broken symmetry phase.\n}\n\\label{fig:KZcartoon}\n\\end{figure}\n\nIn QKZM a system initially prepared in its ground state is smoothly ramped across a critical point to the other side of the quantum phase transition. A distance from the critical point, measured by a dimensionless parameter $\\epsilon$ controlling a Hamiltonian, can be linearized close to the critical point as\n\\begin{equation}\n\\epsilon(t)=\\frac{t}{\\tau_Q}.  \n\\label{epsilont}\n\\end{equation}\nHere $\\tau_Q$ is a quench time. Initially, far from the critical point, the evolution is adiabatic, and the system follows its adiabatic ground state, see Fig.~\\ref{fig:KZcartoon}. The adiabaticity fails at $-\\hat t$ when the reaction time of the system given by the inverse of the gap becomes slower than the timescale $|\\epsilon\/\\dot \\epsilon| = |t|$ on which the transition is being imposed.\nThe gap closes like $\\Delta\\simeq|\\epsilon|^{z\\nu}$, where $z$ and $\\nu$ are the dynamical and correlation length exponents, respectively. From the equation $|t|\\simeq |t\/\\tau_Q|^{-z\\nu}$ we obtain $\\hat t\\simeq \\tau_Q^{z\\nu\/(1+z\\nu)}$ and the corresponding $\\hat\\epsilon=\\hat t\/\\tau_Q\\simeq\\tau_Q^{1\/(1+z\\nu)}$. In the naive ``freeze-out'' version of the impulse approximation the ground state at $-\\hat\\epsilon$, with a corresponding correlation length\n\\begin{equation}\n\\hat\\xi \\simeq \\tau_Q^{\\nu\/(1+z\\nu)},  \n\\label{hatxi}\n\\end{equation}\nis expected to characterize the state of the system until $+\\hat t$, when the evolution can restart. In this way, $\\hat\\xi$ becomes imprinted on the initial state for the final adiabatic stage of the evolution after $+\\hat t$. Simplistic as it is, the adiabatic-impulse-adiabatic approximation correctly predicts the scaling of the characteristic lengthscale $\\hat\\xi$ and the timescale \n\\begin{equation} \n\\hat t\\simeq \\hat\\xi^z,\n\\label{hatt}\n\\end{equation} \nwith the critical exponents and $\\tau_Q$.\nThey both diverge in the adiabatic limit, $\\tau_Q\\to\\infty$, where they become the unique relevant scales in the KZ scaling ansatz~\\cite{KZscaling1,KZscaling2,Francuzetal}. For instance, a two-point correlation function $C_R(t)$, between two sites separated by a distance $R$, should satisfy\n\\begin{equation} \n\\hat\\xi^{\\Delta} C_R(t) = F\\left(t\/\\hat\\xi^z,R\/\\hat\\xi\\right).\n\\label{CRscaling}\n\\end{equation}\nHere $\\Delta$ is a scaling dimension and $F$ a non-universal scaling function. Eq.~\\eqref{CRscaling} is expected to be accurate in the long-wavelength and low-frequency limit.\nIt is worth to observe here, that the crude adiabatic-impulse-adiabatic approximation is consistent with the scaling hypothesis \\eqref{CRscaling}. However, it implies a particular  (time independent) form of the scaling function $F$.\n\n\\begin{figure}[t]\n\\vspace{-0.5cm}\n\\includegraphics[width=0.9\\columnwidth,clip=true]{Fig2.pdf}\n\\vspace{-0.1cm}\n\\caption{  \n{\\bf Sonic horizon view of KZM.} Initially, the correlation length $\\xi$ follows adiabatically the equilibrium healing length that -- in the adiabatic ground state (black) -- diverges at the critical point. Critical slowing down means that the size of the correlation length will begin to lag behind the values dictated by the ground state of the Hamiltonian at about $- \\hat t$. Pre-transition fluctuations reach size $\\hat \\xi$ at that instant and seed subsequent evolution of the system. The new broken symmetry phase is therefore selected by fluctuations in domains if size $\\hat \\xi$ at $-\\hat t$. Broken symmetry spreads within the impulse time interval of $2 \\hat t$ with the velocity $2\\hat v$ in every direction, enlarging the resulting ``sound cone'' to roughly $ 5 \\hat \\xi$ by $\\hat t$. In the freeze-out approximation (blue), after $-\\hat t$ the correlation length freezes, and remains close to the adiabatic correlation length at $-\\hat t$. Both the freeze-out and the sonic horizon views lead to the same scalings, but they result in different estimates of the pre-factors for domain sizes and defect densities.\n}\n\\label{fig:KZreal}\n\\end{figure}\n\n\nAs emphasized already in the early papers, see Ref.~\\onlinecite{Z-a, Z-b, Z-c, Z-d}, the freeze-out is not the complete story, and often not even a good approximation. A simple ``sonic horizon'' argument appealing to causality that goes beyond the impulse approximation is often more accurate. It is illustrated schematically in Fig.~\\ref{fig:KZreal}. As long as the evolution is adiabatic, the rate of growth of the diverging adiabatic correlation length, $\\xi\\simeq|\\epsilon|^{-\\nu}$, is  \n\\begin{equation} \n\\frac{d\\xi}{dt}=\n\\frac{d\\epsilon}{dt}\\frac{d\\xi}{d\\epsilon}=\n\\frac{1}{\\tau_Q}\n\\frac{\\nu}{|\\epsilon|^{\\nu+1}}. \n\\end{equation} \nThis rate diverges at the critical point. Hence there must be time $-\\hat t$ when it exceeds the speed limit set by twice\n\\begin{equation} \n\\hat v\\simeq\\frac{\\hat\\xi}{\\hat t}\\simeq\\tau_Q^{-\\nu(z-1)\/(1+z\\nu)}.\n\\label{hatv}\n\\end{equation} \nThe scaling of $-\\hat t$ obtained in this way is the same as in Eq.~\\eqref{hatt}. \n\nCausality and the KZ velocity $\\hat v$ are also central for the short-cuts to adiabaticity via inhomogeneous KZM. Therein, the external driving field has a smooth position dependence, gradually taking the system across the critical point---one part after another. Velocity of the driven critical front below $\\hat v$ (which in general depends on the shape of the above position dependence) is expected to pave the way to adiabatic dynamics, both for classical\\cite{ inhomo_classical-a, KZnum-c, inhomo_classical-c, inhomo_classical-d, inhomo_classical-e, inhomo_classical-f} and for quantum\\cite{inhomo_quantum-a, *inhomo_quantum-aa, inhomo_quantum-b, *inhomo_quantum-c, inhomo_quantum-d, inhomo_quantum-e, inhomo_quantum-f, inhomo_quantum-g} systems.\n\nIn the QKZM the speed limit is central to the causal argument. It originates from the dispersion of quasiparticles at the critical point: $\\omega\\simeq k^z$. Their speed for a quasimomentum $k$ is $v=d\\omega\/dk\\simeq k^{z-1}$. Between $-\\hat t$ and $\\hat t$ the quench excites quasiparticles with the magnitude of $k$ up to $\\hat k\\simeq \\hat\\xi^{-1}$. The speed of quasiparticles with the largest excited $k$ is therefore $\\hat v\\simeq\\hat k^{z-1}\\simeq\\hat\\xi^{1-z}\\simeq\\hat\\xi\/\\hat t$. When $z\\geq1$, $\\hat v$ is an upper bound on the velocity of quasiparticles. A quench in a translationally invariant system excites entangled pairs of quasiparticles with opposite quasimomenta: $k$ and $-k$. When moving apart, they are spreading correlations across the system. For $z\\geq1$ the rate of correlation spreading is limited by twice the speed $\\hat v$ of the fastest quasiparticles. \n\nIn the crudest version, neglecting in particular dependence of $\\hat v$ on the distance from the critical point, the argument implies that after $-\\hat t$ the correlation range (i.e., the ``sonic cone'') continues to grow at the rate $2\\hat v$ until $+\\hat t$. By this time, the range increases from the initial $\\hat\\xi$ at $-\\hat t$ to a final $\\hat\\xi+2\\hat v \\times 2\\hat t\\approx 5\\hat\\xi$. The growth is roughly five-fold. Even if the $5$ is just a very rough estimate, it shows how much the actual evolution can differ from the impulse approximation. Nevertheless, this argument confirms the role of $\\hat\\xi$ as the key relevant scale of length. With or without the prefactor of $5$, the final correlation length is proportional to $\\hat\\xi$, i.e., it scales with $\\tau_Q$ in the same way as given by Eq.~\\eqref{hatxi}. \n\nFrom another perspective, in the impulse approximation, the adiabatic ground state $|\\psi\\rangle$ corresponding to $-\\hat t$ freezes out as the state of the system. After $-\\hat t$ the adiabatic ground state departs from the frozen state. The frozen state becomes a superposition over adiabatic eigenbasis $|n\\rangle$: $\\sum_n c_n(t) |n\\rangle$, where $c_n(t)=\\langle n|\\psi\\rangle$. As a first step beyond the impulse approximation, we can include approximate dynamical phases: $\\sum_n c_n(t) e^{-i\\omega_n t}|n\\rangle$, where $\\omega_n$ is the adiabatic eigenfrequency at the critical point. In a (non-interacting) translationally invariant system, the eigenstates consist of pairs of excited quasiparticles, $|k,-k\\rangle$, and the eigenfrequencies are sums of $2\\omega_k$. The dynamical phase factors become scrambled -- and the phases begin to appear random -- when the largest of them, $2\\omega_{\\hat k}t\\propto\\hat k^z t$, becomes comparable to $1$ near $\\hat t$. The dephasing begins when the evolution crosses over from the non-adiabatic KZ stage to the post-KZ adiabatic stage. That is when the phases definitely can no longer be ignored, but even before the cross-over the phase factors $e^{-2i\\omega_k t}$ make the quasiparticle phase fronts propagate and let the quasiparticles spread the correlations across the system.\n\nFor $z=1$, when the dispersion is linear in $k$ and the quasiparticles have a definite speed of sound. This effect was termed the quasiparticle event horizon \\cite{EventHorizon}. In the QKZM context, it was considered in Refs. \\onlinecite{KZscaling1,Francuzetal}---see Fig.~\\ref{fig:Ising} for an example of the\nprototypical 1D quantum Ising model.\n\n\n\n\\begin{figure}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\columnwidth]{fig2a.pdf}\n\\vspace{-0.7cm}\n\\caption{ {\\bf Quantum Ising chain.} \nThe correlation length during the quench in the 1D quantum Ising model, $H=-\\sum_n(1-\\epsilon) \\sigma_n^z + \\sigma^x_n \\sigma^x_{n+1} $, where the dynamical critical exponent $z=1$ and the excited quasiparticles posses definite speed of sound. Compare with Fig.~\\ref{fig:KZreal}.\nData from Ref.~\\onlinecite{Francuzetal}.\n}\n\\label{fig:Ising}\n\\end{figure}\n\n\nIn this paper we go beyond $z=1$ and present two examples with $z>1$: the classical Ising model with Glauber dynamics in Section \\ref{sec:Onsager} and the generalized quantum XY chain in Section \\ref{sec:XY}. They both exhibit an effective event horizon with a speed limit that depends on the quench time $\\tau_Q$.\nThe generic scenario is delimited by two examples where the sonic horizon effect cease to manifest because one of its underlying assumptions is not satisfied. The first one is the random Ising model in Section \\ref{sec:Random}, where localization of excited quasiparticles prevents the spreading of correlations, thus yielding in effect a ``freeze-out''. The other is the extended Ising model with long-range interactions in Section \\ref{sec:LR}, where the dynamical exponent $z$ is less than $1$. The excited quasiparticles with $k\\to0$ have infinite velocity, and the speed $\\hat v$ at the maximal excited $\\hat k$ is not an upper but a lower speed limit. Consequently, there is no sonic horizon effect, and the correlations have a long-range power-law tail that can evolve in time. After $-\\hat t$ the tail begins to lag behind its adiabatic evolution. However, instead of completely freezing out, it continues to grow at a finite rate.\n\n\\begin{figure}[b]\n\\vspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig3.pdf}\n\\vspace{-0.7cm}\n\\caption{ {\\bf Classical Ising in 2D on a square lattice.} \nIn (a),\nenergy per site, $E$, as a function of $\\beta(t)$ for different quench times $\\tau_Q$.\nFor reference, \nthe black line is the equilibrium internal energy $U(\\beta)$. \nWith increasing quench time, the quench curves converge to the equilibrium one.\nIn (b),\nexcitation energy per site, $\\Delta E=E-U$, as a function of scaled time for different quench times $\\tau_Q$.\nHere, both $\\Delta E$ and $t$ are rescaled according to the KZM predictions.\nFor large $\\tau_Q\\gg1$ the scaling makes them collapse in the KZ regime between \n$t\\approx-\\hat t$ and $t\\approx\\hat t$, corresponding to the rescaled times $-1$ and $+1$, respectively.\nAt later times the phase ordering kinetics steps in, which goes beyond the KZ physics.\n}\n\\label{fig:onsagerenergy}\n\\end{figure}\n\\section{ Classical Ising model: ${\\mathbf z=2}$ } \n\\label{sec:Onsager}\n\nWe begin with the classic example of the classical Ising model on a periodic square lattice of size $L\\times L$:\n\\begin{equation}\nH~=~-\\sum_{\\langle j,j'\\rangle} \\sigma^z_j \\sigma^z_{j'} ~.\n\\label{Honsager}\n\\end{equation}\nOn an infinite lattice, the critical inverse temperature would be $\\beta_c=\\ln(1+\\sqrt2)\/2\\simeq 0.4407$.\nThe relevant equilibrium exponents are $\\nu=1$ and $\\eta=1\/4$ \\cite{Schultz1964}.\nWe model relaxation to an external heat bath with the Glauber dynamics: \nMonte Carlo update thermalizes one spin (chosen at random) at a time.\nThe time needed for $L^2$ such one-spin updates sets unit of time.   \nFor such simple relaxation, the dynamical exponent is $z=2$, belonging to the universality class of model-A dynamics~\\cite{hohenberg1977theory}. \nWe performed all our numerical simulations on a $4096\\times 4096$ lattice. This lattice size is $100$ times longer than the longest correlation range encountered in the simulations, hence any finite size effects are eliminated with a wide safety margin. All results were averaged over $50$ repetitions of the quench, each of them starting from a different initial random spin configuration at infinite temperature.\n\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth]{fig4.pdf}\n\\vspace{-0.5cm}\n\\caption{  \n{\\bf Classical Ising model in 2D.} The scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$\nfor scaled times $t\/\\hat t=-1,0,1$ (left to right) and different quench times $\\tau_Q$.\nFor each scaled time,\nwhen $\\tau_Q\\gg1$ the plots with different $\\tau_Q$ collapse to a single scaling function $F(t\/\\hat t,R\/\\hat\\xi)$\ndemonstrating the KZ scaling (\\ref{CRscaling}) hypothesis for slow enough quenches.\n}\n\\label{fig:corrcollapse}\n\\end{figure*}\n\n\nThe inverse temperature of the heat bath is ramped linearly in time,\n\\begin{equation} \n\\beta(t)=\\beta_c\\left(1+\\frac{t}{\\tau_Q}\\right),\n\\end{equation}\nstarting with random spin configuration at $\\beta(-\\tau_Q)=0$. Figure~\\ref{fig:onsagerenergy}(a) shows the energy $E$ during the ramp as a function of $\\beta$ for different quench times $\\tau_Q$. Generally, the system is more ordered for slower quenches. For slow enough quenches, the KZ picture emerges. The system evolves adiabatically until it begins to go out of equilibrium around $-\\hat t$, where $\\hat t\\simeq\\tau_Q^{2\/3}$ is the KZ timescale.\n\nIn order to see how the excitation energy should depend on the quench time, let us consider the equilibrium internal energy. Near the critical point it is\n\\begin{equation} \nU(\\beta) = - \\left[ 1 + A (\\beta-\\beta_c) \\ln\\left|\\beta-\\beta_c\\right| \\right]\/\\tanh\\beta_c,\n\\end{equation}\nwhere $A\\simeq 1$ is a constant. In the adiabatic-impulse approximation the state becomes effectively frozen near $-\\hat t$ when $\\beta_c-\\hat\\beta\\simeq\\tau_Q^{-1\/3}$. At $\\beta_c$ the energies of the frozen state (i.e. the instantaneous state at $-\\hat t$) and the equilibrium one differ by\n\\bea\n\\Delta E(\\beta_c) &=&\nU(\\hat\\beta)-U(\\beta_c) \\nonumber\\\\\n&=&\nA (\\beta_c-\\hat\\beta) \\ln\\left|\\hat\\beta-\\beta_c\\right|\/\\tanh\\beta_c \\nonumber\\\\\n&\\simeq &\n\\tau_Q^{-1\/3}\\ln\\left(\\tau_Q\/\\tau_0\\right).\n\\end{eqnarray} \nHere $\\tau_0\\simeq1$ is a constant. We can see that, up to a subleading logarithmic correction, the KZ scale of energy is $\\simeq\\tau_Q^{-1\/3}$. Accordingly, in Figure~\\ref{fig:onsagerenergy}(b) we show scaled excitation energy $\\tau_Q^{1\/3}(E-U)$ in function of scaled time $t\/\\hat t$. For $\\tau_Q\\gg1$ the plots for different $\\tau_Q$ collapse in the KZ regime: $-1<t\/\\hat t<1$. There is no collapse at later times when phase ordering kinetics \\cite{POK} brings in new physics beyond the KZ mechanism.\nHaving seen how the excitations build up, we can have a closer look at the ferromagnetic correlator\n\\begin{equation} \nC_R(t) =\n\\langle \\sigma^z_i \\sigma^z_{j} \\rangle -\n\\langle \\sigma^z_i \\rangle \\langle \\sigma^z_{j} \\rangle.\n\\end{equation}\n\n\\begin{figure}[b]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\columnwidth]{fig5.pdf}\n\\vspace{-0.3cm}\n\\caption{  {\\bf Classical Ising model in 2D.}\nIn (a),\nthe scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$ for $\\tau_Q=2^{15}$ at different scaled times $t\/\\hat t\\in[-1,1]$. The dashed line marks a cut-off $c$ (here $c=0.1$) whose crossing point is a working definition of a scaled correlation range $\\xi_c\/\\hat\\xi$.\nIn (b),\nthe scaled correlation range $\\xi_c\/\\hat\\xi$ as a function of the scaled time $t\/\\hat t$ for the cut-off $c=0.1$.\nThe best linear fit yields a slope $0.72$ as an estimate of the scaled velocity.\nIn (c),\nthe scaled velocity (slope) as a function of the cut-off. A linear extrapolation to zero cut-off yields $0.93$. This number is an estimate for (scaled) velocity at which the furthest correlation tail is spreading.\n}\n\\label{fig:spread}\n\\end{figure}\n\n\\begin{figure*}[t!]\n\\begin{center}\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth]{fig6.pdf}\n\\vspace{-0.7cm}\n\\end{center}\n\\caption{{\\bf Extended quantum XY chain.} Scaled ferromagnetic correlations $C^{xx}_R$ as a function of scaled distance $R\/\\hat{\\xi}$ for the extended XY model in (\\ref{XYHsigma}) plotted for different quench times $\\tau_Q$ and at different scaled times $t\/\\hat{t}=-1,0,1$. For large enough $\\tau_Q$ plots collapse towards a universal scaling function $F(t\/\\hat{t},R\/\\hat{\\xi})$ demonstrating the KZ scaling hypothesis (\\ref{CRscaling}).\n}\n\\label{fig:XYcollapse}\n\\end{figure*}\n\n\n\nHere, the sites $i$ and $j$ are separated by a distance $R$ along one of the axes and the average is taken in the state at time $t$. Figure~\\ref{fig:corrcollapse} shows the scaled correlator $\\hat\\xi^\\eta C_R$ in function of the scaled distance $R\/\\hat\\xi$ for three scaled times $t\/\\hat t=-1,0,1$. Here $\\hat\\xi=\\tau_Q^{1\/3}$ is the scale of length in Eq.~(\\ref{hatxi}) and $\\hat t=\\hat\\xi^z=\\tau_Q^{1\/3}$ is the scale of time in Eq.~(\\ref{hatt}). In accordance with the scaling hypothesis (\\ref{CRscaling}), for each scaled time the plots with different $\\tau_Q$ collapse to a common scaling function when $\\tau_Q$ is large enough. The plot for the longest $\\tau_Q=2^{15}$ is practically equal to the scaling function $F(t\/\\hat t,R\/\\hat\\xi)$.\n\n\nComparing the three panels in Figure~\\ref{fig:corrcollapse} we can see that (at odds with the impulse approximation) the range of correlations increases several times between $-\\hat t$ and $\\hat t$. This observation is further corroborated by Fig.~\\ref{fig:spread}(a) where we collect scaled plots for $\\tau_Q=2^{15}$ at different scaled times. Using $\\hat\\xi$ as a natural unit of length, this figure shows how the correlation spreads in the KZ regime between $-\\hat t$ and $+\\hat t$. \n\nFigure~\\ref{fig:spread}(b) shows how a scaled range of the correlator in Fig.