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+{"text":"\\section{Introduction}\n\nRare-earth orthorhombic perovskites, orthoferrites \\textit{R}FeO$_3$ and orthochromites \\textit{R}CrO$_3$ (where \\textit{R} is a rare-earth ion and yttrium), exhibit many important features such as weak ferro- and antiferromagnetism, magnetization reversal, anomalous circular magnetooptics, and the phenomenon of the spontaneous spin reorientation. The spin reorientation (SR) is one of their unique properties that have attracted a lot of attention back in the 70s of the last century\\,\\cite{BZKL,KP}, though their exact microscopic origin is still a challenge to theorists and experimentalists.\n\nThe revival of interest in the mechanism of the spontaneous spin reorientation and magnetic compensation in rare-earth perovskites in recent years is related with the discovery of the magnetoelectric and the exchange bias effect, which can have a direct application in magnetoelectronics. Along with the emergence of new experimental studies (see, e.g., Refs.\\,\\cite{PRB102,PRB103}), there also appeared theoretical works claiming to modify the mean-field theory of the spontaneous spin-reorientation transitions\\,\\cite{Bazaliy} or to scrutinize the microscopic mechanism responsible for spin reorientations and magnetization reversal\\,\\cite{Luxem}. In fact, these results are not directly related to the microscopic theory of the spontaneous spin reorientation in rare-earth orthoferrites and orthochromites. For instance, the authors of the most recent paper\\,\\cite{Luxem} did not take into account a number of interactions, such as the fourth-order anisotropy for the $ 3d $ sublattice of orthoferrites and the crystal field for $ R $-ions, which play a fundamental role in determining the spontaneous spin reorientation. The spin anisotropy of the second order in the $3d$ sublattice of orthorhombic orthoferrites and orthochromites is generally not reduced to an effective uniaxial form as adopted in Ref.\\,\\cite{Luxem}. Furthermore, the density functional theory does not allow in principle to give an adequate description of such effects of higher orders of perturbation theory as spin anisotropy or antisymmetric exchange\\,\\cite{CM-2019}.\n\nIn this paper, we present the results of a simple but realistic microscopic model of the spontaneous spin reorientation in rare-earth orthoferrites and orthochromites, which takes into account all the main relevant interactions. This model was developed back in the 80s of the last century\\,\\cite{thesis}, but has not been published until now.\n\n\n\\section{Model formulation}\n\nThe most popular examples of systems with the spontaneous SR transitions are magnets based on $3d$ and $4f$ elements such as rare-earth orthoferrites \\textit{R}FeO$_3$, orthochromites \\textit{R}CrO$_3$, intermetallic compounds \\textit{R}Co$_5$, \\textit{R}Fe$_2$ etc. In all cases, an important cause of the spontaneous SR is the $4f-3d$ interaction. Usually this interaction is taken into account by introducing an effective field of the magnetically ordered $3d$ sublattice acting on the $4f$ ions.\n\nTo consider the contribution of the rare-earth sublattice to the free energy at low temperatures, we are developing a model which takes into account either the well isolated lower Kramers doublet of the $4f$ ions (with an odd number of the $4f$ electrons) or the well isolated two lower Stark sublevels with close energies that form a quasi-doublet.\n\nWithin the framework of such ``single-doublet'' approximation we consider the spontaneous SR transition in orthorhombic weak ferromagnets \\textit{R}FeO$_3$ and \\textit{R}CrO$_3$, where the free energy per ion can be represented as follows\n\\begin{equation}\\label{phiRFEO3}\n \\Phi (\\theta) = K_1 \\cos 2 \\theta + K_2 \\cos 4 \\theta - k T \\ln 2 \\, \\cosh \\dfrac{\\Delta (\\theta)}{2 k T},\n\\end{equation}\nwhere $K_1$, $K_2$ are the first and second anisotropy constants of the $3d$ sublattice, which are temperature independent (at least in the SR region), $\\theta$ is the orientation angle of the antiferromagnetic, or N\\'{e}el vector $\\bf G$ of the $3d$ sublattice (e.g. in the $ac$ plane), and $\\Delta (\\theta)$ is the lower doublet (quasi-doublet) splitting of the $4f$ ion in a magnetic field induced by the $3d$ sublattice. \n\nTheoretical estimations\\,\\cite{thesis,M_2021,JETP-1981} of the different\ncontributions to the first constants of the magnetic anisotropy for orthoferrites \\textit{R}FeO$_3$ point to a competition of several main mechanisms with relatively regular (Dzyaloshinskii-Moriya (DM) coupling, magnetodipole interaction) or irregular (single-ion anisotropy, SIA) dependence on the type of \\textit{R}-ion. For instance, the microscopic theory predicts an unexpectedly strong increase in  values of the constant $K_1(ac)$ for LuFeO$_3$ as compared with YFeO$_3$.\nThe SIA contribution to $K_1(ac)$ partially compensates for the large contribution of the DM interaction in YFeO$_3$,  whereas in LuFeO$_3$, they add up. \nThis result is confirmed by experimental data on the measurement of the threshold field $H_{SR}$ of spin reorientation $\\Gamma_4\\rightarrow\\Gamma_2$ ($G_x\\rightarrow G_z$) in the orthoferrite Lu$_{0.5}$Y$_{0.5}$FeO$_3$, in which $H_{SR}$\\,=\\,15\\,T as compared to $H_{SR}$\\,=\\,7.5\\,T in YFeO$_3$\\,\\cite{JETP-1981}.\nThus, one can estimate $K_1(ac)$ in LuFeO$_3$ as around three times as much as $K_1(ac)$ in YFeO$_3$.\n\nLet us pay attention to recent works on the determination of the parameters of the spin Hamiltonian in YFeO$_3$ from measurements of the spin-wave spectrum by the inelastic neutron scattering\\,\\cite{Hahn,Park} and terahertz absorption spectroscopy\\,\\cite{Amelin}. However, these authors started with a simplified spin-Hamiltonian that took into account only Heisenberg exchange, DM interaction, and single-ion anisotropy. Obviously, disregarding the magnetic dipole and exchange-relativistic anisotropy, the ``single-ion anisotropy'' constants found by the authors are some effective quantities that are not directly related to the SIA.\n\nUnfortunately, despite numerous, including fairly recent, studies of the magnetic anisotropy of orthoferrites, we do not have reliable experimental data on the magnitude of the contributions of various anisotropy mechanisms.\n\nAs shown by theoretical calculations\\,\\cite{thesis,M_2021,cubic} the constants $K_2$ of the fourth order spin anisotropy rather smoothly decrease in absolute value, changing by no more than two times on going from La to Lu. But the most interesting was the conclusion about the different signs of these constants, positive for the $ac$ and $bc$ planes and negative for the $ab$ plane, thus indicating a different character of spin-reorientation transitions in the corresponding planes, i.e., second-order transitions in the $ac$ and $bc$ planes and first-order transitions in the $ab$ plane\\,\\cite{KP}. \nIndeed, all currently known spin-reorientation transitions of the $\\Gamma_4 - \\Gamma_2$ ($G_x - G_z$) type\nin orthoferrites \\textit{R}FeO$_3$ (\\textit{R} = Pr, Nd, Sm, Tb, Ho, Er, Tm, Yb) \nare smooth, with two characteristic temperatures of the second-order phase transitions to be a start and finish of the spin reorientation, and the only known jump-like first order SR transition for these crystals is the SR transition $\\Gamma_4 - \\Gamma_1$ ($G_x - G_y$) in the $ab$ plane in DyFeO$_3$\\,\\cite{KP}. \nA unique example that confirms the conclusions about the sign of the second anisotropy constant is a mixed orthoferrite Ho$_{0.5}$Dy$_{0.5}$FeO$_3$\\,\\cite{KP}\nin which two spin-reorientation\ntransitions $G_x - G_y$ ($T$\\,=\\,46\\,K) and $G_y - G_z$ (18\\,$\\div$\\,24\\,K) are realized through one phase transition of the first order in the $ab$ plane and two phase transitions of the second order in the $bc$ plane, respectively.\n\n\nThe splitting value $\\Delta (\\theta)$ for the Kramers doublet in a magnetic field $\\bf H$ has the well-known form\n\\begin{equation}\\label{DeltaKramers}\n \\Delta (\\theta) = \\mu_{B} \\left[ \\left( g_{xx} H_x + g_{xy} H_y \\right)^2 + \\left( g_{xy} H_x + g_{yy} H_y \\right)^2 + g_{zz}^2 H_z^2 \\right]^{1\/2},\n\\end{equation}\nwhere it is taken into account that for the $4f$ ions in \\textit{R}FeO$_3$ the $\\hat{g}$-tensor (with the local symmetry $C_s$) has the form \n\\begin{equation}\\label{g-tensor}\n \\hat{g} = \n \\begin{pmatrix}\n  g_{xx} & g_{xy} & 0\\\\\n  g_{xy} & g_{yy} & 0\\\\\n  0 & 0 & g_{zz}\n\\end{pmatrix}.\n\\end{equation}\n\nThe effective field $\\bf H$ for the SR transition $G_x\\rightarrow G_z$ in the $ac$ plane can be represented as follows\n\\begin{equation}\\label{MagnField}\n H_x = H_x^{(0)} \\cos \\theta, \\,\\, H_y = H_y^{(0)} \\cos \\theta, \\,\\, H_z = H_z^{(0)} \\sin \\theta,\n\\end{equation}\nso in the absence of an external magnetic field, for $\\Delta (\\theta)$ we have the rather simple expression:\n\\begin{equation}\\label{DeltaKramers2}\n \\Delta (\\theta) = \\left( \\dfrac{\\Delta_a^2 - \\Delta_c^2}{2} \\cos 2 \\theta + \\dfrac{\\Delta_a^2 + \\Delta_c^2}{2} \\right)^{1\/2},\n\\end{equation}\nwhere $\\Delta_{a,c}$ are the doublet splitting for the cases of $\\theta = 0$ ($G_z$-phase) and $\\theta = \\pi \/2$ ($G_x$-phase) respectively. The dependence $\\Delta (\\theta)$ from \\eqref{DeltaKramers2} is also valid in the case of quasi-doublet.\n\nA contribution of splitting $\\Delta$ to the free energy $\\Phi (\\theta)$ for the rare-earth sublattice is usually considered in the ``high-temperature'' approximation, when $kT \\gg \\Delta$ and the influence of the $4f$ sublattice are reduced only to renormalization of the first anisotropy constant $K_1$:\n\\begin{equation}\\label{Ku*}\n K_1^* = K_1 \\left( 1 - \\dfrac{1}{\\tau} \\right),\n\\end{equation}\nwhere $\\tau = T\/T_{SR}$ is the reduced temperature and $T_{SR} = (\\Delta_a^2 - \\Delta_c^2)\/ 16k K_1$ is the characteristic transition temperature.\n\nBelow we will consider a specific situation when $K_1 > 0$ and $\\Delta_a > \\Delta_c$, i.e. when the configuration $G_x$ ($\\theta = \\pi\/2$) is realized at high temperatures and a decrease in temperature can lead to the spin reorientation $G_x \\rightarrow G_z$ or $G_x \\rightarrow G_{xz}$ (transition to an angular spin structure). The type of the phase transition of the spin reorientation in the ``high-temperature'' approximation is determined by the sign of the second constant $K_2$: at $K_2 < 0$ it will be realized by one first-order phase transition at $T = T_{SR}$, i.e. $\\tau = 1$, or at $K_2 > 0$ by two second-order phase transitions at $\\tau_s = (1 + \\gamma)^{-1}$ and $\\tau_f = (1 - \\gamma)^{-1}$, where $\\tau_s$ and $\\tau_f$ are the reduced temperatures of the beginning and end of the SR phase transition and $\\gamma = 4 K_2 \/ K_1$.\n\n\n\n\\section{Analysis of the ``single-doublet'' model}\n\nA behavior of a system described by the free energy \\eqref{phiRFEO3} can be analyzed rigorously. The condition $\\partial \\Phi \/ \\partial \\theta = 0$ reduces in this case to two equations:\n\\begin{equation}\\label{SinEq}\n \\sin 2 \\theta = 0,\n\\end{equation}\n\\begin{equation}\\label{mainEq}\n \\alpha \\mu + \\beta \\mu^3 = \\tanh \\dfrac{\\mu}{\\tau};\n\\end{equation}\nwhere the following notations are introduced:\n\\begin{equation}\\label{mainEq_params}\n\\alpha = 1 - \\gamma \\dfrac{\\Delta_a^2 + \\Delta_c^2}{\\Delta_a^2 - \\Delta_c^2},\\ \\beta = \\dfrac{2 \\gamma}{\\mu_f^2 - \\mu_{s}^2},\\ \\mu = \\dfrac{\\Delta (\\theta)}{2k T_{SR}},\\ \\mu_{s} = \\dfrac{\\Delta_c}{2k T_{SR}},\\ \\mu_f = \\dfrac{\\Delta_a}{2k T_{SR}}.\n\\end{equation}\nThis corresponds to three possible magnetic configurations:\n\\begin{itemize}\n\\item The configuration $G_x$: $\\theta = \\pm \\pi\/2$, stable at $\\tanh {\\mu_s}\/{\\tau} \\leq \\alpha \\mu_s + \\beta \\mu_s^3$ .\n\\item The configuration $G_z$: $\\theta = 0, \\ \\pi$, stable at $\\tanh {\\mu_f}\/{\\tau} \\geq \\alpha \\mu_f + \\beta \\mu_f^3$ .\n\\item The angular configuration $G_{xz}$: the temperature dependence of $\\theta (\\tau)$ is determined by solving the equation \\eqref{mainEq} (see Figure\\,\\ref{fig1}), the state is stable at $\\partial \\mu \/ \\partial \\tau \\leq 0$.\n\\end{itemize}\nThe peculiar $\\mu$-$\\tau$ phase diagram which represents solutions of the master equation \\eqref{mainEq} given a fixed value of the $\\alpha$ parameter and different value of the $\\beta$ parameter is shown in Figure\\,\\ref{fig1}, where areas with different character of the SR transition are highlighted in different colors. For the solutions in the FO region, the SR goes through one first-order phase transition, in the SO region we arrive at one or two second-order phase transitions, in the MO$_{1,2}$ regions we arrive at a \"mixture\" of the first and second-order phase transitions. All the lines $\\mu (\\tau)$ on the right side converge to $\\sqrt{|\\alpha\/\\beta|}$ at $\\tau \\rightarrow \\infty$; on the left side, when $\\tau \\rightarrow 0$ the branch point $\\mu = \\frac{3}{2\\alpha}$ is obtained at $\\beta = - \\frac{4}{27}\\alpha^3$,  and the point $\\mu = 1\/\\alpha$ at $\\beta = 0$; all the solutions, where $\\mu$ can reach zero, converge to $\\tau = 1\/\\alpha$. \n\n\n\\begin{figure}[H]\n\\centering\n  \\includegraphics[width=0.7\\textwidth]{classes.pdf}\n\\caption{(Color online) The peculiar $\\mu$-$\\tau$ phase diagram which represents solutions of the master equation \\eqref{mainEq} given a fixed value of the $\\alpha$ parameter and different value of the $\\beta$ parameter (see text for detail). }\n\\label{fig1}\n\\end{figure}\n\nThe character of the SR transition will be determined by the form of the solution of the equation \\eqref{mainEq} in the region $\\mu_s \\leq \\mu \\leq \\mu_f$. Let us analyze this equation starting with the simplest case $K_2 = 0$, i.e. $\\alpha = 1$, $\\beta = 0$. In this case, the main equation transforms into the molecular field equation well known in the basic theory of ferromagnetism:\n\\begin{equation}\\label{MolFieldEq}\n \\mu = \\tanh \\dfrac{\\mu}{\\tau} = B_{\\frac{1}{2}} \\left( \\dfrac{\\mu}{\\tau} \\right) ,\n\\end{equation}\nwhere $B_{1\/2}(x)$ is the Brillouin function. The equation has only one non-trivial solution at $0 \\leq \\tau \\leq 1$, $0 \\leq \\mu \\leq 1$, and the function $\\mu (\\tau)$ has the usual ``Weiss'' form. Thus, with the absence of the cubic anisotropy ($K_2 = 0$) in the ``single-doublet'' model the SR will be realized either through two second-order phase transitions at $\\mu_f \\leq 1$ (the complete spin-reorientation $G_x \\rightarrow G_z$), or through one second-order phase transition at $\\mu_f > 1$, but in this case the SR will be incomplete, i.e. it will end with a transition to the angular spin structure $G_{xz}$. The spin reorientation will begin at a temperature $T_s \\leq T_{SR}$ and $T_s$ is equal to $T_{SR}$ only in the case $\\mu_s = 0$ ($\\Delta_c = 0$), which can be realized in the general case only for Ising ions (e.g. Dy$^{3+}$ in DyFeO$_3$). For this type of ions, the temperature dependence of the ``order parameter'' $\\mu$ (in fact the splitting $\\Delta (\\theta)$ of the doublet) in a close range of $T_{SR}$ will be very sharp: $\\mu (T) \\sim (T - T_{SR})^{-1\/2}$. Nevertheless, the SR will be continuous and the temperature range of the SR $\\Delta T = T_s - T_f$ at $\\mu \\ll 1$ can theoretically reach arbitrarily small values.\n\nThus, the results of the rigorous analysis of the ``single-doublet'' model are fundamentally different from the conclusions of the simplified model (the ``high-temperature'' approximation), according to which for $K_2 = 0$ the spin reorientation always occurs as the first-order phase transition at $T = T_{SR}$.\n\nFor a positive second anisotropy constant ($K_2 > 0$, $\\beta > 0$), the main equation \\eqref{mainEq} has one non-trivial solution in the region $0 \\leq \\tau \\leq 1\/\\alpha$, $0 \\leq \\mu \\leq \\mu_0$ at $\\alpha > 0$, and one in the region $0 \\leq \\tau \\leq \\infty$, $ \\sqrt{|\\alpha\/\\beta|} \\leq \\mu \\leq \\mu_0$ at $\\alpha \\leq 0$, where $\\mu_0$ is determined from the solution of the equation $\\alpha \\mu_0 + \\beta \\mu_0^3 = 1$. The situation in this case is very similar to the previous one, i.e. the beginning of the SR will always be a second-order phase transition, and the reorientation will be complete ($G_x \\rightarrow G_z$) or incomplete ($G_x \\rightarrow G_{xz}$). Note that under the condition $(\\mu_f^2 - \\mu_s^2)\/(\\mu_f^2 + \\mu_s^2) \\geq \\gamma$, i.e. $\\alpha \\leq 0$, the width of the reorientation region becomes very large, even if $\\mu_s$ differs slightly from $\\mu_f$.\n \nFor Ising ions at $\\Delta_c = 0$, the SR beginning temperature is determined in exactly the same way as in the ``high-temperature'' approximation $T_s = T_{SR} \/ (1 + \\gamma)$.\n\nFor a negative second anisotropy constant ($K_2 < 0$, $\\beta < 0$), the several fundamentally different solutions of the main equation \\eqref{mainEq} are possible. For $K_2^* \\geq K_2$, where $K_2^*$ is determined from the condition $\\beta = -\\frac{1}{3} \\alpha^3$, i.e.\n\\begin{equation}\\label{ConditionForK*}\n \\dfrac{2 \\gamma}{\\mu_f^2 - \\mu_{s}^2} = -\\dfrac{1}{3} \\left( 1 - \\gamma \\dfrac{\\mu_f^2 + \\mu_{s}^2}{\\mu_f^2 - \\mu_{s}^2}  \\right)^3,\n\\end{equation}\nthere is one non-trivial solution of the equation \\eqref{mainEq} in the region $1\/\\alpha \\leq \\tau < \\infty$, $\\mu \\leq \\sqrt{\\alpha \/ \\beta}$, but here $\\mu (T)$ decreases with decreasing temperature, i.e. $\\partial \\mu \/ \\partial \\tau > 0$. This solution is unstable and there is no fundamental possibility for a smooth rotation of spins, the SR is always realized through the first-order phase transition.\n\nIn the intermediate range of values $K_2$ ($K_2^* < K_2 < 0$ or $-\\frac{1}{3} \\alpha^3 < \\beta < 0$) the main equation has two non-trivial solutions, and for one of them $\\partial \\mu \/ \\partial \\tau > 0$ (corresponding to bigger values of $\\mu$), and for the second $\\partial \\mu \/ \\partial \\tau < 0$ (corresponding to smaller values of $\\mu$). It is convenient to consider separately three areas of variation $\\beta$.\n\n1. $-\\frac{4}{27} \\alpha^3 < \\beta < 0$:\\\\\na) the first solution:  $0 \\leq \\tau < \\infty, \\,\\, \\mu_\\text{\\scriptsize >} \\leq \\mu < \\sqrt{|\\alpha\/\\beta|}$,\\\\\nb) the second solution:  $0 \\leq \\tau \\leq 1\/\\alpha, \\,\\, 0 \\leq \\mu \\leq \\mu_\\text{\\scriptsize <}$,\\\\\nwhere $\\mu_\\text{\\scriptsize >}$, $\\mu_\\text{\\scriptsize <}$ are the bigger and smaller positive solution of the equation $\\alpha \\mu + \\beta \\mu^3 = 1$.\n\n2. $\\beta = -\\frac{4}{27} \\alpha^3$:\\\\\na) the first solution:  $0 \\leq \\tau < \\infty, \\,\\, 3\/(2 \\alpha) \\leq \\mu < \\sqrt{|\\alpha\/\\beta|}$,\\\\\nb) the second solution:  $ 0 \\leq \\tau \\leq 1\/\\alpha, \\,\\, 0 \\leq \\mu \\leq 3\/(2 \\alpha),$\\\\\nmoreover, in this case we have a branch point of the main equation solution at $\\tau = 0$, $\\mu = 1$.\n\n3. $-\\frac{1}{3} \\alpha^3 < \\beta < -\\frac{4}{27} \\alpha^3$:\\\\\na) the first solution:  $\\tau_0 \\leq \\tau < \\infty, \\,\\, \\mu_0 \\leq \\mu < \\sqrt{|\\alpha\/\\beta|}$,\\\\\nb) the second solution:  $\\tau_0 \\leq \\tau \\leq 1\/\\alpha, \\,\\, 0 \\leq \\mu \\leq \\mu_0$,\\\\\nwhere the quantities $\\mu_0$, $\\tau_0$ correspond to the branch points of the main equation solutions.\n\n\nIllustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple (quasi)doublet model are shown in Figure\\,\\ref{fig2}. The Figure\\,\\ref{fig2}a, built with $K_1 = 1,\\ \\gamma = 0.05,\\ \\Delta_a = 30.84,\\ \\Delta_c = 14.82$, which corresponds to $T_{SR} = 45.73,\\ \\mu_s = 0.162,\\ \\mu_f = 0.337,\\ \\tau_s = 1.04,\\ \\tau_f = 0.91$, describes a typical smooth SR transition\n with two second-order phase transitions $G_x-G_{xz}$  at the beginning ($\\tau_s$) and $G_{xz}-G_{z}$ at the end ($\\tau_f$) of the spin reorientation.\n \nThe Figure\\,\\ref{fig2}b, built with $K_1 = 1,\\ \\gamma = -0.1,\\ \\Delta_a = 33.19,\\ \\Delta_c = 27.1$, which corresponds to $T_{SR} = 22.95,\\ \\mu_s = 0.59,\\ \\mu_f = 0.72,\\ \\tau_s = 0.762,\\ \\tau_f = 0.93$, describes an abrupt first-order  SR transition. For $\\tau > \\tau_f$ there is the $G_x$-phase, which can remain stable up to $\\tau_s$ when cooled. For $\\tau < \\tau_s$ there is the $G_z$-phase, which can remain stable up to $\\tau_f$ when heated. The point $A$ marks a phase transition point when the phases $G_x$ and $G_z$ have equal energies.\n\n\nThe Figure\\,\\ref{fig2}c, built with $K_1 = 1,\\ \\gamma = -0.222,\\ \\Delta_a = 6.72,\\ \\Delta_c = 1.63$, which corresponds to $T_{SR} = 2.65,\\ \\mu_s = 0.307,\\ \\mu_f = 1.266,\\ \\tau_s = 0.778,\\ \\tau_f = 0.523$ and the Figure\\,\\ref{fig2}d, built with  $K_1 = 1,\\ \\gamma = -0.25,\\ \\Delta_a = 6.71,\\ \\Delta_c = 2.02$, which corresponds to  $T_{SR} = 2.56,\\ \\mu_s = 0.396,\\ \\mu_f = 1.31,\\ \\tau_s = 0.73,\\ \\tau_f = 0.545$ describe unconventional \"mixed\" SR transitions. At $\\tau_s$ there is the smooth second-order phase transition $G_x-G_{xz}$. At $\\tau \\leq \\tau_f$ we have two stable phases $G_z$ and $G_{xz}$: at those temperatures the sharp first-order phase transition $G_{xz}-G_{z}$ can happen, or the system could stay in the angular $G_{xz}$-phase.\n\n\\begin{figure}[H]\n\\centering\n  \\includegraphics[width=1.0\\textwidth]{4cases.pdf}\n\\caption{Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple (quasi)doublet model (see text for detail). The arrows indicate the direction of the antiferromagnetic vector $\\bf G$ in the $ac$ plane. The insets in panel (b) show the $\\theta$-dependence of the free energy.}\n\\label{fig2}\n\\end{figure}\n\nThus, there are not only the smooth and abrupt SR transitions, a characteristic feature of the range of intermediate values $K_2$ is the fundamental possibility of the existence of ``mixed'' SR transitions, in which the spins first smoothly rotate through a certain angle and then jump to the position with $\\theta = 0$. For this, it is sufficient that $\\mu_f$ corresponds to a point on the upper branch of solutions, and $\\mu_s$ to a point on the lower branch of solutions at $\\tau_f < \\tau_s$. In this case, the spin reorientation begins with the single second-order transition $G_x \\rightarrow G_{xz}$ and then ends with the first-order phase transition $G_{xz} \\rightarrow G_z$.