~\\ref{fig:spread}(a) depends on the scaled time $s=t\/\\hat t$. Here, the range is defined as the distance at which the scaled correlator falls below a threshold---set here at $0.1$. The nearly linear time dependence is fitted with a line whose slope estimates the scaled velocity of correlation spreading. We repeated the same procedure for thresholds down to $0.02$ (below $0.02$ the tail of the correlator becomes too noisy for an unambiguous estimation of the correlation range). The estimated slopes\/velocities are collected in Fig.~\\ref{fig:spread}(c). The velocity increases rather slowly with a decreasing threshold. In the limit of zero threshold -- probing the longest correlation tail -- it extrapolates to $0.93$. We can conclude that the speed limit in this model is $\\hat v$, which allows the size of the sonic horizon to expand with\n\\begin{equation} \n2\\hat v = \n0.93~ \\frac{\\hat\\xi}{\\hat t} = \n0.93~ \\tau_Q^{-1\/2}.\n\\end{equation}\nIn natural units, in accordance with KZ prediction in Eq.~\\eqref{hatv}, the speed limit becomes slower for slower quenches.\n\\section{ Extended quantum XY chain: ${\\mathbf z=3}$ } \n\\label{sec:XY}\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{fig7.pdf}\n\\caption{{\\bf Extended quantum XY chain.}\nIn (a), \nfunction (\\ref{fitting}) is fitted (dashed lines) to the scaled ferromagnetic correlators $\\hat{\\xi}^{1\/4}C^{xx}_R$ for different scaled times during the non-adiabatic stage of the evolution: $t\/\\hat{t}\\in\\left[-2,2\\right]$.\nHere $\\tau_Q=2^{20}$.\nIn (b),\nthe fits provide the scaled correlation length $\\xi\/\\hat{\\xi}$ plotted against the scaled time $t\/\\hat{t}$. The slope of this plot is an estimate of the scaled velocity $v\/\\hat{v}=0.84$ at which the correlations are spreading. \nIn (c),\na more accurate approximation of the velocity is obtained by taking a derivative of the plot (b) with respect to the scaled time, which yields the scaled velocity $v\/\\hat{v}$ as a function of the scaled time.\nMaximal velocity is $v_{\\rm max}\/\\hat v=0.84$ for $\\tau_Q=2^{20}$. \nIn (d),\nthe maximal velocity is shown as a function of $\\tau_Q$.\nThe red line is a fit, which, for large $\\tau_Q$, saturates at $v_{max}\/\\hat v= 0.868$. \n}\n\\label{fig:XYspeed}\n\\end{figure*}\n\n\nWe step into the quantum realm \nwith the extended quantum XY periodic spin chain,\n\\bea\nH&=&-\\sum_{n=1}^N\n\\left[\n(1-\\epsilon) \\sigma^z_n +\nJ^{xx}\\sigma^x_n\\sigma^x_{n+1}+ \nJ^{yy}\\sigma^y_n\\sigma^y_{n+1}+\n\\right. \\nonumber \\\\\n&&\n\\left.\nJ^{xzx}\\sigma^x_n \\sigma^z_{n+1} \\sigma^x_{n+2}+ \nJ^{yzy}\\sigma^y_n \\sigma^z_{n+1} \\sigma^y_{n+2}\n\\right],\n\\label{XYHsigma}\n\\end{eqnarray} \na generalization of the quantum XY chain. It combines the desired $z>1$ with exact solvability.\nHere we consider an anisotropic model:\n\\bea\n&&\\left(J^{xx},J^{yy},J^{xzx},J^{yzy}\\right)=\\nonumber\\\\\n&&\n\\left(a\\frac{1+\\gamma}{2},a\\frac{1-\\gamma}{2},\n      b\\frac{1+\\delta}{2},b\\frac{1-\\delta}{2}\\right)\n\\end{eqnarray}\nwith the external magnetic field parametrized by $\\epsilon\\in[-1,1]$. The parameter is driven linearly (\\ref{epsilont}) from an initial $\\epsilon=-1$ in the paramagnetic phase, across the critical point at $\\epsilon=0$, to a final $\\epsilon=1$ in the ferromagnetic phase. After the Jordan-Wigner and Fourier transformations (\\ref{JordanWigner},\\ref{Fourier}), the Hamiltonian becomes:\n\\bea\nH &=\\sum_{k>0} &\nA_k(\\epsilon) \\left(c_k^\\dag c_k+c_{-k}^\\dag c_{-k}\\right) \\nonumber \\\\ \n&&\n+B_k \\left(c_kc_{-k}+c_{-k}^\\dag c_k^\\dag\\right),\n\\label{XYHk}\n\\end{eqnarray}\nwhere,\n\\bea \nA_k(\\epsilon) &=& 1-\\epsilon-a\\cos k-b\\cos 2k,  \\nonumber\\\\  \nB_k           &=& a\\gamma\\sin k+b\\delta\\sin 2k.\n\\end{eqnarray}\nIt is diagonalized by eigenmodes of the stationary Bogoliubov-de Gennes equations:\n\\bea\n\\omega_k\n\\left(\n\\begin{array}{c}\nU_k \\\\\nV_k\n\\end{array}\n\\right)&=&\n2\\left[\\sigma^z A_k(\\epsilon)+\\sigma^x B_k \\right] \n\\left(\n\\begin{array}{c}\nU_k \\\\\nV_k\n\\end{array}\n\\right),\n\\label{XYBdGk}\n\\end{eqnarray}\nwith eigenfrequencies\n$\n\\omega_k=2\\sqrt{A_k^2(\\epsilon)+B_k^2}.\n$\nHere, we set $a = 4\/3, b=-1\/3, \\gamma =1\/2, \\delta=1$. For the critical $\\epsilon=0$, we obtain a cubic dispersion relation\n\\begin{equation} \n\\omega_k\\simeq |k|^3,\n\\label{XYomegakz}\n\\end{equation} \nexpanding $\\omega_k$ around $k=0$, and the dynamical exponent $z=3$. On the other hand, setting $k=0$ and expanding in small $\\epsilon$, we obtain the gap opening as $\\omega_0\\simeq|\\epsilon|^1$. Hence $z\\nu=1$ and $\\nu=1\/3$. The dominant static ferromagnetic correlations have oscillatory tails:\n\\bea\nC_R     &=&\\langle\\sigma_i^x\\sigma_{i+R}^x\\rangle-\n           \\langle\\sigma_i^x\\rangle\\langle\\sigma_{i+R}^x\\rangle \\nonumber \\\\\n        &\\sim & R^{-\\eta} e^{-R\/\\xi}\\cos(b~R\/\\xi+c).\n\\end{eqnarray} \nHere $\\xi\\sim|\\epsilon|^{-\\nu}$ is a static correlation length and $\\eta=1\/4$. Given the exponents $z$ and $\\nu$, we can define the dynamical length (\\ref{hatxi}) and time (\\ref{hatt}) scales: $\\hat\\xi=\\tau_Q^{1\/6}$ and $\\hat t=\\tau_Q^{1\/2}$, respectively. The fastest excited quasiparticles have velocity $\\hat v\\simeq\\tau_Q^{-1\/3}$, see Eq.~\\eqref{hatv}. \n\nThe time-dependent quench is solved in appendix \\ref{XYtime} in a standard way \\cite{d2005} by mapping to the Landau-Zener problem, see appendix \\ref{LZ}. The KZ scaling hypothesis (\\ref{CRscaling}) is demonstrated in Fig.~\\ref{fig:XYcollapse} by collapse of the plots for different $\\tau_Q$. A perfect collapse requires very large $\\tau_Q\\simeq2^{20}$, as explained by Eq.~\\eqref{rescuv}. The oscillatory behaviour of $C_R$ in the original paramagnetic phase survives through the transition. In order to estimate the range of the scaled correlators, in Fig.~\\ref{fig:XYcollapse} we fit their tails with oscillatory functions of the form:\n\\begin{equation} \n\\hat{\\xi}^{1\/4} C_R(t)=\na~(R\/\\hat{\\xi})^{-1\/4}~\\mathrm{e}^{-\\frac{R\/\\hat{\\xi}}{\\xi\/\\hat{\\xi}}}\\cos(b~R\/\\hat{\\xi}+c),\n\\label{fitting}\n\\end{equation}\nwhere $a,b,c$ and $\\xi\/\\hat\\xi$ are the fitting parameters. We are interested how fast the scaled correlation length $\\xi\/\\hat\\xi$ grows with the scaled time $t\/\\hat t$. In order to reveal the universal behavior undisturbed by any short-range effects, we perform the fit in the range of scaled distances $R\/\\hat\\xi>2.5$. Ferromagnetic correlations for different scaled times $t\/\\hat{t}$, together with the fits, are shown in Fig.~\\ref{fig:XYspeed}(a) (for $\\tau_Q=2^{20}$). The correlation length $\\xi\/\\hat{\\xi}$ as a function of scaled time is shown in Fig.~\\ref{fig:XYspeed}(b). Its slope, equal to $0.838$, provides an estimate of the scaled velocity at which the correlations are spreading. For completeness, in Fig.~\\ref{fig:XYspeed}(c) we plot the derivative of the plot in Fig. \\ref{fig:XYspeed}(b) with respect to the scaled time. Its maximal value is shown in Fig.~\\ref{fig:XYspeed}(d) as a function of $\\tau_Q\\in [2^{10},2^{20}]$. For large $\\tau_Q$ it extrapolates to\n\\begin{equation} \n2 \\hat v = 0.868~ \\frac{\\hat{\\xi}}{\\hat{t}} = 0.868~ \\tau_Q^{-1\/3}.\n\\end{equation}\nAs in the classical 2D Ising model, where also $z>1$, the correlation spreading becomes slower for slower quenches.\n\n\\section{ Random Ising model } \n\\label{sec:Random}\n\nIn order to break the translational invariance and impede the propagation of excited quasiparticles that could spread correlations, next we consider the random Ising model (RIM) in one dimension defined by the Hamiltonian\n\\bea\nH=\n-\\sum_{n=1}^N\n\\left[ (h_{n} - \\epsilon)\\sigma^z_n \\ +\n J_{n}\\sigma^x_n\\sigma^x_{n+1}\n\\right].\n\\label{Hamil_RIM}\n\\end{eqnarray} \nHere both transverse fields $h_n$'s and nearest-neighbor couplings $J_n$'s are randomly selected from a uniform distribution between $0$ and $1$. The quantum critical point at $\\epsilon=0$ is separating the paramagnetic phase ($\\epsilon<0$) from the ferromagnetic one ($\\epsilon>0$). The critical point in this model is surrounded by Griffiths region \\cite{Griffiths-a, Griffiths-b, Griffiths-c}, where the presence of the so-called rare regions primarily manifests in two features: the activated dynamical scaling (the dynamical exponent $z\\rightarrow\\infty$) at the quantum critical point and the existence of singular regions where the linear susceptibility diverges even away from the critical point. The locally ordered rare regions act as giant spins that flip as a whole and are responsible for the exponentially slow dynamics near the critical point. These features are encapsulated in a dynamical exponent that diverges at the critical point as $z \\sim|\\epsilon|^{-1}$, see Ref. \\onlinecite{Fisher1-a, Fisher1-b} which also shows that the correlation length exponent is $\\nu = 2$. The average correlation function at criticality is a power law\\cite{McCoyWu}:\n\\begin{equation} \nC^{xx}_R=\\langle \\sigma_{i}^{x}\\sigma_{i+R}^{x}\\rangle \\sim R^{-\\eta},\n\\end{equation}\nwhere $\\eta = \\frac{3-\\sqrt{5}}{2}\\approx 0.38$. \n\nWe consider ramping the parameter $\\epsilon$ linearly as a function of time (\\ref{epsilont}) driving the Hamiltonian from the initial paramagnetic ground state, across the critical point at $t=0$, into the ordered phase. Ref. \\onlinecite{dziarmagaRIM-a, dziarmagaRIM-b, dziarmagaRIM-c} showed that the KZ correlation length $\\hat \\xi$ of the model varies logarithmically with the quench rate $\\tau_Q$:\n\\bea\n\\centering\n\\hat{\\xi} = \\ln^2{(\\tau_Q\/a)},\n\\label{hatxiRIM}\n\\end{eqnarray}\nwhen $\\ln{(\\tau_Q\/a)}\\gg 1$. Here $a\\simeq1$ is a non-universal constant. We can see, that the dependence on $\\tau_Q$ is very weak compared to any of the usual KZ power-law scalings (\\ref{hatxi}). Taking into account that the correlation-length exponent near the critical point is $\\nu = 2$, we get an estimate of the characteristic timescale,\n\\bea\n\\centering\n\\hat{t} = \\frac{\\tau_{Q}}{\\ln{(\\tau_{Q}\/a)}},\n\\label{hattRIM}\n\\end{eqnarray}\n for $\\ln{(\\tau_Q\/a)} \\gg 1$. \n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{fig8.pdf}\n\\caption{{\\bf Random Ising model.} Average scaled density of excitations $\\hat\\xi d_{\\rm ex}$ during the quench as a function of scaled time $t\/\\hat t$ for different quench times $\\tau_Q$. Here, we use Eqs.~(\\ref{hatxiRIM},\\ref{hattRIM}) with $a = 0.118$. This $a$ was tuned to obtain the best possible collapse at $t\/\\hat t=0$ but the plots collapse well during the whole quench. The KZ diabatic stage (shaded in green) extends roughly from $-0.25t\/\\hat t$ to $0.25t\/\\hat t$. This is where the excitation grows before it saturates in the last adiabatic stage. Averaging was done over 30 random realizations for a lattice of size $N = 128$.\n}\n\\label{FigDoERIM}\n\\end{figure}\n\\begin{figure*}[t]\n\\vspace{-0.1cm}\n\\includegraphics[width=\\textwidth,clip=true]{fig9.pdf}\n\\vspace{-0.1cm}\n\\caption{ {\\bf Random Ising model.} Average scaled ferromagnetic correlations $\\hat\\xi^\\eta C^{xx}_R$ as a function of scaled distance $R\/\\hat{\\xi}$ for the random Ising model (\\ref{Hamil_RIM}) plotted for different quench times $\\tau_Q$ and at different scaled times $t\/\\hat{t}=-0.25,0,0.25$. For large enough $\\tau_Q$ plots collapse towards a universal scaling function $F(t\/\\hat{t},R\/\\hat{\\xi})$ demonstrating the KZ scaling hypothesis\n(\\ref{CRscaling}).\n}\n\\label{fig:RIMcollapse}\n\\end{figure*}\n\nIn order to solve the dynamics, we map the Hamiltonian by the Jordan-Wigner transformation (\\ref{JordanWigner}) to a quadratic spinless free-fermionic model\n\\bea\nH = \\sum_{n}(h_{n}-\\epsilon)c_{n}^\\dag c_{n} -J_{n}c_{n}^\\dag c_{n+1} - J_{n}c_{n+1} c_{n} + {\\rm h.c}.\n\\label{JW_RIM}\n\\end{eqnarray}\nFollowing the convention of this article, we confine ourselves to the subspace of even parity of $c$-quasiparticles (anti-periodic boundary conditions, i.e., $c_{N+1} = -c_1$) and in this subspace, we diagonalize the Hamiltonian by a Bogoliubov transformation:\n\\bea\n\\centering\nc_{n} = \\sum\\limits_{m=1}^{N}(U_{nm}\\gamma_{m} + V_{nm}^* \\gamma_{m}^\\dag)\n\\label{bogoliubov_RIM}\n\\end{eqnarray}\nThe index $m$ labels the (Bogoliubov) eigenmodes of the stationary Bogoliubov-de Gennes equation\n\\bea\n\\centering\n\\omega_{m}U_{n,m}^{\\pm} = 2h_{n}U_{n,m}^{\\mp} - 2j_{n-1}U_{n-1,m}^{\\mp},\n\\label{stationary_B-DeG}\n\\end{eqnarray}\nwhere $\\omega_{m}>0$, $U_{nm}^{\\pm} = U_{nm}\\pm V_{nm}$ and anti-periodic boundary conditions $(U_{N+1,m}^{\\pm} = -U_{1,m}^{\\pm},U_{0,m}^{\\pm}=-U_{N,m}^{\\pm})$ is implemented. The eigenstates ($U_{nm},V_{nm}$), with positive energy, $\\omega_{m}>0$ normalized so that $\\sum_{n}(|U_{nm}|^2 + |V_{nm}|^2) = 1$, define quasiparticle operators $\\gamma_{m}=U_{nm}^{*}c_{n}+V_{nm}c_{n}^{\\dag}$. We have corresponding negative energy components of the eigenstates labelled ($U_{nm}^{neg},V_{nm}^{neg}$) with energy $-\\omega_{m}$, which defines a quasiparticle operator $\\gamma_{m}^{neg}=(U_{nm}^{neg})^{*}c_{n}+V_{nm}^{neg}c_{n}^{\\dag}$. The Bogoliubov transformation renders the Hamiltonian to be $H = \\sum_{m = 1}^{N}\\omega_{m}(\\gamma_{m}^{\\dag}\\gamma_{m} - \\frac{1}{2})$. In the even parity subspace only states with even number of quasi-particles belong to the spectrum of $H$.\n\nIn order to find whether the RIM fits into our narrative of the uniform scaled velocity of the correlation spreading by quasiparticles,\nwe proceed with the numerical simulation of the quench. \nWe prepare the initial state of the system deep in the paramagnetic phase in the ground state, i.e., in the Bogoliubov vacuum state for quasiparticles at an initial time $t_{0}$ where $\\epsilon(t_{0}) = 5$. As we tune the parameter $\\epsilon(t)$ towards $0$, see Eq.~\\eqref{epsilont}, the state of the systems departs from its adiabatic ground state and gets excited due to closing of the energy gap near the critical point. We work in the Heisenberg picture where we assume that the excited state is a Bogoliubov vacuum while the time-dependence is ascribed to a set of time-dependent quasi-particle operators\n\\bea\n\\centering\n\\gamma_{m}(t)=u_{nm}^{*}(t)c_{n}+v_{nm}^{*}(t)c_{n}^{\\dag}\n\\end{eqnarray}\nThe Bogoliubov modes $u_{nm}$ and $v_{nm}$ solve the time-dependent Bogoliubov-de Gennes  equations:\n\\bea\n\\centering\ni\\frac{du_{nm}^{\\pm}}{dt} = 2(h_{n}-\\epsilon(t))u_{nm}^{\\mp}-2J_{n}u_{n\\mp1,m}^{\\mp}\n\\label{t_B-deG}\n\\end{eqnarray}\nWe integrate equation (\\ref{t_B-deG}) numerically using the 2nd order Suzuki-Trotter method, see Appendix \\ref{tB-deG_RIM}. The density of excited Bogoliubov quasiparticles $d_{ex}(t)$ can be calculated at each time step by projecting the time-dependent Bogoliubov modes ($u_{nm}(t),v_{nm}(t)$) onto the corresponding instantaneous static negative Bogoliubov modes ($U_{nm}^{\\text{neg}},V_{nm}^{\\text{neg}}$):\n\\bea\n\\centering\nd_{ex}(t) = \\frac{1}{N} \\sum_{s}^N\\sum_{p}^N \\left|\\langle U_{p}^{\\text{neg}},V_{p}^{\\text{neg}}|u_{s}(t),v_{s}(t)\\rangle\\right|^2.\n\\label{doe_RIM}\n\\end{eqnarray}\n\nWhile in the initial adiabatic stage, the system remains in the instantaneous ground state and hence the density of quasiparticle excitations is $0$. On tuning the field $\\epsilon(t)$ towards its critical value at $t\/\\hat{t} = 0$, the system gets excited and consequently $d_{ex}$ starts to grow. As we keep increasing the field $\\epsilon(t)$, $d_{ex}$ saturates as the system ends its non-adiabatic journey across the critical point. In Fig.~\\ref{FigDoERIM} we see that the scaled plots (averaged over disorder) for different $\\tau_Q$ collapse in accordance with the dynamical scaling hypothesis (\\ref{CRscaling}). \n\n\\begin{figure}[b]\n\\includegraphics[width=\\columnwidth]{fig10.pdf}\n\\caption{   {\\bf Random Ising model.}\nIn (a),\nthe scaled correlator $\\hat\\xi^{\\eta}C_R$ as a function of the scaled distance $R\/\\hat\\xi$ for $\\tau_Q=2^{11}$\nat different scaled times $t\/\\hat t$ in $[-0.25,0.25]$.\nThe dashed line marks a cut-off (here $1$) whose crossing point \nis a working definition of a scaled correlation range $\\xi_c\/\\hat\\xi$.\nIn (b),\nthe scaled correlation range $\\xi_c\/\\hat\\xi$ as a function of the scaled time $t\/\\hat{t}$ for the cut-off $c=0.5$. The best linear fit yields a slope of $0.113$ as an estimate of the scaled velocity.\nIn (c),\nthe scaled velocity (slope) as a function of the cut-off, with the error bars indicating fitting errors.\n}\n\\label{fig:RIMspread}\n\\end{figure}\n\nThe hypothesis for correlations in Eq.~\\eqref{CRscaling} is verified by a similar collapse of the plots in Fig.~\\ref{fig:RIMcollapse}. We consider large $\\tau_Q$ as the logarithmic KZ scaling laws are only valid in that regime. The correlation function is averaged over $100$ instances of disorder and lattice translations on a lattice of $N=256$. This lattice is a few times longer than the longest correlation range. The details of calculating the correlation functions are given in Appendix \\ref{appcorrRIM}.\n\nA close look at Fig.~\\ref{fig:RIMcollapse} reveals that the scaled dynamical correlation function grows with scaled time. In order to appreciate the growth, we collect the scaled correlators for different scaled times in Fig.~\\ref{fig:RIMspread}(a). In order to estimate the scaled correlation range $\\xi_c\/\\hat{\\xi}$ as a function of scaled time, we choose a cut-off $c$ for the scaled correlation and calculate scaled distances corresponding to the cut-off by spline interpolation. Fig.~\\ref{fig:RIMspread}(b) shows the scaled range in function of the scaled time for the cut-off (selected equal to $0.5$) in Fig.~\\ref{fig:RIMspread}(a).\nNear the critical point, we approximate it with linear dependence where the slope is $0.113\\hat\\xi\/\\hat t$. We repeat the same procedure for other cut-offs. We present the enumerated slopes for various cut-offs in Fig.~\\ref{fig:RIMspread}(c), with the error bars indicating the fitting error. Finally, we estimate the sonic horizon expansion speed in this model with the maximal observed value\n\\bea\n\\centering\n2\\hat{v} = 0.12 \\frac{\\hat{\\xi}}{\\hat{t}} = 0.12 \\frac{\\ln^3(\\tau_Q\/a)}{\\tau_Q}\n\\end{eqnarray}\nCompared to the previous models, with the prefactor $0.12$, the speed limit is significantly below expectations in this case. The entangled Bogoliubov quasiparticles get excited but, due to their localization by disorder, they do not propagate to spread correlations appreciably. In the spin language, the locally ordered regions are excited, but they essentially stay where they are. One may even argue that in this case, the adiabatic-impulse-adiabatic freeze-out approximation captures the essential physics.