\nIn contrast to the ``high-temperature'' approximation,  the ``single-doublet'' model claims the nature of the phase transition is determined not simply by the sign of the second anisotropy constant, but also it depends on the ratio between $K_1$, $K_2$ and the doublet splitting in both phases. Nevertheless, if we apply the simplified model to describe the SR transition, we have to renormalize both the first and the second anisotropy constant, giving the last one sometimes a rather complicated temperature dependence, in particular with a change in sign when considering transitions of the ``mixed'' type. Of course, in this case Fe sublattice alone is not enough to provide the value of the effective second constant.\n\n\n\\section{Conclusion}\nThe model of the spin-reorientation transitions induced by the $4f-3d$ interaction in rare-earth orthoferrites and orthochromites has been investigated. It is shown that both the temperature and the character of the spin-reorientation transition following from the solution of the  transcendental equation \\eqref{mainEq} are the result of competition between the second and fourth order spin anisotropy of the $3d$ sublattice, the crystal field for 4f ions, and the $4f-3d$ interaction. At variance with the ``high-temperature'' approximation,  the ``single-doublet'' model, along with typical smooth and abrupt SR transitions, predicts the appearance of mixed-type SR transitions, with an initial second-order transition  and a final abrupt first-order transition.\n\n\\ \\\\\n\\textbf{Funding:} The research was supported by the Ministry of Education and Science of the Russian Federation, project \u2116 FEUZ-2020-0054, and by Russian Science Foundation, project \u2116 22-22-00682.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\n\nCombinatorics is often of great importance in the study of the moduli space of stable curves of genus $g$, $\\overline M_g$.\nRecent  examples are the combinatorial  computation of its Euler characteristic \\cite{bodo}; \nthe combinatorial aspects in  the study of the Nef cone of $\\overline M_g$  (see in particular question 0.13 of \\cite{GKM}). \nThe importance of combinatorics is evident also in the study of spin curves \\cite{CC} \\cite{CCC}.\n\nThe main result relating the geometry of $\\overline M_g$ to the combinatoric of stable curves is \n is the stratification of $\\overline M_g$ by topological type, \nwhich is governed by the {\\em weighted dual graph} associated  to any stable curve \n (Definition \\ref{dwg}).\nTo any (multi)graph, \nit is naturally associated a group,\nwhich we will call {\\em complexity group} of the graph (Definition \\ref{cg}).\nIn particular, given a stable (or, more generally, nodal) curve $C$ we call $\\Delta_C$ the complexity group of the\ndual graph of the curve.\nThis group has been extensively studied as an invariant of graphs, with applications to\n Physics, Chemistry, and many other areas,  and  it goes under many different names, \n such as critical group \\cite{biggschip} \\cite{C-R}, determinant group \\cite{tatiana}, Picard group \\cite{tatiana}, Jacobian group \\cite{B-N},  \n abelian sandpiles group \\cite{C-E}.\nFrom the point of view of geometry, the complexity  group was introduced in \\cite{cap}, with the name of {\\em degree class group},\nin order to describe and handle the fibres of the compactification of the universal Picard variety $\\overline P_{d,g}$ \nover $\\overline M_g$ (also constructed in the same article). \nMoreover, this group arises naturally in the study of the \ncompactified Jacobians of families of curves.\nMore precisely, let $R$ be a discrete valuation ring with fraction field $K$ and residue field ${\\mbox{\\itshape {\\textbf k}}}$ and denote by $B$ the spectrum of $R$. \nConsider a flat and proper regular curve $X\\longrightarrow B$, such that the closed fiber $X_{\\mbox{\\small \\itshape {\\textbf k}}}$\nis geometrically irreducible.\nUnder some technical assumptions,  it is defined  the  group $\\Phi$ of connected\ncomponents of the  the N\\'eron model of the  Jacobian $J_K$ of the generic fiber $X_K$\n(see \\cite{BLR} sec. 9.6).\nNotice that in this definition the closed fiber doesn't need to be nodal;\nwhen $X_{\\mbox{\\small \\itshape {\\textbf k}}}$ is nodal, $\\Phi$ is precisely $\\Delta_{X_{\\mbox{\\small \\itshape {\\textbf k}}}}$.\nCaporaso in  \\cite{capneron} gave a geometric counterpart of this construction, showing that, under the assumption that $(d-g+1, 2g-2)=1$, there exists a space over $\\overline M_g$ such that, for every regular family of stable curves over $B$, the N\\'eron model of the Picard variety of degree $d$ of $X_K$ is obtained by base change via the moduli map $B\\longrightarrow \\overline M_g$. A yet another geometric interpretation of $\\Phi$ is given by Chiodo in \\cite{chiodo}, where the $r$-torsion points of $\\Phi$ are described as N\\'eron models of $r$-torsion line bundles on $ X_K$.\n\n\n\nThe relationship between the structure of this group and the structure of the graph has been studied, among others, by Lorenzini\nin several papers (see for instance  \\cite{lorgraphs} and \\cite{lorjac}).\nOther important references are \\cite{biggschip},  \\cite{tatiana} and \\cite{C-R}.\nWe addressed this problem in \\cite{BMS}, where in particular we constructed\na family of graphs with cyclic complexity group.\nIt seems that the question of finding a relation between the structure of the complexity group and the geometry of the curve \nis extremely difficult and intricate.\n\n\n\nRather that in the structure, we are interested here in the cardinality of the complexity group.\nKirkoff's Matrix Tree Theorem says that this \ninteger is the number of spanning trees of the dual graph of the curve, usually called the {\\em complexity}; \nthis is the reason for our notation, also suggested by L. Caporaso in \\cite{prag}. \nMoreover,  it can be calculated \nas the determinant of a certain  matrix (Theorem \\ref{MTT}, Remarks \\ref{spanningtrees} and \\ref{bourbaki}).\n\nWe shall call complexity of a curve the corresponding complexity of its dual graph.\nThe complexity being an upper semicontinuous function over $\\overline M_g$ (Lemma \\ref{usc}), \nit defines a weak stratification on $\\overline M_g$.\nIn Section \\ref{pre} we investigate the relationship between this stratification and the (strong) one given by topological type.\nThis relation is also enlighted using the list of possible graphs for curves of genus $3$ given in Section \\ref{lista}.\n\n\nIn particular, we are interested in the classes of curves with {\\em maximal complexity}. Define the function\n$\\psi(g):=\\mbox{max}\\{ |\\Delta_C|,\\,\\, C \\mbox{ stable curve of genus } g\\}$;\nour guiding problems are the following.\\\\\n\n1. Give a characterization of the curves $C$ such that $c(C)=\\psi (g)$, or at least  with ``big'' complexity.\\\\\n\n2. Find  bounds for $\\psi$, depending only on $g$.\\\\\n\nIn Section \\ref{MAX}, we give partial answers to problem 1, \nfinding necessary conditions for curves to have maximal complexity.\nWe show that these curves are all graph curves in the sense of \\cite{B-E}  i.e., they have simple trivalent dual graph \n$\\Gamma$ with $b_1(\\Gamma)=g$; moreover, they have no disconnecting nodes (Theorem \\ref{maxconf} and its combinatorial version Theorem \\ref{combinmaxconf}).\nUsing this reduction, we can apply some results of  the immense literature regarding  the complexity of \nregular graphs.\n\nIn Section \\ref{ub} we give an answer to our second problem, \n using  results of Biggs \\cite{biggs}, McKay \\cite{McK} and Chung-Yau \\cite{C-Y}.\nIn particular, we obtain an asymptotically sharp upper bound  for $g\\gg 0$.\nBy what observed in the beginning, these bounds limit the number of connected components of the \nN\\'eron model of the generalized Jacobian  of stable curves. \nNote that this result is different from the one given by Lorenzini in \\cite{lorner}: see Remark \\ref{confrontoL}.\nIn Section \\ref{lower}, we provide an example of families of graphs corresponding to stable curves with increasing genus,\nand compute explicitly the complexity depending on the genus, thus giving an explicit lower bound.\n\nIn the last section we discuss the conjectural behavior of the curves with maximal complexity.\nIn particular, it seems that the graph of  such a curve should have maximal connectivity (which is 3 in this case).\nMoreover, it seems that the girth has to be big (Conjecture \\ref{connectivity} and \\ref{girth}).\nWe prove in this section some partial results that seem to support the conjectures (Proposition \\ref{bah} and Corollary \\ref{g<10}).\nMoreover, we prove a uniform necessary condition holding for any sequence of graphs of curves  with maximal complexity\n(Theorem \\ref{cicli}), using again a result of McKay.\n\nEven though our point of view is irreparably geometrical, our main results are proven with (very simple) combinatorics methods, \nand  can be rewritten in purely combinatorial terms (see in particular \\ref{combi}).\nWe would like to express the hope that our geometric approach does not discourage non-geometers from the reading of the article;\nand in particular from considering the list of open question given in section \\ref{congetture}.\n\n\n\\medskip\n\n\\noindent {\\bf Acknowledgments}\n\nWe would like to express our gratitude to L. Caporaso for introducing us the problems we are dealing with in the present paper and for following this work very closely\nand to C. Casagrande for the important corrections and suggestions she gave us.\n\nWe also thank A. Machi for transmitting us much encouragement after reading a preliminary version.\n\n\n\n\\section{Combinatorial invariants of nodal curves}\\label{pre}\n\n\nWe work over an algebraically closed field $\\mbox{\\itshape {\\textbf k}} =\\overline \\mbox{\\itshape {\\textbf k}}$.\nA {\\em nodal} curve is a reduced curve which has only ordinary double points as singularities.\n\n\\subsubsection*{The dual graph of a curve}\n\\begin{defi}\\label{dwg}\nTo a nodal curve $C$ we can associate a graph $\\Gamma_C$,\ncalled the \\textit{weighted dual graph}, \ngiven by a triple $(V,E,g)$, where \n$V$ is the set of vertices, $E$ the set of edges, and $g$ a function on the set  $V$ with non-negative integer values.\nThis triple is defined in the following way\n\\begin{itemize}\n\\item to each irreducible component $A$ corresponds a vertex\\footnote{\nSometimes in the literature the vertices of a graph are denoted ``nodes''; of course we will never adopt this notation\nwhich is extremely confusing in our context. A node will always be for us an ordinary double point of a curve!} \n$v_A$; \n\\item to each node intersecting the components $A$ and $B$ (where $A$ and $B$ can coincide) \ncorresponds an edge connecting the vertices $v_A$ and $v_B$;\n\\item $g: V\\longrightarrow \\mathbb Z_{\\geq 0}$ is \n the function that associates to any vertex $v$ the geometric genus of the corresponding component of $C$.\n\\end{itemize}\n\\end{defi}\nCall $\\gamma$ the number of irreducible components of $C$ and \n$\\delta$ the number of nodes.\nThus $\\Gamma_C$ has $\\gamma$ vertices, $\\delta$ edges, \nand among the edges there is a loop \nfor every node lying on a single irreducible component of $C$.\nThe weighted graph encodes all the topological \ninformation about the curve.\nOf course, conversely, any weighted graph can be realized as the dual graph of a nodal curve.\n\n\n\n\n\nIt is important to stress that dual graphs of nodal curves can have more than one edge connecting two nodes, and can also have loops.\nThese are usually called {\\em multigraphs}. \nIn this paper, by graph we will always mean multigraph, while a graph without loops and multiple edges will be called {\\em simple}.\n\nGiven a graph $\\Gamma$ with $\\delta$ edges, $\\gamma$ vertices and $c$ connected components, its first Betti number is\n$b_1(\\Gamma):= \\delta-\\gamma+c$; it corresponds to the number of independent cycles on $\\Gamma$.\n\nLet $\\{C_1,\\ldots,C_{\\gamma}\\}$ be the set of  irreducible components of a nodal curve $C$ \nwith $\\delta$ nodes and $c$ connected components.\nRecall that the arithmetic genus of $C$ can be computed by the following formula\n\\begin{equation}\\label{genere}\ng(C)=\\sum_{i=1}^{\\gamma}g(C_i)+\\delta-\\gamma+c=\\sum_{v\\in V(\\Gamma_C)}g(v)+b_1(\\Gamma_C).\n\\end{equation}\nBy analogy, for any weighted graph $\\Gamma$ given by the triple  $(V,E,g)$, we will call \n$g(\\Gamma):=\\sum_{v\\in V}g(v)+b_1(\\Gamma)$ the {\\em (arithmetic) genus of the graph}.\n\n\n\n\n\\subsubsection*{Complexity group} \n\nLet us consider a connected nodal curve $C$.\nLet $\\left\\{C_i\\right\\}_{i=1,...,\\gamma}$ be the irreducible components of $C$. \nDefine \n$$\n\\begin{matrix}k_{ij}:= & \\left\\{\\begin{array}{l}\\;\\;\\, \\sharp(C_i\\cap C_j) \\, \\mbox{ if }i\\not= j\\\\  \\\\\n-\\sharp(C_i\\cap \\overline{C\\setminus C_i}) \\, \\mbox{ if } i=j\\\\\\end{array}\\right.\\end{matrix}\n$$\nAs $C_i\\cap \\overline{C\\setminus C_i}=\\bigcup_{j\\not= i} C_i\\cap C_j$, we have that, for fixed $i$, $\\sum_j k_{ij}=0$.\nFor every $i$ set \n$$\\underline{c}_i:=(k_{1i},\\ldots,k_{\\gamma i}) \\in \\mathbb Z^\\gamma.$$ \nCall $Z:=\\{\\underline{z}\\in\\mathbb{Z}^\\gamma : |\\underline z|=0\\}$.\nAs observed before, $\\underline c_i\\in Z$. Let us call $\\Lambda_C$ the sublattice of $Z$ spanned by \n$\\{\\underline c_1,\\ldots,\\underline c_\\gamma\\}$. In fact, $\\Lambda_C$ is a lattice in  $Z$ (it\nhas rank $\\gamma -1$) as we will show in a moment (see \\cite{capneron} for a geometric proof of this fact).\n\n\\begin{defi}\\label{cg}\nThe complexity group of $C$ is the finite abelian group $\\Delta_C:=Z\/\\Lambda_C$.\n\\end{defi}\nIt is important to notice that this group depends {\\em only} on the dual (non-weighted) graph\nof the curve; clearly it is defined  for any connected\\footnote{The definition could be easily extended to \nnon connected graphs\/curves.}  graph.\nAs noted in the Introduction, this group arises in many  contexts where graphs are used,\nand it is known with many other names.\n\n\n\nLet $M$ be the $\\gamma\\times\\gamma$ matrix whose columns are the $\\underline c_i$'s. \nWe will call $M$ the {\\em intersection matrix}, the name clearly deriving from its geometrical meaning. \nHowever, in literature, $M$ is known as the {\\em (combinatorial) Laplacian} matrix (cf. e.g. \\cite{bollobas} and \\cite{lorfinite}). \nIt is obtained from the so-called adjacency matrix of $\\Gamma_C$ subtracting the vertex degrees on the diagonal.\n\n\n\n\n\\subsubsection*{Complexity of a graph}\n\n\nA tree is a connected graph $G$ with $b_1(G)=0$.\nLet $\\Gamma$ be a graph. A {\\it spanning tree} of $\\Gamma$ is a subgraph of $\\Gamma$ which is a tree having \nthe same vertices as $\\Gamma$. \n\\begin{defi}\nThe complexity of $\\Gamma$, indicated by the symbol $c(\\Gamma)$, is the number of spanning trees contained \nin $\\Gamma$ (see e.g. \\cite{biggs}, sec 6, \\cite{berge}, cap.3 $\\natural$ 5,  \\cite{west}, sec 2.2).\n\\end{defi}\n\n\nObserve that $c(\\Gamma)=0$ if and only if $\\Gamma$ is not connected, \nand that if $\\Gamma$ is a connected tree  $c(\\Gamma)=1$.\nFor the complexity of the dual graph associated to a curve $C$, we will often use the symbol $c(C)$, instead of\n$c(\\Gamma_C)$.\nThe following theorem, known as Kirkoff's Matrix Tree Theorem, will be a key ingredient for our work. \nThere are at least three different proofs of this result; see \\cite{BMS} for a proof and for the references.\n\n\\begin{teo}{\\upshape (Matrix Tree Theorem)}\\label{MTT}\nLet $s, t \\in \\{1,\\ldots \\gamma\\}$.\nUsing the above notations, if $M^\\star$ is obtained from $M$ by deleting the $t$-th column and the $s$-th row, then \n$$c(\\Gamma)=(-1)^{s+t+\\gamma -1}\\mbox{det}(M^\\star).$$\n\\end{teo}\n\n\nIn particular, the Matrix Tree Theorem assures that, in the case of the \ndual graph of a {\\em connected} curve $C$, the matrix  $M$ has rank $\\gamma-1$\ni.e., $\\Lambda_C$ is indeed a lattice.\n\n\\begin{rem}\\label{spanningtrees} \\upshape{\nFor $r\\in\\{1,\\ldots,\\gamma\\}$, consider the isomorphism $\\alpha_r:Z\\stackrel{\\sim}\\longrightarrow\\mathbb{Z}^{\\gamma - 1}$\nwhich consists of deleting the $r$-th component.\nThe group $\\Delta_C$ is the quotient of $\\mathbb{Z}^{\\gamma - 1}$ by the lattice generated by \n$$\\underline c_i^\\prime:=(k_{1i}\n,\\ldots,\\widehat{k_{ri}},\\ldots,k_{\\gamma i}).$$\nObserve that again $\\sum_i \\underline c_i^\\prime = \\underline 0 \\in \\mathbb{Z}^{\\gamma -1}$.\nTherefore $\\Delta_C$ is presented by the matrix $M^\\star$ obtained from $M$ by deleting a column and the $r$-th row.\nHence, we can compute its cardinality via Theorem \\ref{MTT}\n$$c(\\Gamma_C)=|\\Delta_C|=|\\mbox{det}(M^\\star)|.$$\nSo, we can conclude that {\\em the cardinality of the complexity group of a curve $C$ is the complexity of the dual graph $\\Gamma_C$.}}\n\\end{rem}\n\n\\begin{rem}\\label{bourbaki}\\upshape{\nBy the diagonalization theorem of integer matrices (i.e. the structure theorem for finite abelian groups, \nsee \\cite{ar}), $M$ is equivalent over $\\mathbb Z$ to a diagonal matrix\n$diag (d_1, d_2,\\ldots, d_\\gamma)$, where the $d_i's$ are the invariant factors, i.e. the \nnon-negative integers  obtained in the following way:\nLet $D_i$ be the greatest common divisor of the $i\\times i$ minors of $M$. \nThen  $d_i=D_i\/D_{i-1}$\n(cf. also \\cite{lorarithmetical}, Theorem 1.5). \nNote that $d_\\gamma=0$ and $d_i>0$ for $i\\not = \\gamma$, so \n$\\Delta_C=\\oplus _{i=0}^{\\gamma-1}\\mathbb Z\/d_i \\mathbb Z $.\nIn particular, $ |\\Delta_C|$ is equal to the greatest common divisor of all the $(\\gamma-1)\\times(\\gamma-1)$ \nminors of $M$. \n\nOn the other hand, if we diagonalize $M$ over the real numbers, we get real eigenvalues\n$0=\\lambda_1>\\lambda_2\\geq \\ldots\\geq \\lambda_\\gamma$. \nThese eigenvalues are deeply studied in Combinatorics.\nNote that in general the $\\lambda_i$'s have no relation with  the invariant factors, not even if they happen to be integers; \na nice counterexample can be found in Section 9.2 of \\cite{tatiana}.\nAn attempt to construct the invariant factors from the $\\lambda_i's$, for a particular class of graphs, can be find in \\cite{C-R}.\n\nLet us note that in particular $ | \\Delta_C|=\\gamma^{-1}\\lambda_2\\lambda_3\\ldots\\lambda_\\gamma$ \n(cf. \\cite{biggs} cor.6.5).\n}\n\\end{rem}\n\n\n\n\n\\subsubsection*{Stable curves and their graphs}\n\n\\begin{defi}\\label{stable}\nA curve $C$ of genus $g\\geq 2$ over $\\mbox{\\itshape {\\textbf k}}$ is stable (resp. semistable) if it is  nodal, connected and  such that if \n$D\\subset C$ is a smooth rational component, then $|D\\cap\\overline{C\\setminus D}|\\geq 3$ (resp. $\\geq 2$).\n\\end{defi}\n\nThe moduli space of stable curves of genus $g$, $\\overline M_g$, is a projective variety of dimension $3g-3$, the so-called Deligne-Mumford compactification of the moduli space of smooth curves of genus $g$, $M_g$.\nThe theory of stable curves was first introduced by A. Mayer and D. Mumford and was first developed in \\cite{DM} in order to prove the irreducibility of $M_g$ in any characteristic. In that paper, the authors prove the main properties of stable curves used later by D. Gieseker in \\cite{gie} to establish the existence of $\\overline M_g$.\n\n\\medskip\n\nGiven a graph $\\Gamma$ and a vertex $v$, the degree $d(v)$ of $v$  is the number of half edges touching $v$.\nThe combinatorial version of the definition of a stable curve is the following.\n\n\\begin{defi}\\label{cstable}\nA weighted graph  $\\Gamma$ of arithmetic genus $g\\geq 2$ is stable if\n\\begin{equation}\\label{stability}\n2g(v)-2+d(v)>0 \\,\\,\\,\\,\\,\\,\\mbox{ for any }\\,\\,\\,\\,\\,v\\in V.\n\\end{equation}\n$\\Gamma$ is said to be semistable if the inequality above holds with $\\geq$.\n\\end{defi}\n\n\n\n\\subsubsection*{Topological stratification of $\\overline M_g$}\n\nThere is a natural stratification on $\\overline M_g$ given by topological type.\nEach stratum of codimension $k$ is the subset of $\\overline M_g$ \nconsisting of classes of  stable curves  with weighted graphs of genus $g$  having exactly $k$ edges.\nIn particular, the stratum of codimension $0$ is the open set $M_g\\subset \\overline M_g$ of smooth curves.\nThe strata of codimension $1$ are $[g\/2]+1$, and the corresponding graphs are of the following types.\n\n\\bigskip\n\n\\centerline{\n\\xymatrix@=.2pc{\n*{\\bullet} \\ar@{-}^<{g-1} @(ur,dr)\\\\{}\n}\n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n}\n\n\\bigskip\n\\centerline{\n\\xymatrix@R=.1pc{\n*{\\bullet} \\ar@{-}^<{i}^>{g-i}[r] & *{\\bullet}\n\\\\{}\n}\n\\,  for $i=1,\\ldots, [g\/2]$.\n}\n\n\\bigskip\n\n\\noindent The closure of these strata are effective divisors on $\\overline M_g$, the so-called boundary divisors,\nusually denoted by $\\Delta_i$, for  $i=0,\\ldots, [g\/2]$, according to the preceding list.\nFor instance, $\\Delta_0$ has as generic point  corresponding to an irreducible curve of geometric genus $g-1$ \nwith exactly one node.\nIt can be described as the locus \nof curves having at least one non-disconnecting node.\nThe boundary of $\\overline M_g$ is $\\partial \\overline M_g=\\overline M_g\\setminus M_g=\\cup_{i=0}^{[g\/2]}\\Delta_i$.\n\nThe union of the strata of  codimension $k$ is the variety  of isomorphism classes of stable curves having exactly $k$ nodes; \nits closure is the locus of curves with at least $k$ nodes.\n\n\n\\subsubsection*{Degeneration of nodal curves}\n\nThe following well-known result describes the possible transformations on a dual graph corresponding to the degenerations of a nodal curve.\n\n\\begin{prop}\\label{degenerations}\nLet $\\Gamma=(V,E,g)$ be a weighted graph. \nLet  $C$ be a nodal  curve having $\\Gamma$ as dual weighted  graph.\nThe dual graphs of the possible nodal curves obtained by degenerations of $C$ \nare obtained from $\\Gamma$ via a sequence of the following\ntransformations:\n\\begin{itemize}\n\\item[(I)] given a vertex $v$ such that $g(v)\\geq 1$ add a loop on $v$ and decrease its weight by 1;\n\\item[(II)] given a vertex $v$, and given two nonnegative integers $a$ and $b$ such that $a+b=g(v)$, substitute it with \ntwo vertices $v_a$ and $v_b$ with weights respectively $a$ and $b$  and one edge $l$ connecting them.\n\\end{itemize}\n\\end{prop}\n\\noindent The figure below illustrates operation (II).