\n\n\n\\section{Long-range extended Ising model: $0<z<1$ } \n\\label{sec:LR}\n\n\nThe long-range extended Ising model is a further generalization of the extended XY model that we have discussed in Sec. \\ref{sec:XY}. Now the Hamiltonian of the system includes not only the nearest-neighbor and next-nearest-neighbor interaction, but all possible long-range cluster terms: \n\\bea\nH=\n-\\sum_{n=1}^N\n\\left[\n (1-\\epsilon) \\sigma^z_n +\n\\sum_{r=1}^{N-1} \nJ_r\\sigma^x_n \\sigma^x_{n+r} \\prod_{i=n+1}^{n+r-1} \\sigma^z_i\n\\right].\n\\label{Hamil_LR}\n\\end{eqnarray} \nHere \n\\begin{equation}\nJ_r ~=~ \\frac{1}{\\zeta(\\alpha)} ~ \\frac{1}{r^\\alpha},\n\\end{equation}\nand the normalization by the Riemann zeta function, possible for $\\alpha>1$, is such that $\\sum_r J_r=1$. The case of $\\alpha>2$ is not the most interesting here because the model behaves effectively like the short-range one \\cite{LRKitaev1,LRKitaev2,LR1,LR2,LRBosonic}. Similarly, when $0\\le\\alpha\\le1$ we would need to restrict ourselves to a finite system because the thermodynamic limit does not exist in this case, and the model behaves effectively like the Lipkin-Meshkov-Glick model \\cite{LMG} with infinite-range interactions. Therefore, we focus here on the intermediate $1<\\alpha<2$, where a cross-over between the short and infinite-ranges happens. In this regime, we consider the linear quench (\\ref{epsilont}) driving the system from the initial paramagnetic phase at $t=-\\infty$, across the critical point at $t=0$, to the final ferromagnetic phase. \n\n\\begin{figure}[b]\n\\includegraphics[width=\\columnwidth]{fig11.pdf}\n\\caption{  {\\bf Long-range extended Ising model.}\nIn (a),\nthe static correlation function $C_R$ at the critical $\\epsilon=0$ for $\\alpha=3\/2$. \nThe solid line is a linear fit with a slope $-1.002\\approx -1 $.\nThe same panel shows the same function far from the critical point at $\\epsilon=-2$. \nThe solid line is a linear fit with a slope $-1.45 \\approx -(3 -\\alpha) $.  \nIn (b),\nthe scaled static correlation function $\\xi C_R$ as a function of scaled distance $R\/\\xi$.\nIn accordance with the static scaling hypothesis (\\ref{staticSH}),\nthe plots for different $\\epsilon$ collapse to a unique scaling function.\nWe can see a crossover from the critical $(R\/\\xi)^{-1}$ to the off-critical $(R\/\\xi)^{-3\/2}$\nnear $R\/\\xi=1$. The straight lines are linear fits with slopes $-1.085\\approx-1$ and $-1.42\\approx -(3-\\alpha)$ for smaller and larger $R\/\\xi$, respectively. \n}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=\\textwidth]{fig12.pdf}\n\\caption{ {\\bf Long-range extended Ising model.}\nThe scaled dynamical correlation function $\\hat\\xi C_R$ for different quench times $\\tau_Q$ at different scaled times, $s={t}\/{\\hat{t}}$, during the quench.\nHere $\\alpha=3\/2$.\nFrom left to right: $s=-1,0,0.35,0.4$. \nIn (a) at $s=-1$,\nthe evolution is still adiabatic and the plots for different $\\tau_Q$ collapse according to the static scaling hypothesis which is equivalent to the KZ hypothesis in this early adiabatic regime.\nIn (b) at $s=0$,\nwhen crossing the critical point the evolution is diabatic. Different $\\tau_Q$ collapse to a common scaling function $F(s,R\/\\hat\\xi)$ in accordance with the KZ scaling hypothesis (\\ref{CRscaling}).\nIn (c) at $s=0.35$,\nthe correlation function begins to bend down at distances between $R\/\\hat\\xi=10$ and $100$.\nDifferent $\\tau_Q$ collapse, as the evolution is still in the KZ regime.\nFinally in (d) at $s=0.4$,\nthe collapse fails due to the phase ordering after the system left the KZ regime and entered the adiabatic one. At the distances between $R\/\\hat\\xi=10$ and $100$ the correlations become negative for longer $\\tau_Q$.\n}\n\\label{fig:corrp0m}\n\\end{figure*}\n\n\nThanks to the string operator in the long-range interaction terms, the model can be mapped to a quadratic free fermion model and solved analytically. After the Jordan-Wigner transformation (\\ref{JordanWigner}) the Hamiltonian (\\ref{Hamil_LR}) becomes:\n\\bea\nH &=& -\\sum_n \\left(1-\\epsilon\\right)\\left(c_nc_n^\\dag-c_n^\\dag c_n\\right) \\nonumber \\\\\n&&- \\sum_{n,r} J_r\\left( c_n^\\dag c_{n+r} + c_n^\\dag c_{n+r}^\\dag + {\\rm h.c.}  \\right),\n\\label{HcLR}\n\\end{eqnarray}\nfor the anti-periodic boundary conditions. This representation is also known as the long-range Kitaev model \\cite{LRKitaev1, LRKitaev2}, where the hopping and pairing terms are of equal strength. After Fourier transformation (\\ref{Fourier}),\n\\bea\nH &=&\n-2\\sum_{k>0} \n(1-\\epsilon -  \\Re(\\tilde{J_k})) \\left(c_k^\\dag c_k+c_{-k}^\\dag c_{-k}\\right) \\nonumber \\\\\n&&+  ~\\Im(\\tilde{J_k})\\left(c_{k}^\\dag c_{-k}^\\dag + c_{-k}c_{k}\\right),\n\\label{HkLR}\n\\end{eqnarray}\nwhere  $\\Re(\\tilde{J_k})$ and  $\\Im(\\tilde{J_k})$ are, respectively, real and imaginary parts of the Fourier transform $\\tilde{J_k}=\\sum_r J_r e^{ikr}$. We have $\\tilde{J_k}=\\frac{{\\rm Li}_{\\alpha}(e^{ik})}{\\zeta(\\alpha)}$, where ${\\rm Li}$ is the polylogarithm function: ${\\rm Li}_\\alpha(x)=\\sum_{n=1}^\\infty\\frac{x^n}{n^\\alpha}$. \n\n\n\nWe can now find the stationary Bogoliubov-de Gennes equations (\\ref{XYBdGk}) with\n$\nA_k(\\epsilon)=1-\\epsilon-\\Re(\\tilde{J_k}),~~ B_k=\\Im(\\tilde{J_k})\n$\nand eigenfrequencies\n$\n\\omega_k=2\\sqrt{A_k^2(\\epsilon)+B_k^2}.\n$\nWe have a critical point at $\\epsilon=0$ where the gap closes for $k=0$. Another critical point, \nnot to be considered here, is at $\\epsilon=2(1-2^{-\\alpha})$ and $k=\\pi$. The dispersion relation at the critical $\\epsilon=0$ is\n\\bea\n\\omega_k \\simeq |k|^{\\alpha-1},\n\\end{eqnarray}\nhence the dynamical exponent is $z=\\alpha-1\\in(0,1)$. On the other hand, for small $\\epsilon$, the gap at $k=0$ closes as\n$ \n\\omega_0 = 2|\\epsilon|,\n$\nhence $z\\nu=1$. \n\n\n\nIn a short range model for large $R$ the correlation function decays exponentially with $R$ when the system is away from the critical point but this does not need to hold for long-range interactions. Indeed, our system has a power law scaling even far away from the critical point \\cite{LRKitaev1,LRKitaev2}:\n\\bea \nC_R &=&\n\\langle \\sigma_i^x \\sigma_{i+1}^z \\ldots \\sigma_{i+R-1}^z \\sigma_{i+R}^x\\rangle\n\\sim 1\/R^{3-\\alpha},\n\\label{Corr}\n\\end{eqnarray}\ncompare Fig.~\\ref{fig2}(a). On the other hand, as discussed in more detail in Ref.~\\onlinecite{natun}, at the critical point we expect a critical power law $C_R \\sim 1\/r^{\\eta}$ with $\\eta<\\alpha$. Indeed, in Fig.~\\ref{fig2}(a) we find that $\\eta=1$ for $\\alpha=3\/2$. Similarly as for short-range interactions, for a small $\\epsilon$ we expect a cross-over between the two power laws when $R$ is close to $\\xi\\sim\\epsilon^{-\\nu}$. Note that here $\\xi$ is not the usual exponential correlation length, even though it scales with $\\epsilon$ in the characteristic way. The crossover can be verified with a static scaling hypothesis:\n\\bea \n\\xi^\\eta ~C_R = F_{\\rm st}\\left( R\/\\xi \\right).\n\\label{staticSH}\n\\end{eqnarray} \nWith $\\xi=\\epsilon^{-\\nu}$ the plots of the scaled correlator $\\xi^\\eta C_R$ as a function of the scaled distance $R\/\\xi$ for different $\\epsilon$ should collapse to a common static scaling function $F_{\\rm st}(x)$. We expect $F_{\\rm st}(x)$ to cross-over around $x=1$ between the critical tail $x^{-1}$ for small $x$ and $x^{3-\\alpha}$ for large $x$. Indeed, this is what we see in Fig.~\\ref{fig2}(b). The unscaled leading static tail is, therefore, \n\\begin{equation} \nC_R\\simeq \\xi^{1\/2} R^{-3\/2}\n\\label{lrtailstatic}\n\\end{equation} \nfor $R\\gg\\xi$.\n\n\n\\begin{figure}[b]\n\\vspace{-0.5cm}\n\\includegraphics[width=\\columnwidth]{fig13.pdf}\n\\vspace{-0.5cm}\n\\caption{ {\\bf Long-range extended Ising model.}\nIn (a), \nscaled correlator $\\hat\\xi C_R$ for $\\alpha=3\/2$ as a function of scaled distance for different scaled times in the log-log scale.\nIn (b),\nin the inset the coefficients $A,B,C$ in Eq.~\\eqref{ABC(s)fit} fitted to the tail of the scaled correlator are shown as a function of scaled time $s=t\/\\hat t$. The main picture shows $A(t\/\\hat t)$ together with the adiabatic $A_{\\rm ad}(t\/\\hat t)$.\nIn the initial adiabatic stage $A$ follows $A_{\\rm ad}$. \nIn the KZ stage $A_{\\rm ad}$ diverges to infinity at the critical point forcing $A$ to lag behind.\n}\n\\label{fig:fig2a}\n\\end{figure}\n\n\nHaving verified the static hypothesis, we are encouraged to propose its dynamical version in the same form (\\ref{CRscaling}) as for local interactions. The dynamics is described by the time-dependent Bogoliubov-de Gennes equations (\\ref{tBdGk}) whose solution is presented in Appendix \\ref{AppLR}. The KZ mechanism has recently been verified in a similar model~\\cite{amitdutta}, where defect density was found to scale as $\\tau_Q^{-1\/2(\\alpha-1)}$ in agreement with the KZ prediction $\\tau_Q^{-\\nu\/(1+z\\nu)}$, see also Ref.~\\cite{KZLR2}. Further developments, like scaling of excitation energy (consistent with KZ) or possible experimental implementation, were considered in Refs.~\\onlinecite{MorigiLR,LRPlenio}.\nHere, we are interested in the build-up of correlations during the quench. In Fig.~\\ref{fig:corrp0m}(a) we show scaled correlation functions at different scaled times. For definiteness, in this figure we set $\\hat t=\\tau_Q^{1\/2}$ and $\\hat\\xi=\\tau_Q^{1\/2(\\alpha-1)}$ with both numerical prefactors equal to $1$. We can see that up to $t=0.35\\hat t$ the scaled correlators for different $\\tau_Q$ collapse demonstrating the KZ scaling hypothesis in Eq.~\\eqref{CRscaling}. At $t\/\\hat t\\approx 0.4$ they already fail to collapse, signalling the end of the KZ regime and the beginning of phase ordering in the second adiabatic stage. \n\nIn Fig.~\\ref{fig:fig2a}(a), we collect scaled correlators for different scaled times, $s=t\/\\hat t$, in the KZ regime. In order to gain better insight, in Fig.~\\ref{fig:fig2a}(b) we fit their tails with a function\n\\begin{equation} \n\\hat \\xi C_R(s)=A(s)(R\/\\hat\\xi)^{-3\/2}+B(s)(R\/\\hat\\xi)^{-2}+C(s)(R\/\\hat\\xi)^{-5\/2}.\n\\label{ABC(s)fit}\n\\end{equation} \nIts form is motivated by correlation tails after a sudden quench considered in appendices \\ref{SuddenAppInfinite} and \\ref{SuddenAppFinite}. Here, the coefficients $A,B,C$ are functions of the scaled time shown in Fig.~\\ref{fig:fig2a}(c). On top of the leading $A(s)$ tail, sub-leading dynamical correlations build up in time, with $B(s)$ growing negative and $C(s)$ positive. Their behavior is similar to what is observed after a sudden quench to the critical $\\epsilon=0$ either from $\\epsilon=-\\infty$ or $\\epsilon=-1$, see appendices \\ref{SuddenAppInfinite} and \\ref{SuddenAppFinite}, respectively. The main Fig.~\\ref{fig:fig2a}(b) shows $A(s)$, i.e., the coefficient of the leading long-range tail $\\propto R^{-3\/2}$. \n\n\nThe same figure shows $A_{\\rm ad}(s)$, i.e., the same coefficient but in case the evolution were adiabatic. $A_{\\rm ad}(s)$ was obtained by a fit to the far tail of the static correlation function at $\\epsilon(s)=s\/(\\tau_Q\/\\hat t)$. The same $A_{\\rm ad}(s)$ can be obtained by equating the static tail (\\ref{lrtailstatic}) with the adiabatic tail $C_R=A_{\\rm ad}(s)\\hat\\xi^{1\/2}\/R^{3\/2}$, compare (\\ref{ABC(s)fit}):\n\\begin{equation} \nA_{\\rm ad}(s) \\simeq\n\\left[\\frac{\\xi(s)}{\\hat\\xi}\\right]^{1\/2} \\simeq\n\\left[\\frac{\\epsilon(s)^{-\\nu}}{\\hat\\xi}\\right]^{1\/2} \\sim\ns^{-\\nu\/2}.\n\\end{equation} \n$A_{\\rm ad}(s)\\sim s^{-1}$ for $\\alpha=3\/2$ is consistent with the divergence we observe in Fig.~\\ref{fig:fig2a}(b). We can also see that, as expected, $A(s)$ follows $A_{\\rm ad}(s)$ in the initial adiabatic stage up to $s\\simeq -1$. After that, in the diabatic KZ stage, it lags behind as it cannot catch up with the diverging $A_{\\rm ad}(s)$. If it did not, and the evolution were adiabatic, then the correlation range -- defined by $R$ where $C_R$ falls below a fixed small cutoff -- would diverge near the critical point as $|t|^{-2\/3}$. \n\n\n\nAfter $A(s)$ begins to lag behind the adiabatic evolution, it continues to grow at a finite rate until the critical point. This is the stage where the long-range model is the most reminescent of the standard causal KZ picture. After the critical point $A(s)$ begins to dip down around $s\\approx 0.2$ but this is where the KZ scaling hypothesis (\\ref{CRscaling}) begins to be violated, see Fig.~\\ref{fig:corrp0m}(d).\n\n\\section{ Conclusion } \n\\label{SectionConclusion}\n\nWe recall the causal\/sonic horizon version of the Kibble-Zurek mechanism (KZM).\nIn the initial adiabatic stage of the evolution, \nthe correlation range follows its adiabatic counterpart.\nAt $-\\hat t$ it begins to lag behind the diverging adiabatic range and continues to grow at a finite rate set by the speed limit $\\simeq \\hat\\xi\/\\hat t$.\nThis way, between $-\\hat t$ and $+\\hat t$, the correlation range can increase several times at odds with the impulse approximation where it remains frozen. \n\nThere are notable exceptions, like the random quantum Ising chain, where the increase is small due to localization of excited quasiparticles that prevents them from spreading entanglement. There are interesting generalizations, like the long-range extended Ising model, where it is the power law correlation tail, rather than the exponential correlation length, that lags behind its diverging adiabatic evolution.\n\n\n\\acknowledgements\nAS would like to thank Titas Chanda for useful discussions.\nWe acknowledge funding by National Science Centre (NCN), \nPoland under projects No.~2016\/23\/B\/ST3\/00830 (JS,AF) \nand No. 2016\/23\/D\/ST3\/00384 (MMR), \nNCN together with European Union through QuantERA ERA NET \nprogram No.~2017\/25\/Z\/ST2\/03028 (AS,DS,JD),\nand Department of Energy under the Los Alamos National Laboratory \nLDRD Program (WHZ).\nWHZ was also supported by the U.S. Department of Energy, Office of Science, Basic Energy\nSciences, Materials Sciences and Engineering Division, Condensed Matter Theory Program.\nAF acknowledges financial support by Polish Ministry of Science and Education, project No. DI2015 021345, from the budget funds for science in 2016-2020 under the \"Diamond Grant\" program.\nThis research was carried out with the equipment purchased thanks to the financial support of the European Regional Development Fund in the framework of the Polish Innovation Economy Operational Program (contract No. POIG.02.01.00-12-023\/08). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\nThe theoretical study of anomalously slow relaxation processes in time-dependent force fields \nconstitutes a challenge of current research interest which is not free of ambiguity. \nIt is known that there is no unique physical mechanism responsible for the \noccurrence of subdiffusion in condensed media \\cite{Bouchaud1990}. \nOne possible mechanism, which will be addressed in this work, corresponds to \ndisordered glassy-like media consisting of trapping domains where the traveling particle can dwell\nfor a random time with divergent mean value \\cite{Scher, ScherMontroll, shlesinger1974}. \nThe successive residence times in traps are assumed to be mutually uncorrelated. \nDiffusion is nevertheless a non-Markovian (semi-Markovian) process exhibiting long\n(quasi-infinite) time correlations in the particle positions with a weak ergodicity breaking \\cite{BelBarkai}. \nMathematically this physical picture can be described by a continuous time random walk (CTRW) \nmodel \\cite{ScherMontroll, shlesinger1974} which in the continuous space limit leads\nto the fractional Fokker-Planck equation (FFPE) \\cite{metzler2000R, letter, heinsalu2006b}. \nThis latter formulation is incomplete, as no non-Markovian master equation can define \nthe underlying non-Markovian  stochastic process \\cite{HT}. \nHowever, the FFPE is very useful and has a tightly associated, complete description with (ordinary)  \nLangevin equation in subordinated, random operational time \\cite{Fogedby1994, Stanislavsky, Magdziartz}.\n\nThe generalization of the CTRW and FFPE to time-dependent\nforces is a highly non-trivial matter since the force changes in\nthe real and not in the operational time \\cite{heinsalu2007b, goychuk07}. \nAlso, how a field varying in time affects the distribution of the residence \ntimes in the traps is not clear without\nspecifying a concrete mechanism or some plausible model,\nespecially when the mean residence time does not exist \\cite{PRE04}. \nThe FFPE describing the dynamics in time-dependent force fields $F(x,t)$  \nbecomes ambiguous with a frequently (ab)used, \\textit{ad hoc} version \\cite{sokolov2001, Sample} \nwhich lacks clear theoretical grounds \\cite{heinsalu2007b}. \nThe correct version of the FFPE for time-dependent fields was first given in \nRefs.~\\cite{SokolovKlafter06, heinsalu2007b}: differently from the FFPE for a \ntime-independent force, in the case of a time-dependent field the fractional derivative \ndoes not stand in front of the Fokker-Planck operator but after it. \nAs we explain with this work in more detail, such a FFPE can be justified beyond \nthe linear response approximation within a CTRW approach only for a special \nclass of dichotomously fluctuating fields.\nThe derivation of the FFPE for subdiffusion in such time-dependent fields \nis presented in Sec.~\\ref{derivation}. \n\nIn Sec.~\\ref{results} we apply the derived FFPE to study the influence of \ntime-periodic rectangular fields on subdiffusive motion. \nAnalytical solutions of the FFPE are confirmed by stochastic \nMonte Carlo simulations of the underlying CTRW.  \nIn particular, we show with this work that the universal scaling relation between \nthe biased anomalous diffusion and sub-current \n\\cite{ScherMontroll, shlesinger1974, letter, heinsalu2006b} is not affected by the periodic driving.  \nNeither current nor diffusion are influenced asymptotically by the time-periodic field. \nThis is in spite of the fact that unbiased subdiffusion of the studied kind can be  strongly\nenhanced in the time-periodic field \\cite{SokolovKlafter06, heinsalu2007b}.  \n\n\n\n\n\\section{Derivation of the FFPE for time-dependent fields from the underlying CTRW} \\label{derivation}\n\n\n\nSince the FFPE does not define the underlying stochastic non-Markovian process, \nits generalization to include the influence of a time-dependent field should \nstart from the underlying CTRW \\cite{metzler2000R, letter}.\nFollowing the general picture of the CTRW we introduce a\none-dimensional lattice $\\{ x_i = i \\Delta x\\}$ with a lattice period\n$\\Delta x $ and $i = 0, \\pm 1, \\pm 2, \\ldots$\nLet us first assume that there is no time-dependent field.  \nAfter a random trapping time $\\tau$ a particle at site $i$ hops with  probability\n$q _i ^\\pm $ to one of the nearest neighbor sites $i \\pm 1$; $q_i^+ + q_i^-=1$.  \nThe random time $\\tau$ is extracted from a site-dependent \nresidence time distribution (RTD) $\\psi_i (\\tau)$. \nThe corresponding generalized master equation for\n populations $P_i(t)$ reads \\cite{Kenkre1973,Hughes,Weiss}\n\\begin{eqnarray} \\label{GME}\n\\dot P_i(t) &=& \\int_0^t \\{K_{i-1}^{+}(t-t') P_{i-1}(t') +\nK_{i+1}^{-}(t-t') P_{i+1}(t') \\nonumber \\\\\n&-& [K_i^{+}(t-t') + K_i^{-}(t-t')] P_i(t')\\} \\, \\mathrm{d} t' \\, .\n\\end{eqnarray} \nThe Laplace-transform of the kernel $K_i^{\\pm}(t)$ is related to the Laplace-transform \nof the RTD $\\psi_i (\\tau)$ via $\\tilde K_i^{\\pm}(s) = q_i^{\\pm} s\\tilde \\psi_i(s) \/ [1 - \\tilde \\psi_i(s)]$. \nIn the presence of a time-dependent field the kernels become generally functions \nof both instants of time and not only of their difference, i.e. $K_i^{\\pm}(t-t')\\to K_i^{\\pm}(t,t')$. \nOne can relate $K_i^{\\pm}(t,t')$ with the corresponding time-inhomogeneous RTDs\n $\\psi_i^{\\pm}(t+\\tau,t)\\equiv \\psi_i^{\\pm}(\\tau|t)$, which are conditioned on the entrance time $t$. \nHowever, one always needs a concrete and physically meaningful model to proceed further \\cite{PRE04}. \nA simple example is a Markovian process with time-dependent rates $g_i^{\\pm}(t)$, where \n$\\psi_i^{\\pm}(\\tau|t) = g_i^{\\pm}(t+\\tau) \\exp\\left \\{ -\\int_{t}^{t+\\tau}[g_i^{+}(t') + g_i^{-}(t')]dt'\\right \\}$ \nand $K_i^{\\pm}(t,t') = 2g^{\\pm}_i(t) \\, \\delta(t-t')$. \nThis yields in Eq.~(\\ref{GME}) the standard master equation for a \ntime-inhomogeneous Markovian process. \nHow the non-exponential RTDs will be modified for a time-inhomogeneous \nprocess is generally not clear \\cite{PRE04}.\nIn the present case, one can assume that the trapping occurs due to the \nexistence of  direction(s) orthogonal to the $x$-coordinate. \nAccording to the modeling in Refs.~\\cite{SokolovKlafter06, Magdziartz}, \nan external field directed along $x$ \nwould not affect the motion in the orthogonal direction(s).\nHowever, it is not correct to think that the RTD in the trap will not be \ninfluenced by the field acting in the direction of $x$, as it will change the rates \n(let us assume this simplest, tractable model) \nfor moving left or right when escaping from the trap. \nTherefore, the RTD will generally be affected, see e.g. Ref.~\\cite{PRE05}.\nObviously, the only situation when the RTD in the trap will not be changed is \nwhen the sum of the rates to escape from the trap, either left or right, is constant. \nIn that case, only the probabilities $q_i^{\\pm}(t)$ acquire additional \ntime-dependence and not the RTDs $\\psi_i(\\tau)$. \nThis corresponds to the special class of dichotomously fluctuating force fields \n$F (x_i, t) = F (x_i) \\xi (t)$, where $\\xi (t) = \\pm 1$. \nBeyond this class, at most the linear response approximation can work \\cite{SokolovKlafter06}. \nTherefore, we restrict our treatment to the above class of fluctuating potentials.\nIn this case, we can write $K_i^{\\pm}(t,t') = q^{\\pm}_i(t)K_i(t-t')$,\nwhere $\\tilde K_i(s) = s\\tilde \\psi_i(s) \/ [1 - \\tilde \\psi_i(s)]$.\nFurthermore, we use the Mittag-Leffler distribution for the residence times \\cite{metzler2000R},\n\\begin{equation} \\label{ML}\n\\psi_i(\\tau) = -\\frac{\\mathrm{d}}{\\mathrm{d} \\tau} E_{\\alpha}[\n-(\\nu_i \\tau)^{\\alpha}] \\, .\n\\end{equation}\nHere $E_{\\alpha}(z) = \\sum_{n = 0}^{\\infty} z^n \/ \\Gamma(n \\alpha + 1)$ denotes \nthe Mittag-Leffler function, $\\alpha \\in (0,1)$ is the index of subdiffusion, and \n$\\nu_i = [g_i^+(t) + g_i^-(t)]^{1\/\\alpha}$ the time scaling parameter; \n$g_{i}^\\pm (t) = q^{\\pm}_i(t) \\nu_i^{\\alpha}$.\nThen $\\tilde K_i(s) = \\nu_i^{\\alpha} s^{1-\\alpha}$ and we get\n\\begin{eqnarray} \\label{FME}\n\\dot P_i(t) &=& g_{i-1}^+(t)\\sideset{_0}{_t} {\\mathop{\\hat\nD}^{1-\\alpha}} P_{i-1}(t) + g_{i+1}^-(t) \\sideset{_0}{_t} {\\mathop{\\hat\nD}^{1-\\alpha}} P_{i + 1}(t)  \\nonumber \\\\\n&-& [g_i^+(t) + g_i^-(t)]\\sideset{_0}{_t} {\\mathop{\\hat\nD}^{1-\\alpha}} P_i(t) \\, ,\n\\end{eqnarray}\nwhere the symbol $\\sideset{_0}{_t}{\\mathop{\\hat D}^{1-\\alpha}}$ stands for the \nintegro-differential operator of the Riemann-Liouville fractional derivative \nacting on a generic function of time $\\chi (t)$ as \n\\begin{equation} \\label{RL}\n\\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} \\chi (t) \n= \\frac{1}{\\Gamma (\\alpha)} \\frac{\\partial}{\\partial t} \\int_{0}^{t} \\mathrm{d} t' \\, \\frac{\\chi (t')}{(t - t')^{1 - \\alpha}} \\, ;\n\\end{equation}\n$\\Gamma (\\alpha)$ is the gamma-function.\nIn a time-dependent potential $U(x,t)$, one can set\n\\begin{eqnarray} \\label{f-rate}\ng_i^\\pm(t) &=& (\\kappa_{\\alpha} \/ \\Delta x ^2)\n\\exp \\{ -\\beta[U_{i \\pm 1\/2}(t) - U_i(t)] \\} \\nonumber \\\\\n&\\approx & (\\kappa_{\\alpha} \/ \\Delta x ^2)\n\\exp [\\pm \\beta F(x_i,t)\\Delta x\/2] \\, ,\n\\end{eqnarray}\nso that the Boltzmann relation $g_{i-1}^+(t)\/g_i^-(t) = \\exp \\{ \\beta[U_{i-1}(t)-U_i(t)]\\}$\nis satisfied exactly and the time-independence of $g_i^+(t) + g_i^-(t) = \\nu_i^{\\alpha} = \\mathrm{const}$\nis also maintained for small $\\Delta x$ and a sufficiently smooth potential.\nWe have used here the notation $U_i(t) \\equiv U(i \\Delta x,t)$ and \n$U_{i \\pm 1\/2}(t) \\equiv U(i \\Delta x \\pm \\Delta x \/2,t)$; \n$\\beta = k_B T$ is the inverse of temperature and $\\kappa_{\\alpha}$ free fractional \ndiffusion coefficient with dimension $\\mathrm{cm}^2 \\mathrm{s}^{-\\alpha}$.\nBy passing to the continuous space limit $\\Delta x\\to 0$ \nas in Ref.~\\cite{letter}, one finally obtains,\n\\begin{eqnarray} \\label{FFPEmod}\n\\frac{ \\partial }{\\partial t} P (x, t) = \n\\left [ - \\frac{\\partial}{\\partial x} \\frac{F (x, t)}{\\eta _\\alpha} \n+ \\kappa _\\alpha \\frac{\\partial ^2}{\\partial x^2} \\right ] \n\\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} P (x, t) \\, .\n\\end{eqnarray}\nIn the latter equation $\\eta _\\alpha = ( \\beta \\kappa_{\\alpha})^{-1}$ \nis the fractional friction coefficient.\n\nIn the following we use Eq.~(\\ref{FFPEmod}) to study analytically \nthe subdiffusion in time-periodic rectangular fields. \nOur study is complemented by stochastic simulations of the\nunderlying CTRW using the algorithm detailed in Ref.~\\cite{heinsalu2006b}.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figure1.eps}\n\\caption{(Color online) \nAverage particle position $\\langle x (t) \\rangle$ for various values of \nthe parameter $r$ (average force $\\bar{F}$) and anomalous exponent $\\alpha $.\nSymbols represent the results from the numerical simulations of the CTRW obtained by \naveraging over $10 ^5$ trajectories (for $r = 0.5$ over $10 ^6$ trajectories). \nContinuous lines represent the analytical solution (\\ref{xx}) of the FFPE (\\ref{FFPEmod}). \nThe time-period of the rectangular force (\\ref{Fper}) is \n$\\tau _0 = 1$ and the fractional exponent $\\alpha = 0.5$;  \nin numerical simulations $F_0 \/ (\\eta _{\\alpha} \\sqrt{\\kappa _{\\alpha}}) = 1$ is used. \nThe value $r = 0.9$ corresponds to $\\bar{F} = 0.8 F_0$ and $\\sigma = 0.6 F_0$;\n$r = 0.8$ corresponds to $\\bar{F} = 0.6 F_0$ and $\\sigma = 0.8 F_0$; \n$r = 0.7$ corresponds to $\\bar{F} = 0.4 F_0$ and $\\sigma\\approx 0.92 F_0$; \n$r = 0.6$ corresponds to $\\bar{F} = 0.2 F_0$, $\\sigma\\approx 0.98 F_0$; \nand $r = 0.5$ to $\\bar{F} = 0$, $\\sigma = F_0$.} \n\\label{Fig1}\n\\end{figure}\n\n\n\n\n\\section{Driven subdiffusion} \\label{results}\n\n\n\nWe consider a dichotomous modulation  of a\nbiased subdiffusion where the absolute value of the bias is\nfixed but its direction flips periodically in time, i.e.\n\\begin{equation} \\label{Fper}\nF (t) = F_0 \\xi (t)\n\\end{equation}\nwith\n\\begin{eqnarray} \\label{FperPM}\n\\xi (t) = \\left\\{ \\begin{array}{ll} + 1 & \\textrm{~for}~~ n \\tau_0 < t < (n+r) \\tau_0 \\\\\n- 1 & \\textrm{~for}~~ (n+r) \\tau_0 < t < (n+1) \\tau _0\n\\end{array} \\right. .\n\\end{eqnarray}\nHere $\\tau _0$ is the period of the time-dependent force and $n = 0, 1, 2 \\ldots $ \nThe quantity $r \\in (0,1)$ determines the value of the average force: \n\\begin{equation}\n\\bar{F} = \\langle F (t) \\rangle _{\\tau _0} = F_0 (2r - 1) \\, .\n\\end{equation}\nFor $r = 0.5$ the average bias is zero and we recover the model investigated\nin Ref.~\\cite{heinsalu2007b}. \nNotice that the force $F(t)$ can be decomposed in the following way: $F(t) = \\bar{F} + \\tilde F(t)$. \nThe asymmetric driving, \n\\begin{eqnarray} \\label{Fper2}\n\\tilde F (t) \\!= \\!\n\\left\\{\\!\\begin{array}{ll} 2 F_0 (1-r) & \\textrm{~for}~~ n \\tau_0 < t < (n+r) \\tau_0 \\\\\n- 2 F_0 r & \\textrm{~for}~~ (n+r) \\tau_0 < t < (n+1) \\tau _0\n\\end{array} \\right. \\!,\n\\end{eqnarray}\nhas a zero mean value, $\\langle \\tilde F (t) \\rangle _{\\tau _0} = 0$, \nand the driving root-mean-squared (rms) amplitude is \n$\\sigma = \\langle \\tilde F^2 (t) \\rangle^{1\/2} _{\\tau _0} = 2F_0 \\sqrt{r(1-r)}$.\nFor a fixed average bias $\\bar{F}$, this yields\n\\begin{eqnarray}\n\\sigma = 2 \\bar{F} \\frac{\\sqrt{r(1-r)}}{2r-1}\n\\end{eqnarray}\nand therefore one can vary the ratio $\\sigma\/ \\bar{F}$ between $0$ for\n$r=1$ and $\\infty$ for $r=1\/2+\\epsilon$, $\\epsilon\\to 0$.\nThis offers the way to study the influence of an asymmetric, zero-mean\ndriving $\\tilde{F}(t)$ with period $\\tau_0$ and rms amplitude $\\sigma$ \non the subdiffusion under constant bias $\\bar{F}$.\n\nLet us begin by finding the recurrence relation  for the moments $\\langle x ^n (t) \\rangle$.  \nAssuming in Eq.~(\\ref{FFPEmod}) the force of the form \n(\\ref{Fper}) with (\\ref{FperPM}), multiplying both sides of Eq.~(\\ref{FFPEmod}) by \n$x ^n$, and integrating over the $x$-coordinate one obtains,\n\\begin{eqnarray} \\label{a2}\n\\frac{d \\langle x ^n (t) \\rangle}{d t} \n&=& n v _\\alpha \\xi (t) \\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} \\langle x^{n-1} (t) \\rangle \\nonumber \\\\\n&+& n (n - 1) \\kappa _\\alpha \\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} \\langle x ^{n - 2} (t) \\rangle \\, ,\n\\end{eqnarray}\nwith subvelocity $v _{\\alpha} = F_0 \/ \\eta _{\\alpha}$ ($n > 1$). \nFor $n = 1$ the last term on the right hand side of Eq.~(\\ref{a2}) is absent,\n\\begin{eqnarray} \\label{xav}\n\\frac{d \\langle x(t) \\rangle}{d t} = \\frac{v _\\alpha}{\\Gamma (\\alpha)} \n\\, \\xi (t) \\, t ^{\\alpha - 1} \\, .\n\\end{eqnarray}\nEquations (\\ref{a2}) and (\\ref{xav}) will be used to calculate the \naverage particle position and the mean square displacement.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figure2.eps}\n\\caption{(Color online) \nThe analytical solution (\\ref{xx}) for average particle position \n$\\langle x (t) \\rangle$ obtained from the FFPE (\\ref{FFPEmod}) is presented \nfor various values of the parameter $r$ (average bias $\\bar{F}$) and anomalous exponent $\\alpha $. \nThe time-period of the force is $\\tau _0 = 1$, however, in the long time limit \nthe same asymptotic value is obtained for any value of $\\tau _0$. \nThe relation between $r$ and $\\bar{F}$ and $\\sigma$  is the same as in Fig.~\\ref{Fig1}.} \n\\label{Fig2}\n\\end{figure}\n\n\n\n\n\\subsection{Average particle position}\n\n\n\nUpon integrating Eq.~(\\ref{xav}) in time with $\\xi (t)$ given by (\\ref{FperPM}), \nthe solution for the average particle position reads:\n\\begin{eqnarray} \\label{xx}\n&& \\langle x (t) \\rangle \\nonumber \\\\\n&& =\\left\\{ \n\\begin{array}{l@{\\quad \\quad}}\nx _N + \\frac{v _{\\alpha} t ^\\alpha}{\\Gamma (\\alpha + 1)} \\, , \\quad N \\tau _0 \\le t < (N + r) \\tau _0 \\\\\nx _N' - \\frac{v _{\\alpha} t ^\\alpha}{\\Gamma (\\alpha + 1)} \\, , \\quad (N + r) \\tau _0 \\le t < (N + 1) \\tau _0 \n\\end{array} \n\\right.\n\\end{eqnarray}\nwith\n\\begin{eqnarray} \\label{xn1}\nx _N &=& \\langle x (0) \\rangle - \\frac{v _{\\alpha} (N \\tau _0) ^\\alpha}{\\Gamma (\\alpha + 1)} \n+ \\frac{v _{\\alpha} \\tau _0 ^\\alpha}{\\Gamma (\\alpha + 1)} \\nonumber \\\\\n&\\times& \\sum_{n = 0} ^{N - 1} \\left [2(n + r) ^\\alpha -n ^\\alpha - (n + 1) ^\\alpha \\right ] \\, , \\\\\nx _N' &=& x _N + \\frac{2 v _{\\alpha} \\tau _0 ^\\alpha}{\\Gamma (\\alpha + 1)} (N + r) ^\\alpha \\, ;\n\\end{eqnarray}\n$N$ counts the number of time periods passed.\n\nWhen the average bias is zero, i.e., $r = 0.5$, in the long time limit\nthe mean particle position approaches the constant value\n\\begin{eqnarray} \\label{asympt}\n\\langle x (\\infty) \\rangle = v _{\\alpha} \\tau _0 ^{\\alpha} b (\\alpha ) \/ \\Gamma(\\alpha + 1),\n\\end{eqnarray}\nwith $b (\\alpha ) = \\sum _{n = 0} ^{\\infty}[2 (n + 1\/2) ^{\\alpha} - n ^{\\alpha} - (n + 1) ^{\\alpha}]$. \nThe function $b (\\alpha )$ changes monotonously from $b (0) = 1$ to $b (1) = 0$. \nIt describes the initial field phase effect which the system remembers forever when \n$\\alpha \\in (0, 1)$ (see also Ref.~\\cite{heinsalu2007b}). \nThis is one of the main differences between the anomalous motion in the absence \nof a force and in the presence of a time-dependent field with zero average value. \n\nIn Fig.~\\ref{Fig1} the analytical solution (\\ref{xx}) for the mean\nparticle position $\\langle x (t) \\rangle$, obtained from the\nFFPE (\\ref{FFPEmod}), is compared with the numerical solution of the CTRW\nfor different values of $r$, i.e. for different values of the average bias $\\bar{F}$.  \nIn Fig.~\\ref{Fig2} the solution (\\ref{xx}) is presented in the \nlong time limit for various values of $r$ and $\\alpha $. \nFigures \\ref{Fig1} and \\ref{Fig2} demonstrate that in the presence of an\naverage bias the mean particle position grows as $t ^\\alpha$.\nFor all values of $r$, the asymptotic value of subvelocity corresponds\nto the averaged bias $\\bar{F} = F_0 (2r-1)$, indicating that the periodic\nunbiased field does not affect the subdiffusion current for different values\n$\\bar{F}$ and the field rms-amplitude $\\sigma$.\n\nFurthermore, the results depicted in Figs.~\\ref{Fig1} and \\ref{Fig2} clearly show the \nphenomenon of the ``death of linear response'' of the fractional kinetics to time-dependent \nfields in the limit $t \\to \\infty$: the amplitude of the oscillations decays to\nzero as $1 \/ t ^{1 - \\alpha}$, Eq.~(\\ref{xav}) (see also Refs.~\\cite{Barbi, SokolovKlafter06}).  \nThe amplitude of the oscillations is larger for larger values of \n$\\alpha $ and of $\\tau _0$.\nHowever, $\\Gamma (1+\\alpha) \\langle x(t) \\rangle \/ t^\\alpha$ reaches asymptotically \nthe same value for any $\\alpha$ and $\\tau _0$.\n\n\n\n\n\\subsection{Mean square displacement}\n\n\n\nLet us now study the mean square displacement, defined as\n\\begin{equation} \\label{MSD}\n\\langle \\delta x ^2 (t) \\rangle = \\langle x ^2 (t) \\rangle - \\langle x (t) \\rangle ^2 \\,.\n\\end{equation}\nFor $n = 2$ one obtains from Eq.~(\\ref{a2}),\n\\begin{equation} \\label{AP1}\n\\frac{d \\langle x ^2 (t) \\rangle}{d t} \n= 2 v _\\alpha \\xi (t) \\sideset{_0}{_t}{\\mathop{\\hat D}^{1 - \\alpha}} \\langle x (t) \\rangle \n+ \\frac{ 2 \\kappa _\\alpha }{\\Gamma (\\alpha)} \\, t ^{\\alpha - 1} \\, .\n\\end{equation}\nIn order to find the analytical solution for the mean square displacement, \nwe use the Laplace-transform method and the Fourier series expansion for \n$\\xi (t) = \\xi (t + \\tau _0)$ given wuth Eq.~(\\ref{FperPM}),\n\\begin{equation} \\label{AP3}\n\\xi (t) = \\sum_{n = - \\infty}^{\\infty} f _n \\exp (i n \\omega _0 t) \\, ,\n\\end{equation}\nwith\n\\begin{eqnarray} \\label{AP4}\nf _n &=& \\frac{1}{\\tau _0} \\int _{0} ^{\\tau _0} \\xi (t) \\exp ( - in \\omega _0 t) dt \\nonumber \\\\\n&=& [1 - \\exp ( - inr 2 \\pi )] \/ ( in \\pi )\n\\end{eqnarray}\nand $\\omega _0 = 2 \\pi \/ \\tau _0$. \nApplying them to Eq.~(\\ref{AP1}) and assuming $\\langle x (0) \\rangle = 0$ and \n$\\langle x ^2 (0) \\rangle = 0$ we obtain that in the long time limit \n(see Appendix~\\ref{APPENDIX-1}),\n\\begin{eqnarray} \\label{MSD-1}\n\\langle x ^2 (t) \\rangle &=& \\frac{2v _\\alpha ^2 (2r - 1) ^2 }{\\Gamma (2 \\alpha + 1)} t ^{2 \\alpha } \n+ \\frac{2 \\kappa _\\alpha}{\\Gamma (\\alpha + 1)} t ^\\alpha \\nonumber \\\\\n&+& \\frac{2 v _\\alpha ^2 (2r - 1) S_1}{\\omega _0 ^\\alpha \\Gamma (\\alpha + 1)} t ^\\alpha\n + \\frac{8v _\\alpha ^2 \\cos (\\alpha \\pi \/ 2)}{\\pi ^2 \\omega _0 ^\\alpha \\Gamma (\\alpha + 1)}  \\nonumber \\\\\n&\\times & \\left [\\zeta (2 + \\alpha ) - \\sum _{n = 1} ^{\\infty } \\frac{\\cos (nr 2 \\pi )}{n ^{2 + \\alpha }} \\right ] t ^\\alpha \\, ;\n\\end{eqnarray}\nhere $\\zeta (x)$ is the Riemann's zeta-function and $S_1$ is a function of \n$\\alpha$ and $r$ as given by Eq.~(\\ref{AP13}) in Appendix~\\ref{APPENDIX-1}.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figure3.eps}\n\\caption{(Color online) Scaled effective fractional diffusion coefficient $\\kappa _\\alpha ^{\\mathrm{(eff)}}$ \nversus fractional exponent $\\alpha$ for different driving periods $\\tau _0$. \nThe average bias is zero ($r = 0.5$). \nThe analytical prediction (\\ref{basic}) (continuous lines) is compared with the results (symbols) \nobtained from the numerical simulation of the CTRW by averaging over $10 ^5$ trajectories. \nFor $\\tau _0 > \\tau _0^* \\approx 8.818$ \\cite{heinsalu2007b} the effective fractional diffusion coefficient \n$\\kappa _\\alpha ^{\\mathrm{(eff)}}(\\alpha )$ exhibits a maximum.}\n\\label{Fig3}\n\\end{figure}\n\n\nFor $r = 0.5$ (average zero bias) the first and third term in Eq.~(\\ref{MSD-1}) are equal to zero. \nFurthermore, in the long time limit the average particle position $\\langle x (\\infty ) \\rangle $ is a finite constant. \nThe asymptotic behavior of the mean square displacement is thus proportional to $t ^{\\alpha}$ as in the force\nfree case, however, characterized by an effective fractional diffusion coefficient \n$\\kappa _{\\alpha} ^\\mathrm{(eff)}$ instead of the free fractional diffusion coefficient $\\kappa _\\alpha$, i.e.\n$\\langle \\delta x ^2 (t) \\rangle = 2\\kappa _{\\alpha} ^\\mathrm{(eff)} t^ {\\alpha} \/ \\Gamma (1 + \\alpha )$ for $t \\to \\infty$. \nThe effective diffusion coefficient is \\cite{heinsalu2007b},\n\\begin{eqnarray} \\label{basic}\n\\kappa _\\alpha ^{\\mathrm{(eff)}} &=& \\kappa _\\alpha \n+ \\frac{8 F_0^2}{\\pi ^2 \\eta _\\alpha ^2 \\omega _0 ^\\alpha } \\, \\zeta (\\alpha + 2) \\nonumber \\\\\n&\\times& \\left(1- \\frac{1}{2 ^{\\alpha + 2}} \\right ) \\cos (\\alpha \\pi \/ 2) \\, .\n\\end{eqnarray}\nThe driving-induced part of the effective subdiffusion coefficient is directly proportional \nto the square of driving amplitude $F_0$ and inversely proportional to $\\omega _0 ^\\alpha $. \nFor slowly oscillating force fields this leads to a profound \nacceleration of subdiffusion compared with the force free case: \nan optimal value of the fractional exponent $\\alpha$ exists, at which the driving-induced part \nof the effective fractional diffusion coefficient possesses a maximum (see Fig.~\\ref{Fig3}).\n\nWhen $r \\neq 0.5$ (finite average force) we obtain in the long time limit\nfor $\\langle x (t) \\rangle ^2$ (see Appendix~\\ref{APPENDIX-2}),\n\\begin{equation} \\label{MSD-2}\n\\langle x (t) \\rangle ^2 = \\frac{v _\\alpha ^2 (2r - 1) ^2}{\\Gamma ^2 (\\alpha + 1)} t ^{2 \\alpha } + \n\\frac{2v _\\alpha ^2 (2r - 1) S_1}{\\omega _0 ^\\alpha \\Gamma (\\alpha + 1)} t ^\\alpha\n\\end{equation}\n[$S_1$ is given by Eq.