\n\n\n\\vspace{0.5cm}\n\\centerline{\n\\begin{tabular}{cc}\n{}\n&\n\\xymatrix@=1pc{\n& &\\ar@{-}[dl]_>v &&&& &&&&\\ar@{-}[dl]\\\\\n& *{\\bullet} && \\ar@{:>}[rr] && & &*{\\bullet} \\ar@{-}[rr]^l_<{v_a}_>{v_b}&& *{\\bullet}& \\\\\n\\ar@{-}[ur]& &\\ar@{-}[ul] &&&& \\ar@{-}[ur] &&&&\\ar@{-}[ul]\n}\n\\end{tabular}}\n\\vspace{0.5cm}\n\n\n\n\\noindent Of course, there are many ways to perform operation (II) (see also Definition \\ref{operazione} below).\nNotice that this is the opposite operation of contracting an edge, in the sense that if we contract the \nnew edge $l$ we get the original graph.\n\nNote also that if our given curve $C$ is stable, degenerations of $C$ obtained by performing operation (I) are still stable, while if we perform operation (II) this is the case only if\n$2a-2+d(v_a)\\geq 1$ and $2b-2+d(v_b)\\geq 1$.\n\n\n\n\n\n\n\\subsubsection*{Polygonal curves}\n\nWe present here an elementary combinatorial proof of the following well-known fact for stable curves. \nA geometric proof can be obtained using the above results about the topological stratification of $\\overline M_g$.\n\\begin{lem} \\label{lemcomb}\nLet $C$ be a stable curve of genus $g\\ge 2$. \nThen\n\\begin{enumerate}\n\\item $C$ has at most $3g-3$ nodes and $2g-2$ irreducible components.\n\\item Assume that $C$ has $3g-3$ nodes. Then $C$ has $2g-2$ components $C_1,\\ldots,C_{2g -2}$ and, \nif $\\nu_i:C_i^\\nu\\longrightarrow C_i$ is the normalization of $C_i$, then \n$C_i^\\nu\\simeq \\mathbb{P}^1$ and $|\\nu_i^{-1}(C_i\\cap C_{sing})|=3$ for all $i$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nLet $\\Gamma=(V,E,g)$ be the weighted graph associated to $C$.\nNote that $\\sum_{v\\in V}d(v)=2\\delta$, hence we can rewrite formula (\\ref{genere}) as\n\\begin{equation}\\label{genere2}\ng=\\sum_{v\\in V}\\left(g(v)+\\frac{d(v)}{2}\\right)-\\gamma +1.\n\\end{equation}\nBy the connectedness of $C$, $d(v)\\geq 1$ for any $v$. \nUsing also the stability condition, we have that \n$\ng\\geq \\sum_{v\\in V}\\frac{3}{2}-\\gamma +1=\\frac{\\gamma}{2}+1,\n$\nhence, $\\gamma \\leq 2g-2$. Now, using (\\ref{genere}) again, we have that $\\delta \\leq 3g-3$, and (1) follows.\n\nNow, if $\\delta =3g-3$, again by (\\ref{genere}), we see that necessarily $\\gamma =2g-2$ and \n$g(v)=0$ for any $v\\in V$.\nMoreover, using again (\\ref{genere2}), we see that $d(v)=3$ for any vertex $v$, so also (2) is proved.\n\\end{proof}\n\nStable curves with $3g-3$ nodes (and $2g-2$ components), \nare the $0$-strata of the topological stratification of $\\overline M_g$; \nthey are rigid, in the sense that \nany deformation of such curves in a family of stable curves must have necessarily at least one node smoothed. \nIndeed, from  Proposition \\ref{degenerations}  we see that both operations (I) and (II) cannot be performed, \ni.e. there is no possible further degeneration for such curves.\nWe will call such curves {\\em polygonal curves}\n(also called large limit curves, see \\cite{tyurin}).\nPolygonal curves with simple graph are called graph curves (\\cite{B-E}).\n\nWe refer to \\cite{B-E} for an ample discussion on the importance and the properties of these curves.\nLet us just make a couple of remarks. \nThe associated dual graphs are trivalent (multi)graphs such that the weight function $g$ is $0$ on every vertex.\nDue to the fact that $\\mathbb P^1$ with $3$ fixed points has no non-trivial automorphisms, \nthe automorphism group of such curves coincides with the automorphism group of the graph.\nArtamkin in \\cite{artamkingen} has given a recursive rule that computes the number\n$$\n\\sum_{\\begin{array}{c}\\Gamma \\,\\,\\,trivalent,\\\\\nb_1(\\Gamma)=g\n\\end{array}}\n\\frac{1}{|Aut(\\Gamma)|} \\mbox{ ,}\n$$\nwhich is the ``stacky top self-intersection'' of the boundary divisor $\\partial \\overline M_g$.\n\n\n\\subsubsection*{Stratification by complexity of $\\overline M_g$}\n\n\nWe can define a  function $c\\colon \\overline M_g\\longrightarrow \\mathbb Z_{\\geq 0}$ \nassociating to every $[C]\\in \\overline M_g$ its complexity $c(C)$.\n\nOf course, $c$ is bounded from above, i.e., there is a bound for the complexity of a stable curve of given genus;\nindeed, by Lemma \\ref{lemcomb}, there is only a finite number of possible graphs.\nIn section \\ref{ub} we provide upper bounds, which are asymptotically sharp for $g\\gg 0$.\n\\begin{rem}\\upshape{\nClearly, this wouldn't make sense for nodal curves (not even for semistable ones). \nIndeed, blowing up a node an arbitrary number of times doesn't change the genus of the curve, \nbut it can increase arbitrarily the complexity of a nodal curve (see for instance \\cite{BMS} Proposition 3.3.).}\n\\end{rem}\n\n\n\\begin{lem}\\label{usc}\nThe function $c\\colon \\overline M_g\\longrightarrow \\mathbb Z_{\\geq 0}$  is upper semi-continuous.\n\\end{lem}\n\\begin{proof}\nFollows from Proposition \\ref{increasing} below.\n\\end{proof}\nThis result implies in particular that we could define a stratification ``by complexity'' of $\\overline M_g$.\nThis is clearly a much rougher stratification than the one by topological type; \n{\\em the set of curves in $\\overline{M}_g$ with given complexity is a (maybe empty) union of components of different codimension strata \nof the topological type stratification, of different codimension} (see the case of genus $3$ in the next section).\n\n\nFor instance, the set $\\overline M_g^{c=1}$ of curves with complexity one is the set of curves whose dual graph is a tree with loops.\nIt contains the curves of compact type $\\overline M_g^{ct}=\\overline M_g\\setminus \\Delta_0$, but also the interior of $\\Delta_0$,\nand other strata of bigger codimension; it also contains $0$-strata, i.e. isolated points (see next section).\nThe complement $\\overline M_g\\setminus \\overline M_g^{c=1}$ of curves with complexity greater than one is a closed \nsubset of codimension $2$.\n\n\n\n\n\n\\section{List of graphs for $\\overline M_3$}\\label{lista}\n\nIn this section, we list all the possible weighted graphs for stable curves of genus 3, \nas well as their complexity and their complexity group. \nThis list (with one error, now corrected, which was kindly pointed out to us by A. Chiodo) appeared in \\cite{BMS}.\nWe will use $\\mathbb{Z}_n$ to denote the quotient group $\\mathbb{Z}\/n\\mathbb{Z}$. \nThe graphs are ordered by increasing the number of nodes (i.e. according to the codimension of corresponding \nstratum of the stratification by topological type).\nIn the graphs we will indicate the weight of each vertex only if it is not zero.\nRecall that if $C$ is a stable curve of genus 3, then $C$ has at most $6$ nodes and $4$ components.\n\n\\begin{equation*}\n\\begin{tabular}{||c|c|c|c|c||}\n\\hline \\hline\nGraph configuration & Nodes & Components & Complexity & DCG\\\\\n\\hline\n\\xymatrix@=.2pc{\n{}\\\\\n*{\\bullet}\\\\{}\n\\scriptstyle{3}}\n&\n\\xymatrix@=.001pc{\n\\\\\n0}\n&\n\\xymatrix@=.001pc{\n\\\\\n1}\n&\n\\xymatrix@=.001pc{\n\\\\\n1}\n&\n\\xymatrix@=.001pc{\n\\\\\n0}\\\\\n\n\\xymatrix{\n*{2 \\,\\bullet} \\ar@{-}@(ur,dr) & {}\n}\n&\n1\n&\n1\n&\n1\n&\n0 \\\\\n&&&&\\\\\n\n\\xymatrix{\n*{2\\, \\bullet} \\ar@{-}[r] & *{\\bullet \\, 1}\n}\n&\n1&2&\n1\n&\n0\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}_<{1}@(ul,dl) \\ar@{-}@(dr,ur)\n}\n&\n2&1&\n1\n&\n0\n\\\\\n&&&&\\\\\n\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r]^<{1}^>{1}& *{\\bullet}\n}\n&\n2&2&\n1\n&\n0\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r]^>{2}& *{\\bullet}\n}\n&\n2&2&\n1\n&\n0\n\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-} @\/_\/[r]_>{1}  & *{\\bullet} \\ar@{-} @\/_\/[l]_>{1}\n}\n&\n2&2&\n2\n&\n$\\mathbb{Z}_2$\\\\\n&&&&\\\\\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}[r]^<{1} & *{\\bullet}\\ar@{-}[r]^<{1}^>{1} & *{\\bullet}\\\\\n{}\n}\n&\n2&3&\n1\n&\n0\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{tabular}{||c|c|c|c|c||}\n\\hline \\hline\nGraph configuration & Nodes & Components & Complexity & DCG\\\\\n\\hline \n{}&&&&\\\\\n{}&&&&\\\\\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}@(ur,dr) \\ar@{-}@(ur,ul)\\\\\n{}\n}\n&\n3&1\n&\n1&\n0\\\\\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}@(ur,ul) \\ar@{-}@(dr,dl) \\ar@{-}[r]^>{1}& *{\\bullet}\\\\\n{}\n}\n&\n3&2&\n1\n&\n0\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r]_<{1} & *{\\bullet} \\ar@{-} @(ur,dr)\n}\n&\n3&2&\n1\n&\n0\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-}@(ul,dl) \\ar @{-}@\/_\/[r] & *{\\bullet} \\ar @{-}_<{1}@\/_\/[l]\n}\n&\n3&2&\n2\n&\n$\\mathbb Z_2$\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-} @\/_\/[r] \\ar@{-}[r] & *{\\bullet} \\ar@{-}_>{1} @\/_\/[l]\n}\n&\n3&2&\n3\n&\n$\\mathbb Z_3$\\\\\n&&&&\\\\\n\\xymatrix@R=.5pc{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r] & *{\\bullet}\\ar@{-}[r]^<{1}^>{1} & *{\\bullet}\\\\\n{}\n}\n&\n3&3\n&\n1\n&\n0\\\\\n\\xymatrix@R=.2pc{\n*{\\bullet} \\ar@{-}[r]^<{1} & *{\\bullet}\\ar@{-}@(ul,ur) \\ar@{-}[r]^>{1} & *{\\bullet}\\\\\n{}\n}\n&\n3&3&\n1&\n0\\\\\n\\xymatrix{\n *{\\bullet} \\ar@{-}^<{1}[r]& *{\\bullet} \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}_<{1}@\/_\/[l]\n}\n&\n3&3&\n2\n&\n$\\mathbb Z_2$\\\\\n&&&&\\\\\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}@(ur,ul) \\ar@{-}@(dr,dl) \\ar@{-}[r]& *{\\bullet} \\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\n4&2&\n1&\n0\\\\\n\\xymatrix{\n*{\\bullet} \\ar @{-}@(ul,dl) \\ar @{-}@\/_\/[r] & *{\\bullet} \\ar @{-}@\/_\/[l] \\ar @{-}@(ur,dr)\n}\n&\n4&2&\n2\n&\n$\\mathbb Z_2$\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar @{-} @\/_\/[r] \\ar@{-}[r] & *{\\bullet} \\ar@{-}@\/_\/[l]\n}\n&\n4&2&\n3\n&\n$\\mathbb Z_3$\\\\\n&&&&\\\\\n\\xymatrix{\n*{\\bullet}  \\ar @{-} @\/_.3pc\/[r] \\ar @{-}@\/_.8pc\/[r]\/& *{\\bullet} \\ar@{-}@\/_.3pc\/[l] \\ar@{-}@\/_.8pc\/[l]\n}\n&\n4&2&\n4\n&\n$\\mathbb Z_4$\\\\\n&&&&\\\\\n\\xymatrix@R=.5pc{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r] & *{\\bullet}\\ar@{-}[r]^<{1} & *{\\bullet}\\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\n4&3\n&\n1\n&\n0\\\\\n\\xymatrix@R=.2pc{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl) & *{\\bullet}\\ar@{-}@(ul,ur) \\ar@{-}[r]^>{1} & *{\\bullet}\\\\\n{}\n}\n&\n4&3&\n1&\n0\\\\\n\\xymatrix@R=.1pc{\n{} & *{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}[r]& *{\\bullet} \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}_<{1}@\/_\/[l]\\\\\n{}\n}\n&4&3\n&\n2\n&\n$\\mathbb Z_2$\\\\\n\\xymatrix@=1.2pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}^<{1}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] \\ar@{-}@\/_\/[rr]& & *{\\bullet}\n}\n&\n\\xymatrix@=.3pc{\n\\\\\n4}\n&\n\\xymatrix@=.3pc{\n\\\\\n3}\n&\n\\xymatrix@=.3pc{\n\\\\\n5\\\\\n{}\n}\n&\n\\xymatrix@=.3pc{\n\\\\\n{\\mathbb Z_5}\n}\n\\\\\n\\xymatrix@R=.8pc{\n{} & *{\\bullet} \\ar@{-}^<{1}[r]& *{\\bullet} \\ar@{-}[r] \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}@\/_\/[l]&\n}\n&\n4&3&\n3\n&\n$\\mathbb Z_3$\n\\\\\n\\xymatrix@=.5pc{\n*{\\bullet}\\ar@{-}^<{1}[dr]&&&\\\\\n&*{\\bullet}\\ar@{-}[rr]&&*{\\bullet}\\ar@{-}@(ur,dr)\\\\\n*{\\bullet} \\ar@{-}_<{1}[ur]&&\n}&\n\\xymatrix@=.3pc{\n\\\\\n4\n\\\\\n{}\n\\\\\n{}}\n&\\xymatrix@=.3pc{\n\\\\\n4}\n&\n\\xymatrix@=.3pc{\n\\\\\n1}\n&\n\\xymatrix@=.3pc{\n\\\\\n0}\n\\\\\n\\xymatrix@R=.3pc{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl) & *{\\bullet}\\ar@{-}@(ul,ur) \\ar@{-}[r] & *{\\bullet}\n\\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\n5&3&\n1&\n0\\\\\n\\xymatrix@R=.3pc{\n{}&&\\\\\n*{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl) & *{\\bullet} \\ar@{-}@\/^\/[r] \\ar@{-}@\/_\/[r] & *{\\bullet}\n\\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\n\\xymatrix@R=.3pc{\n{}\\\\\n5}\n&\n\\xymatrix@R=.3pc{\n{}\\\\\n3}\n&\n\\xymatrix@R=.3pc{\n{}\\\\\n2}\n&\n\\xymatrix@R=.3pc{\n{}\\\\\n\\mathbb{Z}_2}\\\\\n\n\\hline \\hline\n\\end{tabular}\n\\end{equation*}\n\n\\begin{equation*}\n\\begin{tabular}{||c|c|c|c|c||}\n\\hline \\hline\nGraph configuration & Nodes & Components & Complexity & DCG\\\\\n\\hline \n{}&&&&\\\\\n\\xymatrix@R=.2pc{\n{}\\\\\n{} & *{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl)& *{\\bullet} \\ar@{-}[r] \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}@\/_\/[l]&\\\\&&&&\n}\n&\n\\xymatrix@R=.2pc{\n{}\\\\\n5}\n&\n\\xymatrix@R=.2pc{\n{}\\\\\n3}\n&\n\\xymatrix@R=.2pc{\n{}\\\\\n3}\n&\n\\xymatrix@R=.2pc{\n{}\\\\\n\\mathbb Z_3}\n\\\\\n\\xymatrix@=1.2pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}@\/_\/[dl] \\ar@{-}@\/^\/[dr] \\ar@{-}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] & & *{\\bullet}\\\\{}\n}\n&\n\\xymatrix@=.3pc{\n\\\\\n5}\n&\n\\xymatrix@=.3pc{\n\\\\\n3}\n&\n\\xymatrix@=.3pc{\n\\\\8\n\\\\\n{}\n}\n&\n\\xymatrix@=.3pc{\n\\\\\n{\\mathbb Z_8}}\\\\\n\\xymatrix@=1.2pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr] \\ar@{-}@(ul,ur)& \\\\\n*{\\bullet} \\ar@{-}[rr] \\ar@{-}@\/_\/[rr]& & *{\\bullet}\n}\n&\n\\xymatrix@=.5pc{\n\\\\5}\n&\n\\xymatrix@=.5pc{\n\\\\3\n\\\\\n{}\n}\n&\n\\xymatrix@=.5pc{\n\\\\5\n\\\\\n{}\n}\n&\n\\xymatrix@=.5pc{\n\\\\\n{\\mathbb Z_5}}\n\\\\\n\\xymatrix@=1.2pc{\n&*{\\bullet} \\ar@{-}^<{1}[d]&\\\\\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr] & \\\\\n*{\\bullet} \\ar@{-}[rr] \\ar@{-}@\/_\/[rr]& & *{\\bullet}\n}\n&\\xymatrix@=.5pc{\n{}\\\\\n{}\\\\\n5\n}\n&\n\\xymatrix@=.5pc{\n{}\\\\\n{}\\\\\n4}\n&\n\\xymatrix@=.5pc{\n{}\\\\\n{}\\\\\n5\n\\\\\n{}\\\\{}\n}\n&\n\\xymatrix@=.5pc{\n{}\\\\\n{}\\\\\n{\\mathbb Z_5}}\\\\\n\\xymatrix@R=.2pc{\n{} & *{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl)& *{\\bullet} \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}@\/_\/[l] \\ar@{-}^>{1}[r]& *{\\bullet}\n\\\\&&&&\n}\n&5&4&\n2\n&\n$\\mathbb Z_2$\n\\\\\n\\xymatrix@=.5pc{\n*{\\bullet}\\ar@{-}^<{1}[ddd]&&&\\\\\n&&*{\\bullet}\\ar@{-}@(ur,dr)&\\\\\n&&&\\\\\n*{\\bullet}\\ar@{-}[rrr]\\ar@{-}[rruu]&& &*{\\bullet}\\ar@{-}@(ur,dr)\\\\\n{}\n}\n&\\xymatrix{\n\\\\5}\n&\\xymatrix{\n\\\\4}\n&\n\\xymatrix{\n\\\\1}\n&\n\\xymatrix{\n\\\\\n0}\n\\\\\n\\xymatrix@R=.2pc{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}@(ul,dl)& *{\\bullet} \\ar @{-} @\/_\/[r] & *{\\bullet} \\ar@{-}@\/_\/[l] \\ar@{-}[r]& *{\\bullet} \\ar@{-}@(ur,dr)\n\\\\{}\n}\n&6&4&\n2\n&\n$\\mathbb Z_2$\n\\\\\n\\xymatrix@=1pc{\n*{\\bullet} \\ar@{-}[dd]\\ar@{-}@\/_.5pc\/[dd]\\ar@{-}[drr]&&&\\\\\n&&*{\\bullet}\\ar@{-}[r]&*{\\bullet} \\ar@{-}@(ur,dr)\\\\\n*{\\bullet} \\ar@{-}[urr]&&&\n}\n&\n\\xymatrix@=.6pc{\n\\\\6}\n&\\xymatrix@=.6pc{\n\\\\4}\n&\n\\xymatrix@=.6pc{\n{}\\\\\n5\n\\\\\n{}\\\\{}\n}\n&\n\\xymatrix@=.6pc{\n{}\\\\\n{\\mathbb Z_5}}\\\\\n\n\\xymatrix@=.5pc{\n&*{\\bullet}\\ar@{-}[ddd] \\ar@{-}@(ur,ul)&\\\\\n*{\\bullet} \\ar@{-}@(ul,dl)&&*{\\bullet}\\ar@{-}@(ur,dr)&\\\\\n&&&\\\\\n&*{\\bullet}\\ar@{-}[ruu]\\ar@{-}[luu] & \\\\\n{}\n}\n&\\xymatrix@=.5pc{\n\\\\6}\n&\\xymatrix@=.5pc{\n\\\\4}\n&\n\\xymatrix@=.5pc{\n\\\\1}\n&\n\\xymatrix@=.5pc{\n\\\\\n0}\n\\\\\n\\xymatrix{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}[d] \\ar@{-}@\/_\/[d] & *{\\bullet} \\ar@{-}[d] \\ar@{-}@\/^\/[d]\\\\\n*{\\bullet} \\ar@{-}[r] & *{\\bullet}\n}\n&\\xymatrix@=.5pc{\n\\\\6}\n&\\xymatrix@=.5pc{\n\\\\4}\n&\n\\xymatrix@=.5pc{\n\\\\12}\n&\n\\xymatrix@=.5pc{\n\\\\\n\\mathbb Z_{12}\n}\n\\\\\n\\xymatrix@=1.2pc{\n&*{\\bullet}\\ar@{-}[dd] \\ar@{-}[dl] \\ar@{-}[dr] &\\\\\n*{\\bullet} \\ar@{-}[rr] |!{[r]}\\hole & &*{\\bullet}\\\\\n&*{\\bullet}\\ar@{-}[ru]\\ar@{-}[lu] & \n}\n&\\xymatrix@=.5pc{\n\\\\6\\\\\n{}}\n&\\xymatrix@=.5pc{\n\\\\4}\n&\n\\xymatrix@=.5pc{\n\\\\16}\n&\n\\xymatrix@=.5pc{\n\\\\\n\\mathbb Z_4\\times\\mathbb Z_4}\n\\\\\n\n\\hline \\hline\n\\end{tabular}\n\\end{equation*}\n\n\n\\bigskip\n\n\\bigskip\n\n\\bigskip\n\n\\noindent Note that the set of curves of genus $3$ with complexity $1$  contains at least one stratum \nof the topological stratification of any codimension.\nThe set of curves with complexity $2$ contains at least one stratum among the ones of codimension\ngreater or equal to $2$.\nThe set of curves with complexity $6$ is empty.\n\n\n\n\n\n\n\n\\section{Stable curves with maximal complexity}\\label{MAX}\n\nWe will now focus our interest in curves with maximal complexity.\nWe can see that in case $g=3$ above, the curves reaching the maximal complexity are polygonal curves. \nHowever, as already observed, there are also polygonal curves with small, even $0$, complexity.\nIn what follows we prove that, indeed, any curve with maximal complexity is necessarily a polygonal curve\nand, moreover, it is a graph curve without disconnecting nodes.\nIn Section \\ref{congetture} we shall make some remarks and conjectures on sufficient conditions.\n\nWe shall need the following well-known  result, whose proof is elementary.\n\n\\begin{prop}\\label{comb}\nLet $\\Gamma$ be a graph. If $e$ is an edge of $\\Gamma$ which is not a loop,\ncall $\\Gamma -e$ the graph obtained from $\\Gamma$ by removing $e$, \nand $\\Gamma\\cdot e$ the one obtained by contracting $e$. \nThen, we have the following relation between the complexities of these three graphs:\n\\begin{equation}\\label{fond}\nc(\\Gamma)=c(\\Gamma - e)+c(\\Gamma\\cdot e).\n\\end{equation}\n\\end{prop}\n\\noindent See \\cite{BMS} for a geometric interpretation and discussion of this proposition.\\\\\n\n\n\n\nThe main result of this section is the following\n\n\\begin{teo} \\label{maxconf}\nLet $C$ be a stable curve of genus $g\\geq 3$ with maximal complexity. \nThen, $C$ is a curve without disconnecting nodes and with trivalent simple dual graph.\nIn particular, $C$ is a graph curve.\n\\end{teo}\n\n\n\nFrom a geometrical point of view, this result implies in particular that the curves with maximal complexity \nlie on those $0$-strata of the topological stratification\nwhich are contained in  $\\partial \\overline M_g\\setminus \\cup_{i=1}^{[g\/2]} \\Delta_i$.\n\nSee the end of this section for a purely combinatorial statement.\n\n\\bigskip\n\nOur strategy is the following: we consider a generic stable curve $C$ of genus $g$ and its weighted dual graph, $\\Gamma_C$;\nthen, if $\\Gamma_C$ is not as stated in Theorem \\ref{maxconf}, we modify the graph obtaining a new weighted graph $\\Gamma'$, which is the dual graph of another stable curve \nof genus $g$ with the desired properties, and we prove that $c(\\Gamma_C)<c(\\Gamma')$.\nSo, we need to perform operations on the graph increasing strictly \nthe number of spanning trees.\nA. Kelmans made a deep study  of operations increasing the complexity of a graph, \neven though from a different point of view; see for instance \\cite{Kelmans}.\n\nRecall the two operations associated to the degeneration of a curve described in Proposition \\ref{degenerations}.\nWe now study when do these operations increase the complexity of the graph. \n\n\\begin{defi}\\label{operazione}\nWe shall distinguish two different kinds of operation (II):\n\\begin{itemize}\n\\item[(II)a] if the new edge $l$ is a disconnecting edge;\n\\item[(II)b] otherwise, i.e. if there is  a cycle $\\mathcal C$ that contains $v$ such that \n$\\mathcal C\\cup l$ is a cycle for the new graph.\n\\end{itemize}\n\\end{defi}\n\n\\begin{prop}\\label{increasing}\nLet $R$ be a discrete valuation ring with fraction field $K$ and residue field { $\\mbox{\\itshape {\\textbf k}}$}, \nand $f\\colon X \\longrightarrow \\,\\mbox{spec}\\, R$ a family of nodal curves.\nLet $\\Gamma_K$, resp. $\\Gamma_{\\small\\mbox{\\itshape {\\textbf k}}}$, be the weighted graph associated to the generic fiber $X_K$, \nresp. to the closed fiber $X_{\\small\\mbox{\\itshape {\\textbf k}}}$. \n\nThen, the complexity function $c\\colon \\,\\mbox{spec}\\, R \\longrightarrow \\mathbb Z_{\\geq 0}$ is constant if and only if \n$\\Gamma_{\\small\\mbox{\\itshape {\\textbf k}}}$ is obtained from $\\Gamma_K$\nvia a sequence of operations of type (I) and (II)a.\nOtherwise, $c(\\Gamma_{\\small\\mbox{\\itshape {\\textbf k}}})>c(\\Gamma_{K})$.\n\n\\end{prop}\n\\begin{proof}\nClearly, the weights on the graphs do not interfere with the complexity, as well as adding loops.\n\nWe now prove that, applying operation (II) the complexity remains the same only in case (II)a, and it increases strictly in case (II)b.\nCall $\\Gamma'$ the graph obtained by applying operation (II) to a vertex $v$ of degree $d$, and, accordingly to the notation of Proposition \\ref{degenerations}, call \n$d_a$ and $d_b$ the vertex degrees of $v_a$ and $v_b$, respectively,  in $\\Gamma'$. \nClearly \n$$\nd=d_a+d_b -2,\n$$\nSince we are adding a vertex and a component, the first Betti number of the graph, $b_1$, remains the same. \nHowever, the complexity can increase, but not decrease. \nIndeed, as observed above,  we have that $\\Gamma=\\Gamma'\\cdot l$. \nSo, by formula (\\ref{fond}),\n$$\nc(\\Gamma')=c(\\Gamma)+c(\\Gamma'-l)\\geq c(\\Gamma).\n$$\nMoreover, we see from the above formula that the complexity remains the same if and only if $c(\\Gamma'-l)=0$, \ni.e., if and only if  $l$ \nis a disconnecting edge for $\\Gamma'$. \n\\end{proof}\n\n\n\n\n\\subsubsection*{The switching of two edges}\nAs we will often make use of another operation on graphs, it is convenient to describe it separately.\nLet us consider a graph $\\Gamma$, and fix two distinct edges $l$ and $m$, with associated vertices $v$, $v'$ and $w$, $w'$ respectively. Let us suppose that $v\\not = w$ (but we do not ask that $v\\not = v'$ or $w\\not = w'$, i.e. $l$ and $m$ can be loops). \nWe shall construct a new graph  $\\Gamma'$ which has the same vertices as $\\Gamma$, \nsuch that $\\Gamma\\setminus\\{l,m\\}$ is a subgraph of $\\Gamma'$, and the edges $l$, $m$, are substituted by \n$l'$ and $m'$ as illustrated in figure below.\n\n\n\n\\bigskip\n\n\\centerline{\n\\begin{tabular}{rcl}\n\\xymatrix@R=2pc{\n*{\\bullet}  &   &*{\\bullet} \\\\\n& &\\\\\n *{\\bullet}  \\ar @{-}[uu]^{l}^>{v}^<{v'}   &   & *{\\bullet} \\ar @{-}[uu]_{m}_>{w}_<{w'}\\\\}\n&\n\\xymatrix@R=2pc{\n\\\\\n=\\Rightarrow\\\\\n\\\\ \n\\\\}\n&\n\\xymatrix@R=2pc{\n*{\\bullet} \\ar @{-}[ddrr]^{l^\\prime } &   & *{\\bullet} \\\\\n& &\\\\\n *{\\bullet}  \\ar @{.}[uu]^{l}^>{v}^<{v'}   \\ar @{-}[uurr]^{m^\\prime} &   & *{\\bullet} \\ar @{.}[uu]_{m}_>{w}_<{w'}\\\\}\n\\end{tabular}}\n\n\\bigskip\n\n\n\\noindent So, the new edges $l'$ and $m'$ connect $v$ and $w'$, $v'$ and $w$ respectively. \nWe will call this process {\\em the switching of  $l$ and $m$ with respect to $v$ and $w$\\footnote{Note that this operation\n is the $\\widetilde X$-transformation described in \\cite{tsukui}.}.}\nNote that the vertex degrees of $\\Gamma$ and $\\Gamma'$ are the same.