~(\\ref{AP13}) in Appendix~\\ref{APPENDIX-1}].\nClearly, the leading term in Eq.~(\\ref{MSD-2}) corresponds to subvelocity in constant \nfield $\\bar{F}$ (averaged bias), i.e. the influence of periodic, unbiased driving \n$\\tilde F(t)$ dies asymptotically out, as illustrated in Fig.~\\ref{Fig2}.\n\nThe results (\\ref{MSD-1}) and (\\ref{MSD-2}) indicate that in the \npresence of a rectangular time-periodic force with a finite average value the\ngeneral behavior of the mean square displacement is similar to the\ncase of a constant force, i.e. the mean square displacement \n$\\langle \\delta x ^2 (t)\\rangle $ consists of terms proportional to \n$t ^\\alpha $ and $t ^{2 \\alpha }$.\nIn fact, for the leading term proportional to $t ^{2 \\alpha }$ in the \nmean square displacement one obtains the coefficient\n\\begin{equation*} \\label{AP22}\n\\frac{{\\bar{F}}^2}{\\eta _\\alpha ^2} \\left [ \\frac{2}{\\Gamma (2 \\alpha + 1)} \n- \\frac{1}{\\Gamma ^2 (\\alpha + 1)} \\right ] \\, .\n\\end{equation*}\nThis coefficient is the same as in the case of the subdiffusive motion under the influence \nof a constant force if the value of the constant force would be $\\bar{F}$.\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figure4.eps}\n\\caption{(Color online) \nThe asymptotic scaling relation (\\ref{anomal}). \nSymbols correspond to the numerical results obtained from the CTRW \nfor different values of $r$ ($r \\neq 0.5$) and $\\tau_0$.\nSolid curve corresponds to the analytical result (\\ref{anomal}). \nDifferent values of the parameter $r$ correspond to different\nvalues of the bias $\\bar{F}$ and the periodic field rms $\\sigma$, as described in\nFig.~\\ref{Fig1}. \nThe field parameters do not influence the results within the statistical errors.} \n\\label{Fig4}\n\\end{figure}\n\n\nFurthermore, similarly to the case of a constant bias, the asymptotic scaling relation holds \n($r \\neq 0.5 $, $\\bar{F} \\neq 0 $) between the mean square displacement and\naverage particle position.\nIn the limit $t \\to \\infty$ the mean square displacement grows as \n$\\langle \\delta x ^2 (t) \\rangle \\propto t ^{2 \\alpha }$  and\n\\begin{eqnarray} \\label{anomal}\n\\lim_{t \\to \\infty} \\frac{\\langle \\delta x^2 (t) \\rangle}{ \\langle x (t) \\rangle ^2} \n= \\frac{2 \\Gamma ^2 (\\alpha + 1)}{\\Gamma (2 \\alpha + 1)} - 1 \\, , \\quad \\bar{F} \\neq 0 \\, .\n\\end{eqnarray}\nIt is illustrated by Fig.~\\ref{Fig4},  where the analytical curve\n[Eq.~(\\ref{anomal})] is compared with the numerical results. \nThe universality of relation (\\ref{anomal}) under the unbiased driving $\\tilde F(t)$\nmeans that the biased diffusion is not affected by the driving. \nThis is in a sharp contrast with the unbiased diffusion in Fig.~\\ref{Fig3}.\n\n\n\n\n\\section{Conclusion}\n\n\n\nWith this work we presented the derivation of the FFPE (\\ref{FFPEmod}) for a \nspecial class of space- and time-dependent force fields from the underlying CTRW picture. \nOur derivation shows along with the corresponding discussion\nthat it is difficult to justify this equation for time-dependent forces \ndifferent from $F(x, t) = F(x) \\xi (t)$ with $\\xi (t) = \\pm 1$\nbeyond the linear response approximation. \nUsing the FFPE (\\ref{FFPEmod}) we demonstrated that the universal \nscaling relation (\\ref{anomal}) for a biased subdiffusion is not affected \nby the additional action of a time-periodic zero-mean rectangular driving;\nneither is the asymptotic anomalous current nor the anomalous biased diffusion.\nWe argue that this result is general and it is valid for other driving forms\nas well.\nThis driving-immunity is due to the fact that the CTRW subdiffusion occurs \nin a random operational time which lacks mean value, whereas any physical field\nchanges in the real, physical time. \nThe CTRW-based subdiffusion fails to respond \nasymptotically to such time-dependent fields while on its intrinsic random \noperational time scale any real, alternating field is acting infinitely fast\nand it makes effectively no influence in a long run \\cite{heinsalu2007b,goychuk07},\nunless the rate of its change is precisely zero. \nThis is the main reason for the observed anomalies. \nThe remarkable enhancement of the unbiased subdiffusion within the CTRW framework \nby time-periodic rectangular fields is rather an exception than the rule. \n\n\n\n\n\\begin{acknowledgements}\n\n\n\nThis work has been supported by the targeted financing project SF0690030s09, \nEstonian Science Foundation via grant no. 7466 (M.P., E.H.), \nSpanish MICINN and FEDER through project FISICOS (FIS2007-60327) (E.H.), \nthe EU NoE BioSim, LSHB-CT-2004-005137 (M.P.),\nthe DFG-SFB-486, and by the Volks\\-wagen Foundation, no. I\/80424 (P.H.).\n\\end{acknowledgements}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminaries}\n\n\n\n\n\\begin{abstract}\n\n\nThe paper studies the containment companion of a logic $\\vdash$. This consists of the consequence relation $\\vdash^{r}$ which satisfies all the inferences of $\\vdash$, where the variables of the conclusion are \\emph{contained} into those of the (set of) premises. In accordance with the work started in \\cite{BonzioMorascoPrabaldi}, we show that a different generalization of the  P\\l onka sum construction, adapted from algebras to logical matrices, allows us to provide a matrix-based semantics for containment logics. In particular, we provide an appropriate completeness theorem for a wide family of containment logics, and we show how to produce a complete Hilbert style axiomatization.\n\\end{abstract}\n\n\n\\section{introduction}\n\nIt is a recent discovery (see \\cite{Bonzio16}) that the algebraic counterparts of weak Kleene logics are formed by a regularized variety, whose members coincide with the \\emph{P\\l onka sum} of Boolean algebras, the algebraic semantics of propositional classical logic. Subsequently, the abstract construction of the P\\l onka sum of algebras has been generalized to logical matrices \\cite{BonzioMorascoPrabaldi}. The main outcome is that the suggested notion provides an algebra-based semantics for a class of propositional logics, called \\emph{logics of left variable inclusion}, of which paraconsistent weak Kleene represents the most prominent example. \n\nThe logics in the weak Kleene ``family'' -- essentially, Bochvar \\cite{BochvarBergmann} and paraconsistent weak Kleene \\cite{Hallden} --  are syntactically characterized by imposing certain limitations on the inclusions of variables to classical propositional logic \\cite{Urquhart2001,CiuniCarrara}.   \n\nThe extension of the construction of P\\l onka sums to logical matrices, introduced in \\cite{BonzioMorascoPrabaldi}, allows for an insightful investigation into the algebraic features of those logics, where the inclusion of variables runs from premises to conclusions. However, this is just one side of the coin of the \\emph{logics of variable inclusion}; the other side consisting of those logics verifying inferences in which the variables occurring in the conclusion are contained into the ones occurring in the premises. Consequence relations satisfying this feature are usually know as \\emph{containment logics}; the syntactic requirement which they share is a strengthen form of what Ferguson \\cite{ferguson2017meaning} understands as \\emph{Proscriptive Principle}, which also resembles the one defining logics of \\emph{analytic containment} \\cite{ledda2019algebraic, epstein1990semantic}. \n\n\nThe most famous example of containment logic is \\emph{Bochvar logic} $\\mathsf{B_{3}}$ \\cite{BochvarBergmann}, defined by a single matrix which features the presence of an infectious truth-value (a peculiarity shared by the twin-sister paraconsistent weak Kleene). $\\mathsf{B_{3}}$ has been successfully applied in different contexts: avoiding paradoxes in set-theory \\cite{BochvarBergmann}, modeling computer programs affected by errors \\cite{Ferguson} and non-sensical information databases \\cite{Ciuni1}, capturing the notion of truth in relations with on\/off topic arguments \\cite{beall2016off}. \n\n  \nThe main condition defining containment logics mirrors the syntactic requirement defining logics of left variable inclusion. For this reason, the present paper may be understood as an ideal continuation of the path started in \\cite{BonzioMorascoPrabaldi}.\nThis amounts to answer the very natural question on whether it is possible to build a new generalization of the P\\l onka sum construction, suitable for obtaining a matrix semantics for containment logics. Due to the intrinsic difference between the  variable inclusion constraints at stake, the solution is not self-evident.\n\n\nThe paper is structured as follows.\n\n\nIn Section \\ref{sec: preliminari}, we recall all the preliminary notions needed to go trough the reading of the whole paper. They basically consist of the basic notions of abstract algebraic logic and of the theory of P\\l onka sums.\n \nIn Section \\ref{sec:compl}, we formally introduce containment logics. By providing an adequate notion of P\\l onka sum for logical matrices, we obtain soundness and completeness for the \\emph{containment companion} $\\vdash^{r}$ of an arbitrary (finitary) logic $\\vdash$, with respect to the P\\l onka sum of the matrix models of $\\vdash$.\n\n\nIn Section \\ref{sec:hilbert}, we focus on a specific (though very wide) class of logics, namely those  possessing a binary term called partition function (a property shared by the vast majority of known logics). We provide a method for obtaining an Hilbert-style axiomatization for a logic $\\vdash^{r}$ (Theorem \\ref{th: completezza calcolo Hilbert}) out of an axiomatization for (a finitary) logic $\\vdash$.\n\nFinally, in Section \\ref{sec: examples}, we put at work our machinery and characterize the axiomatization of containment companions of some well-known logics, namely of classical propositional logic, Belnap-Dunn and the Logic of Paradox.\n\n\n\\section{Preliminaries}\\label{sec: preliminari}\n\n\n\nFor standard background on universal algebra and abstract algebraic logic, we refer the reader, respectively, to \\cite{Be11g,BuSa00} and \\cite{Font16}. In this paper, algebraic languages are assumed not to contain constant symbols. \nMoreover, unless stated otherwise, we work within a fixed but arbitrary algebraic language. We denote algebras by $\\A, \\B, \\C\\dots$ respectively with universes $A, B, C \\dots$.\nLet $\\Fm$ be the algebra of formulas built up over a countably infinite set $\\Var$ of variables (which we indicate by $x,y,z,\\dots$). Given a formula $\\varphi\\in Fm$, we denote by $\\Var(\\varphi)$ the set of variables really occurring in $\\varphi$. Similarly, given $\\Gamma\\subseteq Fm$, we set\n\\[\n\\Var(\\Gamma)=\\bigcup \\{\\Var(\\gamma)\\colon \\gamma\\in\\Gamma\\}.\n\\]\nA \\emph{logic} is a substitution invariant consequence relation $\\vdash \\?\\? \\subseteq \\mathcal{P}(Fm) \\times Fm$ meaning that for every substitution $\\sigma \\colon \\Fm \\to \\Fm$,\n\\[\n\\text{if }\\Gamma \\vdash \\varphi \\text{, then }\\sigma [\\Gamma] \\vdash \\sigma (\\varphi).\n\\]\nGiven formulas $\\varphi, \\psi$, we write $\\varphi \\sineq \\psi$ as a shorthand for $\\varphi \\vdash \\psi$ and $\\psi \\vdash \\varphi$.\nA logic $\\vdash$ is \\emph{finitary} when for all $\\Gamma\\cup\\varphi\\subseteq Fm$:\n\\begin{align*}\n\\Gamma\\vdash\\varphi \\Longleftrightarrow \\exists \\Delta \\subseteq\\Gamma \\text{ such that } \\Delta \\text{ is finite and } \\Delta\\vdash\\varphi.\n\\end{align*}\n\n A \\emph{matrix} is a pair $\\langle \\A, F\\rangle$ where $\\A$ is an algebra and $F \\subseteq A$. In this case, $\\A$ is called the \\textit{algebraic reduct} of the matrix $\\langle \\A, F \\rangle$.\n \n \n Every class of matrices $\\mathsf{M}$ defines a logic as follows:\n\\begin{align*}\n\\Gamma \\vdash_{\\mathsf{M}} \\varphi \\Longleftrightarrow& \\text{ for every }\\langle \\A, F \\rangle \\in \\mathsf{M} \\text{ and homomorphism }h \\colon \\Fm \\to \\A,\\\\\n& \\text{ if }h[\\Gamma] \\subseteq F\\text{, then }h(\\varphi) \\in F.\n\\end{align*}\nWe say that a logic $\\vdash$ is \\emph{complete} with respect to a class of matrices $\\mathsf{M}$ when  $\\vdash_{\\mathsf{M}}\\;=\\;\\vdash$.\nSometimes, we will refer to such homomorphisms $h$ as \\emph{evaluations}.\n\n A matrix $\\langle \\A, F\\rangle$ is a \\emph{model} of a logic $\\vdash$ when\n\\begin{align*}\n\\text{if }\\Gamma \\vdash \\varphi, &\\text{ then for every homomorphism }h \\colon \\Fm \\to \\A,\\\\  \n&\\text{ if }h[\\Gamma] \\subseteq F\\text{, then }h(\\varphi) \\in F.\n\\end{align*}\nA set $F \\subseteq A$ is a (deductive) \\textit{filter} of $\\vdash$ on $\\A$, or simply a $\\vdash$-\\textit{filter}, when the matrix $\\langle \\A, F \\rangle$ is a model of $\\vdash$. We denote by $\\mathcal{F}i_{\\vdash}\\A$ the set of all filters of $\\vdash$ on $\\A$\n\n Although the present paper does not address the study of reduced models (for containment logics), in order to make it self-contained, we recall those notions, concerning reduced models, that will be used. Let $\\A$ be an algebra and $F \\subseteq A$. A congruence $\\theta$ of $\\A$ is \\emph{compatible} with $F$ when for every $a,b\\in A$,\n\\[\n\\text{if }a\\in F\\text{ and }\\langle a, b \\rangle \\in \\theta\\text{, then }b\\in F.\n\\]\nThe largest congruence of $\\A$ which is compatible with $F$ always exists, and is called the \\emph{Leibniz congruence} of $F$ on $\\A$. It is denoted by $\\Leibniz^{\\A}F$. \nThe \\emph{Suszko congruence} of $F$ on $\\A$, is defined as\n\\[\n\\@ifnextchar _  {\\mathchoice{\\tarskidsp\\kern-.07em}{\\tarskitxt\\kern-.07em} {\\tarskiscr\\kern-.07em}{\\tarskiscrscr\\kern-.07em}} {\\mathchoice{\\tarskidsp}{\\tarskitxt}{\\tarskiscr}{\\tarskiscrscr}}^{\\A}_{\\vdash}F \\coloneqq \\bigcap \\{ \\Leibniz^{\\A}G : F \\subseteq G \\text{ and }G \\in \\mathcal{F}i_{\\vdash}\\A \\}.\n\\]\n\n\n \n\n\n\n\nThe Leibniz and Suszko congruences allow to single out a distinguished class of models of logics. More precisely, given a logic $\\vdash$, we set\n\\begin{align*}\n\\Mod(\\vdash) & \\coloneqq \\{ \\langle \\A, F \\rangle : \\langle \\A, F \\rangle \\text{ is a model of }\\vdash \\};\\\\\n\\Modstar(\\vdash) & \\coloneqq \\{ \\langle \\A, F \\rangle \\in \\Mod(\\vdash) : \\Leibniz^{\\A}F \\text{ is the identity} \\};\\\\\n\\mathsf{Mod}^{\\textup{Su}}(\\vdash) & \\coloneqq \\{ \\langle \\A, F \\rangle \\in \\Mod(\\vdash) : \\@ifnextchar _  {\\mathchoice{\\tarskidsp\\kern-.07em}{\\tarskitxt\\kern-.07em} {\\tarskiscr\\kern-.07em}{\\tarskiscrscr\\kern-.07em}} {\\mathchoice{\\tarskidsp}{\\tarskitxt}{\\tarskiscr}{\\tarskiscrscr}}_{\\vdash}^{\\A}F \\text{ is the identity} \\}.\n\\end{align*}\nThe above classes of matrices are called, respectively, the classes of \\text{models}, \\textit{Leibniz reduced models} (or, simply reduced models), and \\textit{Suszko reduced models} of $\\vdash$.\n \n\n\nGiven a logic $\\vdash$, we set\n\\[\n\\Algstar(\\vdash) = \\{ \\A : \\text{ there is }F \\subseteq A \\text{ s.t. }\\langle \\A, F \\rangle \\in \\Modstar(\\vdash) \\},    \\text{ and}\n\\]\n\\[\n\\Alg(\\vdash) = \\{ \\A : \\text{ there is }F \\subseteq A \\text{ s.t. }\\langle \\A, F \\rangle \\in \\mathsf{Mod}^{\\textup{Su}}(\\vdash) \\}.\n\\]\n\n$\\Alg(\\vdash)$ is the class of algebraic reducts of matrices in $\\mathsf{Mod}^{\\textup{Su}}(\\vdash)$.\nThe class $\\Alg(\\vdash)$ is called the \\textit{algebraic counterpart} of $\\vdash$ as, for the vast majority of logics $\\vdash$, $\\Alg(\\vdash)$  is the class of algebras intuitively associated with $\\vdash$.\n\n\n\nTrivial matrices have a central role in the whole paper. We say that a matrix $\\langle \\A, F \\rangle$ is \\textit{trivial} if $F = A$. We denote by $\\langle \\boldsymbol{1}, \\{ 1 \\} \\rangle$ the trivial matrix, whose algebraic reduct $\\boldsymbol{1}$ is the trivial algebra.\nObserve that the latter matrix is a model (resp. reduced, Suszko reduced model) of every logic. Moreover, if $\\vdash$ is a logic and $\\langle \\A, F\\rangle \\in \\mathsf{Mod}^{\\textup{Su}}(\\vdash)$ is a trivial matrix, then $\\langle \\A, F\\rangle = \\langle \\boldsymbol{1}, \\{ 1 \\}\\rangle$.\n\nA set of models of a logic $\\vdash$ is said to be non trivial, if it does not contain trivial matrices. We indicate by $\\Mod_{+}(\\vdash)$ the set of non trivial models of a logic $\\vdash$.\n \n\n\n \n\\subsection*{P\\l onka sums}\n\n\nAs standard references on P\\l onka sums we mention \\cite{Plo67,Plo67a,Romanowska92}. A \\textit{semilattice} is an algebra $\\A = \\langle A, \\lor\\rangle$, where $\\lor$ is a binary associative, commutative and idempotent operation. Given a semilattice $\\A$ and $a, b \\in A$, we set\n\\[\na \\leq b \\Longleftrightarrow a \\lor b = b.\n\\]\nIt is easy to see that $\\leq$ is a partial order on $A$.\n\\begin{definition}\\label{Def:Directed system of algebras}\nA \\textit{direct system of algebras} consists of: \n\\benroman\n\\item a semilattice $I = \\langle I, \\lor\\rangle$;\n\\item a family of similar algebras $\\{ \\A_{i} : i \\in I \\}$ with pair-wise disjoint universes;\n\\item a homomorphism $f_{ij} \\colon \\A_{i} \\to \\A_{j}$, for every $i, j \\in I$ such that $i \\leq j$.\n\\eroman\nMoreover, $f_{ii}$ is the identity map for every $i \\in I$, and $f_{ik} = f_{jk} \\circ f_{ij}$, for $i \\leq j \\leq k$.\n\\end{definition}\n\nLet $X$ be a direct system of algebras as defined above. The \\textit{P\\l onka sum} of $X$, in symbols $\\PL(X)$ or $\\PL(\\A_{i})_{i \\in I}$\\footnote{When no confusion shall occur, we will write $\\PL(\\A_{i})$ instead of $\\PL(\\A_{i})_{i\\in I}$.}, is the algebra in the same type defined as follows: the universe of $\\PL(\\A_{i})_{i \\in I}$ is the union $\\bigcup_{i \\in I}A_{i}$. Moreover, for every $n$-ary basic operation $g$ and $a_{1}, \\dots, a_{n} \\in \\bigcup_{i \\in I}A_{i}$, we set\n\\[\ng^{\\PL(\\A_{i})_{i \\in I}}(a_{1}, \\dots, a_{n}) \\coloneqq g^{\\A_{j}}(f_{i_{1} j}(a_{1}), \\dots, f_{i_{n} j}(a_{n})),\n\\]\nwhere $a_{1} \\in A_{i_{1}}, \\dots, a_{n} \\in A_{i_{n}}$ and $j = i_{1} \\lor \\dots \\lor i_{n}$.\\ \n\nObserve that if in the above display we replace $g$ by any complex formula $\\varphi$ in $n$-variables, we still have that\n\\[\n\\varphi^{\\PL(\\A_{i})_{i \\in I}}(a_{1}, \\dots, a_{n}) = \\varphi^{\\A_{j}}(f_{i_{1} j}(a_{1}), \\dots, f_{i_{n} j}(a_{n})).\n\\]\n\\noindent \\textbf{Notation:} Given a formula $\\varphi$, we will often write $\\varphi^{\\PL}$ instead of $\\varphi^{\\PL(\\A_{i})_{i \\in I}}$ when no confusion shall occur. \n\n\\vspace{5pt}\n\n\n\nThe theory of P\\l onka sums is strictly related with a special kind of binary operation, called partition function.\n\n\\begin{definition}\\label{def: partition function}\nLet $\\A$ be an algebra of type $\\nu$. A function $\\cdot\\colon A^2\\to A$ is a \\emph{partition function} in $\\A$ if the following conditions are satisfied for all $a,b,c\\in A$, $ a_1 , ..., a_n\\in A $ and for any operation $g\\in\\nu$ of arity $n\\geqslant 1$.