\n\nIn general, this operation doesn't increase the complexity; counterexamples are easy to construct.\nHowever, we shall prove that, for certain graph configurations, it does.\n\n\n\n\\begin{prop}\\label{1}\nLet $C$ be as in Theorem \\ref{maxconf}. \nThen $C$ has no disconnecting nodes.\n\\end{prop}\n\\begin{proof}\nSuppose that $C$ has a disconnecting node, which corresponds in $\\Gamma=\\Gamma_C$ to a disconnecting edge $r$.\nConsider $l$, $m$ two  edges adjacent to $r$. \nThe only case in which we cannot find two different edges, is if one of the vertex joined by  $r$, say $v$, has degree one.\nIn this case, as the curve $C$ is stable, necessarily $ g(v)\\geq 1$; we therefore modify the graph by adding a loop on $v$, and \ndecreasing by one the weight on $v$.\nWe still have a graph associated to a stable curve of genus $g$ with the same complexity, which we will call again $\\Gamma$, and we can choose $l$ and $m$ as above.\nNow, we perform the switching of  $l$ and $m$ w.r.t. $v$ and $w$:\\footnote{this operation is the \n``slide transformation of $l$ and $m$ at $v$ and $w$ along $r$'' with the terminology of \\cite{tsukui}.}\n\n\\bigskip\n\n\\centerline{\n\\xymatrix@R=1pc{\n\\ar @{}[r]|->{\\ldots} &*{\\bullet}  \\ar @{-}[dr]^>{v} &  & &  & *{\\bullet} \\ar@{-}[dl]_>{w}& \\ar @{}[l]|->{\\ldots} &\\ar @{}[r]|->{\\ldots} &*{\\bullet}  \\ar @{-}[dr]^>{v} &  & &  & *{\\bullet} \\ar@{-}[dl]_>{w}& \\ar @{}[l]|->{\\ldots} \\\\\n&& *{\\bullet} \\ar@{-}[rr]^{r} & & *{\\bullet} & &\\ar@{:>}[r] &&& *{\\bullet} \\ar@{-}[rr]^{r} & & *{\\bullet} & \\\\\n*{} \\ar @{}[r]_>{}|->{\\ldots}& *{\\bullet}  \\ar @{-}[ur]|-{l} &  & &  & *{\\bullet} \\ar @{-}[ul]|-{m}  & \\ar @{}[l]^>{}|->{\\ldots}&\n*{} \\ar @{}[r]_>{}|->{\\ldots}& *{\\bullet}  \\ar @{.}[ur]|-{l} \\ar @{-}[urrr]|-{l^\\prime} &  & &  & *{\\bullet} \\ar @{.}[ul]|-{m} \\ar @{-}[ulll]|-{m^\\prime} & \\ar @{}[l]^>{}|->{\\ldots}\\\\}\n}\n\n\\bigskip\n\n\n\nNote that $\\Gamma\\cdot r=\\Gamma' \\cdot r$.\nNow, $\\Gamma'$ is the dual graph of a stable curve (indeed, we can think of the modification made as if we have \nresolved two nodes of the curve, and attached the components where they belong in a different way).\n\nNow, if the edge $r$ is still a disconnecting edge for $\\Gamma'$, then both $l$ and $m$ should be also disconnecting edges in $\\Gamma$. In this case, we must have $g(v)\\geq 1$ and we perform operation (I) in $v$. Then, if we choose $l$ to be the new loop, after the switching of $l$ and $m$ with respect to $v$, the edge $r$ will not be diconnecting anymore.\n\nSo, we can suppose that $r$ is no longer a disconnecting edge for $\\Gamma'$, and it is immediate to check that we have \nintroduced no new disconnecting edges, i.e. if $f$ is a disconnecting edge for $\\Gamma'$, then it is also a disconnecting edge for $\\Gamma$.\n\nNow, applying Proposition \\ref{comb} and observing  that $c(\\Gamma-r)=0$, while $c(\\Gamma'-r)>0$, we get:\n$$\nc(\\Gamma')=c(\\Gamma'\\cdot r)+c(\\Gamma'-r)=c(\\Gamma\\cdot r)+c(\\Gamma'-r)>c(\\Gamma\\cdot r)=c(\\Gamma).\n$$\nThis proves the statement.\n\\end{proof}\n\n\\begin{rem}\n\\upshape{In \\cite{DJ-IR}, sec.3, it is observed that a general graph (not necessarily regular) has to be\nfree of disconnecting edges in order to have maximal complexity. \nProposition \\ref{1}, as stated, could be indeed proven in the same way.\nHowever, the operation used by the two authors is the switching \nof {\\em one} edge, which modifies the vertex degrees; as we will see, it is necessary for us to remain in the class \nof cubic graphs, that's why we apply our switching.}\n\\end{rem}\n\n\n\\begin{prop}\\label{2}\nLet $C$ be as in Theorem \\ref{maxconf}. \nThen $C$ is a loopless graph curve.\n\\end{prop}\n\\begin{proof}\nFrom Proposition \\ref{1} we can suppose that $C$ has no disconnecting nodes.\nLet $\\Gamma$ be the weighted graph associated to $C$ and suppose $\\Gamma$ is not a loopless trivalent graph.\nUsing the degeneration operations described in Definition \\ref{degenerations},\nwe will describe an algorithm in two steps that, given $\\Gamma$ with no disconnecting nodes,\nproduces a loopless graph $\\Gamma_3$ \nwith strictly bigger complexity than $\\Gamma$, such that $b_1(\\Gamma_3)=g$ and each vertex has degree $3$ and weight  $0$. \nTherefore $\\Gamma_3$ is the dual graph associated to a loopless graph curve of genus $g$.\n\n\\medskip\n\n\\noindent FIRST STEP (Reduction to a curve having only rational components): \nWe replace any vertex $v_i$ with strictly positive weight $g(v_i)$ with a bouquet given by a vertex of weight $0$ and $g(v_{i})$ \nloops attached to it. \nThis is a reiterate application of operation (I) and does not change the arithmetic genus, nor the complexity. \nCall $\\Gamma_1$ the graph obtained in this way.\n\n\\medskip\n\n\\noindent SECOND STEP (Reduction to a loopless graph curve):\nsuppose that  in $\\Gamma_1$ there is a vertex $v$ with degree $d$  greater or equal to $4$. \nAs $\\Gamma_1$ is obtained from $\\Gamma$ by adding loops, it has no disconnecting edges. \nWe can therefore perform operation (II)b on $v$, with the request that $\\deg(v_a)\\geq 3$, and $\\deg(v_b)\\geq 3$\n(i.e. remaining in the class of graphs of stable curves).\nAs proved in Proposition \\ref{increasing}, this operation increases strictly the complexity.\nMoreover, the degrees of $v_a$ and $v_b$ are strictly smaller than the degree of $v$.\nWe perform this transformation until all the vertices of the new graph have exactly degree 3. \nCall $\\Gamma_2$ the resulting graph. \nSo by the genus formula $\\Gamma_2$ has $2g-2$ vertices of weight 0, $3g-3$ edges, and $b_1(\\Gamma_2)=g$; \nequivalently, $\\Gamma_2$ is the graph associated to a graph curve.\nMoreover, $\\Gamma_2$ has no disconnecting edges,  and no loops. \nIndeed, if there were a loop, $\\Gamma_2$  would contain a subgraph  of the form:\n\n\\bigskip\n\n\\begin{equation*}\n\\xymatrix@=1.2pc{\n&*{\\bullet} \\ar@{-}@(ur,ul) \\ar@{-}[d]\\\\\n&*{\\bullet} & \\\\\n\\ar@{-}[ur] && \\ar@{-}[ul] & .\n}\n\\end{equation*}\nhence with a disconnecting node.\nMoreover, by what observed above, $c(\\Gamma_2)> c(\\Gamma_1)$.\n\\end{proof}\n\n\n\n\\begin{ex}\n\\upshape{\nLet us consider the dual  graph of a smooth genus $3$ curve; this is just one vertex with weight $3$.\nLet us apply the above algorithm to this graph.\nWith the first step, we obtain  the following `` bouquet\" of $3$ loops,\n\n\\medskip\n\n\\begin{equation*}\n\\xymatrix@R=1pc{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar@{-}@(ur,dr) \\ar@{-}@(ur,ul)\n}\n\\end{equation*}\n\n\\medskip\n\nwhich is the graph of a stable curve with a single irreducible rational component with $3$ nodes. \nAs the irreducible component  has geometric genus $0$, we apply the second step. \nWe have several choices. \nSo, we can obtain either one of the following configurations.\n\n\\begin{equation*}\n\\begin{tabular}{cccccccccc}\n\\xymatrix{\n*{\\bullet}  \\ar @{-} @\/_.3pc\/[r] \\ar @{-}@\/_.8pc\/[r]\/& *{\\bullet} \\ar@{-}@\/_.3pc\/[l] \\ar@{-}@\/_.8pc\/[l]\n}\n&{}&{}&\n\\xymatrix{\n*{\\bullet} \\ar@{-}@(ul,dl) \\ar @{-} @\/_\/[r] \\ar@{-}[r] & *{\\bullet} \\ar@{-}@\/_\/[l]\n}\n&{}&{}&\n\\xymatrix{\n*{\\bullet} \\ar @{-}@(ul,dl) \\ar @{-}@\/_\/[r] & *{\\bullet} \\ar @{-}@\/_\/[l] \\ar @{-}@(ur,dr)\n}\n\\end{tabular}\n\\end{equation*}\n}\n\n\nNotice that all these graphs have still vertices with degree greater then 3. So, we therefore must go on applying step 2. \nFor example, one of the possibilities for the first graph would be the following.\n\n\\begin{equation*}\n\\begin{tabular}{lcr}\n\\xymatrix{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}@\/_\/[dl] \\ar@{-}@\/^\/[dr] \\ar@{-}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] & & *{\\bullet} & \\ar@{:>}[r] &\n}\n&\n&\\xymatrix@=1.2pc{\n&*{\\bullet}\\ar@{-}[dd] \\ar@{-}[dl] \\ar@{-}[dr] &\\\\\n*{\\bullet} \\ar@{-}[rr] |!{[r]}\\hole & &*{\\bullet}\\\\\n&*{\\bullet}\\ar@{-}[ru]\\ar@{-}[lu] & \n}\n\\end{tabular}\n\\end{equation*}\n\nIf we start from the third graph, we can perform the following chain of transformations\n(indeed, in this case, these operations are forced).\n\n\\begin{equation*}\n\\begin{tabular}{lcccr}\n\\xymatrix@R=1pc{\n{}\\\\\n *{\\bullet} \\ar @{-}@(ul,dl) \\ar @{-}@\/_\/[r] & *{\\bullet} \\ar @{-}@\/_\/[l] \\ar @{-}@(ur,dr)&\\ar@{:>}[r]&\n}\n&&\n\\xymatrix@=1pc{\n*{\\bullet} \\ar@{-}[dd]\\ar@{-}@\/_.5pc\/[dd]\\ar@{-}[drr]&&&&\\\\\n&&*{\\bullet} \\ar@{-}@(ur,dr)&&\\ar@{:>}[r]&\\\\\n*{\\bullet} \\ar@{-}[urr]&& \n}\n&\n\\xymatrix{\n*{\\bullet} \\ar@{-}[r] \\ar@{-}[d] \\ar@{-}@\/_\/[d] & *{\\bullet} \\ar@{-}[d] \\ar@{-}@\/^\/[d]\\\\\n*{\\bullet} \\ar@{-}[r] & *{\\bullet}\n}\n\\end{tabular}\n\\end{equation*}\n\\end{ex}\n\nAs also the simple example above shows, the algorithm of Proposition \\ref{2} doesn't give a unique output.\nOn the contrary, if we start from a bouquet of $g$ loops we can obtain \\emph{all} the possible cubic graphs without disconnecting edges  using step 2,  as one can see in the following way. \nConsider a cubic graph without disconnecting edges  $\\Sigma$ with $b_1(\\Sigma)=g$. \nLet $T$ be a  spanning tree for $\\Sigma$. To contract all the edges of $T$ is the reverse operation to step 2 of the algorithm, and the result is the bouquet of $g$ loops.\n\n\n\n\\begin{proof} ({\\it of Theorem \\ref{maxconf}})\nBy Proposition \\ref{1} and \\ref{2} we can suppose that $\\Gamma=\\Gamma_C$  is trivalent,\nloopless and without disconnecting edges.\nHence, the only possible multiple edges are of the form\n\n\\medskip\n\n\\centerline{\n\\xymatrix{ &&&&\\ar @{-}[dl]\\\\\n*{\\bullet}  \\ar @{-}[r]_s^>{v_2}^<{v_1}&*{\\bullet}  \\ar @{-}@\/_\/[r]_l& *{\\bullet} \\ar @{-}@\/_\/[l]_f & *{\\bullet}  \\ar @{-}[l]^r_>{v_3}_<{v_4}&\\\\\n&&&& *{\\bullet}\\ar @{-}[ul]^m^<{v_5}\\\\\n}\n}\n\n\\bigskip\n\nNote that $l$ and $m$ have to be distinct as well as $v_1$ and $v_4$ (otherwise $\\Gamma$ would have disconnecting edges).\nWith the notations adopted in the figure, we apply the switching of  $l$ and $m$ w.r.t. $v_3$ and $v_4$:\n\n\n\n\\centerline{\n\\xymatrix{ &&&&\\ar @{-}[dl]\\\\\n& & *{\\bullet} \\ar @{-}@\/_\/[dl]_{f} & *{\\bullet}  \\ar @{-}[l]_r_>{v_3}_<{v_4}&\\\\\n*{\\bullet}  \\ar @{-}[r]_s^<{v_1}^>{v_2}&*{\\bullet}  \\ar @{.}@\/_\/[ur]_{l^\\prime}\\ar @{-}[urr]&&&*{\\bullet} \\ar @{.}[ul]\\ar @{-}[ull]^{m^\\prime}^<{v_5}\\\\\n}\n}\n\n\\bigskip\n\n\\bigskip\n\nWe have constructed a new trivalent graph $\\Gamma'$ which has one less couple of multiple edges: indeed, \nit is immediate to check that we have not created any other multiple edges.\n\n\nLet us prove that $c(\\Gamma ')>c(\\Gamma)$.\nWe prove it by giving an injective map from the spanning trees of $\\Gamma$ to the spanning trees of $\\Gamma'$, as follows.\n\nLet $T$ be a spanning tree of $\\Gamma$. If $r\\in T$, the switching of $\\Gamma$ we performed transforms $T$ into a spanning tree $T^\\prime$ of $\\Gamma^\\prime$, with $r\\in \\Gamma^\\prime$.\n\nNow, suppose $r \\not\\in T$. We shall distinguish between $4$ different situations.\n\\begin{itemize}\n\\item $l,m\\not\\in T$ ($\\Rightarrow f\\in T$). $T$ is transformed into a tree $T^\\prime$ of $\\Gamma^\\prime$ with $l^\\prime\\not\\in T^\\prime,m^\\prime\\not\\in T^\\prime,r\\not\\in T^\\prime$ and $f\\in T^\\prime$. \n\\item $l\\in T$, $m\\not\\in T$ ($\\Rightarrow f\\not\\in T$). Then, in $T$, the path connecting $v_2$ and $v_4$ passes by $l$. So, the switching of $l$ and $m$ along $r$ restricted to $T$ is not a spanning tree of $\\Gamma^\\prime$ since $l^\\prime$ would create a cycle containing $l^\\prime$ and $v_3$ and $v_5$ get disconnected. So, we consider the subgraph of $\\Gamma^\\prime$ obtained by the union of $m^\\prime$ with the switching of $T$ minus $l^\\prime$. It is easy to see that this is a spanning tree $T^\\prime$ of $\\Gamma^\\prime$, with $l^\\prime\\not\\in T^\\prime$, $m^\\prime\\in T$, $r\\not\\in T^\\prime$ and $f\\not\\in T^\\prime$.\n\\item $l\\not\\in T$, $m\\in T$ ($\\Rightarrow f\\in T$). In this case we must do a further distinction. \n\\begin{itemize}\n\\item In $T$ the path connecting $v_2$ and $v_4$ does not contain $m$. \n\nIn this case $T$ is transformed into a spanning tree $T^\\prime$ of $\\Gamma^\\prime$ with $l^\\prime\\not\\in T^\\prime$, $m^\\prime\\in T^\\prime$, $r\\not\\in T^\\prime$ and $f\\in T^\\prime$.\n\\item In $T$ the path connecting $v_2$ and $v_4$ contains $m$.\n\nIn this case the switching we performed restricted to $T$ is not a spanning tree of $\\Gamma^\\prime$, since $m^\\prime$ creates a cycle and $v_4$ and $v_2$ get disconnected. So, we consider $T^\\prime$ as the union of $l^\\prime$ with the switching of $\\Gamma$ restricted to $T$ minus $m^\\prime$. Then $T^\\prime$ is a spanning tree of $\\Gamma^\\prime$ with $l^\\prime\\in T^\\prime$, $m^\\prime\\not\\in T^\\prime$, $r\\not\\in T^\\prime$ and $f\\in T^\\prime$.\n\\end{itemize}\n\\item $l,m\\in T$ ($\\Rightarrow f\\not\\in T$). Also in this case we have to distinguish between the following situations:\n\\begin{itemize}\n\\item In $T$ the path connecting $v_2$ and $v_4$ does not contain $m$.\n\nThen the switching of $l$ and $m$ along $r$ restricted to $T$ is not a spanning tree of $\\Gamma^\\prime$ since it creates a cycle containing $l^\\prime$ and $v_4$ and $v_5$ get disconnected. So if we consider $T^\\prime$ as the union of $f$ with the switching of $T$ minus $s$, then $T^\\prime$ is a spanning tree of $\\Gamma^\\prime$ with $l^\\prime\\in T^\\prime$, $m^\\prime\\in T^\\prime$, $r\\not\\in T^\\prime$ and $f\\in T^\\prime$.\n\\item In $T$ the path connecting $v_2$ and $v_4$ contains $m$.\n\nIn this case the switching we performed restricted to $T$ is a spanning tree $T^\\prime$ of $\\Gamma^\\prime$ with $l^\\prime\\in T^\\prime$, $m^\\prime\\in T^\\prime$, $r\\not\\in T^\\prime$ and $f\\not\\in T^\\prime$.\n\\end{itemize}\n\\end{itemize}\n\nSo, the association of each spanning tree $T$ of $\\Gamma$ with $T^\\prime$ gives an injective map from the spanning trees of $\\Gamma$ into the spanning trees of $\\Gamma^\\prime$.\n\nNow, to prove that, indeed strict inequality holds, we observe that $\\Gamma^\\prime$ has at least one spanning tree $T^\\prime$ such that $l'$, $f$ and $r$ are not in $T^\\prime$ ($\\Rightarrow m^\\prime\\in T^\\prime$), and this kind of spanning trees are not in the image of the map constructed above. \n\\end{proof}\n\n\n\n\n\n\n\\subsection{Combinatorial version}\\label{combi}\nWe can give a purely combinatorial translation of the above results, as follows.\nRecall that a graph $\\Gamma$ is said to be {$k$-connected} if  for any  set $S$ of $k-1$ edges of \n $\\Gamma$, $\\Gamma\\setminus S$ is still connected. \n $\\Gamma$ is said to be {\\it strictly} $k$-connected if it is $k$-connected but not $k+1$-connected. \nSo, a graph is $1$-connected  if and only if it is connected; it is strictly $1$-connected if and only if it is connected and \n it has at least one disconnecting node.\n Of course, a trivalent graph can never be $4$-connected (any triple of edges incident in one vertex is a disconnecting set).\n \n Given a positive integer $g$, let $\\mathcal G_g$ be the set of all weighted multigraphs of genus $g$ satisfying \n condition (\\ref{stability}). Theorem \\ref{maxconf} can now be written in the following way.\n \n \\begin{teo}\\label{combinmaxconf}\n The graphs $\\Gamma\\in \\mathcal G_g$ reaching the maximal complexity\n are trivalent, simple and $2$-connected.\n \\end{teo}\n \n \nWe shall call a simple trivalent  graph a {\\em cubic graph}.  \n\n\n\n\n\n\n\n\n\n\n\n\\section{Bounds on the maximal complexity of stable curves}\\label{ub}\n\nThe question of an upper bound on the complexity depending on the genus has several geometrical meanings: for instance, it implies that there is a bound for the irreducible components of the compactified Jacobian of a stable curve, and a bound for the group of components of the N\\'eron model of the relative Jacobian of a family of curves having $C$ as special fiber; see also Remark \\ref{confrontoL} .\nLet us define \n$$\\psi(g):=\\mbox{max}\\{c(C), C \\mbox{ stable curve of genus } g\\}.$$\n\n\\begin{rem}\\upshape{\nNotice that $\\psi$ is strictly increasing with respect to the genus $g$. \nIn fact, if $C$ is a curve of genus $g$ such that $\\psi(g)=|\\Delta_C|$, let $C^\\prime$ be the stable curve obtained from $C$ by adding an extra node connecting two of the components of $C$. Then, using the above formula for the genus of $C^\\prime$, one gets $g_{C^\\prime}=g+1$ and, clearly, $c(C^\\prime)>c(C)$.}\n\\end{rem}\n\n\n\nAccording to the results of the preceding section, we know that curves achieving maximal complexity are in particular graph curves. \nUsing this fact, we can find a first rough bound:\n$$\\psi(g)\\le \\binom{3g-3}{g}\\le 2^{3g-4}.$$  \nFor the proof, see \\cite{lorjac}, Lemma 2.7.\nNote that  the first inequality follows immediately from the fact that any spanning tree in a cubic\ngraph of order $2g-2$ is obtained by removing $g$ edges. \n\n\nApplying the results on the surjectivity of the ``Abel-Jacobi map'' in \\cite{B-N},\nwe could derive a better bound: $$\\psi(g)\\le \\binom{2g-2}{g}.$$\n\nBoth these bounds are not sharp, even for low genus.\nUsing a result of Biggs \\cite{biggs}, we can prove the following.\n\\begin{equation}\\label{disbiggs}\nc(\\Gamma)\\leq  \\frac{1}{2g-2}\\left(\\frac{6g-6}{2g-3}\\right)^{2g-3}=:\\alpha(g).\n\\end{equation}\n\nFor $g=3$, this bound is optimal.\nHowever, the bound is not asymptotically sharp.\n\nThe complexity of $k$-regular graphs has been studied also  by McKay \\cite{McK}, and Chung-Yau \\cite{C-Y}.\nWe can apply their results obtaining a sharper bound. \n\\begin{teo}\\label{bound}\nLet $C$ be a stable curve of genus g, and let $\\Gamma$ be its dual graph. \nThen \n\\begin{equation}\\label{disCY}\nc(\\Gamma)\\leq  \\frac{2 \\ln{(2g-2)}}{3(2g-2) \\ln{(9\/8)}}\n\\exp{\\left(\\frac{12}{\\sqrt\\pi}\\left(\\frac{\\ln{9\/8}}{\\ln{(2g-2)}}\\right)^{5\/2}\\right)}\n\\left(\\frac{4}{\\sqrt{3}}\\right)^{2g-2}=:\\beta(g).\n\\end{equation}\nMoreover, this bound is asymptotically sharp for $g\\gg 0$.\n\\end{teo}\n\\begin{proof}\nBy making explicit the computations in Theorem 4 of \\cite{C-Y}, we obtain the bound (\\ref{disCY})\nfor $3$-regular graphs with $2g-2$ vertices. The thesis is now straightforward because of the reduction to\ngraph curves of Theorem \\ref{maxconf}.\nMcKay proves in \\cite{McK} that there is a sequence of cubic graphs \n$\\{\\Gamma_i\\}_{i\\in\\mathbb N}$ with increasing orders $n_i$'s, such that, if we set \n\\begin{equation}\\label{tau}\n\\tau(\\Gamma_i):=\\left(n_ic(\\Gamma_i)\\right)^{\\frac{1}{n_i}},\n\\end{equation}\nthen $\\tau(\\Gamma_i)\\longrightarrow \\frac{4}{\\sqrt 3}$ for $i\\rightarrow \\infty$.\nIndeed, he proves much more than the existence, but that a ``random'' sequence of cubic\ngraphs satisfies this property, see also \\cite{lyons}.\nHence, the constant $ \\frac{4}{\\sqrt 3}$ in the bound (\\ref{disCY}) is the best possible.\n\\end{proof}\n\n\n\nIn the next picture we describe the graphs with maximal complexity for $g=4$, $g=5$ and $g=6$, respectively.\n\n\\bigskip\n\n\\centerline{\n\\begin{tabular}{ccc}\n\\xymatrix@C=1.7pc@R=3.3pc{\n& *{\\bullet} \\ar@{-}[rr] \\ar@{-}[ddrr] & & *{\\bullet} \\ar@{-}[ddll]|!{[ll];[dd]}\\hole &\\\\\n*{\\bullet} \\ar@{-}[rrrr]|!{[rr]}\\hole \\ar@{-}[ur] \\ar@{-}[dr]  & & & & *{\\bullet} \\ar@{-}[ul] \\ar@{-}[dl] \\\\\n& *{\\bullet} \\ar@{-}[rr] & & *{\\bullet}\n}\n&\n\\xymatrix@=.8pc{\n& & & *{\\bullet} \\ar@{-}[dll] \\ar@{-}[drr] \\ar@{-}[dddddd]|!{[ddd]}\\hole & & & \\\\\n& *{\\bullet} \\ar@{-}[ddl] \\ar@{-}[ddddrrrr] & & & & *{\\bullet} \\ar@{-}[ddr]\n\\ar@{-}[ddddllll]|!{[llll];[dddd]}\\hole & \\\\\n&&&&&&\\\\\n*{\\bullet} \\ar@{-}[rrrrrr]|!{[rrr]}\\hole & & &&& & *{\\bullet} \\\\\n&&&&&&\\\\\n& *{\\bullet} \\ar@{-}[uul] & & & & *{\\bullet} \\ar@{-}[uur] & \\\\\n& & &  *{\\bullet} \\ar@{-}[ull] \\ar@{-}[urr]\n}\n&\n\\xymatrix@C=.9pc@R=.8pc{\n&  *{\\bullet} \\ar@{-}[rrrr] \\ar@{-}[dddl] \\ar@{-}[ddddr] &&&& *{\\bullet} \\ar@{-}[dddr] \\ar@{-}[ddddl]|!{[ddd];[lll]}\\hole & \\\\\n&&&&&&\\\\\n&  & & *{\\bullet} \\ar@{-}[d] \\ar@{-}[dlll]|!{[d];[lll]}\\hole \\ar@{-}[drrr]&&&\\\\\n*{\\bullet} \\ar@{-}[dddr] &&  &*{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr]& && *{\\bullet} \\ar@{-}[dddl] \\\\\n&&*{\\bullet} \\ar@{-}[ddrrr]|!{[d];[rrr]}\\hole &&*{\\bullet} \\ar@{-}[ddlll]&&\\\\\n&&&&&&\\\\\n& *{\\bullet} \\ar@{-}[rrrr] &&&&*{\\bullet}\n}\\\\\n$\\Gamma_1 , g=4$\n&\n$\\Gamma_2 , g=5$\n&\n$\\Gamma_3 , g=6$\n\\end{tabular}\n}\n\n\\bigskip\n\nNote that $\\Gamma_3$ is the famous Peterssen graph. \nThese graphs are proved to be of maximal complexity among simple cubic graphs of their respective orders in \\cite{DJ-IR}, sec.5. \n\n\n\n\\begin{rem}\\label{confrontoL}\n\\upshape{As already noticed, the geometric meaning of this result is that it gives a bound on the group of connected components of the N\\'eron model of the degree-$d$ Picard variety for families of {\\em stable}\ncurves (\\cite{BLR}, Theorem 1, sec.9.6), and as well on the number of irreducible components of the fibres of the scheme\n$P_g^d$ constructed in \\cite{capneron} and of $\\overline{P_{d,g}}$ of \\cite{cap}. \nOf course this is also a bound on the cardinality of the group of components of the N\\'eron models of the Jacobians of such families.\nHowever, notice that this bound is different from the ones found  by Lorenzini in \\cite{lorner} (see also \\cite{BLR}, \nTheorem 9 sec.6.9). \nIndeed, given any strictly henselian discrete valuation ring $R$, with algebraically closed residue field $\\mbox{\\itshape {\\textbf k}}$ and field of fractions $K$,  any regular family of curves $f\\colon X\\longrightarrow \\mbox{spec }R$ such that the {\\em general} fiber $X_K$ is of genus $g$, and the Jacobian $J_K$  has {\\em potential good reduction}, Lorenzini finds a bound, depending only on \n$g$, for the group of components of the N\\'eron model of $f$.\nWe obtain instead a bound for the group of components of the N\\'eron model of \\emph{any} smooth family $f\\colon X\\rightarrow \\mbox{spec }R$ such that the \\emph{closed} fiber $X_{\\mbox{\\small \\itshape {\\textbf k}}}$ is a  stable curve.\nSo one could say that his approach is dynamic, in the sense that it depends on the generic fiber, while our approach is static, as it starts from a given curve.\nMoreover, the boundedness comes on the one hand from the assumption of potential good reduction on the general fiber, on the other from the assumption of stability of the special one.\nAs it is proven in \\cite{lorgraphs}, a  family has potential good reduction if and only if the closed fiber has a {\\em tree} as dual graph, \nhence complexity 1.\n}\n\\end{rem}\n\n\n\n\n\n\\subsection{Example and lower bound}\\label{lower}\n\n\n\n\n\n\n\n\n\n\nWe present here an example of a family of cubic graphs of increasing order and increasing complexity.\nIn particular, this gives explicit lower bounds on $\\psi(g)$ (even though not sharp, given the above result).\n\nLet $\\Gamma_m$ be the graph with $4m$ vertices of valency $3$ formed from  $m$ \npairwise disjoint graphs $G_i$ of the following form:\n\n\\centerline{\n\\begin{tabular}{c}\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ul] \\ar@{-}[ur]\n}\n\\end{tabular}\n}\n\n\\noindent by adding $m$ edges $l_1,\\ldots ,l_m$ to link them as a ring, as shown in the figure below for $m=6$. \nClearly, $\\Gamma_m$ is the dual graph of a graph curve of genus $g=2m+1$.\n\n\\centerline{\n\\xymatrix@=.2pc{\n&&&&&*{\\bullet} \\ar@{-}[drr] \\ar@{-}[ddl] \\ar@{-}[dddr] &&&&&&& *{\\bullet} \\ar@{-}[dll] \\ar@{-}[ddr] \\ar@{-}[dddl]&&& \\\\\n&&&&&&&*{\\bullet} \\ar@{-}[ddl] \\ar@{-}[rrr] &&&  *{\\bullet} \\ar@{-}[ddr] &&&&& \\\\\n&&&&*{\\bullet} \\ar@{-}[drr] \\ar@{-}[dddll] &&&&&&&&& *{\\bullet} \\ar@{-}[dll] \\ar@{-}[dddrr]\\\\\n&&&&&&*{\\bullet} &&&&& *{\\bullet} &&&& \\\\\n\\\\\n&&*{\\bullet} \\ar@{-}[ddll] \\ar@{-}[ddrr] &&&&&&&&&&&&& *{\\bullet} \\ar@{-}[ddrr]\\ar@{-}[ddll]& \\\\\n\\\\\n*{\\bullet} \\ar@{-}[rrrr] \\ar@{-}[ddrr] &&&& *{\\bullet} \\ar@{-}[ddll] &&&&&&&&& *{\\bullet}  \\ar@{-}[ddrr] \\ar@{-}[rrrr] &&&& *{\\bullet} \\ar@{-}[ddll]\\\\\n\\\\\n&&*{\\bullet} \\ar@{-}[dddrr] &&&&&&&&&&&&& *{\\bullet} \\ar@{-}[dddll] & \\\\\n\\\\\n&&&&&& *{\\bullet} \\ar@{-}[dll] \\ar@{-}[ddr] \\ar@{-}[dddl] &&&&& *{\\bullet} \\ar@{-}[drr] \\ar@{-}[ddl] \\ar@{-}[dddr]\\\\\n&&&& *{\\bullet} \\ar@{-}[ddr] &&&&&&&&& *{\\bullet} \\ar@{-}[ddl]\\\\\n&&&&&&&*{\\bullet} \\ar@{-}[dll] &&& *{\\bullet} \\ar@{-}[lll] \\ar@{-}[drr]\\\\\n&&&&& *{\\bullet} &&&&&&& *{\\bullet}\n}\n}\n\n\\vspace{.1cm}\n\n\n\\begin{prop}\n$\nc(\\Gamma_m)=2m8^m.\n$\n\\end{prop}\n\\noindent {\\bf Proof}: To prove this formula we will proceed by hands counting the number of spanning trees for $\\Gamma_m$.\nNotice that the complexity of the subgraphs $G_i$'s is $8$. \n\nChoosing one of the edges $l_i$, the number of spanning trees not containing it is $8^m$. \nIndeed, all the others $l_j$'s have to be included in any spanning tree, while for any $G_k$, we have to count the $8$ possibilities for the spanning trees. So, we have $m8^m$ of these.\n\nOn the other hand, if $T$ is a spanning tree that contains all the $l_i$, then there is one and only one $j$ such that $T\\cap G_j$ is disconnected, and it has to be of  one of the $8$ following forms:\n\n\\vspace{.2cm}\n\\begin{tabular}{cccccccc}\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dl] \\ar@{-}[dr]& \\\\\n*{\\bullet}  && *{\\bullet} \\\\\n& *{\\bullet}\n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet}& \\\\\n*{\\bullet} && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ul] \\ar@{-}[ur]\n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dl] & \\\\\n*{\\bullet} \\ar@{-}[rr] && *{\\bullet} \\\\\n& *{\\bullet} \n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet}  \\ar@{-}[dr]& \\\\\n*{\\bullet} \\ar@{-}[rr] && *{\\bullet} \\\\\n& *{\\bullet} \n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} & \\\\\n*{\\bullet}  \\ar@{-}[rr]&& *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ul] \n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} & \\\\\n*{\\bullet} \\ar@{-}[rr] && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ur]\n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet} \\ar@{-}[dl] & \\\\\n*{\\bullet}  && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ur]\n}\n&\n\\xymatrix@=1pc{\n& *{\\bullet}  \\ar@{-}[dr]& \\\\\n*{\\bullet}  && *{\\bullet} \\\\\n& *{\\bullet} \\ar@{-}[ul] \n}\n\\end{tabular}\n\n\\vspace{.2cm}\n\nFor $i\\not =j$, $T\\cap G_i$ is a spanning tree for $G_i$ as above.\nTherefore, we have $8m8^{m-1}=m8^m$ possible spanning trees of this form.\n\nSo, summing all up, we have $2m8^m$ spanning trees, as required.\\hfill $\\Box$\n\n\\bigskip\n\nNotice that $$c(\\Gamma_m) =(g-1)8^{\\frac{g-1}{2}}=(g-1)(2\\sqrt{2})^{g-1}.$$\nSo, for odd $g$, we have this lower bound on $\\psi$. \nFor even $g$, remembering that $\\psi$ is an increasing function of the genus, we have\n$\\psi(g)>\\psi(g-1)$. \nHence, $\\psi$ is bounded from below by $(g-2)(2\\sqrt{2})^{g-2}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\\section{Further results and conjectures}\\label{congetture}\n\n\\subsubsection*{3-connectedeness}\n\nNote that a trivalent graph with loops, or with double edges is strictly $2$-connected. \nWith Theorem \\ref{MAX}, we have therefore excluded the strictly $1$-connected case, \nand some of the cases of strict $2$-connectedness. \n>From this, and from the observation of graphs with maximal complexity for low genus, \nit is natural to ask\n\n\\begin{conj}\\label{connectivity}\nA cubic graph with maximal complexity is $3$-connected.\n\\end{conj}\nThis seems to be generally believed, also for bigger classes of graphs (see e.g.\\cite{DJ-IR} sec.3), \nbut no proof is known.\nLet us note that $3$-connected graph curves have also interesting geometrical properties; \nindeed, they are precisely those graph curves for which the canonical bundle is very ample, yielding an embedding morphism \\cite{B-E}.\n\nWith similar techniques to the ones used in Section \\ref{MAX}, we can prove a partial result.\n\\begin{prop}\\label{bah}\nLet $C$ be as in Theorem \\ref{maxconf}. \nThen \n\\begin{enumerate}\n\\item $\\Gamma_C$ has no couple of disconnecting edges that lay on a cycle of length $\\leq 6$.\n\\item  $\\Gamma_C$ has no couple of disconnecting edges such that at least one of them is adjacent to a cycle of length $\\leq 4$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{cor}\\label{g<10}\nConjecture \\ref{connectivity} holds for $g\\leq 8$.\n\\end{cor}\n\n\n>From a result on the so-called ``Abel-Jacobi map'' for graphs proved on \\cite{B-N} (Theorem 1.8),\nwe can derive the following \n\\begin{prop}[Baker-Norine]\nIf $\\Gamma$ is any $k$-connected graph of order $n$, \n$$\nc(\\Gamma)\\geq \\binom{n}{k-1}.\n$$\n\\end{prop}\nThis proves that the complexity grows with the connectivity.\nIn the case of cubic graphs, however, the bound obtained for $3$-connected graphs is just \nquadratic in the order.\n\n\n\\subsubsection*{Maximal girth}\nThe explicit sequences of cubic graphs with large complexity \nhave all {\\em large girth}. \nLet us explain the terminology.\nThe {\\em girth} of a graph is the length of the shortest cycle.\nA simple argument shows that in a cubic graph the girth cannot exceed $2\\ln n\/\\ln2$ ($n$\nbeing the order of the graph).\nHence, the girth of a family of cubic graphs can grow at most as the rate of the logarithm of \nthe number of vertices.\n\n\\begin{defi}\\label{largegirth}\nA sequence $\\{\\Gamma_i\\}_{i\\in \\mathbb N}$  of cubic graphs with increasing \norders $n_i$ is called {\\em sequence of large girth} if\n$$\n\\lim_{i\\to \\infty}\\frac{\\ln(n_i)}{\\ln2 \\,\\, girth(\\Gamma_i)} \\,\\,\\,\\mbox{ is finite. }\n$$\n\\end{defi}\nMcKay proves in \\cite{McK} that sequences with large girth satisfy  condition (\\ref{tau}).\n\n\\begin{conj}\\label{girth}\nA sequence $\\{\\Gamma_i\\}_{i\\in \\mathbb N}$  of cubic graphs with maximal complexity has large girth. \n\\end{conj}\nSee \\cite{DJ-IR}, sec.3 for an heuristic argument supporting this conjecture.\n \n\n\nFrom a result of McKay we can derive the following property of sequences of trivalent graphs of maximal complexity.\nGiven a graph $\\Gamma$, and a positive integer $m$, let $C_{\\Gamma}(m)$ be the number of \ncycles  of length $\\leq m$  in $\\Gamma$.\n\n\\begin{teo}\\label{cicli}\nLet $m$ be a positive integer. \nFor any $\\epsilon>0$ there exist a positive integer $j$ such that: \ngiven any infinite sequence of trivalent graphs\n $\\{\\Gamma_i\\}_{i\\in \\mathbb N}$  such that \n$\\Gamma_i$ has $2i$ vertices, and reaches the maximal complexity between cubic graphs of order $2i$,\nthe following inequality holds\n$$\n\\frac{C_i(m)}{2i}\\leq \\epsilon \\,\\,\\mbox{ for any }i\\geq j,\n$$\nwhere $C_i(m)=C_{\\Gamma_i}(m)$.\n\\end{teo}\n\\begin{proof}\nTheorem 4.5 of \\cite{McK} states that, given a sequence of cubic graphs $\\{\\Gamma_i\\}_{i\\in \\mathbb N}$ \n satisfying (\\ref{tau}), then, for each fixed $m$,\n $$\n\\frac{C_i(m)}{n_i}\\rightarrow 0 \\,\\,\\mbox{ for } i\\to \\infty.\n$$\nThis does not directly imply the statement, which is uniform, in the sense that the constant $j$ depends only on $m$ and not \non the chosen sequence.\nHowever, let us argue as follows: given $m$, consider a sequence of cubic  graphs with maximal complexity {\\em and} with\nmaximal cardinality of $C(m)$.\nClearly this sequence satisfies condition (\\ref{tau}).\nNow, by assumption, the statement holds uniformly  on every sequence of graphs of maximal complexity.\n\\end{proof}\nNote that a sequence of graphs of large girth satisfies trivially the condition of Theorem \\ref{cicli}.\n\n\n\n\n\n\\subsubsection*{Further speculations}\n\nOther open questions are: \n\\begin{itemize}\n\\item Are cubic graphs of maximal complexity \n{\\em strongly regular}?\n\\item Do cubic graphs of maximal complexity have {\\em maximal automorphism group}? \n\\item Is  a cubic graph with maximal complexity and given genus {\\em unique}?\n\\item Let $C$ be a graph curve  with maximal complexity; suppose that it is indeed $3$-connected. \nThen the canonical map is an embedding realizing $C$ as a configurations of lines in $\\mathbb{P}^{g-1}$.\nWe conjecture that these lines are in ``general position'', in a sense that can be made precise.\nThis property seems to be connected with the large girth property.\n\\item   Let $\\Gamma$ be a cubic graph with maximal complexity; suppose that it is indeed $3$-connected. \nWe think that the ``Clifford index'' of this graph, as defined for instance by Bayer and Eisenbud, has to be maximal as well. \n\\item Let $\\Gamma$ be a cubic graph with maximal complexity; suppose that it is indeed $3$-connected. \nWe wonder if  any cutset of $3$ edges is made of adjacent edges.\n\\end{itemize}\n\n\n\\bibliographystyle{amsalpha}\n  \\nocite{*}\n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA majority of modern Silicon Photomultipliers (SiPMs) is designed as an array of independent Single-Photon Avalanche Diode (SPAD) cells with individual quenching resistors connected to a common electrode. An active photosensitive part of the cell is formed by planar pn-junctions. The inactive cell periphery is typically formed by guard rings or trenches to provide an independent operation of the cells in the Geiger breakdown processes. This design has been initially developed in Russia \\cite{A,B,1}. Since the mid-2000s, the planar SiPM design concept has been utilized with some modifications by Hamamatsu \\cite{2}, SensL \\cite{C}, STM \\cite{3}, FBK \\cite{4}, Excelitas \\cite{5}, and KETEK \\cite{6}. It was worldwide recognized as the SiPM, a new photon-number-resolving avalanche detector of outstanding performance \\cite{7}.\n\nNowadays, the planar SiPMs are approaching the physical limits of the design performance with its main inherent trade-off between photon detection efficiency (PDE) and dynamic range (DR) because of the dead space at the cell periphery \\cite{Vinogradov2011}. \nThis trade-off becomes challenging when developing a SiPM which should be sensitive for a wide spectral range, especially for red and near-infrared (NIR) wavelengths.\nThe development of a \"NIR-SiPM\" is anticipated to be a breakthrough in a number of applications \u2013 first and foremost in LIDARs \u2013 as well as in medicine and health sciences, biology and physics, environmental monitoring and quantum telecommunications. To achieve efficient absorption of photons and a high PDE for the red part of the light spectrum, a thicker photosensitive layer is required due to the low absorption coefficient of silicon at these wavelengths. But, the available increase of the thickness is eliminated by the so-called \"border effect\". \nThe thicker the active layer of a SiPM for efficient absorption of NIR photons, the higher the losses of a sensitive volume adjacent to trenches \\cite{9}. Further, the breakdown voltage increases proportionally to the depletion depth for planar technologies. \nThis leads to high operating voltages in the Geiger-Mode with high temperature coefficients. Simultaneously, the lateral distance of the active area to the micro-cell edge has to be increased in order to prevent edge breakdowns at high voltages. \nDespite these limitations, further progress in planar SiPM technology has been made with respect to the sensitivity at \\SI{905}{\\nano m}. For example, FBK started from \\SI{11}{\\%} for \\SI{35}{\\micro m} cells in 2017 \\cite{9} and Broadcom improved this result to \\SI{18}{\\%} (cell size is claimed as \"smallest\") in 2020 \\cite{Broadcom2020}.\n\\IEEEpubidadjcol\n\\\\\nIn contrast, there are also several photoelectric devices based on non-planar configurations.\nThe most relevant examples are Geiger-Mode APDs developed as predecessors or alternatives to the planar SiPMs. Spherical avalanche diodes with a radius of the pn-junction of \\SI{2}{\\micro m} and a breakdown voltage of \\SI{50}{V} seems to be the earliest device of this type operated in a photon counting mode \\cite{12}. Geiger-Mode APDs with negative feedback, also known as Metal-Resistor-Semiconductor (MRS) APDs, were designed as an array of avalanche micro-channels. A few micrometer-size n+ diffusion dots or \"needle\" junctions on p-Si wafer are covered by a thin resistive SiC layer as quenching resistor \\cite{G,13}. The first NIR-GM-APD with negative feedback has also been developed with a similar design \\cite{14}. Another kind of the microchannel GM-APD, the Micropixel APD (MAPD), is based on deep n-type micro-dots buried into a p-type epi-layer \\cite{H,15}. The MAPDs demonstrated a unique pixel-density of up to $\\SI{4e4}{mm^{-2}}$ with the highest geometric efficiency of almost \\SI{100}{\\%}. However, the MAPD design is also associated with a long pixel recovery time eliminating its dynamic range for continuous illumination and long light pulse detection.\n\nFor years, KETEK has been providing state-of-the-art SiPMs optimized for a blue range of the light spectrum, which is commonly used in medical and high energy physics applications.\nTo overcome the above mentioned limitations of the planar technology, we developed a new SiPM concept according to the patent application \\cite{Patent_TAPD}, the \"Tip Avalanche Photodiode\" (TAPD).\nThe concept is based on utilizing the properties of tip-like electrodes to focus and enhance the electric field, to reduce the breakdown voltage and cell capacitance and to eliminate the needs in a peripheral separation of the SiPM cells (avalanche regions). \n\nIn this article, we give an in-depth overview of the physical model of our new SiPM design and explain the working principle of the device. Further, we present the metrological characterization of our prototype samples and put them in relation to our simulations.\n\n\n\\section{Concept and Physical Model}\nThe simplified cross section of the TAPD is shown in Fig.~\\ref{fig:schematic_concept}.\nA spherical tip which consists of high doped n-silicon is placed in a low doped epitaxial layer.\nThe tip is connected to the bias supply through a quenching resistor $R_q$ placed on the surface.\nThe pn-junction on the surface of the tip causes a depletion and therefore an electric field around the sphere.\n\n\\subsection{Analytic Description}\n\nIn this article, the basic properties of the new SiPM concept are derived using a simplified model of an n-doped sphere inside an infinite p-doped bulk. The transition from n-doped to p-doped region is first approximated as an abrupt junction where the depleted charge has a box profile (in 3D spherical shells).\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\columnwidth]{Model_with_electricfield_plot.pdf}\n\\caption{Schematic of the spherical SiPM concept}\n\\label{fig:schematic_concept}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=7.5cm]{Model_with_electricfield_plot3.pdf}\n\\caption{Dimensions and depletion layers of the spherical pn-junction model.}\n\\label{fig:concept_model}      \n\\end{figure}\n\nIn thermal equilibrium, the charge carrier currents of drift and diffusion cancel out and the Fermi level $E_F$ is constant throughout the junction:\n\\begin{equation}\n\\begin{split}\nJ_{n} = &  q \\mu_n \\left( n \\mathcal{E} + \\frac{kT}{q} \\frac{dn}{dr} \\right) = 0  \\\\\nJ_p = & q \\mu_p \\left( p \\mathcal{E} + \\frac{kT}{q} \\frac{dp}{dr} \\right) = 0\n\\end{split}\n\\label{eq:zero_net_current}\n\\end{equation}\nThe depletion approximation considers complete impurity ionization in the n- and p-region. At thermal equilibrium (subscript '$o$') the free charge carrier concentration can be simplified to $n_{no} \\approx N_D$ in the n-region and $p_{po} \\approx N_A $ in the p-region. \nThe condition of Eq.(\\ref{eq:zero_net_current}) requires a constant Fermi level and therefore a built-in voltage is present in the pn-junction. The built-in potential $\\Psi_{bi}$ can be written as: \n\\begin{equation}\n\\begin{split}\n\\Psi_{bi}\t& =\t\\frac{kT}{q} \\ln \\left( \\frac{n_{no}}{n_i}  \\right) + \\frac{kT}{q} \\ln \\left( \\frac{p_{po}}{n_i}  \\right) \\\\\n\t\t\t& \\approx \\frac{k_B T}{q} \\ln \\left(\\frac{N_D N_A}{n_i^2}\\right)\n\\end{split}\n\\end{equation}\nThis expression of the built-in potential is valid for planar and spherical junctions.\nThe electric field and the potential of the space charge region can be obtained solving the Poisson equation.\nThe one dimensional Poisson equation in spherical coordinates for any arbitrary charge density $\\rho(r)$ is \\cite{JayantBaliga1976}:\n\\begin{equation}\n\t\\frac{1}{r^2} \\frac{d}{d r} \\left( r^2 \\frac{d \\Psi_i(r)}{d r} \\right)  = - \\frac{\\rho(r)}{\\varepsilon \\varepsilon_0}\n\\label{eq:poisson}\n\\end{equation}\n\n\nIn the first case of an abrupt junction, the charge carrier densities are given by the completely ionized acceptor and donor impurities, $N_A$ and $N_D$. \nThe ionized regions and the notation of the dimensional variables are shown in Fig.~\\ref{fig:concept_model}. \nThe analytic solution for the electric field in the n-region and p-region ($\\mathcal{E}_n$ and $\\mathcal{E}_p$) can be obtained from integration of Eq.(\\ref{eq:poisson}) with the boundary condition of zero electric field outside the depletion region:  \n\\begin{align}\n\\begin{split}\n\\mathcal{E}_n(r)& = \\frac{ e N_D}{3\\varepsilon \\varepsilon_0} \\frac{\\left(r^3 - r_D^3\\right)}{r^2},\\\\\n\\mathcal{E}_p(r) &= \\frac{ e N_D}{3\\varepsilon \\varepsilon_0} \\frac{\\left(r_j^3 - r_D^3\\right)}{r^2} - \\frac{e N_A}{3\\varepsilon \\varepsilon_0} \\frac{\\left(r^3 - r_j^3\\right)}{r^2}\n\\end{split}\n\\label{eq:Efield_ana}\n\\end{align}\nThe electric field in the n-region is zero at the inner depletion edge $r_D$ and will increase linear if the term $r_D^3 \/ r^2$ is small.\nIn the p-region, the expression of the electric field has two terms.\nThe first term is dominant if $N_D \\gg N_A$ and the electric field will decrease proportional to $1\/r^2$. \nThe second term becomes dominant if the first term is small enough with increasing radius and the electric field will equal zero at the outer depletion edge $r_A$.\n\nIn this concept, the donor concentration in the sphere is much higher than the acceptor concentration in the epitaxial layer.\nA exemplary electric field distribution according to Eq.(\\ref{eq:Efield_ana}) for different metallurgical junction radii is presented in Fig.~\\ref{fig:Efield_ana}. Here, the donor and acceptor concentrations were arbitrarily set to $N_A = \\SI{1e14}{\\per \\cm}$, $N_D = \\SI{1e18}{\\per \\cm}$.\nThe depletion width of each sphere size was adapted to create a depletion potential of \\SI{40}{\\V}.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{ElectricField_AnalyticSolution.pdf}\n\\caption{Electric field distribution in respect with the analytic solution for different sphere radii at a depletion potential of \\SI{40}{\\V}.}\n\\label{fig:Efield_ana}\n\\end{figure}\n\n\n\\subsection{Numerical Simulation}\n\nThe electric field in the previous section is derived using an approximation of the depletion regions as box profiles.\nIn a processed device, the dopant concentration will vary along the radius of the sphere.\nA dopant distribution is typically created by ion implantation and thermal annealing during different process steps. \nThe diffusion of impurities is thermally activated and occurs in direction of the concentration gradient.\nIn the case of the TAPD, especially the donor impurities of the tip will start to diffuse into the p-region.\nThree different structure sizes were processed and tested. The nominal junction radii of these devices are \\SI{0.6}{\\micro \\m}, \\SI{0.8}{\\micro \\m} and \\SI{1.0}{\\micro \\m}.\nThe measured dopant profile of these devices is used for the following simulations.\n\nThe complete impurity ionization (as assumed in Eq.(\\ref{eq:Efield_ana})) is prevented by diffusion of free charge carriers into the depleted regions \\cite{Sze2007}.\nWhile Eq.(\\ref{eq:zero_net_current}) deals with the steady-state, the continuity equations describe the net current flowing in and out of a region of interest. Here, the generation and recombination in the depletion region is neglected. The divergence operator has to be adapted for one dimensional spherical coordinates. The simplified continuity equations and the differential equation of the electric field from the system of coupled equation which has to be solved:\n\\begin{equation}\n\\frac{\\partial n}{\\partial t}\t= \\frac{1}{r^2} \\frac{\\partial}{\\partial r} r^2  \\left( \\mu_n \\mathcal{E}\tn + D_n \\frac{\\partial n}{\\partial r} \\right)\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial p}{\\partial t} = \\frac{1}{r^2} \\frac{\\partial }{\\partial r} r^2  \\left( - \\mu_p \\mathcal{E} p + D_p \\frac{\\partial p}{\\partial r} \\right)\n\\end{equation}\n\\begin{equation}\n\\label{eq:depl}\n\t\\frac{1}{r^2} \\frac{d}{d r} r^2 \\mathcal{E}(r)   = \\frac{q}{\\varepsilon_s} \\left( p(r) - n(r) + D(r) \\right)\n\\end{equation}\nwhere $\\mu_n$, $\\mu_p$ are the electron and hole mobility and $D_n$, $D_p$ are the electron and hole diffusivities.