\n\\begin{enumerate}[label=\\textbf{P\\arabic*}., leftmargin=*]\n\\item $a\\cdot a = a$;\n\\item $a\\cdot (b\\cdot c) = (a\\cdot b) \\cdot c $;\n\\item $a\\cdot (b\\cdot c) = a\\cdot (c\\cdot b)$;\n\\item $g(a_1,\\dots,a_n)\\cdot b = g(a_1\\cdot b,\\dots, a_n\\cdot b)$;\n\\item $b\\cdot g(a_1,\\dots,a_n) = b\\cdot a_{1}\\cdot_{\\dots}\\cdot a_n $.\n\\end{enumerate}\n\\end{definition}\n\nDifferent definitions of partition function appeared in literature. We adopted the one which uses the minimal number of defining conditions (see \\cite{Romanowska92}). \n\nThe next result underlines the connection between P\\l onka sums and partition functions:\n\n\\begin{theorem}\\cite[Thm.~II]{Plo67}\\label{th: Teorema di Plonka}\nLet $\\A$ be an algebra of type $\\nu$ with a partition function $\\cdot$. The following conditions hold:\n\\begin{enumerate}\n\\item $A$ can be partitioned into $\\{ A_{i} : i \\in I \\}$ where any two elements $a, b \\in A$ belong to the same component $A_{i}$ exactly when\n\\[\na= a\\cdot b \\text{ and }b = b\\cdot a.\n\\]\nMoreover, every $A_{i}$ is the universe of a subalgebra $\\A_{i}$ of $\\A$.\n\\item The relation $\\leq$ on $I$ given by the rule\n\\[\ni \\leq j \\Longleftrightarrow \\text{ there exist }a \\in A_{i}, b \\in A_{j} \\text{ s.t. } b\\cdot a =b\n\\]\nis a semilattice order.\n\\item For all $i,j\\in I$ such that $i\\leq j$ and $b \\in A_{j}$, the map $f_{ij} \\colon A_{i}\\to A_{j}$, defined by the rule $f_{ij}(x)= x\\cdot b$ is a homomorphism. The definition of $f_{ij}$ is independent from the choice of $b$, since $a\\cdot b = a\\cdot c$, for all $a\\in A_i$ and $c\\in A_j$.\n\\item $Y = \\langle \\langle I, \\leq \\rangle, \\{ \\A_{i} \\}_{i \\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}\\rangle$ is a direct system of algebras such that $\\PL(Y)=\\A$.\n\\end{enumerate}\n\\end{theorem}\n\n\nThe statement of Theorem \\ref{th: Teorema di Plonka} displayed above relies on the assumption that the algebraic language contains no constant symbols\\footnote{When considering types containing constants, then additional conditions should be added to the definition of partition function. This results into a decomposition over a semilattice having a least element: constants of the P\\l onka sum will belong to the algebra whose index is the least element. For further details, see \\cite{plonka1984sum}.}. \nIt is worth remarking that the construction of Plonka sums preserves the validity of \\textit{regular identities}, i.e. identities of the form $ \\varphi \\thickapprox \\psi $ such that $\\Var(\\varphi) = \\Var(\\psi)$.\n\n\n\n\n\\section{Algebraic completeness}\\label{sec:compl}\n\nThe usual presentations of Kleene three-valued logics divide them into two families, depending on the meaning given to the connectives $\\land,\\lor$: \\emph{strong logics} -- including Strong Kleene and the Logic of Paradox \\cite{Priestfirst} -- and \\emph{weak logics} -- Bochvar logic ($\\mathrm{B_3}$) and paraconsistent weak Kleene (or Halld\\'en logic). Logics in each family differentiate upon the choice of the truth-set: $\\{1\\}$ in Strong Kleene and Bochvar, $\\{1,\\ant \\}$ in the logic of Paradox and paraconsistent weak Kleene.\n\nBochvar \\cite{BochvarBergmann} is the logic induced by the matrix $\\pair{\\WK,\\{1\\}}$ of the so-called weak Kleene tables\\footnote{In accordance with \\cite{Urquhart2001}, here we are actually considering Bochvar ``internal calculus'', which is only one of the two logics introduced in \\cite{BochvarBergmann}. The ``external calculus'' consists of a linguistic extension of the weak Kleene tables, with a connective $\\mathrm{t}$, interpreted (for every evaluation $h$) as $h(\\mathrm{t}\\varphi)= 1$ if and only if $h(\\varphi)=1$ (for further details, see \\cite{BochvarBergmann,KarpenkoTomova,FinnGrigolia}).} displayed in Figure \\ref{fig: tavole weak}. \n\n\\vspace{5pt}\n\\begin{figure}[h]\n\\begin{center}\\renewcommand{\\arraystretch}{1.25}\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\land & 0 & \\ant & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & \\ant & 0 \\\\\n       \\ant & \\ant & \\ant & \\ant \\\\          \n       1 & 0 & \\ant & 1\n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\lor & 0 & \\ant & 1 \\\\[.2ex]\n \\hline\n     0 & 0 & \\ant & 1 \\\\\n     \\ant & \\ant & \\ant & \\ant \\\\          \n     1 & 1 & \\ant & 1\n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}}\n  \\lnot &  \\\\[.2ex]\n\\hline\n  1 & 0 \\\\\n  \\ant & \\ant \\\\\n  0 & 1 \\\\\n\\end{tabular}\n\n\\caption{The algebra $\\mathbf{WK}$ of weak Kleene tables.}\\label{fig: tavole weak}\n\\end{center}\n\\end{figure}\n\\vspace{10pt}\n\\noindent\nIt is not difficult to check that the algebra $\\mathbf{WK}$ is the P\\l onka sum of the two-element Boolean algebra $\\two$ and the trivial (Boolean) algebra $\\mathbf{\\ant}$ (over the index set given by the two-element semilattice)\\footnote{We refer the reader interested in further details directly to \\cite{Bonzio16}.}.\n\nBochvar logic can be equivalently presented as follows:\n\n\\begin{theorem}\\cite[Theorem 2.3.1]{Urquhart2001}\\label{teorema: Urquhart}\nThe following are equivalent: \n\\begin{enumerate}\n\\item $\\Gamma\\vdash_{\\mathrm{B_3}}\\varphi$;\n\\item $\\Gamma\\vdash_{\\ensuremath{\\mathsf{CL}}}\\varphi $ with $ \\Var(\\varphi)\\subseteq\\Var(\\Gamma)$ or $\\Gamma$ is an inconsistent set of formulas.  \n\\end{enumerate}\n\\end{theorem}\n \nIn words, Bochvar logic is the consequence relation obtained out of classical logics, imposing the constraint that variables of the conclusion (formula) shall be included into those of the (set of) premises. For this reason, $\\mathrm{B_{3}}$ is often referred to as a \\emph{containment logic}, see for e.g. \\cite{ferguson2017meaning,Parry}. \n\n\nThe following definition originates in \\cite{ThomasLavtesi}, but see also \\cite{CampercholiRaftery,JGR13}.\n\n\\begin{definition}\\label{def inconsistency terms}\nA set of formulas $\\Sigma$ is an antitheorem of a logic $\\vdash$ if\n$\\sigma[\\Sigma] \\vdash \\varphi$, for every substitution $\\sigma:\\Fm\\to\\Fm$ and formula $\\varphi$.\n\\end{definition}\n\nObserve that, if the set $\\Sigma(y_{1},\\dots,y_{n})$, where the variables $y_{1},\\dots,y_{n}$ really occur, is an antitheorem for $\\vdash$, then, by substitution, also $\\Sigma(x)$ (where only $x$ occurs) is an antitheorem for $\\vdash$. In other words, if a logic $\\vdash$ possesses an antitheorem $\\Sigma$, then it possesses an antitheorem in one variable only and we will write $\\Sigma(x)$.  \n\nThe most intuitive example one can keep in mind is the following: for any formula $\\varphi$, the set $\\{\\varphi,\\neg\\varphi\\}$ is an antitheorem of Intuitionistic, Classical and both local and global modal logics\n\n\n\nThe above characterization of Bochvar logic suggests that a logic $\\vdash^{r}$ satisfying an analogous criterion on the inclusion of variables as that of Theorem \\ref{teorema: Urquhart} can be associated to any arbitrary logic $\\vdash$. \n\n\\begin{definition}\\label{def: containment companion}\nLet $\\vdash$ be a logic. $\\vdash^{r}$ is the logic defined as \n\n\\[\n\\Gamma\\vdash^{r}\\varphi\\iff \\left\\{ \\begin{array}{ll}\n\\Gamma\\vdash\\varphi  \\ \\text{and} \\ \\Var(\\varphi)\\subseteq\\Var(\\Gamma)& \\text{or}\\\\\n\\Sigma(x)\\subseteq\\Gamma, & \\\\\n  \\end{array} \\right.  \n\\]\n\nwhere $\\Sigma(x)$ is an antitheorem of $\\vdash$.\n\\end{definition}\n\nWe will refer to $\\vdash^{r}$ as the \\emph{containment companion} of the logic $\\vdash$. It follows from the definition that $\\vdash^{r}$ and $\\vdash$ have the same antitheorems. \nBochvar logic is not the unique example of containment logic that can be found in literature. Indeed, the four-valued logic $\\mathrm{\\mathbf{S}_{fde}}$, introduced by Deutsch \\cite{Deutsch} (see also \\cite{BelikovPetrukhin, Szmuch2016, ciunilast}), can be counted as the containment companion of the Logic of Paradox (this follows from our analysis, see Subsection \\ref{Subsec: relevance}). Also one of the four-valued logics introduced by Tomova (see Example \\ref{ex: containment di PWK}) is a containment logic, more precisely the containment companion of \\ensuremath{\\mathsf{PWK}}.\n\n\nThe containment companion ($\\vdash^{r}$) of a logic $\\vdash$ is, somehow, ``opposite'' to the logic $\\vdash_{l}$ considered in \\cite{BonzioMorascoPrabaldi}, as the inclusion of variables works from conclusions to premises,  namely from ``right to left'', and not vice-versa (as for $\\vdash_{l}$).\n\\begin{lemma}\\label{lemma: finitarieta di r}\nLet $\\vdash$ be a finitary logic. Then $\\vdash^{r}$ is finitary.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\Gamma\\vdash^{r}\\varphi$. By definition of $\\vdash^{r}$, two cases are to be considered: \n\\begin{itemize}\n\\item[i)] $\\Gamma\\vdash\\varphi$ with $\\Var(\\varphi)\\subseteq\\Var(\\Gamma)$;\n\\item[ii)] $\\Gamma $ contains an antitheorem $\\Sigma(x)$ of $\\vdash$.\n\\end{itemize}\ni) Clearly, $\\mid\\Var(\\varphi)\\mid < \\infty$, hence there exist formulas $\\gamma_{1},\\dots,\\gamma_{n}\\in \\Gamma$ such that $\\Var(\\varphi)\\subseteq\\Var(\\gamma_{1})\\cup\\dots\\cup\\Var(\\gamma_{n})$. Since $\\vdash$ is finitary, then there exists a finite set $\\Gamma'\\subseteq\\Gamma$ such that $\\Gamma'\\vdash\\varphi$. If $\\Var(\\varphi)\\subseteq\\Var(\\Gamma')$, then also $\\Gamma'\\vdash^{r}\\varphi$, i.e. $\\vdash^{r}$ is finitary. So, suppose that $\\Var(\\varphi)\\not\\subseteq\\Var(\\Gamma')$. Consider $\\Delta=\\Gamma'\\cup\\{\\gamma_{1},\\dots,\\gamma_{n}\\}$. Obviously, $\\Delta$ is finite, $\\Var(\\varphi)\\subseteq\\Var(\\Delta)$ and $\\Delta\\subseteq\\Gamma$. By monotonicity of $\\vdash$, $\\Delta\\vdash\\varphi$, hence $\\Delta\\vdash^{r}\\varphi$, i.e. $\\vdash^{r}$ is finitary. \\\\\n\\noindent\nii) Since $\\Sigma(x)$ is an antitheorem for $\\vdash$, then $\\Sigma(x)\\vdash\\varphi$. Hence \\\\\n\\noindent\n$\\Sigma(x)\\vdash_{r}\\varphi$ and $\\Sigma(x)$ is finite (as $\\vdash$ is finite). \n\\end{proof}\n\n\n\n\nSince the algebra $\\WK$ is a P\\l onka sum (of Boolean algebras), it makes sense to ask whether the matrix $\\pair{\\WK,\\{1\\}}$ can be constructed as P\\l onka sum of (two) matrices. To the best of our knowledge, the construction of P\\l onka sums between matrices has been developed exclusively in \\cite{BonzioMorascoPrabaldi}. However, it is not difficult to check that the mentioned construction, when applied to the matrices $\\pair{\\two, \\{1\\}}$ and $\\pair{\\mathbf{\\ant}, \\emptyset}$ (where $\\two$ and $\\mathbf{\\ant}$ stand for the two-element Boolean algebra and the trivial algebra, respectively), does not result in  $\\pair{\\WK,\\{1\\}}$. This suggests that a different notion of direct system of logical matrices shall be introduced. \n\n\n\\begin{definition}\\label{Def:Directed-System-Matrices}\nAn \\textit{r-direct system} of matrices consists of: \n\\benroman\n\\item A semilattice $I = \\langle I, \\lor\\rangle$.\n\\item A family of matrices $\\{ \\langle\\A_{i},F_{i}\\rangle : i \\in I \\}$ such that \\\\ $I^{+}\\coloneqq\\{i\\in I: F_{i}\\neq\\emptyset\\} $ is a  sub-semilattice of $I$.\n\\item a homomorphism $f_{ij} \\colon \\A_{i} \\to \\A_{j}$, for every $i, j \\in I$ such that $i \\leq j$, satisfying also that: \n\\begin{itemize}\n\\item $f_{ii}$ is the identity map, for every $i \\in I$;\n\\item if $i \\leq j \\leq k$, then $f_{ik} = f_{jk} \\circ f_{ij}$;\n\\item if $F_{j}\\neq\\emptyset$ then $ f_{ij}^{-1}[F_{j}]= F_{i}$, for any $i\\leq j$.\n\\end{itemize}\n\\eroman\n\\end{definition}\n\nAs the nomenclature highlights, the above introduced notion of direct system of matrices is essentially different from the one in \\cite{BonzioMorascoPrabaldi}. The main difference concerns the interplay between homomorphisms of the system and matrices' filters.\n\n\nGiven a r-direct system of matrices $X$, we define a new matrix as\n\\[\n\\PL(X) \\coloneqq \\langle \\PL(\\A_{i})_{i \\in I}, \\bigcup_{i \\in I}F_{i}\\rangle.\n\\]\n\\vspace{5pt}\n\n\\noindent\nWe will refer to the matrix $\\PL(X)$ as the \\textit{P\\l onka sum} over the r-direct system of matrices $X$. Given a class $\\mathsf{M}$ of matrices, $\\PL(\\mathsf{M})$ will denote the class of all P\\l onka sums of r-direct systems of matrices in $\\mathsf{M}$.\n\n\n\nLet $h\\colon\\Fm\\to\\PL(\\A_i)$ be a homomorphism from the formula algebra into a generic P\\l onka sum of algebras. Then, for any formula $\\varphi\\in Fm$, we set \n\n\\[\n i_{h}(\\varphi)\\coloneqq\\bigvee\\{i\\in I:h(x)\\in A_{i}, x\\in\\Var(\\varphi)\\}.\n \\]\n \n \\vspace{2pt}\n\nIn words, $i_{h}(\\varphi)$ indicates the index where the formula $\\varphi$ is interpreted by the homomorphism $h$, into a P\\l onka sum. Moreover, for any $\\Gamma\\subseteq Fm$,\nwe set $i_{h}(\\Gamma)\\coloneqq\\bigvee \\{i_{h}(x)\\colon x\\in\\Var(\\Gamma)\\}$. \n\n\\begin{remark}\\label{rem: Gamma finito}\nNotice that the index $i_{h}(\\Gamma)$ is defined provided that the set $\\Var(\\Gamma)$ is finite. \nFor several results, we will assume that the logic $\\vdash$ is finite. Hence, by Lemma \\ref{lemma: finitarieta di r}, also $\\vdash^{r}$ is finitary, and this allows us to consider finite sets $\\Gamma\\subseteq Fm$, for which the existence of $i_{h}(\\Gamma)$ is assured.\nMoreover, observe that, for every homomorphism $h\\colon\\Fm\\to\\PL(X)$ from the formula algebra into a generic P\\l onka sum over an r-direct system of matrices $X$, and every $\\Gamma\\cup\\{\\varphi\\}\\subseteq Fm$, it is immediate to check that $\\Var(\\varphi)\\subseteq\\Var(\\Gamma)$ implies $i_{h}(\\varphi)\\leq i_{h}(\\Gamma)$.\n\\end{remark}\n\n\n\n\n\\begin{lemma}\\label{lemma: soundness}\nLet $X$ be an r-direct system of non trivial models of a finitary logic $\\vdash$. Then $\\PL(X)=\\langle\\PL(\\A_i),\\bigcup_{i\\in I}F_i\\rangle$ is a model of $\\vdash^{r}$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $X$ be an r-direct system of non trivial models of $\\vdash$. Assume $\\Gamma\\vdash^{r}\\varphi$. Since $\\vdash $ is finitary, so it is also $\\vdash^{r}$ (by Lemma \\ref{lemma: finitarieta di r}), there exists a finite subset $\\Delta\\subseteq\\Gamma $, such that $\\Delta\\vdash^{r}\\varphi$.\nWe distinguish the following cases:\n\n\\begin{itemize}\n\\item[(a)] $\\Sigma(x)\\subseteq\\Delta$, where $\\Sigma(x)$ is an antitheorem of $\\vdash$;\n\\vspace{3pt}\n\\item[(b)] $\\Delta\\vdash\\varphi$ with $\\Var(\\varphi)\\subseteq\\Var(\\Delta)$. \n\\end{itemize}\n\n\n\n\\noindent\nSince $X$ contains non-trivial models only, the case (a) easily follows by noticing that, for any homomorphism $h\\colon \\Fm\\to\\PL(\\A_i)$, $h[\\Sigma(x)]\\not\\subseteq F=\\bigcup_{i\\in I}F_i$.\nTherefore $\\Sigma(x)\\vdash_{\\PL(X)}\\varphi$, hence also $\\Delta\\vdash_{\\PL(X)}\\varphi$. \n\\vspace{5pt}\n\n\\noindent\nSuppose (b) is the case, i.e. $\\Delta\\vdash\\varphi$ with $\\Var(\\varphi)\\subseteq\\Var(\\Delta)$. Let $h\\colon\\Fm\\to\\PL(\\A_{i})$ be a homomorphism such that $h[\\Delta]\\subseteq F$.\nSince $\\Delta$ is a finite set, then we can fix $j\\coloneqq i_{h}(\\Delta)$ and, for any formula $\\delta\\in \\Delta$, we have $h(\\delta)\\in F_{i_{h}(\\delta)}$. This implies that each $i_{h}(\\delta)\\in I^{+}$ and, as $I^{+}$ forms a  sub-semilattice of $I$, we have that $j\\in I^{+}$.\n\nNow, define $g\\colon\\Fm\\to\\A_{j}$ as \n\\[\ng(x)\\coloneqq f_{i_{h}(x)j}\\circ h(x),\n\\]\nfor every $x\\in\\Var(\\Delta)$. For any $\\delta\\in \\Delta$, we have $g(\\delta)=f_{i_{h}(\\delta) j}\\circ h(\\delta)$, hence $g[\\Delta]\\subseteq F_{j}$. From the fact that $\\Delta\\vdash\\varphi$ and $\\langle\\A_{j},F_{j}\\rangle\\in\\Mod(\\vdash)$, it follows that $g(\\varphi)\\in F_{j}$. Setting $k\\coloneqq  i_{h}(\\varphi)$, by Remark \\ref{rem: Gamma finito}, we have $k\\leq j$ and this, together with the observation that $F_{j}\\neq\\emptyset$, implies $f_{kj}^{-1}[F_{j}]=F_{k}$. Moreover, we claim that $F_{k}\\neq\\emptyset$. Suppose, by contradiction, that $F_{k}=\\emptyset$. Then, by definition of r-direct system of matrices, we have that $f_{kj}^{-1}[F_{j}]=\\emptyset$, that is: there exists no $a\\in A_{k}$ such that $f_{kj}(a)\\in F_{j}$. On the other hand, since $\\Var(\\varphi)\\subseteq \\Var(\\Delta)$, then $g(\\varphi)=f_{kj}\\circ h(\\varphi)\\in F_{j}$, a contradiction.  \n\nFrom the fact that $g(\\varphi)\\in F_{j}$ together with $f_{kj}^{-1}[F_{j}]=F_{k}$, we conclude $h(\\varphi)\\in F_{k}$. This proves that $h(\\varphi)\\in F_{k}\\subseteq\\bigcup_{i\\in I}F_{i}$. \n\\end{proof}\n\n\\begin{remark}\\label{rem su inconsistency}\n\nObserve that the assumption on the non-triviality of models of the logic $\\vdash$ is crucial in Lemma \\ref{lemma: soundness}, as witnessed by the following example. Let $\\vdash$ be a theoremless logic possessing an anti-theorem $\\Sigma(x)$ (an example is the almost inconsistent logic). \n Set $X=\\pair{\\A\\oplus \\mathbf{1}, A}$ to be the r-direct system of models of $\\vdash$, consisting of the two algebras $\\mathbf{A}$ and $\\mathbf{1}$ with the unique homomorphism $f\\colon\\A\\to\\mathbf{1}$ (plus the identity homomorphisms).\nThen $\\Sigma(x)\\vdash y$, for an arbitrary variable $y$, and therefore $\\Sigma(x)\\vdash^{r} y$. However, $\\PL(X)$ is not a model of the latter inference (consider, for instance, an evaluation $v\\colon\\Fm\\to \\PL(\\A\\oplus \\mathbf{1})$ such that $v(x)=a\\in A$ and $v(y)=1$). \n\\end{remark}\n\nObserve that, if the logic $\\vdash$ does not possess an antitheorem, then the following holds:\n\\begin{corollary}\nLet $X$ be an r-direct system of models of a finitary logic  $\\vdash$ possessing no antitheorems. Then $\\PL(X)$ is a model of $\\vdash^{r}$.\n\\end{corollary}\n\n\n\nGiven a logic $\\vdash$ which is complete with respect to a class $\\mathsf{M}$ of matrices, we set  $\\mathsf{M^{\\emptyset}}\\coloneqq\\mathsf{M}\\cup\\langle\\A,\\emptyset\\rangle$, for any arbitrary $\\A\\in\\Alg(\\vdash)$.\n\n\n\n\\begin{theorem}\\label{completeness}\nLet $\\vdash$ be a finitary logic which is complete with respect to a class of non trivial matrices $\\mathsf{M}$. Then $\\vdash^{r}$ is complete with respect to $\\PL(\\mathsf{M^{\\emptyset}})$. \n\\end{theorem}\n\n\\begin{proof}\n\nWe aim at showing that $\\vdash^{r}=\\;\\vdash_{\\PL(\\mathsf{M^{\\emptyset}})}$. \\\\\n\\noindent\n($\\vdash^{r}\\;\\subseteq \\;\\vdash_{\\PL(\\mathsf{M^{\\emptyset}})}$). Consider $\\Gamma\\vdash^r\\varphi$ and a P\\l onka sum  $\\langle\\PL (\\A_i),\\bigcup_{i\\in I}F_{i}\\rangle$ of matrices in $\\mathsf{M^{\\emptyset}}$. Set $\\A=\\PL(\\A_i)$ The cases in which $\\Gamma$ is an antitheorem of $\\vdash$ or $\\langle\\A,\\emptyset\\rangle$ is a model of $\\vdash$ follow by Lemma \\ref{lemma: soundness}. \n\nSo, assume $\\langle\\A,\\emptyset\\rangle$ is not a model of $\\vdash$ and that $\\Gamma$ is not an antitheorem of $\\vdash$. \nLet $h\\colon Fm\\to \\A$ be a homomorphism such that $h[\\Gamma]\\subseteq \\bigcup_{i \\in I}F_{i}$. Suppose, in view of a contradiction, that $h(\\varphi)\\not\\in \\bigcup_{i \\in I}F_{i}$.