\nThe spatial dependent doping profile $D(r)$ defines the p and n-region as:\n\\begin{equation}\nD(r) = \n\\begin{cases}\nr \\leq r_j; &N_D(r) \\\\\nr \\geq r_j;  &-N_A(r)\n\\end{cases}\n\\end{equation}\n\nThe presented governing system of equations is considered in a spherical domain.\nThe boundary conditions in the center and on the spherical shell of the region of interest were of the Dirichlet type. \nWe used the finite difference method with a centred difference stencil for the spatial numerical approximation. \nThe approximation in time of the continuity equations was done using the Crank Nicolson method \\cite{crank_nicolson_1947}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Model_with_electricfield_plot2.pdf}\n\\caption{Schematic 2D electric field distribution due to the spherical depletion around the tip.}\n\\label{fig:2D-field}\n\\end{figure}\n\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\columnwidth]{ElectricField_Ion_at_Breakdown.pdf} \n\\caption{Spatial distribution of the electric field and the ionization rates of electrons and holes for the smallest structure S06.}\n\\label{fig:plot_Efield}      \n\\end{figure}\n\n\n\n\\begin{figure*}[bp]\n\\centering\n\\includegraphics[width=14cm]{TAPD_layout.pdf}\n\\caption{High density layout and bias supply of a TAPD array.}\n\\label{fig:layout}\n\\end{figure*}\n\nIn Fig.~\\ref{fig:2D-field}, a schematic 2D distribution of the normalized electric field around the tip is illustrated. The highest field strength is located close to tip surface and decreases with increasing distance to the junction.\nThe electric field is present up to the passivated surface and a spherical active volume is available for electron attraction in direction of the n-region.\nHowever, impact ionization multiplication due to a high electric field occurs just close to tip surface.\nThe solution of the numerical approximation at the breakdown voltage (here \\SI{43.4}{V}) is presented in Fig.~\\ref{fig:plot_Efield} for a nominal junction radius of \\SI{0.6}{\\micro \\m}.\nThe ionization rates $\\alpha_n$ and $\\alpha_p$ for electrons and holes depend on the electric field:\n\\begin{equation}\n\\alpha(\\mathcal{E}) = \\alpha_\\infty \\exp\\left(\\frac{-b}{\\mathcal{E}} \\right)\n\\end{equation}\nwith $\\alpha_\\infty$ and $b$ at room temperature taken from \\cite{VanOverstraeten1970} for electrons and holes, respectively.\nIn order to estimate the breakdown voltage of different sphere sizes, the multiplication of charge carriers due to impact ionization has to be evaluated using the spatial dependent ionization rates at different bias voltages.\nA breakdown occurs if the impact ionization multiplication becomes infinite which is equivalent to the condition that the ionization integral \\cite{McINTYRE1966} equals one:\n\\begin{equation}\n\\int_{r_D}^{r_A} \\alpha_n \\exp\\left[  - \\int_{r}^{r_A}  \\left( \\alpha_n - \\alpha_p  \\right) dr' \\right] dr = 1\n\\end{equation}\nThe results of this evaluation are presented in Tab.~\\ref{tab:structures}. The breakdown voltage increases with the size of the tip due to a lower electric field at the same bias voltage.\n\\begin{table}[h]\n\\caption{Tested device structures and nominal radii of the metallurgical junction.}\n\\label{tab:structures}      \n\\begin{tabular}{lll}\n\\hline\\noalign{\\smallskip}\nStructure Name &  Nominal Radius ($r_j$) & Breakdown Voltage \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\nS06 & \\SI{0.6}{\\micro \\m} & \\SI{43.4}{V} \\\\\nS08 & \\SI{0.8}{\\micro \\m} & \\SI{50.7}{V} \\\\\nS10 & \\SI{1.0}{\\micro \\m} & \\SI{53.9}{V} \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\nThe range of the electric field directly affects the photon detection capability regarding light with increasing wavelength.\nThe absorption of a photon inside of a material at a certain depth $x$ can be described by the Beer-Lambert law.\nThe probability for the generation of charge carriers at a certain depth inside the epitaxial layer decreases exponentially with increasing distance to the surface.\nAdditionally, in the range of visible light, the absorption coefficient decreases with increasing wavelength.\nA SiPM with a high charge collection efficiency aims to absorb as much of the incoming photon flux as possible. \nConsequently, the electric field has to be present as deep as possible.\nPhotoelectrons generated at a distance to the tip are first accelerated due the drift field in direction of the multiplication region (high electric field) and a Geiger discharge is triggered.\n\n\nThe simulated range of the electric field into the epitaxial layer for the three structure sizes at different bias voltages is shown in Fig.~\\ref{fig:depletion_width}.\nAll structures reach at least \\SI{8}{\\micro \\m} at their respective breakdown voltage.\nThe maximal active volume is reached placing the center of the sphere at a depth of the maximal depletion range. In this configuration, photoelectrons which are generated close to the surface and up to the maximal active depth are detected. This advantage of the new concept allows for a high photon detection efficiency in a wide spectral range. Theoretically, it is possible to reach a total active depth of \\SI{20}{\\micro m} with the current technology (see Fig.~\\ref{fig:depletion_width}) and consequently \na photon detection efficiency above \\SI{30}{\\%} at \\SI{905}{nm}.\nOur prototypes were processed in an epitaxial layer of \\SI{12}{\\micro m}, which limits the maximal depleted volume for all structure types. \n\n\n\n\\subsection{Cell Placement}\n\nThe operation of a SiPM above the breakdown voltage in Geiger-Mode requires a serial connected quenching resistor to keep the device in a quasi-stable state. \nThe potential of the epitaxial layer during operation is set to ground while the tip is biased through the resistor.\nThe spherical shape of the active volume allows a high density placement of multiple single cells in an array.\nIn the top view, the cells are placed in a hexagonal lattice achieving a theoretical packing density of $\\eta \\approx \\SI{90.7}{\\percent}$.\nThe bias voltage is supplied through a metal grid connected to the quenching resistor of each cell.\nThe schematic layout is shown in Fig.~\\ref{fig:layout}.\nThe aim of the layout is to cover the smallest possible area with metal. \nThe metal on the surface blocks visible light therefore reducing the active area of the SiPM for light detection.\nThe presented placement offers the advantage to use just one metal line for two rows of SiPM cells.\n\nThe relation of uncovered area to total area is called geometric efficiency (see also Sec.~\\ref{sec:PDE}).\nCompared to planar devices, the new concept offers the advantage of a frameless layout. The TAPD cells are biased through the center and can be placed as close as possible next to each other without losing active area.\n\n\nA typical parameter for SiPMs is the cell pitch which equals the cell size for planar devices. We defined the cell pitch for the TAPD as the distance of pillars (see Fig.~\\ref{fig:layout}).\nA small cell pitch offers the advantage of a higher dynamic range \\cite{Vinogradov2011}.\nThe absorption of a photoelectron in the active volume triggers a Geiger-discharge of a single cell. During the discharge and the following recharge of the cell, the cell is partially inactive for incoming light \\cite{Oide2007,Renker2009}.\nConsequently, the SiPM array shown in Fig.~\\ref{fig:layout} could only detect a maximum of 24 photons during a fast light pulse (e.g. shorter than recharge time).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{DepletionRadius_vs_Bias_Graph.pdf}\n\\caption{Range of the electric field for the three structure sizes.}\n\\label{fig:depletion_width}\n\\end{figure}\n\nThe aim of a high dynamic range and a high geometric efficiency is in general contradictory.\nRegarding the results of Fig.~\\ref{fig:depletion_width}, we chose a cell pitch of \\SI{15}{\\micro \\m} for the processed devices.\nThe active areas of two neighbouring cells for the smallest structure S06 are just in contact.\nThe geometric efficiency of our device is only determined by the opaque materials like metal lines and semi-transparent materials like the quenching resistors, which are located on top of the active area. \nFor the prototypes with a \\SI{15}{\\micro m} pitch, the nominal geometric efficiency of \\SI{83}{\\%} was realized. The value was calculated from the layout, accounting metal lines and quenching resistors as opaque.\n\n\n\\subsection{Depletion Layer Capacitance}\n\nThe depletion layer around the tip creates a certain amount of charge on each side of the junction whereas the total charge is zero.\nThe incremental charge $dQ_D$ of one side of the junction upon an incremental change of the bias voltage defines the depletion layer capacitance $C_D = dQ_D \/ dV$.\nThe spatial depletion of the respective structures was obtained with the solution of the continuity equations (see Eq.(\\ref{eq:depl})).\nThe simulated depletion layer capacitance dependent on the bias voltage is presented in Fig.~\\ref{fig:depletion_cap}.\nAll three structures show a similar curve progression while the increased tip size leads to a higher capacitance.\n\nThe simulated depletion layer capacitance is part of the total cell capacitance $C_{\\mathrm{cell}}$, which includes additionally the parasitic capacitance due to the pillar and the connection to the quenching resistor.\nThe gain of a single cell is proportional to the cell capacitance and can be expressed as:  \n\\begin{equation}\nG = \\frac{Q}{e} = \\frac{\\Delta V \\cdot C_{\\mathrm{cell}} }{e}\n\\end{equation}\nwhere $\\Delta V$ stands for the excess bias voltage.\nThe main goal for the presented new SiPM design was to a achieve a high photon detection efficiency over a wide spectral range.\nThe recovery time (recharge time) should be short to provide a fast device with a high dynamic range. Consequently, a low cell capacitance is beneficial.\nIn respect with the simulations of this section, we chose the smallest device S06 as the most promising structure. \n\n\n\\section{Metrological Characterization}\n\\subsection{Single electron response}\n\\label{sec:SER}\nTo achieve a high photon count rate and a strong ambient light immunity, a fast recovery of the micro-cells is required. In Fig.~\\ref{fig:SER}, the single electron response (SER) of the 3D-SiPM is shown at $\\Delta V=4\\text{\\,V}$.\nIt was measured as the voltage drop across a \\SI{25}{\\Omega} load resistor. \nThe decay part of the pulse consists of two exponential components with time constants of $\\tau_{1}\\approx \\SI{0.5}{\\nano s}$ and $\\tau_{2}\\approx \\SI{4.3}{ns}$.\nAfter approximately \\SI{9.5}{\\nano s}, the micro-cells are recovered to \\SI{90}{\\%} of their maximum charge. Using the double-light-pulse method, as proposed in \\cite{Popova2012}, comparable results were achieved.\nThis confirms that the SER shape reflects the true recovery process in this case. With this result, the TAPD has the fastest recovery time with respect to state-of-the-art planar SiPMs with enhanced red-sensitivity \\cite{Broadcom2020,RB_Series2020}. \n\\begin{figure}[h!]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{SER.pdf}}\n\\caption{Normalized pulseshape of the S06 structure at an excess bias voltage of \\SI{4}{V}. }\n\\label{fig:SER}      \n\\end{figure}\n\nFig.~\\ref{fig:SPE_Spectrum} provides an example of a single photoelectron charge spectrum. For this spectrum, we used a pulsed laser illumination with a pulse width of \\SI{70}{ps}. The acquisition was synchronous with the light pulses. The charge was integrated within a time-window of \\SI{10}{\\nano s}. The peaks up to 4 photoelectrons are well separated, which makes precise single photon counting possible.\n\\begin{figure}\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{Spectrum_EPJC.pdf}}\n\\caption{Single photoelectron charge spectrum of the S06 structure at an excess bias voltage of \\SI{4}{V}.}\n\\label{fig:SPE_Spectrum}      \n\\end{figure}\n\n\\subsection{Photon detection efficiency}\n\\label{sec:PDE}\nThe photon detection efficiency (PDE) describes the capability of the sensor to detect light as the ratio of the average number of incident and the average number of detected photons. In Eq.(\\ref{eq:PDE_Def}), the PDE is described as a product of three quantities:\n\n\\begin{equation}\n\\label{eq:PDE_Def}\nPDE=\\varepsilon \\cdot QE \\cdot P_{trigg}\n\\end{equation}\n\n(i) The geometric efficiency $\\varepsilon$ describes the fraction of the SiPM area, which is able to detect photons. The area of the SiPM which is not sensitive to light is mainly due to the metal lines for signal readout, the quenching resistors, guard rings for electric field attenuation towards the micro-cell edges and trenches for the suppression of optical crosstalk.\n\n(ii) The quantum efficiency $QE$ describes the efficiency to collect a fraction of charge carriers that a photon generated within the active volume of a micro-cell.\nTo maximize the PDE for blue light, the depletion region has to be extended as close to the surface as possible. Lower energetic photons are also absorbed at larger depths. To reach an enhanced detection of red and near-infrared light, the active region has to reach deeper inside the silicon. With our devices, we reach a spherical depletion volumes with radii of about \\SI{8}{\\micro m} to \\SI{9}{\\micro m} (see Fig.~\\ref{fig:depletion_width}).\n\n\\begin{figure}\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{PDE_vs_Overvoltage_EPJC.pdf}}\n\\caption{Photon detection efficiency vs. the excess bias voltage for the S06 structure. The lines are drawn as eye guides.}\n\\label{fig:PDE_vs_EBV}      \n\\end{figure}\n\n\n(iii) The avalanche triggering probability $P_{trigg}$ describes the probability that a generated e-h pair will successfully initiate an avalanche breakdown by impact ionization. \n   \nIn this work, the PDE was measured by using the continuous low-level light method, as reported in \\cite{Piemonte2012,Vachon2018,Vinogradov2016,Engelmann2018}. The incident photon rate was determined by a calibrated reference PIN-diode \\cite{Thorlabs2019}. Both, the SiPM and the reference diode were homogeneously illuminated trough a lens. The homogeneous part of the light spot was significantly larger than the active area of the photosensors. The photon rate was determined as the difference of the SiPM pulse count rate during illumination and at dark conditions. The count rates were measured according to the method described in Sec.~\\ref{sec:DCR_PCP}.\nIn Fig.~\\ref{fig:PDE_vs_EBV}, the PDE is shown as a function of the excess bias voltage for several wavelengths from \\SI{460}{\\nano m} to \\SI{905}{\\nano m}. In this spectral range, the PDE reaches \\SI{90}{\\%} of its saturation value at excess bias voltages between \\SI{4}{V} and \\SI{5}{V}. This is comparable to state-of-the-art planar SiPMs which are optimized for the detection of blue light and hence have smaller depletion volumes \\cite{Otte2017}.\n\n\n\\begin{figure}[b]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{PDE_vs_WL_EPJC.pdf}}\n\\caption{Photon detection efficiency vs. wavelength for the S06 structure at an excess bias voltage of \\SI{5}{V}.}\n\\label{fig:PDE_vs_WL}      \n\\end{figure}\n\n\n\nIn Fig.~\\ref{fig:PDE_vs_WL}, the PDE is shown at different wavelengths. This result was obtained by converting the spectral response measurement with a monochromator into an absolute PDE measurement as described in \\cite{Otte2017}.\nThe TAPD demonstrates the highest peak PDE value of \\SI{73}{\\%} at \\SI{600}{\\nano m} with respect to state of the art devices with a \\SI{15}{\\micro m} pitch size \\cite{Gola2019,Yamamoto2018}. \nAdditionally, the PDE curve does not show the typical fast decrease with increasing wavelengths and remains above a value of \\SI{45}{\\%} up to a wavelength of $800\\text{\\,nm}$. The measured PDE is in good agreement with the expected values when assuming a Geiger-efficiency of \\SI{90}{\\%} (see Eq.(\\ref{eq:PDE_Def}) and dashed curve in Fig.~\\ref{fig:PDE_vs_WL}).\nFor wavelengths in the NIR-regime, we expect the geometric efficiency to increase approximately to \\SI{90}{\\%} due to the decreasing absorption by the semi-transparent quenching resistors.\nThe oscillations in the PDE curve are caused by destructive interference due to the protective $Si\/SiO_2$ stack on top of the entrance window of the prototype devices.\nThe results of the applied method are in agreement with the well-known method based on pulsed laser illumination \\cite{Eckert2010,Otte2017}.\nThe applied method offers the possibility to directly measure the absolute PDE for a larger number of wavelengths due to the easy access to LEDs of different wavelengths. \n\n\n\\subsection{Dark count rate and delayed correlated pulses}\n\\label{sec:DCR_PCP}\n\n\\begin{figure}[b]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{DCR_vs_EBV_EPJC.pdf}}\n\\caption{Dark count rate vs. the excess bias voltage for the S06 structure. The error bars represent the standard deviation of 5 samples. The line is drawn as an eye guide.}\n\\label{fig:DCR_vs_EBV}      \n\\end{figure}\n\nIn this work, the dark count rate and the probability of delayed correlated pulses is determined by using the method proposed in \\cite{Vinogradov2016}. The method is based on the analysis of the complementary cumulative distribution function (CCDF) of pulses subsequent to a primary dark pulse. Contrary to the pulse counting approach, the applied method provides the benefit that delayed correlated pulses which exceed the detection threshold do not contribute to the dark count rate of the device \\cite{Engelmann2018}. This advantage is of special significance for SiPMs with a fast recovery time, like the TAPD, where delayed correlated pulses reach the maximum charge after a few ns.\nThe CCDF-method is also suited to measure the count rate of a continuous Poissonian photon output from a light source. In this way the photon detection efficiency can be measured (cf. Sec.~\\ref{sec:PDE}).\n\nAs delayed correlated pulses, we understand pulses which may be caused by two kinds of effects: \n\\\\\nThe first one is afterpulsing. Here, trapping centres present as energy states within the bandgap may capture electrons or holes from the conduction or valence band and re-emit them after a certain delay-time $\\Delta t$ into the same band. If the trapping centre is located in the active region of a micro-cell, the re-emitted charge carrier has a finite probability to trigger a subsequent avalanche in the same micro-cell. The delay-times depend on the respective trap type and may vary by many orders of magnitude.\n\\\\\nThe second effect is the so called delayed optical crosstalk.\nHere, one or more photons which are emitted during the avalanche breakdown are absorbed outside the high field region. The generated minority charge carriers then diffuse towards the active region and are able to trigger consecutive breakdowns of the original or a neighbouring micro-cell. This process is significantly slower compared to the prompt optical crosstalk.\n\nThe applied method is only valid under the assumption that the time constants of the delayed correlated effects are smaller with respect to the reciprocal of the dark count rate. In \\cite{Engelmann_PhD_2018}, the applied measurement procedure is described in detail.\n\n\\begin{figure}[b]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{PCP_T0P6_EPJC.pdf}}\n\\caption{Probability of delayed correlated pulses vs. the excess bias voltage for the S06 structure. The error bars represent the standard deviation of 5 samples.}\n\\label{fig:PCP_T0P6}      \n\\end{figure}\n\nIn Fig.~\\ref{fig:DCR_vs_EBV}, the dark count rate is shown as a function of the excess bias voltage.\nAt recommended excess bias voltages between $\\Delta V=\\SI{4}{V}$ and $\\Delta V=\\SI{5}{V}$, the prototypes show a DCR between $\\SI{700}{kHz\/mm^2}$ and $\\SI{1.3}{MHz\/mm^2}$. In comparison, KETEK's planar \\SI{15}{\\micro m} pitch SiPM shows a DCR of typically $125\\text{\\,kHz}\/\\text{mm}^2$ at $\\Delta V=\\SI{5}{V}$ \\cite{KETEK2019}.\n\nThis result matches our expectations, since the active volume of the TAPD is about a factor 10 larger with respect to the planar structures. The state of the art red-sensitive SiPMs from other manufacturers show dark count rates between $\\SI{600}{kHz\/mm^2}$ \\cite{Broadcom2020} and $\\SI{3.5}{MHz\/mm^2}$ \\cite{RB_Series2020} at the recommended operation voltages.\nIn Fig.~\\ref{fig:PCP_T0P6}, the probability of delayed correlated pulses is plotted as a function of the excess bias voltage. It is below \\SI{3}{\\%} at excess bias voltages up to $\\Delta V=\\SI{5}{V}$.\n\n\n\n\\subsection{Prompt optical crosstalk}\nDuring an avalanche breakdown, optical photons are generated by a variety of processes. These photons are able to propagate to neighbouring micro-cells and initiate further avalanche breakdowns. The propagation may occur by a direct path or by several reflections at the top and bottom side of the device. In either case, the time difference between the first and the consecutive pulse is not sufficient for a distinction of the two pulses. \nFor this reason, only one pulse with an amplitude of multiple p.e. (photoelectron equivalent) is registered. This effect is called \"prompt optical crosstalk\" (CT). The prompt optical crosstalk probability scales with the number of generated photons during an avalanche breakdown, the geometric cross-section for the interaction between two micro-cells and the avalanche triggering probability.\nThe prompt optical crosstalk probability $P_{CT}$ is estimated as shown in Eq.(\\ref{eq:P_CT_estimate}). Here, $DCR_{n}$ is the dark count rate measured by the pulse counting method with a discriminator threshold of $n\\text{ p.e.