\n Set $i_{h}(\\varphi) = j$ and $ i_{h}(\\Gamma) = k$; since $\\Var(\\varphi)\\subseteq\\Var(\\Gamma)$ then $j\\leq k$, by Remark \\ref{rem: Gamma finito}. We define a homomorphism $v\\colon \\Fm\\to \\A_k$, as follows\n\\[\nv(x)\\coloneqq f_{lk}\\circ h (x),\n\\]\nwhere $l=i_{h}(x)$. Clearly, $v[\\Gamma]=f_{kk}\\circ h[\\Gamma] = h[\\Gamma]\\subseteq F_{k}$ and $v(\\varphi)=f_{jk}\\circ h (\\varphi)\\in A_{k}\\smallsetminus F_k$, since $h(\\varphi)\\in A_{j}\\smallsetminus F_{j}$ and $F_{j}=f^{-1}_{jk}[F_k]$ (as we know that $F_k\\neq\\emptyset$). Therefore, we have $\\Gamma\\not\\vdash \\varphi$, which is a contradiction. \\\\\n\\noindent\n($\\vdash_{\\PL(\\mathsf{M^{\\emptyset}})}\\;\\subseteq\\;\\vdash^{r}$). By contraposition, we prove that $\\Gamma\\nvdash^{r}\\varphi $ implies $\\Gamma\\nvdash_{\\PL(\\mathsf{M^{\\emptyset}})}\\varphi$. To this end, assume $\\Gamma\\nvdash^{r}\\varphi$. If $\\Var(\\varphi)\\subseteq\\Var(\\Gamma)$, clearly $\\Gamma\\nvdash\\varphi$. Therefore there exists a matrix $\\langle\\A_{i},F_{i}\\rangle\\in\\mathsf{M}$ and a homomorphism $h\\colon\\Fm\\to\\A_{i}$ such that $h[\\Gamma]\\subseteq F_i$ and $h(\\varphi)\\notin F_i$. Upon considering the r-direct system $X=\\pair{\\langle\\A_{i},F_{i}\\rangle, \\{i\\}, id}$ and the homomorphism $h$, we immediately obtain $\\Gamma\\nvdash_{\\PL(\\mathsf{M^{\\emptyset}})}\\varphi$.\n\nThe only other case to consider is $\\Var(\\varphi)\\nsubseteq\\Var(\\Gamma)$. Preliminarily, observe that the assumption $\\Gamma\\nvdash^{r}\\varphi$ implies that $\\Gamma$ contains no anti-theorem $\\Sigma(x)$ for $\\vdash$. Therefore, since  $\\mathsf{M}$ is a class of models complete with respect to $\\vdash$, there exists a matrix $\\langle\\B,G\\rangle\\in\\mathsf{M}$ and a homomorphism $v\\colon\\Fm\\to\\B$ such that $v[\\Gamma]\\subseteq G$ and $v(\\varphi)\\notin G$. Consider any r-direct system formed by the matrices $\\langle\\B,G\\rangle$ and  $\\langle \\A,\\emptyset \\rangle$ for an appropriate $\\A\\in\\Alg(\\vdash)$  (the choice $\\A=\\boldsymbol{1}$ is always appropriate), with $\\langle\\A,\\emptyset\\rangle$  indexed as top element. Denote by $\\B \\oplus \\A^{\\emptyset}$ a P\\l onka sum over the r-direct system just described. \n\nThe homomorphism $g\\colon\\Fm\\to\\B \\oplus \\A^{\\emptyset}$ defined as\n\\[\ng(x)\\coloneqq \\left\\{ \\begin{array}{ll}\nv(x) \\ \\text{if} \\ \\ x\\in\\Var(\\Gamma),& \\\\\na \\ \\text{otherwise.} & \\\\\n  \\end{array} \\right.  \n\\]\nfor arbitrary $a\\in A$\n\\noindent\neasily witnesses $\\Gamma\\nvdash_{\\PL(\\mathsf{M^{\\emptyset}})}\\varphi$, as desired.\n\\end{proof}\n\n\\begin{example}\\label{ex: containment di PWK}\nLet $\\mathbf{K_{4}}=\\pair{\\{0,1,\\ant, n \\}, \\neg,\\land,\\vee}$ be the algebra given by the following tables \n\\vspace{5pt}\n\\begin{center}\\renewcommand{\\arraystretch}{1.25}\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}}\n  \\lnot &  \\\\[.2ex]\n\\hline\n  1 & 0 \\\\\n  \\ant & \\ant \\\\\n  n & n \\\\\n  0 & 1 \\\\\n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\land & 0 & \\ant & n & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & \\ant & n &  0 \\\\\n       \\ant & \\ant & \\ant & n & \\ant \\\\\n       n & n & n & n & n \\\\          \n       1 & 0 & \\ant & n & 1 \n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\lor & 0 & \\ant & n & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & \\ant & n &  1 \\\\\n       \\ant & \\ant & \\ant & n & \\ant \\\\\n       n & n & n & n & n \\\\          \n       1 & 1 & \\ant & n & 1 \n\\end{tabular}\n\n\\end{center}\n\\vspace{10pt}\n\nThe logic $\\mathbf{K^{w}_{4n}}= \\pair{\\mathbf{K_4},\\{1,\\ant\\}}$ is included among the four-valued regular logics counted by Tomova (see \\cite{Tomova,Petrukhin2}). \nObserve that $\\pair{\\mathbf{K_4},\\{1,\\ant\\}}$ is the P\\l onka sum (over the r-direct system) of the matrices $\\pair{\\mathbf{WK},\\{1,\\ant\\}}$ and $\\pair{\\mathbf{n},\\emptyset}$. Since \\ensuremath{\\mathsf{PWK}} is complete with respect to $\\pair{\\mathbf{WK},\\{1,\\ant\\}}$, then, it follows by Theorem \\ref{completeness}, that $\\mathbf{K^{w}_{4n}} =\\; \\vdash^{r}_{\\ensuremath{\\vDash_\\mathsf{pwk}}}$, i.e. $\\mathbf{K^{w}_{4n}}$ is the containment companion of \\ensuremath{\\mathsf{PWK}}. \n\\qed\n\\end{example}\n\nAs \\ensuremath{\\mathsf{PWK}} is the \\emph{left variable inclusion} companion of classical logic (see \\cite{BonzioMorascoPrabaldi}), the above example shows that the constructions yielding the (two) companions (left variable inclusion and containment) can actually be iterated, in alternation, starting from an arbitrary logic $\\vdash$ (for further details see \\cite{prabaldisub}). \n\n\\begin{remark}\nObserve that if $\\vdash$ is a logic which is complete with respect to a finite set of finite matrices, then so is $\\vdash^r$ (by Theorem \\ref{completeness}). This means that the containment companion of a logic $\\vdash$ preserves ``finite valuedness'', a notion introduced and studied in \\cite{CALEIROfinite}. \n\\end{remark}\n\n   \n\n\nTheorem \\ref{completeness} provides a complete class of matrices for an arbitrary logic of right variable inclusion. This class is obtained performing P\\l onka sums over r-direct systems of models of $\\vdash$ together with the matrices $\\langle \\A,\\emptyset\\rangle$ for any $\\A\\in\\Alg(\\vdash)$. Obviously, it is not generally the case that the matrix $\\langle \\A,\\emptyset\\rangle $ is a model of a logic $\\vdash$.\nFor this reason, it is not always true that P\\l onka sums over an r-direct systems of models of $\\vdash$ provide a complete matrix semantics for $\\vdash^{r}$. In this sense, the right variable inclusion companion of a logic is a logic of ``P\\l onka sums'' (of matrices) in weaker sense compared to the case of the left variable inclusion companion, fully described in \\cite{BonzioMorascoPrabaldi}.\nNonetheless, if $\\langle \\boldsymbol{1},\\emptyset\\rangle\\in\\Mod(\\vdash)$, the correspondence between $\\vdash^{r}$ and P\\l onka sums is fully recovered. This is actually the case of every theoremless logic, such as Strong Kleene Logic, $\\vdash_{\\ensuremath{\\mathsf{CL}}}^{\\land,\\lor}$.\n\n\\begin{corollary}\\label{cor completezza con incons}\nA finitary containment logic $\\vdash^{r}$ is complete w.r.t. any of the following classes of matrices:\n\\begin{center}\n$\\PL(\\Mod_{+}(\\vdash)\\cup\\langle \\A,\\emptyset\\rangle)$, $\\PL(\\Modstar_{+}(\\vdash)\\cup\\langle \\A,\\emptyset\\rangle)$, $\\PL(\\mathsf{Mod}^{\\textup{Su}}_{+}(\\vdash)\\cup\\langle \\A,\\emptyset\\rangle),$\n\\end{center}\nfor $\\A\\in\\Alg(\\vdash)$.\n\\end{corollary}\n\n\\noindent\nMoreover, observing that if $\\langle \\mathbf{1},\\emptyset\\rangle\\in\\Mod(\\vdash)$ then $\\langle \\mathbf{1},\\emptyset\\rangle\\in\\Modstar(\\vdash)$, the following holds\n\n\\begin{corollary}\nLet $\\vdash$ be a finitary logic such that $\\langle \\mathbf{1},\\emptyset\\rangle\\in\\Mod(\\vdash)$. Then $\\vdash^{r}$ is complete w.r.t. any of the following classes of matrices:\n\\begin{center}\n$\\PL(\\Mod_{+}(\\vdash))$, $\\PL(\\Modstar_{+}(\\vdash))$, $\\PL(\\mathsf{Mod}^{\\textup{Su}}_{+}(\\vdash)).$\n\\end{center}\n\\end{corollary}\n\nIn case $\\vdash$ does not possess anti-theorems, then the above corollaries can be restated as follows\n\n\\begin{corollary}\\label{cor da usare per calcolo hilbert no inc}\nLet $\\vdash$ be a finitary logic without antitheorems. Then $\\vdash^{r}$ is complete w.r.t. any of the following classes of matrices:\n\\begin{center}\n$\\PL(\\Mod(\\vdash)\\cup\\langle\\A,\\emptyset\\rangle)$, $\\PL(\\Modstar(\\vdash)\\cup \\langle\\A,\\emptyset\\rangle)$, $\\PL(\\mathsf{Mod}^{\\textup{Su}}(\\vdash)\\cup\\langle\\A,\\emptyset\\rangle),$\n\\end{center}\nfor any $\\A\\in\\Alg(\\vdash)$.\n\\end{corollary}\n\n\n\\begin{corollary}\nLet $\\vdash$ a finitary logic without antitheorems such that $\\langle \\boldsymbol{1},\\emptyset\\rangle\\in\\Mod(\\vdash)$, then $\\vdash^{r}$ is complete w.r.t. any of the following classes of matrices:\n\\begin{center}\n$\\PL(\\Mod(\\vdash))$, $\\PL(\\Modstar(\\vdash))$, $\\PL(\\mathsf{Mod}^{\\textup{Su}}(\\vdash)).$\n\\end{center}\n\\end{corollary}\n\n\n\n\n\n\n\n\n\\section{Hilbert style calculi (for logics with r-partition function)}\\label{sec:hilbert}\n\nPartitions functions, which have been defined for algebras (see Definition \\ref{def: partition function}), can be defined also for logics. \n\n\\begin{definition}\\label{def: logic with p-func.}\nA logic $\\vdash$ has a \\emph{r-partition function} if there is a formula $x\\ast y$, in which the variables $x$ and $y$ really occur, such that: \n\\benroman\n\\item$x,y\\vdash x\\ast y$, \n\\item$x\\ast y \\vdash x$,\n\\item $\\varphi(\\varepsilon,\\vec{z}\\?\\?)\\sineq\\varphi(\\delta,\\vec{z}\\?\\?)$,\n\\eroman\nfor every formula $\\varphi(v,\\vec{z}\\?\\?)$ and every identity of the form $\\varepsilon \\thickapprox \\delta$ in Definition \\ref{def: partition function}.\n\\end{definition}\n\n\nCondition (iii) in the Definition of r-partition function is actually equivalent to say that the term operation $\\ast$ is a partition function in every algebra $\\A\\in\\Alg(\\vdash)$. This is the consequence of the following (known) fact in abstract algebraic logic. \n\n\\begin{lemma}\\label{lemma su equazioni p-function}\nLet $\\vdash$ be a logic and $\\varepsilon,\\delta\\in Fm $. The following are equivalent:\n\\benroman\n\\item $\\Alg(\\vdash)\\vDash\\varepsilon\\approx\\delta$;\n\\item $\\varphi(\\varepsilon,\\vec{z}\\?\\?)\\sineq\\varphi(\\delta,\\vec{z}\\?\\?)$, for every formula $\\varphi(v,\\vec{z}\\?\\?)$.\n\\eroman\n\\end{lemma}\n\\begin{proof}\nSee \\cite[Lemma 5.74(1)]{Font16} and \\cite[Theorem 5.76]{Font16}.\n\\end{proof}\n\n\n\n\nFrom now on, we will denote both the formula $x\\ast y$ and the term operation $\\ast$ as r-partition functions with respect to a logic $\\vdash$.\n\nThe definition of partition functions for an arbitrary logic is introduced also in \\cite[Definition 16]{BonzioMorascoPrabaldi}. It shall be noticed that Definition \\ref{def: logic with p-func.} is essentially different (this also motivates the choice of the terminology $r$-partition function). Nevertheless, in most cases (for instance, all substructural logics, classical and modal logics) the very same formula plays both the role of a r-partition function and of a partition function in the sense of \\cite[Definition 16]{BonzioMorascoPrabaldi}. \n\n\\begin{example}\nLogics with a r-partition function abound in the literature. Indeed, the term $x\\ast y\\coloneqq x\\land(x\\lor y)$ is a partition function for every logic $\\vdash$ such that $\\Alg(\\vdash)$ has a lattice reduct. Such examples include all modal and substructural logics \\cite{GaJiKoOn07}. On the other hand, the term $x\\ast y\\coloneqq (y \\to y) \\to x$ as a r-partition function for all the logics $\\vdash$ whose class $\\Alg(\\vdash)$ possesses a Hilbert algebra (see \\cite{Di65}) or a BCK algebra (see \\cite{Iseki}) reduct.\n\\end{example}\n\n\\begin{remark}\\label{rem: vdash e vdashr hanno le stessa pf}\nIt is easily checked that a logic $\\vdash$ has r-partition function $\\ast$ if and only if $\\vdash^{r}$ has r-partition function $\\ast$.\n\\end{remark}\n\nIn the following, we extend P\\l onka representation theorem to r-direct systems of logical matrices.\n\n\n\n\n\\begin{theorem}\\label{th: plonka sum of r- matrices}\nLet $\\vdash$ be a logic with $r$-partition function $\\ast$, and $\\langle\\A,F\\rangle$ be a model of $\\vdash$ such that $\\A\\in\\Alg(\\vdash) $. Then Theorem \\ref{th: Teorema di Plonka} holds for $\\A$.\nMoreover, by setting $F_{i} \\coloneqq F\\cap A_{i}$ for every $i \\in I$, the triple\n\\[\nX=\\langle \\langle I,\\leq\\rangle , \\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}\\rangle\n\\]\nis an r-direct system of matrices such that $\\PL(X)=\\langle\\A,F\\rangle$.\n\\end{theorem}\n\n\\begin{proof}\nTheorem \\ref{th: Teorema di Plonka} holds for $\\A$, by simply observing that $\\ast$ is a partition function for $\\A$. \nFor the remaining part, it will be enough to show: \n\\begin{itemize}\n\\item[(a)] for every $i,j\\in I$ such that $i\\leq j$, if $F_{j}\\neq\\emptyset$ then $f_{ij}^{-1}[F_{j}]=F_{i}$;\n\\item[(b)]  $I^{+}$ is a sub-semilattice of $I$.\n\\end{itemize}\n\n\nIn order to prove (a), consider $i,j\\in I$ such that $i\\leq j$ and let $F_{j}$ be non-empty. Assume, in view of a contradiction, that $f_{ij}^{-1}[F_{j}]\\neq F_{i}$. This implies that $F_{i}\\nsubseteq f_{ij}^{-1}[F_{j}]$ or that $f_{ij}^{-1}[F_{j}]\\nsubseteq F_{i}$. The first case immediately leads to the contradiction that $x,y\\nvdash x\\ast y$, while the second case contradicts $x\\ast y\\vdash x$. This proves (a).\n\n\nIn order to prove (b), consider $i,j\\in I^{+}$ and let $k= i\\lor j$, with $i,j,k\\in I$.  As $\\ast$ is a $r$-partition function  for $\\vdash$, $x,y\\vdash x\\ast y$. Since $i,j\\in I^{+}$, then $F_i$ and $F_j$ are non-empty, therefore there exist two elements $a\\in F_{i}$, $b\\in F_{j}$. We have $a\\ast^{\\A}b=f_{ik}(a)\\ast^{\\A_{k}}f_{jk}(b)\\in A_{k}$. This, together with the fact that $\\langle\\A,F\\rangle\\in\\Mod(\\vdash)$ implies $a\\ast b\\in F_{k}$, i.e. $F_{k}\\neq\\emptyset$. So $k\\in I^{+}$ and this proves (b).\n\\end{proof}\n\nGiven a logic $\\vdash$ with a r-partition function $\\ast$ and a model $\\langle\\A,F\\rangle$ of $\\vdash$ such that $\\A\\in\\Alg(\\vdash)$, we call \\emph{P\\l onka fibers} of $\\langle \\A, F \\rangle$ the matrices $\\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}$ given by the decomposition in Theorem \\ref{th: plonka sum of r- matrices}. From now on, when considering a model $\\pair{\\A,F}$ of a logic $\\vdash$ with $r$-partition function, we will assume that $\\pair{\\A, F}= \\PL(X)$, for a given direct system $\nX=\\langle \\langle I,\\leq\\rangle , \\{\\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\} \\rangle\n$, without explicitly mentioning the r-direct system $X$.\n\n\n\n\\begin{lemma}\\label{lemma su filtri non vuoti che sono modelli della logica iniziale}\nLet $\\vdash^{r}$ be a logic with $r$-partition function $\\ast$, and $\\langle\\A,F\\rangle\\in\\Mod(\\vdash^{r})$, with $\\A\\in\\Alg(\\vdash^{r})$. Then, the P\\l onka fibers $ \\langle\\A_{i},F_{i}\\rangle $, such that $i\\in I^{+}$, are models of $\\vdash$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $\\Gamma\\vdash\\varphi$ and suppose, by contradiction, that there exists a matrix $\\pair{\\A_j, F_j}$, with $j\\in I^{+}$, and a homomorphism $h\\colon\\Fm\\to\\A_{j}$ such that $h[\\Gamma]\\subseteq F_{j}$ and $h(\\varphi)\\notin F_{j}$. Preliminarily, observe that $\\Var(\\varphi)\\nsubseteq\\Var(\\Gamma)$ and, moreover, if $\\vdash$ has an antitheorem $\\Sigma(x)$, then $\\Sigma(x)\\nsubseteq\\Gamma$, for otherwise $\\Gamma\\vdash^{r}\\varphi$, which is in contradiction with our assumption that  $\\langle\\A,F\\rangle\\in\\Mod(\\vdash^{r})$. Denote by $X$ the (non-empty) set of variables occurring in $\\varphi$ but not in $\\Gamma$ and, for $\\gamma\\in \\Gamma$, let $X_{\\gamma}\\coloneqq\\{\\gamma\\ast x: x\\in X\\}$ and $\\Gamma^{-}_{\\gamma}\\coloneqq \\Gamma\\smallsetminus\\{\\gamma\\}$. Since $\\ast$ is a r-partition function for $\\vdash^{r}$, we have $\\gamma\\ast x\\vdash^{r} \\gamma$. Therefore $\\gamma\\ast x\\vdash \\gamma$ and $X_{\\gamma}\\vdash \\gamma$, which implies $X_{\\gamma},\\Gamma^{-}_{\\gamma}\\vdash\\varphi$, for any $\\gamma\\in\\Gamma$. Observe that $\\Var(\\varphi)\\subseteq\\Var (X_{\\gamma})\\cup\\Var(\\Gamma^{-}_{\\gamma})$, hence $X_{\\gamma},\\Gamma^{-}_{\\gamma}\\vdash^{r}\\varphi$.\n\nSince $h(\\gamma),h(\\varphi)\\in A_{j}$ and $x\\in\\Var(\\varphi)$, for every $x\\in X$, we have that $h(\\gamma\\ast x)=h(\\gamma)$, whence $h[X_{\\gamma}]= h(\\gamma)$. Now, for an arbitrary $a\\in A$, we define a homomorphism $g\\colon\\Fm\\to\\A$, as follows \n\\[\ng(x)\\coloneqq \\left\\{ \\begin{array}{ll}\nh(x) \\ \\text{if} \\ \\ x\\in\\Var(\\Gamma)\\cup\\Var(\\varphi)& \\\\\na \\ \\text{otherwise.} & \\\\\n  \\end{array} \\right.  \n\\]\n\nWe have $g[X_{\\gamma}] = h[X_{\\gamma}]= h(\\gamma)\\in F_j$, $g[\\Gamma^{-}_{\\gamma}]=h[\\Gamma^{-}_{\\gamma}]\\in F_j$ and $g(\\varphi)=h(\\varphi)\\not\\in F_j$. A contradiction.\n\\end{proof}\n\n\n\nThe following example provides a simple instance of the above Lemma \\ref{lemma su filtri non vuoti che sono modelli della logica iniziale}.\n\n\\begin{example}\\label{example lemma 24}\nConsider the matrix $\\pair{\\B, F}$ constructed as P\\l onka sum of the (non-trivial) Boolean algebras $\\A_{i},\\A_{j}\\A_{k},\\A_{s}$ over the r-direct system depicted below (the index set is the four-element Boolean algebra): circles indicate filters, consisting of the top elements $1_i, 1_j$ of $\\A_i,\\A_j$, respectively. Thus, $F = \\{1_{i},1_{j}\\}$. Dotted lines represent arbitrary P\\l onka homomorphisms.  \n\\begin{center}\n\n\\begin{tikzpicture}\n\\draw (-2,6) node {$\\bullet$};\n\\draw (-2,7) node {$ \\A_{s}$};\n\\draw (-2,8) node {$\\bullet$};\n\n\\draw (-4,5) node {$\\bullet$};\n\\draw (-4,4) node {$ \\A_{k}$};\n\\draw (-4,3) node {$\\bullet$};\n\n\\draw (0,5) node {\\circled{$\\bullet$}};\n\\draw (0,4) node {$ \\A_{j}$};\n\\draw (0,3) node {$\\bullet$};\n\n\\draw (-2,0) node {$\\bullet$};\n\\draw (-2,1) node {$ \\A_{i}$};\n\\draw (-2,2) node {\\circled{$\\bullet$}};\n\n\n\\draw (-2,6) edge[bend left=70] (-2,8) ;\n\\draw (-2,6) edge[bend right=70] (-2,8);\n\t\\draw (-4,5) edge[ dotted] (-2,8) ;\n\t\t\\draw (-4,3) edge[dotted] (-2,6) ;\t\n\t\t\\draw (0,5) edge[dotted] (-2,8) ;\n\t\t\\draw (0,3) edge[dotted] (-2,6) ;\n\t\t\n\\draw (-4,5) edge[bend left=70] (-4,3) ;\n\\draw (-4,5) edge[bend right=70] (-4,3) ;\n\t\\draw (-2,2) edge[dotted] (-4,5) ;\n\t\t\\draw (-2,0) edge[dotted] (-4,3) ;\n\n\\draw (0,5) edge[bend left=70] (0,3) ;\n\\draw (0,5) edge[bend right=70] (0,3) ;\n\n\\draw (-2,0) edge[bend left=70] (-2,2) ;\n\\draw (-2,0) edge[bend right=70] (-2,2) ;\n\t\\draw (-2,2) edge[dotted] (0,5) ;\n\t\t\\draw (-2,0) edge[dotted] (0,3) ;\n\n\\end{tikzpicture}\n\\end{center}\n\nBy Theorem \\ref{completeness}, $\\pair{\\B, F}$ is a model of Bochvar logic $\\mathsf{B_{3}}$. It can be easily checked that $\\B\\in \\Alg(\\mathsf{B}_3)$.\nMoreover, $I^+=\\{i,j\\}$ and clearly $\\langle\\A_i,1_i\\rangle,\\langle\\A_j,1_j\\rangle\\in\\Mod(\\ensuremath{\\mathsf{CL}})$.\n\\end{example}\n\n\nThe presence of a r-partition function yields an important syntactic consequence: it allows to adapt a Hilbert style calculus of a logic $\\vdash$ into a calculus, for its contaiment companion $\\vdash^{r}$. Despite $\\vdash^{r}$ is defined via a linguistic restriction constraint (on the inclusion of variables), the axiomatization that we obtain is free of any (linguistic) restriction. \n\nThroughout the remaining part of this section, we implicitly assume that the logic $\\vdash$ possesses an antitheorem. Our analysis can be easily adapted to the case where $\\vdash$ does not have antitheorems (see Remark \\ref{rem: Hilbert senza inconsistency terms}).\n\n\n In what follows, by a \\textit{Hilbert-style calculus with finite rules}, we understand a (possibly infinite) set of Hilbert-style rules, each of which has finitely many premises.\n\n\\begin{definition}\\label{def: hilbert calc per r}\nLet $\\mathcal{H}$ be a Hilbert-style calculus with finite rules, which determines a logic $\\vdash$ with a $r$-partition function $\\ast$ and an antitheorem $\\Sigma(x)$.  Let $\\mathcal{H}^{r}$ be the Hilbert-style calculus given by the following rules: \n\n\\begin{align}\nx\\ast \\varphi &\\rhd \\varphi \\tag{H0}\\label{Eq:Axiom0}\\\\\nx,y &\\rhd x\\ast y \\tag{H1}\\label{Eq:Axiom1}\\\\\nx\\ast y &\\rhd  x\\tag{H2}\\label{Eq:Axiom2}\\\\\n\\{\\gamma_{1},\\dots,\\gamma_{n}\\}\\smallsetminus\\{\\gamma_{i}\\}, \\gamma_{i}\\ast\\psi &\\rhd \\psi\\tag{H3}\\label{Eq:Axiom3} \\\\\n\\Sigma(x) &\\rhd \\alpha \\tag{H4}\\label{Eq:Axiom4}\\\\\n\\chi(\\delta, \\vec{z}\\?\\?) \\?\\?\\?\\lhd&\\rhd \\chi(\\varepsilon, \\vec{z}\\?\\?)\\tag{H5}\\label{Eq:Axiom5}\n\\end{align}\n\nfor every\n\\begin{enumerate}[(i)]\n\\item $\\rhd\\varphi$ axiom in $\\mathcal{H}$ ;\n\\item $\\gamma_{1},\\dots,\\gamma_{n}\\rhd \\psi$ rule in $\\mathcal{H}$ (and $\\gamma_i$ such that $i\\in\\{i,\\dots,n\\}$);\n\\item $\\delta \\thickapprox\\varepsilon$ equation in the definition of partition function, and formula $\\chi(v, \\vec{z}\\?\\?)$.\n\\end{enumerate}\n\\end{definition}\n\n\n\\begin{lemma}\\label{lemma: spezza dimostrazione completezza Hilbert}\nLet $\\vdash$ be a logic with a $r$-partition function $\\ast$, an antitheorem $\\Sigma(x)$ and let $\\langle\\A,F\\rangle\\in\\mathsf{Mod}^{\\textup{Su}}(\\vdash_{\\mathcal{H}^{r}})$. Then:\n\\benroman\n\\item $\\pair{\\A,F}=\\PL(X)$, where $X=\\pair{\\langle \\langle I,\\leq\\rangle, \\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}}$ is an r-direct system of matrices;\n\\item if $X$ contains a trivial matrix then $\\A = \\mathbf{1}$.\n\\eroman\n\\end{lemma}\n\\begin{proof}\n(i) Since $\\langle\\A,F\\rangle\\in\\mathsf{Mod}^{\\textup{Su}}(\\vdash_{\\mathcal{H}^{r}})$, $\\A\\in\\Alg(\\vdash_{\\mathcal{H}^{r}})$. Moreover, observe that $\\ast $ is a $r$-partition function for $\\vdash_{\\mathcal{H}^{r}}$ (thanks to conditions \\eqref{Eq:Axiom1}, \\eqref{Eq:Axiom2}, \\eqref{Eq:Axiom5}). These facts, together with Theorem \\ref{th: plonka sum of r- matrices}, implies that $\\langle\\A,F\\rangle=\\PL(X)$, where $X=\\langle\\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}, \\langle I,\\leq\\rangle\\rangle$ is an r-direct system of matrices.\\\\\n\\noindent\n(ii) Suppose that, for some $j\\in I$, $\\langle\\A_{j},F_{j}\\rangle$ is a trivial fiber of $\\langle\\A,F\\rangle$, i.e. $F_j = A_j$. Since $\\Sigma(x)$ is an anti-theorem (for $\\vdash$) and \\eqref{Eq:Axiom4}  is a rule of $\\mathcal{H}^{r}$, then, for every $i\\in I$, we have $A_{i}=F_{i}$, i.e. each fiber is trivial. Indeed, if there exists a non trivial fiber $\\langle\\A_{k},F_{k}\\rangle$ and an  element $c\\in A_{k}\\smallsetminus F_{k}$, then the evaluation $h\\colon\\Fm\\to\\A$, defined as $h(x)=a$, $h(y)=c$ (for an arbitrary $a\\in\\A_{j}$) is such that $h[\\Sigma(x)]\\subseteq F$ while $h(y)\\notin F$, against the fact that $\\Sigma(x)\\vdash_{\\mathcal{H}^{r}}y$. Moreover, the facts that each fiber is trivial and that $\\@ifnextchar _  {\\mathchoice{\\tarskidsp\\kern-.07em}{\\tarskitxt\\kern-.07em} {\\tarskiscr\\kern-.07em}{\\tarskiscrscr\\kern-.07em}} {\\mathchoice{\\tarskidsp}{\\tarskitxt}{\\tarskiscr}{\\tarskiscrscr}}^{\\A}F=id$ immediately implies $\\A=\\mathbf{1}$.\n\\end{proof}\n\n \n \n \\begin{theorem}\\label{th: completezza calcolo Hilbert}\n Let $\\vdash$ be a finitary logic with a $r$-partition function $\\ast$ and an antitheorem $\\Sigma(x)$. Let, moreover, $\\mathcal{H}$ be a Hilbert style calculus  with finite rules. If $\\mathcal{H}$ is complete for $\\vdash$, then $\\mathcal{H}^{r}$ is complete for $\\vdash^{r}$.\n \\end{theorem}\n \n \\begin{proof}\n \n Let us denote with $\\vdash_{\\mathcal{H}^{r}}$ the logic defined by $\\mathcal{H}^{r}$. We show that $\\vdash_{\\mathcal{H}^{r}}\\?=\\?\\vdash^{r}$.\n \n $(\\subseteq)$. It is immediate to check that every rule of $\\mathcal{H}^{r}$ holds in $\\vdash^{r}$.  \n \n $(\\supseteq)$. We now show that $\\mathsf{Mod}^{\\textup{Su}}(\\vdash_{\\mathcal{H}^{r}})\\subseteq\\Mod(\\vdash^{r})$. So let $\\langle\\A,F\\rangle\\in\\mathsf{Mod}^{\\textup{Su}}(\\vdash_{\\mathcal{H}^{r}})$. By Lemma \\ref{lemma: spezza dimostrazione completezza Hilbert}-(i), we know that $\\langle\\A,F\\rangle\\cong\\PL(X)$, where $X=\\langle\\{ \\langle \\A_{i},F_{i}\\rangle\\}_{i\\in I}, \\{ f_{ij} \\! : \\! i \\leq j \\}, \\langle I,\\leq\\rangle\\rangle$ is an r-direct system of matrices.\n \nThe fact that the matrix $\\langle\\A_{i},F_{i}\\rangle\\in\\Mod(\\vdash_{\\mathcal{H}})$ for each $i\\in I^{+}$ can be proved on the ground of \\eqref{Eq:Axiom0} and \\eqref{Eq:Axiom3} by adapting the proof of Lemma \\ref{lemma su filtri non vuoti che sono modelli della logica iniziale} to the calculus $\\mathcal{H}^{r}$. Recalling that $\\mathcal{H}$ is complete for $\\vdash$ we obtain that $\\langle\\A_{i},F_{i}\\rangle\\in\\Mod(\\vdash)$, for each $i\\in I^{+}$. \n\n\n  \n \n\\noindent \n\nMoreover, by Lemma \\ref{lemma: spezza dimostrazione completezza Hilbert}-(ii), we know that if $X$ contains a trivial matrix $\\pair{\\A_j, F_j}$, then $\\A = \\mathbf{1}$. \n\nTherefore, two cases may arise: (1) $\\A=\\mathbf{1}$, (2) $X$ contains no trivial fibers.\nIf (1) then clearly $\\langle\\A,F\\rangle\\in\\{\\langle\\mathbf{1},\\emptyset\\rangle, \\langle\\mathbf{1},\\{1\\}\\rangle \\}$. As $\\vdash^{r}$ is a theoremless logic \n$\\{\\langle\\mathbf{1},\\emptyset\\rangle, \\langle\\mathbf{1},1\\rangle \\}\\subseteq \\Mod(\\vdash^{r})$. If (2), then we can apply Lemma \\ref{lemma: soundness}, so $\\pair{\\A, F}=\\PL(X)\\in\\Mod(\\vdash^{r})$. \n \\end{proof}\n \n\n\n\n\n\n\n \n\n\n \n\\begin{remark}\\label{rem: Hilbert senza inconsistency terms}\nIt is easy to check that, if the logic $\\vdash$ does not possess antitheorems, then a Hilbert-style calculus for $\\vdash^{r}$ can be defined by simply dropping condition (\\ref{Eq:Axiom4}) from Definition \\ref{def: hilbert calc per r}. The completeness of $\\vdash^{r}$ with respect to such calculus can be proven by adapting the strategy in the proof of Theorem \\ref{th: completezza calcolo Hilbert}.\n\\end{remark} \n\n\n\\section{Examples of axiomatizations}\\label{sec: examples}\n\nIn this last section, we show how to obtain Hilbert-style axiomatizations of some containment logics.  \n\n\\subsection{Bochvar logic}\n\n\nBochvar logic is the containment companion of classical logic. Consider the following Hilbert-style axiomatization of classical propositional logic:\n\\benroman\n\\item[($\\mathbf{H}_{1}$)] $\\rhd\\varphi\\to\\varphi$\n\\item [($\\mathbf{H}_{2}$)]$\\rhd\\varphi\\to(\\psi\\to\\varphi)$\n\\item [($\\mathbf{H}_{3}$)]$\\rhd\\varphi\\to(\\psi\\to\\chi)\\to(\\varphi\\to\\psi)\\to(\\varphi\\to\\chi)$\n\\item [($\\mathbf{H}_{4}$)]$\\rhd(\\neg\\varphi\\to\\neg\\psi)\\to(\\psi\\to\\varphi)$\n\\item [($\\mathbf{H}_{5}$)]$\\varphi,\\varphi\\to\\psi\\rhd\\varphi$\n\\eroman\n\n Theorem \\ref{th: completezza calcolo Hilbert} allows to provide the following complete Hilbert style calculus for Bochvar logic $\\mathsf{B_{3}}$.\n\n\n\\begin{itemize}\n\\item[($\\mathbf{H}^{r}_{1}$)] $\\alpha\\ast(\\varphi\\to\\varphi)\\rhd\\varphi\\to\\varphi$\n\\item [($\\mathbf{H}^{r}_{2}$)]$\\alpha\\ast(\\varphi\\to(\\psi\\to\\varphi)\\rhd\\varphi\\to(\\psi\\to\\varphi)$\n\\item [($\\mathbf{H}^{r}_{3}$)]$\\alpha\\ast(\\varphi\\to(\\psi\\to\\chi)\\to(\\varphi\\to\\psi)\\to(\\varphi\\to\\chi))\\rhd\\varphi\\to(\\psi\\to\\chi)\\to(\\varphi\\to\\psi)\\to(\\varphi\\to\\chi)$\n\\item[($\\mathbf{H}^{r}_{4}$)] $\\alpha\\ast((\\neg\\varphi\\to\\neg\\psi)\\to(\\psi\\to\\varphi))\\rhd(\\neg\\varphi\\to\\neg\\psi)\\to(\\psi\\to\\varphi)$\n\\item [($\\mathbf{H}^{r}_{5}$)]$\\varphi\\ast\\psi,\\varphi\\to\\psi\\rhd\\psi$\n\\item [($\\mathbf{H}^{r}_{6}$)] $\\alpha,\\neg \\alpha\\rhd\\varphi$\n\\item[($\\mathbf{H}^{r}_{7}$)] $\\chi(\\delta, \\vec{z}\\?\\?) \\?\\?\\?\\lhd\\rhd\\chi(\\varepsilon, \\vec{z}\\?\\?)$ for every formula $\\chi(x,\\vec{z})$ and equation $\\delta\\thickapprox\\varepsilon$ in  Definition \\ref{def: partition function},\n\\end{itemize}\n\nwhere $\\varphi\\ast\\psi$ is an abbreviation for $\\varphi\\land(\\varphi\\lor\\psi)$.\n\n \n\\subsection{Belnap-Dunn logic}\n\nBelnap-Dunn is a four-valued logic $\\mathsf{B}$, originally introduced as \\emph{First Degree Entailment} within the research enterprise on relevance and entailment logic \\cite{AndersonBelnap,Belnap1977}.   \n\nConsider the algebraic language of type $1,2,2$, containing $\\neg,\\vee,\\land$. Recall that a \\emph{De Morgan lattice} is an algebra $\\A=\\pair{A,\\neg,\\vee,\\land}$ of type $1,2,2$ such that: \n\\benroman\n\\item $\\pair{A,\\land,\\vee}$ is a distributive lattice; \n\\item $\\neg $ satisfies the following equations: \n\\[\nx\\approx\\neg\\neg x, \\hspace{1cm} \\neg(x\\land y)\\approx \\neg x\\lor\\neg y, \\hspace{1cm} \\neg(x\\lor y)\\approx \\neg x\\land\\neg y.\n\\]\n\\eroman\n\nDe Morgan lattices, originally introduced by Moisil \\cite{Moisil} and, independently, by Kalman \\cite{Kalman58} (under the name of \\emph{distributive i-lattices}) form a variety, which is generated by the four element algebra $\\M=\\pair{\\{0,b,n,1\\},\\neg,\\vee,\\land}$, whose lattice reduct is displayed in Figure \\ref{fig: M4} and negation in the following table:\n\\begin{center}\\renewcommand{\\arraystretch}{1.25}\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}}\n  \\lnot &  \\\\[.2ex]\n\\hline\n  1 & 0 \\\\\n  b & b \\\\\n  n & n \\\\\n  0 & 1 \\\\\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n\\begin{figure}[h]\n\\begin{tikzcd}[row sep = tiny, arrows = {dash}]\n & 1 & \\\\\n & & \\\\\n & & \\\\\n b \\arrow[uuur, dash] & & n \\arrow[uuul] \\\\\n & & \\\\\n & & \\\\\n & 0\\arrow[uuul]\\arrow[uuur] &\n  \\end{tikzcd}\\caption{Hasse\n  diagram of the De Morgan lattice $\\M$.}\n \\end{figure}\\label{fig: M4}\n \n  \\end{center}\n\n$\\mathsf{B}$ is the logic induced by the matrix $\\pair{\\M,\\{1,b\\}}$ (or, equivalently, by $\\pair{\\M,\\{1,n\\}}$, see \\cite[Proposition 2.3]{font1997belnap}). $\\mathsf{B}$ is finitary and theoremless (purely inferential). Moreover, the class $\\Alg(\\mathsf{B})$ coincides with the variety of De Morgan lattices \\cite[Theorem 4.1]{font1997belnap}. Observe that the set $\\{\\varphi,\\neg\\varphi\\}$ is not an antitheorem of $\\mathsf{B}$ (indeed $\\varphi,\\neg\\varphi\\not\\vdash_{\\mathsf{B}}\\psi$). It is not difficult to check that $\\mathsf{B}$ does not possess antitheorems. \n\nRecall that a \\emph{lattice filter} of a De Morgan lattice $\\A$ is a subset $F\\subseteq A$ such that $x\\land y\\in F$ if and only if $x\\in F$ and $y\\in F$. The class of matrices \n\\[\n\\mathsf{M}=\\{\\pair{\\A, F} : \\; \\A \\text{ De Morgan algebra, } F\\subseteq A \\text{ lattice filter}\\}\n\\]\n\nis complete for $\\mathsf{B}$ \\cite[Corollary 2.6]{font1997belnap}. Observe that the matrices $\\pair{\\A,\\emptyset}\\in\\mathsf{M}$, for any De Morgan lattice $\\A$. It follows from our analysis (see Theorem \\ref{completeness}) that the containment companion $\\vdash^{r}_{\\mathsf{B}}$ of $\\vdash_{\\mathsf{B}}$ is complete with respect to $\\PL{(\\mathsf{M})}$, i.e. the class of all P\\l onka sums over r-direct systems of matrices in $\\mathsf{M}$.\n  \n\n\nWe present the Hilbert-style axiomatization for $\\mathsf{B}$ which is introduced in \\cite{font1997belnap} (and, independently in \\cite{Pynko95}). Since $\\mathsf{B}$ is theoremless, the calculus has no axioms and the following rules: \n\n\\benroman\n\\item[($\\mathsf{B}_1$)] $\\varphi \\land \\psi \\rhd \\varphi $;\n\\item [($\\mathsf{B}_{2}$)] $\\varphi \\land \\psi \\rhd \\psi$;\n\\item [($\\mathsf{B}_{3}$)]$\\varphi, \\psi\\rhd \\varphi  \\land \\psi $;\n\\item [($\\mathsf{B}_{4}$)]$\\varphi \\rhd \\varphi\\lor \\psi$;\n\\item [($\\mathsf{B}_{5}$)]$\\varphi\\lor \\psi\\rhd \\psi\\lor \\varphi$;\n\\item [($\\mathsf{B}_{6}$)]$\\varphi\\lor \\varphi \\rhd \\varphi$;\n\\item [($\\mathsf{B}_{7}$)]$\\varphi\\lor (\\psi\\lor\\chi) \\rhd (\\varphi\\lor\\psi)\\lor\\chi$;\n\\item [($\\mathsf{B}_{8}$)]$\\varphi\\lor (\\psi\\land\\chi) \\rhd (\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi)$;\n\\item [($\\mathsf{B}_{9}$)]$(\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi) \\rhd  \\varphi\\lor (\\psi\\land\\chi)$;\n\\item [($\\mathsf{B}_{10}$)]$\\varphi\\lor\\psi  \\rhd \\neg\\neg\\varphi\\lor\\psi$;\n\\item [($\\mathsf{B}_{11}$)]$\\neg\\neg\\varphi\\lor\\psi  \\rhd \\varphi\\lor\\psi$\n\\item [($\\mathsf{B}_{12}$)]$\\neg(\\varphi\\lor\\psi)\\lor\\chi  \\rhd (\\neg\\varphi\\land\\neg \\psi)\\lor\\chi$;\n\\item [($\\mathsf{B}_{13}$)]$ (\\neg\\varphi\\land\\neg \\psi)\\lor\\chi  \\rhd \\neg(\\varphi\\lor\\psi)\\lor\\chi$;\n\\item [($\\mathsf{B}_{14}$)]$\\neg(\\varphi\\land\\psi)\\lor\\chi  \\rhd (\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi$;\n\\item [($\\mathsf{B}_{15}$)]$ (\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi  \\rhd \\neg(\\varphi\\land\\psi)\\lor\\chi$;\n\\eroman\n\nA Hilbert-style axiomatization of $\\vdash^{r}_{\\mathsf{B}}$, (see Definition \\ref{def: hilbert calc per r} Theorem \\ref{th: completezza calcolo Hilbert}) is given by the following ($\\varphi\\ast\\psi$ is an abbreviation for $ \\varphi\\land(\\varphi\\lor\\psi)$): \n\n\\benroman\n\\item[($\\mathsf{B}^{r}_{1}$)] $\\varphi,\\psi \\rhd \\varphi \\ast \\psi $;\n\\item[($\\mathsf{B}^{r}_{2}$)] $\\varphi\\ast \\psi \\rhd \\psi $;\n\\item[($\\mathsf{B}^{r}_{3}$)] $(\\varphi \\land \\psi)\\ast\\varphi \\rhd \\varphi $;\n\\item [($\\mathsf{B}^{r}_{4}$)] $(\\varphi \\land \\psi)\\ast\\psi \\rhd \\psi$;\n\\item [($\\mathsf{B}^{r}_{5}$)]$\\varphi\\ast (\\varphi  \\land \\psi) , \\psi\\rhd \\varphi  \\land \\psi $;\n\\item [($\\mathsf{B}^{r}_{6}$)]$\\varphi, \\psi\\ast(\\varphi  \\land \\psi)\\rhd \\varphi  \\land \\psi $;\n\\item [($\\mathsf{B}^{r}_{7}$)]$\\varphi\\ast(\\varphi\\lor \\psi) \\rhd \\varphi\\lor \\psi$;\n\\item [($\\mathsf{B}^{r}_{8}$)]$(\\varphi\\lor \\psi)\\ast (\\psi\\lor \\varphi)\\rhd \\psi\\lor \\varphi$;\n\\item [($\\mathsf{B}^{r}_{9}$)]$(\\varphi\\lor \\varphi)\\ast\\varphi \\rhd \\varphi$;\n\n\\item [($\\mathsf{B}^{r}_{10}$)]$(\\varphi\\lor (\\psi\\lor\\chi))\\ast ((\\varphi\\lor\\psi)\\lor\\chi) \\rhd (\\varphi\\lor\\psi)\\lor\\chi$;\n\n\\item [($\\mathsf{B}^{r}_{11}$)]$\\varphi\\lor (\\psi\\land\\chi)\\ast((\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi)) \\rhd (\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi)$;\n\n\\item [($\\mathsf{B}^{r}_{12}$)]$((\\varphi\\lor\\psi)\\land(\\varphi\\lor\\chi))\\ast(\\varphi\\lor (\\psi\\land\\chi)) \\rhd  \\varphi\\lor (\\psi\\land\\chi)$;\n\n\\item [($\\mathsf{B}^{r}_{13}$)]$(\\varphi\\lor\\psi)\\ast (\\neg\\neg\\varphi\\lor\\psi)  \\rhd \\neg\\neg\\varphi\\lor\\psi$;\n\n\\item [($\\mathsf{B}^{r}_{14}$)]$(\\neg\\neg\\varphi\\lor\\psi)\\ast (\\varphi\\lor\\psi)\\rhd \\varphi\\lor\\psi$;\n\n\\item [($\\mathsf{B}^{r}_{15}$)]$(\\neg(\\varphi\\lor\\psi)\\lor\\chi)\\ast ((\\neg\\varphi\\land\\neg \\psi)\\lor\\chi)  \\rhd (\\neg\\varphi\\land\\neg \\psi)\\lor\\chi$;\n\n\\item [($\\mathsf{B}^{r}_{16}$)]$ ((\\neg\\varphi\\land\\neg \\psi)\\lor\\chi) \\ast (\\neg(\\varphi\\lor\\psi)\\lor\\chi)  \\rhd \\neg(\\varphi\\lor\\psi)\\lor\\chi$;\n\n\\item [($\\mathsf{B}^{r}_{17}$)]$(\\neg(\\varphi\\land\\psi)\\lor\\chi)\\ast ((\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi)  \\rhd (\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi$;\n\n\\item [($\\mathsf{B}^{r}_{18}$)]$ ((\\neg\\varphi\\lor\\neg \\psi)\\lor\\chi)\\ast (\\neg(\\varphi\\land\\psi)\\lor\\chi)  \\rhd \\neg(\\varphi\\land\\psi)\\lor\\chi$;\n\n\\item[($\\mathsf{B}^{r}_{19}$)] $\\chi(\\delta, \\vec{z}\\?\\?) \\?\\?\\?\\lhd\\rhd\\chi(\\varepsilon, \\vec{z}\\?\\?)$ for every formula $\\chi(x,\\vec{z})$ and equation $\\delta\\thickapprox\\varepsilon$ in  Definition \\ref{def: partition function}.\n\\eroman\n\n\n\n\\subsection{The Relevance logic $\\mathrm{\\mathbf{S}_{fde}}$}\\label{Subsec: relevance}\n\nThe logic $\\mathrm{\\mathbf{S}_{fde}}$ has been introduced by Deutsch \\cite{Deutsch}: it is induced the matrix $\\pair{\\mathbf{S}_{4}, \\{1,\\ant\\}}$, whose algebraic reduct $\\mathbf{S}_{4}=\\pair{\\{0,\\ant,m,1\\}, \\neg, \\land, \\vee}$ is given in the following tables.\n\\vspace{5pt}\n\\begin{center}\\renewcommand{\\arraystretch}{1.25}\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}}\n  \\lnot &  \\\\[.2ex]\n\\hline\n  1 & 0 \\\\\n  \\ant & \\ant \\\\\n  m & m \\\\\n  0 & 1 \\\\\n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\land & 0 & \\ant & m & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & 0 & m &  0 \\\\\n       \\ant & 0 & \\ant & m & \\ant \\\\\n       m & m & m & m & m \\\\          \n       1 & 0 & \\ant & m & 1 \n\\end{tabular}\n\\qquad\n\\begin{tabular}{>{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}>{$}c<{$}}\n   \\lor & 0 & \\ant & m & 1 \\\\[.2ex]\n \\hline\n       0 & 0 & \\ant & m &  1 \\\\\n       \\ant & \\ant & \\ant & m & 1 \\\\\n       m & m & m & m & m \\\\          \n       1 & 1 & 1 & m & 1 \n\\end{tabular}\n\n\\end{center}\n\\vspace{10pt}\n\\noindent\nRecall that a \\emph{Kleene lattice} is a De Morgan lattice satisfying $x\\land\\neg x\\leq y\\lor\\neg y$. Kleene lattices form a variety ($\\mathsf{KL}$), generated by the 3-element algebra $\\mathbf{SK}=\\pair{\\{0,1,\\ant\\}, \\neg, \\vee, \\land}$, which is a subalgebra of $\\mathbf{S}_{4}$ (and also isomorphic to the two three-element subalgebras of $\\M$).\n\nThe logic of Paradox $\\mathsf{LP}$ (see \\cite{Priestfirst,BlRiVe01,Pynko}) is defined by the matrix $\\langle\\mathbf{SK},\\{1,\\ant\\}\\rangle$. The algebraic counterpart of $\\mathsf{LP}$ is exactly the variety of Kleene lattice, i.e. $\\mathsf{KL}=\\Alg(\\mathsf{LP})$.\nIt is immediate to check that the matrix $\\pair{\\mathbf{S}_{4}, \\{1,\\ant\\}}$ is the P\\l onka sum over the r-direct system of the two matrices $\\pair{\\mathbf{SK},\\{1,\\ant\\}}$ and $\\pair{\\mathbf{m},\\emptyset}$. Thus, it follows from Theorem \\ref{completeness} that $\\mathrm{\\mathbf{S}_{fde}}$ is the containment companion of the logic of Paradox. \n\nA finite Hilbert style calculus for $\\mathsf{LP}$ (see for instance \\cite{Superbelnap}) can be obtained by adding the axiom\n\n\\benroman\n\\item[$(\\mathsf{LP}_{1})$] $\\rhd \\varphi\\lor\\neg\\varphi$\n\\eroman\nto the calculus for the logic $\\mathsf{B}$ described above. Therefore, by Theorem \\ref{th: completezza calcolo Hilbert}, the calculus consisting of $(\\mathsf{B}_1^r)-(\\mathsf{B}_{19}^r)$ and\n\\benroman\n\\item [$(\\mathsf{LP}^r_{1})$] $\\alpha\\ast \\varphi\\lor\\neg\\varphi\\rhd \\varphi\\lor\\neg\\varphi $\n \n\\eroman\n\nis complete for $\\mathrm{\\mathbf{S}_{fde}}$.\n\n\n \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}