}$ \\cite{Renker2009,Eckert2010,ONeill2015,Otte2017}:\n\\begin{equation}\n\\label{eq:P_CT_estimate}\nP_{CT}\\approx \\frac{DCR_{1.5}}{DCR_{0.5}}\n\\end{equation}\nAt typical excess bias voltages between \\SI{4}{V} and \\SI{5}{V}, we measure crosstalk probabilities between \\SI{27}{\\%} and \\SI{35}{\\%}. For comparison: The reported values from other state-of-the-art devices with a red-enhanced sensitivity are between \\SI{20}{\\%} and \\SI{43}{\\%} \\cite{Broadcom2020,RB_Series2020}.\n\n\\begin{figure}[h!]\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{CTP_vs_EBV_EPJC.pdf}}\n\\caption{Prompt optical crosstalk probability of the S06 structure.}\n\\label{CTP_vs_EBV}      \n\\end{figure}\n\n\n\\FloatBarrier\n\\subsection{Gain}\n\\label{sec:Gain}\n\n\\begin{figure}\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{GAIN.pdf}}\n\\caption{Gain vs. the excess bias voltage for the S06, S08 and the S10 structure.}\n\\label{fig:Gain_vs_EBV}      \n\\end{figure}\n\nThe fast recovery time, described in Sec.~\\ref{sec:SER}, is realized by a strongly reduced micro-cell capacity. As a consequence, the intrinsic gain is about a factor of 20 lower with respect to the KETEK's planar SiPM products \\cite{KETEK2019}. In Fig.~\\ref{fig:Gain_vs_EBV}, the absolute gain is shown as a function of the excess bias voltage for two different structures.\nThe higher gain of the S10 structure can be attributed to the larger ball radius and the larger contribution from the parasitic capacitance of the quenching resistor.\nTo measure the gain, we applied a combination of the single photoelectron charge spectrum and the dark current \\cite{Pagano2012,Engelmann_PhD_2018}. \nIn this procedure, the single photoelectron spectrum was recorded in dark environment with a trigger threshold set to \\SI{0.5}{p.e.}.\nTo receive the probability density function (PDF) of the primary and secondary events, the spectrum was normalized to the total number of detected events.\nThe expected number of firing micro-cells per initial photoelectron is described by the mean value of the PDF which is equivalent to the excess charge factor (ECF) \\cite{Klanner2019}. The dark count rate was determined as described in Sec.~\\ref{sec:DCR_PCP}. In this case, the contribution from afterpulses is neglected. Since the investigated devices have a low afterpulsing probability (see Fig.~\\ref{fig:PCP_T0P6}), this approach is reasonable. The gain $G$ is then determined by using Eq.(\\ref{eq:Gain}). $I_{dark}$ is the dark current at the respective operation voltage:\n\\begin{equation}\n\\label{eq:Gain}\nG(V)=\\frac{I_{dark}(V)}{q\\cdot DCR(V) \\cdot ECF(V)}\n\\end{equation}\nFrom the slopes of the linear fits in Fig.~\\ref{fig:Gain_vs_EBV}, we extract the micro-cell capacitances. The experimentally determined micro-cell capacities differ from the simulated ones by about \\SI{3.3}{\\femto F} to \\SI{3.8}{\\femto F} (see Fig.~\\ref{fig:depletion_cap}). We attribute this discrepancy to the parasitic capacitance of the quenching resistor and variation of the real geometry from the simulated spherical one. Here, we would like to point out that we expect the parasitic capacitance to be larger than the micro-cell capacitance and hence strongly contribute to the charge output of the TAPD.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{Capacitance_vs_Bias_Graph.pdf}\n\\caption{Simulated depletion layer capacitance and measured total cell capacitance (plotted at $V_{bd}$).}\n\\label{fig:depletion_cap}\n\\end{figure}\n\\subsection{Temperature dependence of the breakdown voltage}\nEspecially in automotive applications, the systems must operate in a wide temperature range. Independent of whether the photosensor has a temperature stabilization\/compensation or not, a low temperature coefficient of the breakdown voltage ($V_{bd}$) is beneficial. For our TAPD, we measure a linear decrease of $V_{bd}$ with temperature (see Fig.~\\ref{fig:VBD_vs_Temp}).\nThe slopes increase with increasing pillar diameter from $\\SI{26}{mV\/^\\circ C}$ for the S06 structure to $\\SI{31.5}{mV\/^\\circ C}$ for the S10 structure. These values are comparable with respect to $\\SI{22}{mV\/^\\circ C}$ for planar KETEK SiPMs \\cite{KETEK2019}, despite the clearly larger depletion width and the increased breakdown voltage.\nThe breakdown voltage was determined from the inverse logarithmic derivative (ILD) of the reverse current-voltage-characteristic with low-level light illumination, as described in \\cite{Klanner2017}.\nThe simulated breakdown voltages (see Tab.~\\ref{tab:structures}) overestimate the experimental values at room temperature. The discrepancy decreases from \\SI{5}{V} for the S10, to \\SI{4.7}{V} for the S08 and to \\SI{1}{V} for the S06 structure.\nWe attribute this observation to the fact that the deviation of the processed structures from a perfect sphere increases with increasing tip size.\nFor the S10 structure, the shape of the tip is closer to the one of an ellipse than a sphere. Here, the breakdown voltage is defined by the point with the lowest curvature.\n\n\\begin{figure}\n\\centering\n\\resizebox{\\columnwidth}{!}{\\includegraphics{VBD_vs_Temp_EPJC.pdf}}\n\\caption{Breakdown voltage vs. temperature. The error bars are within the symbols.}\n\\label{fig:VBD_vs_Temp}       \n\\end{figure}\n\n\\newpage\n\\section{Conclusion}\n\\label{sec_Conclusion}\nIn this work, we presented a new Silicon Photomultiplier, the Tip Avalanche Photodiode. The TAPD concept is based on quasi-spherical electrodes (tips) which are placed inside an epitaxial layer.\nWe gave a theoretical overview of the physical models and presented the metrological characterization of the existing prototypes.\nOur simulations are in a good agreement with the obtained experimental data.\nWith the TAPD we achieved a record photon detection efficiency over a wide spectral range from \\SI{400}{\\nano m} to \\SI{905}{\\nano m} with a small micro-cell pitch of 15 \u00b5m. In combination with a high dynamic range, a fast recovery time (\\SI{4}{\\nano s}), and a low temperature coefficient of the breakdown voltage (\\SI{26}{mV\/K}), this SiPM is a promising detector for a large variety of applications.\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\noindent Modulational instability of high-frequency nonlinear \nwaves is a common process in a variety of circumstances involving \nwave propagation in continuous systems. Modulational processes can be seen \nto occur in a wide range of physical situations, from nonlinear waves in \nplasmas \\cite{dter78} to nonlinear electromagnetic waves propagating in optical \nfibers \\cite{mal97}. What usually happens in all those cases is that due to \ngeneric nonlinear interactions, the amplitude of a high-frequency carrier \ndevelops slow modulations in space and time. If the modulations are indeed much \nslower than the high-frequencies involved, one can obtain simplified equations \ndescribing the dynamics of the slowly varying amplitudes solely, the amplitude \nequations \\cite{lili91,novo99,olig96,riz98,erich98}. In the present analysis \nwe consider systems that become integrable in this modulational limit, a \nfeature often displayed. Should this be indeed the case, \nno spatiotemporal chaos would be observed there. The basic interest \nthen would be to see what happens when the approximations leading to \nmodulational approximations cease to be satisfied. The paper is organized as \nfollows: in \\S 2 we introduce our model equation and discuss \nhow and when it can be approximated by appropriate \namplitude equations; in \\S 3 we investigate the modulational process from the \npoint of view of nonlinear dynamics; in \\S4 we perform full spatiotemporal \nsimulations and compare the results with those obtained in \\S 3; and in \\S 5 \nwe conclude the work. \n\n\\section{Model equation, modulational approximations, and amplitude equations}\n\nIn the present paper we \nfocus attention on a nonlinear variant of the Klein-Gordon equation (NKGE) to \ninvestigate the breakdown of modulational approximations in the context of \nnonlinear wave fields. The NKGE used here reads\n\\begin{equation}\n\\partial_t^2 A(x,t) - \\partial_x^2 A(x,t) + {\\partial \\Phi \\over \\partial A}=0,\n\\label{eqzero}\n\\end{equation}\n($\\partial_t \\equiv \\partial \/ \\partial t, \\> \n\\partial_x \\equiv \\partial \/ \\partial x$) \nwhere we write the generalized nonlinear potential as \n\\begin{equation}\n\\Phi(A) = \\omega^2 \\> {A(x,t)^2 \\over 2} - {A(x,t)^4 \\over 4} \n+ {A(x,t)^6 \\over 6},\n\\label{phi}\n\\end{equation}\n$\\omega$ playing the role of a linear frequency which will set the \nfast time scale. The remaining coefficients on the right-hand-side \nwere chosen to \nallow for modulational instability and saturation; we shall see that \nwhile the negative sign of the second fulfills the condition for \nmodulational instability, the positive sign of the third provides saturation. \nThe choice of their numerical values is arbitrary, but our results are \nnevertheless generic. The NKGE is known to describe wave propagation in \nnonlinear media and the idea here is to see how the dynamics changes as a \nfunction of the parameters of the theory: wave amplitude, and time and length \nscales. \n\nLet us first derive the conditions for slow modulations. We start by supposing \nthat the field $A(x,t)$ be expressed in the form \n\\begin{equation}\nA(x,t) = \\tilde A (x,t) e^{i \\omega t} + {\\rm complex\\>\\>conjugate}. \n\\label{fatora}\n\\end{equation}\nThen, if one assumes slow modulations and discards terms like \n$\\partial_t^2 \\tilde A$ and the highest-order power of the potential \n$\\Phi$, one obtains \n\\begin{equation}\n2 i \\omega \\partial_t \\tilde A(x,t) - \\partial_x^2 \\tilde A(x,t) - \n3 |\\tilde A(x,t)|^2 \\tilde A(x,t) = 0, \n\\label{schroclassica}\n\\end{equation}\nwhich is, apart from some rescalings, the Nonlinear Schr\\\"odinger Equation - \nwe shall refer to it as NLSE here - an integrable equation. \n\nThis is no novelty; it is known that the modulational approximation is \nobtainable \nwhen there is a great disparity between the time scales of the high-frequency \n$\\omega$ and the modulational frequency, we call it $\\Omega$, such that \nterms of order $\\Omega^2 \\tilde A$, when compared to $\\omega^2 \\tilde A$, can be \ndropped from the governing equation. The magnitude of the modulational frequency \ncan be estimated as follows. Consider Eq. (\\ref{schroclassica}) and suppose a \ndemocratic balance among the magnitude of its various terms; \n$\\omega \\partial_t \\tilde A \\sim \\partial_x^2 \\tilde A \\sim {\\tilde A}^3$. Then one \nobtains for $\\partial_t \\rightarrow \\Omega$,\n\\begin{equation}\n{\\Omega \\over \\omega} \\sim ({\\tilde A \\over \\omega})^2.\n\\label{condition}\n\\end{equation}\nIt is thus clear that the modulational approximation is valid only \nwhen $\\tilde A \\ll \\omega$, since this condition slows down the modulational \nprocess causing $\\Omega \\ll \\omega$. The next question would be on what is to \nbe expected when the modulational approach ceases to be valid. Before \nproceeding along this line, let us mention that a stability analysis \ncan be performed on  Eq. (\\ref{schroclassica}). One perturbs an homogeneous \nself-sustained state with small fluctuations of a given wavevector $k>0$ (we \nchoose $k>0$ here, but the theory is invariant when $k \\rightarrow - k$) and \nafter some algebra one concludes that: (i) the field in the homogeneous state, \nlet us call this field $A_h$, is given by \n\\begin{equation}\nA_h = a_o e^{- i {3 a_o^2 \\over 2 \\omega} t}\n\\label{comp1}\n\\end{equation}\nwhere $a_o$ is an arbitrary \namplitude parameter and where the exponential term should be seen as providing a \nsmall nonlinear correction to the linear frequency $\\omega$; and (ii) the \nperturbation is unstable when \n\\begin{equation}\nk < k_{tr} \\equiv \\sqrt{6} a_o.\n\\label{modinst}\n\\end{equation}\nwith maximum growth rate at\n\\begin{equation}\nk_{max} \\equiv {k_{tr} \\over \\sqrt{2}}.\n\\label{maxgrowth}\n\\end{equation}\nWhen unstable, the homogeneous state typically evolves towards a state \npopulated by regular structures which can be formed precisely because \nthe underlying governing NLSE, Eq. (\\ref{schroclassica}), is of the \nintegrable type as mentioned earlier. \n\n\\section{Beyond the modulational approximation}\n\nTo advance the analysis beyond the modulational regimes we start from \nthe basic equation, Eq. (\\ref{eqzero}), but do not use approximation \n$\\partial_t^2 \\tilde A \\ll \\omega^2 \\tilde A$ leading to Eq. (\\ref{schroclassica}) \nand eventually to condition $\\tilde A \\ll \\omega$. The idea is precisely to \nexamine what happens as the ratio $\\tilde A \/ \\omega$ of Eq. (\\ref{condition}) \ngrows from values much smaller, up to values comparable to the unit.\n\nOur first task is to examine how the regular and well known modulational \ninstability analyzed in the previous section comes directly from Eq. (\\ref{eqzero}). \nTo do that, let us write a truncated solution as the sum of an \nhomogeneous term plus fluctuations with wavevector $k$, \n$A = A_h(t) + A_1(t) (e^{i k x} + e^{-i k x})$, $A_h$ and $A_1$ real. The \ntruncation, that discards higher harmonics, is legitimate within linear \nregimes, but fortunately we shall see that it is not as restrictive as it might \nappear even in nonlinear regimes. The general idea favouring truncation here \nis that in the reasonable situation where modes with the fastest growth rates \nare more strongly excited, relation (\\ref{maxgrowth}) indicates that second \nharmonics are already outside the instability band since \n$2 k_{max} = \\sqrt{2} k_{tr} > k_{tr}$. Under these circumstance \none would be led to think that the most important modes would be the \nhomogeneous and those at the fundamental spatial harmonic. We shall actually \nsee that the truncation provides a nice and representative approach to the \ncase of regular regimes. \n\nNote that we are already considering the amplitudes of the exponential functions as \nequal. This results from a simplified symmetrical choice of initial conditions and is \ntotally consistent with the real character of Eq. (\\ref{eqzero}). After some lengthy \nalgebra, one finds out that the coupled nonlinear dynamics of the fields $A_h$ and \n$A_1$ is governed by the Hamiltonian\n$$\nH={p_h^2 \\over 2} + {\\omega^2 q_h^2 \\over 2} - {q_h^4 \\over 2} + \n{2 q_h^6 \\over 3} + \n{p_1^2 \\over 2} + {\\chi^2 q_1^2 \\over 2} - {3 q_1^4 \\over 4} + \n{5 q_1^6 \\over 3} - \n$$\n\\begin{equation}\n3 q_h^2 q_1^2 + 10 q_h^4 q_1^2 + 15 q_h^2 q_1^4,\n\\label{hamiltonian}\n\\end{equation}\nwhere $\\chi^2 \\equiv \\omega^2+k^2$, $q_h = A_h\/\\sqrt{2}$, $q_1 = A_1$, and \nwhere the $p$'s denote the two momenta conjugate to the respective \n$q$-coordinates.\n\nThe Hamiltonian (\\ref{hamiltonian}) can be informative. As a first \ninstance it can be used to determine the stability properties of the homogeneous \npump, as mentioned before. To see this, assume that in average \n$q_h \\gg q_1$ and solve the dynamics perturbatively. In zeroth order, \none would have the following Hamiltonian governing the $(p_h,q_h)$ \ndynamics:\n\\begin{equation}\nh_o = {p_h^2 \\over 2} + {\\omega^2 q_h^2 \\over 2} - {q_h^4 \\over 2} + \n{2 q_h^6 \\over 3}.\n\\label{ho}\n\\end{equation}\nWith help of action-angle variables ($\\rho,\\Theta$) for the \nzeroth-order part and conventional perturbative techniques \nthe solution reads \n\\begin{equation}\nq_h = \\sqrt{2 \\rho \\over \\omega} \\cos(\\omega t - {3 \\over 2} \n{\\rho \\over \\omega^2} t), \n\\label{ordemzero}\n\\end{equation}\nif $\\rho$, the amplitude parameter, is not too large. Note that the \noscillatory frequency undergoes a small nonlinear correction which \nis determined by the quartic term in $q_h$ of the Hamiltonian $h_o$. \n\nNext we consider the driven Hamiltonian controlling the dynamics of \nthe canonical pair $(p_1,q_1)$\n\\begin{equation}\nh_1 = {p_1^2 \\over 2} + {\\chi^2 q_1^2 \\over 2} + 3\\>q_h^2 q_1^2,\n\\label{h1}\n\\end{equation}\nwhere we recall that the pair $(p_1,q_1)$ describes the inhomogeneity \nof the system. One again introduces action-angle variables $(I,\\theta)$ \nto rewrite the linear Hamiltonian (\\ref{h1}) in the form\n\\begin{equation}\nh_1 = \\chi I + 12 {\\rho \\over \\omega} {I \\over \\chi} \\cos^2 \\theta \n\\cos^2 (\\omega t - {3 \\over 2} {\\rho \\over \\omega^2} t),\n\\label{h11}\n\\end{equation}\nfrom which we obtain the resonant form\n\\begin{equation}\nh_{1,r} = {1 \\over 2 \\omega^2}(k^2 - 3\\>\\rho) I + \n{3\\>\\rho I \\over 2 \\omega^2} \\cos (2 \\phi),\n\\label{hres}\n\\end{equation}\nwith $\\phi = \\theta -(\\omega - 3\\rho \/ 2\\omega^2)t$, and where use is made of the \napproximation $\\chi \\approx \\omega + k^2\/(2 \\omega)$ valid when \n$k \\ll \\omega$. Now consider the stability of a \nsmall perturbation $I \\sim 0$. For this initial condition \n$h_{1,r} \\rightarrow 0$. \nSince $h_{1,r}$ is a constant of motion, solutions for arbitrarily large \nvalues of $I$, what would indicate instability, are possible only \nwhen $|\\cos(2 \\phi)| \\le 1$. \nFrom the resonant Hamiltonian (\\ref{hres}), \nthis demands \n\\begin{equation}\nk < \\sqrt{6 \\rho \\over \\omega},\n\\label{cond2}\n\\end{equation}\nand also indicates that maximum growth rate for $I$ occurs at \n$k_{max}=\\sqrt{3 \\rho \/ \\omega}$. Comparisons of temporal dependence \nof Eqs. (\\ref{fatora}), (\\ref{comp1}) and (\\ref{ordemzero}) shows that \n$\\rho \/ \\omega = a_o^2$, and that conditions (\\ref{modinst}) and (\\ref{cond2}) \nare therefore one and the same as they should be. In other words, starting \nfrom the full nonlinear wave equation, we recover the typical results \nnaturally yielded by the NLSE.\n\nBut the Hamiltonian (\\ref{hamiltonian}) gives further information because \napart the truncation, it is not an adiabatic approximation like \nEq. (\\ref{schroclassica}); it can therefore tell us whether the reduced \ndynamics, if unstable, is of the regular or chaotic type. The interest on this \nissue comes from the fact that the reduced dynamics usually helps to determine \nthe spatiotemporal patterns of the full system: while regular reduced dynamics \nis associated with regular structures - frequently a collection of \nsolitons or soliton-like structures, chaotic reduced \ndynamics is associated with spatiotemporal chaos. The correlation has its roots \non the so called stochastic pump model \\cite{lili91}. The model states that \nintense chaos in a low-dimensional subsystem of a multidimensional environment can \nmake the subsystem act like a thermal source, irreversibly delivering energy into \nothers degrees-of-freedom. In the limit of a predominantly regular dynamics \nundergoing in the reduced system, \nirreversibility is greatly reduced and energy tends to remain confined \nwithin the subsystem. On the other hand, in the limit of deeply chaotic \ndynamics with no periodicity at all, energy flow out of the subsystem is fast \nand there is not even much sense in defining the subsystem as an approximately \nisolated entity. The intermediary cases are those more amenable to a \ndescription in terms of the stochastic pump \\cite{lili91,novo99}. We point out, \nhowever, that in all cases, even in the chaotic one, analysis of the reduced \nsubsystem serves as an orientation on what to be expected in the full \nspatiotemporal dynamics. In our case, the appropriate subsystem is precisely \nthe one we have been using. This is so because it is the smallest subsystem \ncomprising the most important ingredients of the dynamics: the homogeneous \nand the only linearly unstable modes.\n\nWe now examine these points in some detail. Let us consider $\\omega = 0.1$. When\n$A_{h,o} \\ll \\omega$, $A_{h,o} \\equiv A_h(t=0)$, the modulational\napproximation can be used to determine the instability range, $k < k_{tr}$. \nA surface of section plot based on the two-degrees-of-freedom \nHamiltonian (\\ref{hamiltonian}) produces Fig. (\\ref{fig1.ps}), where we record \nthe values of $(q_1,p_1)$ whenever $p_h = 0$ with $dp_h\/dt > 0$, and \nwhere we take $q_1 = 0.01 q_h \\ll q_h$ and $p_h = p_1 = 0$ to determine the \ntotal unique energy of the various initial conditions we launch in the \nsimulations. Note that the particular ``seed'' initial condition introduced \nabove is included in the ensemble of initial conditions launched and represents \nsmall perturbations to an homogeneous background.\n\\begin{figure}[h]\n\\vspace{4in}\n\\special{psfile=fig1.ser.eps angle=-0 voffset=0 hoffset=40 hscale=30 vscale=40}\n\\caption{Low-dimensional Poincar\\'e plots on the projected\nphase-space $(p_1,q_1)$. $\\omega=0.1$ in all cases, and $k=k_{max}$ in panels \n(b)-(d). Initial conditions discussed in the text.}\n\\label{fig1.ps}\n\\end{figure}\nIn Fig. (\\ref{fig1.ps}a) \nwe take $A_{h,o}\/\\omega = 0.5$ and $k = 2 k_{max} > k_{tr}$. One lies outside \nthe instability range and the figure reveals that the origin, which \nrepresents the purely homogeneous state $q_1=A_1=0$, is indeed stable. \nThe remaining panels are all made for $k=k_{max}$ and increasing values of \nthe ratio  $A_{h,o}\/\\omega$. One sees that for such values of $k$ and \n$A_{h,o}$ not only the origin is rendered unstable, as it also becomes \nprogressively surrounded by chaotic activity. The inner chaotic trajectory \nissuing from the origin - i.e., the trajectory representing the \nmodulational instability - is encircled by invariant curves, but the last panel \nalready shows that some external chaos, here represented by scattered points \naround the invariant curves, is also present. At some amplitude \n$A_{h,o} = A_{critical}$ slightly larger than the one used in panel (d), invariant \ncurves are completely destroyed. Above the critical field, inner orbits are no \nlonger restricted to move within confined regions of phase-space - these \norbits are in fact engulfed by the external chaos seen in panel (d). External \nchaos here is a result of the hyperbolic point positioned at the local \nmaximum of the generalized potential $\\Phi(A)$, at $A_1 \\sim 0.1$.\nOne gross determination of the critical field based on simple \nnumerical observation yields $A_{critical}\/\\omega \\sim 1\/1.62$, although \nthe corresponding transition from order to disorder in the full simulations \nmay not be so sharply defined. Yet, one may expect that in both \nlow-dimensional and full simulations, the transition should occur at \ncomparable amplitudes.    \n\n\\section{Full spatiotemporal simulations}\n\nTo make the appropriate comparisons, we now look into the full simulation of \nEq. (\\ref{eqzero}). Full simulations are made through a discretization of the \nspatial domain via a finite difference method. The dynamics is resolved temporally \nby means of a sympletic integrator, as was the purely temporal Hamiltonian. The \nresults are quite robust and energy is conserved up to $10^{-6}$ parts in one. \nIn Fig. (\\ref{fig2.ps}) we display the spacetime history of the quantity \n$Q = \\sqrt{\\omega^2 A(x,t)^2 + \\dot A(x,t)^2}$ - we plot this quantity \nbecause it becomes a constant of motion in the limit where we discard nonlinearities \nand inhomogeneities. Initial conditions are the same as the seed initial \ncondition used in Fig. (\\ref{fig1.ps}), but the control parameters differ. In \nthe case of Fig. (\\ref{fig2.ps}a) where we choose $A_{h,o}\/\\omega = 0.1$ so as \nto safely satisfy $A_{h,o} \\ll A_{critical}$ and $A_{h,o} \\ll \\omega$, \nit is seen that the \nspatiotemporal dynamics is very regular. The homogeneous state is unstable, but \nonly periodic spatiotemporal spikes can be devised. This is the regular \nspatiotemporal dynamics so typical of the integrable NLSE and this kind of \ndynamics agrees very well with the low-dimensional predictions of the reduced \nHamiltonian (\\ref{hamiltonian}). The fact that only one single structure can be \nfound along the spatial axis at any given time, indicates that the dynamics is \nsingly periodic (period-1) along this axis and thus basically understood in \nterms of the reduced number of active modes (homogeneous plus fundamental \nharmonics) of Hamiltonian (\\ref{hamiltonian}). We now move into the vicinity of \nthe critical amplitude $A_{critical}$ discussed earlier. Under such conditions, \none may expect to see the effects of spatiotemporal chaos. The value \n$A_{h,o}\/\\omega = 1\/1.625$ is chosen in Fig. (\\ref{fig2.ps}b), where full \nsimulations indeed display a highly disorganized state after a short regular \ntransient. Regularly interspersed \nspikes can no longer be seen and spatial and temporal periodicities are lost, \nwhich characterizes spatiotemporal chaos. In this regime many modes become \nactive (we shall return to this point later) and the reduced Hamiltonian \n(\\ref{hamiltonian}) fails to provide an accurate description of the full \ndynamics. However it still provides a good estimate on the point of transition. \nA more throughful examination of the transition in the full simulations \nsuggests that the critical field there is a bit smaller - a value close to \n$\\omega \/ 1.95$; for smaller values we have not observed noticeable \nsignals of spatiotemporal chaos even for much longer runs than those \npresented here. The much larger oscillations \nexecuted by $Q$ in chaotic cases (see the legend of Fig. (\\ref{fig2.ps})) \nis a direct result of the destruction of the invariant curves seen in \nFig. (\\ref{fig1.ps}). In the absence of invariant curves, initial conditions \nare no longer restricted to move near the origin.  \n\\begin{figure}[h]\n\\vspace{4.25in}\n\\special{psfile=fig2.mat.eps angle=-0 voffset=-55 hoffset=0 hscale=50 vscale=50\n}\n\\caption{Full spatio-temporal simulations for $\\omega = 0.1$ and $k=k_{max}$.\n$A_{h,o}\/\\omega=0.1$ in (a) and $A_{h,o}\/\\omega=1\/1.625$ in (b).\nThe intensity plots are made for the quantity\n$Q = \\sqrt{\\omega^2 A(x,t)^2 + {\\dot A}(x,t)^2}$; lighter shades are associated\nwith larger values of $Q$, which varies within the range $0 < Q < 0.0014$ in\n(a) and $0 < Q < 0.8$ in (b). $\\tau \\equiv \\omega t \/ 2 \\pi$ here and in all\nthe remaining simulations}\n\\label{fig2.ps}\n\\end{figure}\nIt is thus seen that there are limits to an integrable modulational \ndescription of the dynamics of a wave field. The limits are essentially \nset by the parameter $A_{h,o} \/ \\omega$. If it is much smaller than \nthe unit, the modulational description is valid and one can \nexpect to see a collection of spatiotemporal periodic structures \nbeing formed as asymptotic states of the dynamics. On the other hand, as the \nparameter approaches the unit, nonintegrable features are likely to be \nseen. In particular, regularity does not survive for very long, and \nfluctuations with various length scales appear in the system. This \nis a regime of spatiotemporal chaos which fundamentally involves the \npresence of nonlinear resonances between the frequency of the \ncarrier, $\\omega$, and the intrinsic nonlinear modulational frequency, \n$\\Omega$.\n\nDue to the presence of chaos, one is suggested that the transition involves an \nirreversible energy flow out of the reduced subsystem. If one computes the average \nnumber of active modes\n\\begin{equation}\n<N^2> \\equiv {\\sum_n n^2 |A_n|^2 \\over \\sum_n |A_n|^2},\n\\label{soma}\n\\end{equation}\nwhere the amplitudes are defined in the form\n\\begin{equation}\nA_n = \\sum_j A(x_j,t) e^{i n k x_j},\n\\label{aene}\n\\end{equation}\none obtains the plots shown in Fig. (\\ref{fig3.ps}). \n\\begin{figure}[b]\n\\vspace{4.25in}\n\\special{psfile=fig3.kg.eps angle=-0 voffset=-0 hoffset=0 hscale=37 vscale=37}\n\\caption{Average number of active modes as a function of time. Parameters\nrespectively equal to those of the previous figure with exception of\npanel (d) where we take $A_{h,o}\/\\omega = 1\/1.95$.}\n\\label{fig3.ps}\n\\end{figure}\n\\noindent In Eq. (\\ref{aene}), \n\\lq\\lq$j$'' is the discretization index. In Fig. (\\ref{fig3.ps}a) we use the \nsame conditions as in Fig. (\\ref{fig2.ps}a). This panel \nshows that in rough terms, energy keeps periodically migrating between the \nhomogeneous mode (when $\\sqrt{<N^2>} \\sim 0$) and the fundamental harmonic \n(when $\\sqrt{<N^2>} \\sim 1$). Conditions of Fig. (\\ref{fig3.ps}b) are the same \nas those of Fig. (\\ref{fig2.ps}b); one sees that as the ratio \n$A_{h,o}\/\\omega$ grows, periodicity is lost, and that energy flow out of the \ninitial subsystem into other modes becomes clearly irreversible. In \nFig. (\\ref{fig3.ps}c) we use the same previous conditions with exception \nof $A_{h,o}$ which we take $A_{h,o} = \\omega \/ 1.95$. This slightly smaller, \nbut not too small, value of the initial amplitude allows to observe the \nslow diffusive transit of energy during the initial stages of the \ncorresponding simulations. During this stage one can actually look at \nthe subsystem as an energy source adiabatically delivering energy into other \nmodes - the concept of the stochastic pump applies more appropriately in those \nsituations.\n\n\\section{Final conclusions}\n\nTo summarize, in this paper we have studied the breakdown of modulational \napproximations in nonlinear wave interactions. We have analyzed \na nonlinear Klein-Gordon equation to draw the following \nconclusions. Adiabatic or modulational approximations are accurate while the \nhigh-frequency of the carrier wave keeps much larger than the modulational \nfrequency. Under these circumstances the \nfull spatiotemporal patterns are regular as is the dynamics \nin the reduced subsystem where the system energy is initially injected. There \nis no net flow of energy out of the reduced subsystem into the remaining \nmodes.\n\nOn the other hand, \nwhen both frequencies become of the same order of magnitude, the reduced \nsubsystem undergoes a transition to chaos. Correspondingly, the spatiotemporal \npatterns of the full system become highly disordered and energy spreads \nout over many modes. The correlation between the low-dimensional and high-dimensional \nspatiotemporal chaos has its roots on the stochastic pump model \\cite{lili91}. \nAccording to the model, a low-dimensional chaotic subsystem may act like a thermal \nsource, delivering energy in a irreversible fashion to other degrees-of-freedom \nof the entire system. Spectral simulations performed here indicates \nthat this seems to be the case with the present setting. \n\nThe transition to \nchaos involves a noticeable increase in terms of wave amplitude. This \ntakes place when the invariant curves of Fig. (\\ref{fig1.ps}) are destroyed, \nallowing for the merge of the external and internal chaotic bands into one \nextended chaotic sea. \nThis merging of chaotic bands is actually a result of reconnections involving the \nmanifolds of the unstable fixed point at the origin, and other hyperbolic points \nassociated with the curvature of the generalized potential $\\Phi$ \\cite{corso98}. \nWhen both manifolds reconnect, inner trajectories issuing from the origin start to \nexecute the large and irregular oscillations seen in Fig. (\\ref{fig2.ps}). More \ndetailed studied of the process is under current analysis.  \n\n\\acknowledgments\nThis work was partially supported by \nFinanciadora de Estudos e Projetos (FINEP), Conselho Nacional de \nDesenvolvimento Cient\\'{\\i}fico e Tecnol\\'ogico (CNPq), and \nFunda\\c{c}\\~ao da Universidade Federal do Paran\\'a - FUNPAR, Brazil. S.R. Lopes \nwishes to express his thanks for the hospitality at the Plasma Research \nLaboratory, University of Maryland. Part of the numerical work was performed on \nthe CRAY Y-MP2E at the Supercomputing Center of the Universidade Federal do \nRio Grande do Sul.  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:1}Introduction}\nThe physics of low-dimensional spin systems has attracted much \ninterest for past decades.\nAlthough the quantum fluctuation is expected to be large in two dimensions,\nthere exist strong evidences that the quantum Heisenberg antiferromagnet\n(QHAF) with nearest-neighbor coupling in a square lattice\nexhibits the N\\'eel long-range order at zero temperature.\\cite{Chakravarty89}\nUnder the presence of the long-range order, the linear spin-wave (LSW) \ntheory works rather well.\\cite{Anderson52,Kubo52}\nA natural way to include quantum fluctuations is\nan expansion in terms of $1\/S$, where $S$ is the magnitude of spin,\nbecause the LSW theory is made up of a leading-order in the $1\/S$ expansion.\nSuch attempts have been done and turned out to be \nuseful.\\cite{Oguchi60,Igarashi92-1,Igarashi92-2,Igarashi93,Canali92,Hamer92}\n\nIn our previous paper,\\cite{Igarashi92-1} basing on the Holstein-Primakoff \ntransformation,\\cite{Holstein40} \nwe calculated corrections up to the second order \nof $1\/S$ in various physical quantities such as the spin-wave dispersion,\nthe sublattice magnetization, the perpendicular susceptibility,\nand the spin-stiffness constant.\nWe also calculated the dynamical structure\nfactors of the transverse and the longitudinal components up to the \nsecond order of $1\/S$.\\cite{Igarashi92-2}\nAt that time, however, available experimental data of inelastic \nneutron scattering (INS) were limited to the momentum transfer \nin a narrow region of the Brillouin zone (BZ).\nTherefore, our study was just a demonstration of usefulness\nof the $1\/S$ expansion.\nNow that INS experiments provide us the information of the excitation \nspectra in the whole BZ, it may be interesting to calculate the excitation\nspectra in the whole BZ in comparison with recent experiments.\nWe show that the second-order correction makes the spin wave energy\nat momentum $(\\pi,0)$ about $2\\%$ smaller than that at $(\\pi\/2,\\pi\/2)$.\nThis difference is smaller than the value $7\\sim 9\\%$ obtained by\nthe series expansion\\cite{Singh95,Zheng04} \nand the Monte Carlo simulation.\\cite{Sandvik01}\nWe have so far not been able to find why the $1\/S$ expansion \nwithin the second order gives different results, since the higher-order \ncorrections is expected to be quite small. \nNote that the spin-wave dispersion was recently measured by\nthe INS experiment for Cu(DCOO)$_2\\cdot$4D$_2$O (CFTD),\nrevealing the $6\\%$ difference.\\cite{Ronnow01,Christensen04}\nThis material is believed to be well described by the $S=1\/2$\nHeisenberg model within the nearest-neighbor coupling.\\cite{Com1}\nIn addition to the spin-wave energy, we calculate the transverse dynamical \nstructure factor up to the second order of $1\/S$. \nIt consists of the $\\delta$-function-like peak\nof one spin-wave excitation and the continuum of three spin-wave excitations.\nThe second-order correction is found quite small in the spin-wave-peak\nintensity due to a cancellation of various second-order processes.\nThe result is compared with the recent experiment\\cite{Christensen04} \nas well as other $1\/S$-expansion study\\cite{Canali93} based on the \nDyson-Maleev transformation and the series expansion \nstudies.\\cite{Singh95,Zheng04}\nIn the three-spin-wave continuum, such a cancellation in the second-order\nprocesses is not severe, and the substantial intensities come out.\nThis is consistent with our previous study\\cite{Igarashi92-2}\nand others.\\cite{Canali93,Singh95,Zheng04}\n\nAnother purpose of this paper is to study the crossover behavior in\nthe QHAF from two dimensions to one dimension with weakening \nexchange coupling in one direction.  In purely one dimension,\nof course, the antiferromagnetic long-range order\ndisappears due to quantum fluctuation \nand therefore the concept of spin waves breaks down.\nCarrying out the $1\/S$ expansion, we demonstrate that\nthe second-order corrections increase with approaching\nthe quasi-one dimensional situation.\nThe second-order correction works to increase considerably \nthe sublattice magnetization, although the first-order correction makes\nit decrease, in the quasi-one dimensional situation. \nInterestingly, the spin-wave dispersion is found to approach to the curve\nknown as the des Cloizaux-Pearson boundary in the $S=1\/2$ QHAF \nchain.\\cite{desCloizeaux62}\nAt the same time, for the spin-wave peak in the transverse dynamical \nstructure factor, the first-order correction makes the intensity\ndecrease but the second-order correction makes it increase.\nIn the quasi-one dimensional situation, the former is much larger than\nthe latter, and the net intensity is strongly reduced from the LSW value.\nOn the other hand, the intensity of three-spin-wave continuum \nby the second-order correction increases and exceeds the spin-wave-peak\nintensities in the quasi-one dimensional situation. \nThis contrasts with the description of using spinon\\cite{Ho01}\nto describe the large intensity of the spectral continuum.\nThe above three characteristics in the quasi-one dimension, \n(a) the spin-wave energy approaches to the des Cloizaux-Pearson boundary,\n(b) the spin-wave-peak intensity decreases, and (c) the intensity of\nthree-spin-wave continuum increases, are suggesting a close relation to\nthe purely-one dimensional behavior that the spectra are described by\ncontinuum of two spinons.\n\nThe present paper is organized as follows.\nIn Sec. \\ref{sect.2}, the Heisenberg Hamiltonian is expressed in terms of \nthe 1\/S-expansion. The Green's functions for spin waves are introduced\nin Sec. \\ref{sect.3}. The sublattice magnetization is calculated \nwith the help of the Green's functions in Sec. \\ref{sect.4}.\nThe spin-wave dispersion is calculated in Sec. \\ref{sect.5},\nand the transverse dynamical structure factor is calculated\nin Sec. \\ref{sect.6}.\nSection \\ref{sect.7} is devoted to the concluding remarks.\n\n\\section{\\label{sect.2}Hamiltonian}\n\nWe consider the Heisenberg Hamiltonian on the square lattice\nwith directional anisotropy of exchange couplings:\n\\begin{equation}\n H= J\\sum_{\\bf\\ell}{\\bf S}_{\\bf\\ell}\\cdot{\\bf S}_{{\\bf\\ell}+{\\bf a}}\n  + J'\\sum_{\\bf\\ell}{\\bf S}_{\\bf\\ell}\\cdot{\\bf S}_{{\\bf\\ell}+{\\bf b}},\n\\label{eq.heis}\n\\end{equation}\nwhere ${\\bf\\ell}$ runs over all lattice sites and\n${\\bf\\ell}+{\\bf a}$ and ${\\bf\\ell}+{\\bf b}$ indicate\nthe nearest neighbors to the ${\\bf\\ell}$th site in the positive $x$ and $y$\ndirections, respectively. \nQuasi-one dimensional situations are realized by weakening the exchange \ncoupling $J'$ in the $y$ direction.\n\nIntroducing the Holstein-Primakoff transformation,\\cite{Holstein40}\nwe express the spin operators in terms of boson annihilation operators \n$a_i$ and $b_j$ (and their Hermite conjugates),\n\\begin{eqnarray}\n S_i^z &=& S - a_i^\\dagger a_i ,  \n \\label{eq.boson1}\\\\\n S_i^+ &=& (S_i^-)^\\dagger = \\sqrt{2S}f_i(S)a_i ,\\\\\n S_j^z &=& -S + b_j^\\dagger b_j ,\\\\\n S_j^+ &=& (S_j^-)^\\dagger = \\sqrt{2S}b_j^\\dagger f_j(S) ,\n \\label{eq.boson2}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n    f_\\ell (S) = \\left(1 - \\frac{n_\\ell}{2S}\\right)^{1\/2}\\\\\n               = 1 - \\frac{1}{2}\\frac{n_\\ell}{2S} -\n                  \\frac{1}{8}\\left(\\frac{n_\\ell}{2S}\\right)^2 + \\cdots .\n\\end{equation}\nwith $n_\\ell=a_i^\\dagger a_i$ and $b_j^\\dagger b_j$.\nIndices $i$ and $j$ refer to sites on the \"up\" and \"down\" sublattices, \nrespectively.\nSubstituting Eqs.~(\\ref{eq.boson1})-(\\ref{eq.boson2}) into Eq.~(\\ref{eq.heis})\nwe expand the Hamiltonian in powers of $1\/S$ as\n\\begin{equation}\n H = -S^2N(J+J')+H_0 + H_1 + H_2 + \\cdots, \n\\end{equation}\nwith $N$ the number of lattice sites.\n\nThe leading term $H_0$ is expressed as\n\\begin{eqnarray}\n H_0 &=& JS\\sum_{i}(2 a_i^\\dagger a_i \n           + 2 b_{i+{\\bf a}} b_{i+{\\bf a}}\n           +  a_i b_{i+{\\bf a}}\n           +  a_i b_{i-{\\bf a}}\n           + a_i^\\dagger b_{i+{\\bf a}}^\\dagger\n           + a_i^\\dagger b_{i-{\\bf a}}^\\dagger ) \\nonumber \\\\\n     &+& J'S\\sum_{i} (2 a_i^\\dagger a_i \n           + 2 b_{i+{\\bf b}} b_{i+{\\bf b}}\n           +  a_i b_{i+{\\bf b}}\n           +  a_i b_{i-{\\bf b}}\n           + a_i^\\dagger b_{i+{\\bf b}}^\\dagger\n           + a_i^\\dagger b_{i-{\\bf b}}^\\dagger ).\n\\label{eq.h0}\n\\end{eqnarray}   \nWe diagonalize $H_0$ by rewriting the boson operators in the momentum \nspace as\n\\begin{eqnarray}\n a_i &=& \\left(\\frac{2}{N}\\right)^{1\/2}\\sum_{\\bf k} a_{\\bf k}\n  \\exp(i{\\bf k\\cdot r}_i), \\\\\n b_j &=& \\left(\\frac{2}{N}\\right)^{1\/2}\n       \\sum_{\\bf k} b_{\\bf k} \\exp(i{\\bf k\\cdot r}_j),\n\\end{eqnarray}\nand by introducing the Bogoliubov transformation,\n\\begin{equation}\n a_{\\bf k}^\\dagger = \\ell_{\\bf k}\\alpha_{\\bf k}^\\dagger\n          + m_{\\bf k}\\beta_{-{\\bf k}}, \\quad\n b_{-{\\bf k}} = m_{\\bf k}\\alpha_{\\bf k}^\\dagger\n          + \\ell_{\\bf k}\\beta_{-{\\bf k}}, \n\\end{equation}\nwhere\n\\begin{equation}\n \\ell_{\\bf k} = \\Bigl[\\frac{1+\\epsilon_{\\bf k}}\n {2\\epsilon_{\\bf k}}\\Bigr]^{1\/2},\\quad\n  m_{\\bf k} = -{\\rm sgn}(\\gamma_{\\bf k})\\Bigl[\\frac{1-\\epsilon_{\\bf k}}\n  {2\\epsilon_{\\bf k}}\\Bigr]^{1\/2} \\equiv - x_{\\bf k}\\ell_{\\bf k},\n\\end{equation}\nwith\n\\begin{eqnarray}\n \\epsilon_{\\bf k} &=& \\left(1-\\gamma_{\\bf k}^2\\right)^{1\/2},\\\\\n   \\gamma_{\\bf k} &=& (\\cos k_x+\\zeta\\cos k_y)\/(1+\\zeta), \\\\\n           \\zeta  &=& J'\/J. \n\\end{eqnarray}\nMomentum ${\\textbf k}$ is defined in the first magnetic Brillouin zone\n(BZ). The ${\\rm sgn}(\\gamma_{\\textbf k})$ denotes the sign of \n$\\gamma_{\\textbf k}$,\nwhich is absorbed into the definition of $x_{\\bf k}$.\nFor the study of the isotropic exchange coupling ($\\zeta=1$), \nwe have neglected this factor \nbecause $\\gamma_{\\bf k}$ is always positive in the first BZ.\nFor the anisotropic coupling, however, this redefinition of $x_{\\bf k}$ \nis necessary, because $\\gamma_{\\bf k}$ is negative in a certain region \nof the first BZ. After this transformation, we have\n\\begin{eqnarray}\n H_0 &=& 2JS(1+\\zeta)\\sum_{\\bf k}(\\epsilon_{\\bf k}-1) \\nonumber \\\\\n     &+& 2JS(1+\\zeta)\\sum_{\\bf k} \\epsilon_{\\bf k}\n   (\\alpha_{\\bf k}^\\dagger \\alpha_{\\bf k}\n   + \\beta_{\\bf k}^\\dagger\\beta_{\\bf k}). \n\\end{eqnarray}\nThis expression is the same as that for the isotropic coupling,\nexcept for the first factor $2JS(1+\\zeta)$.\n\nThe first-order term $H_1$ can be expressed in terms of \nspin-wave operators through the same procedure as above.\nThe result for the anisotropic coupling is given by\nthe previous expression in Ref.~\\onlinecite{Igarashi92-1} with \nsimply replacing \n$JSz$ by $2JS(1+\\eta)$:\n\\begin{eqnarray}\n H_1 &=& \\frac{2JS(1+\\zeta)}{2S} A\\sum_{\\bf k}\\epsilon_{\\bf k}\n (\\alpha_{\\bf k}^\\dagger \\alpha_{\\bf k} + \\beta_{\\bf k}^\\dagger\\beta_{\\bf k})\n \\nonumber \\\\\n     &+&\\frac{-2JS(1+\\zeta)}{2SN}\\sum_{1234}\\delta_{\\bf G}(1+2-3-4)\n     \\ell_1\\ell_2\\ell_3\\ell_4  \\nonumber \\\\\n     &\\times& \\biggl[\\alpha_1^\\dagger\\alpha_2^\\dagger\\alpha_3\\alpha_4 \n     B_{1234}^{(1)}+\\beta_{-3}^\\dagger\\beta_{-4}^\\dagger\\beta_{-1}\\beta_{-2} \n     B_{1234}^{(2)}+4\\alpha_1^\\dagger\\beta_{-4}^\\dagger\\beta_{-2}\\alpha_3 \n     B_{1234}^{(3)}  \\nonumber \\\\\n     &+&\\bigl(2\\alpha_1^\\dagger\\beta_{-2}\\alpha_3\\alpha_4 B_{1234}^{(4)}\n      +2\\beta_{-4}^\\dagger\\beta_{-1}\\beta_{-2}\\alpha_3 B_{1234}^{(5)}\n      +\\alpha_1^\\dagger\\alpha_2^\\dagger\\beta_{-3}^\\dagger\\beta_{-4}^\\dagger\n     B_{1234}^{(6)} + {\\rm H.c.}\\bigr)\\biggr],\n\\label{eq.intham}\n\\end{eqnarray}\nwith \n\\begin{equation}\n A=\\frac{2}{N}\\sum_{\\bf k}(1-\\epsilon_{\\bf k}) .\n \\label{eq.A}\n\\end{equation}\nMomenta ${\\bf k}_1$, ${\\bf k}_2$, ${\\bf k}_3$, $\\cdots$ are abbreviated\nas $1,2,3,\\cdots$.\nThe first term arises from setting the products of four boson operators \ninto normal product forms with respect to spin-wave operators.\nThe second term in Eq.~(\\ref{eq.intham}) represents the scattering\nof spin waves. The Kronecker delta $\\delta_{\\bf G}(1+2-3-4)$ represents\nthe conservation of momenta within a reciprocal lattice vector ${\\bf G}$.\nThe vertex functions $B^{(i)}$'s in a symmetric\nparameterization are the same as those given by Eqs.~(2.16)-(2.20) \nin Ref.~\\onlinecite{Igarashi92-1}, so that they are omitted here.\n\nThe second-order term $H_2$ is composed of products of six boson operators.\nWriting it in a normal product form with respect to spin-wave operators,\nwe have \n\\begin{equation}\n H_2 = \\frac{2JS(1+\\zeta)}{(2S)^2}\\sum_{\\bf k}\\left[ C_1({\\bf k})\n  (\\alpha_{\\bf k}^\\dagger \\alpha_{\\bf k} + \\beta_{\\bf k}^\\dagger\\beta_{\\bf k})\n       + C_2({\\bf k})(\\alpha_{\\bf k}^\\dagger\\beta_{\\bf -k}^\\dagger\n             + \\beta_{\\bf -k}\\alpha_{\\bf k})+\\cdots \\right].\n\\end{equation}\nNeglected terms are unnecessary for calculating corrections up to the\nsecond order. The explicit forms of $C_1({\\bf k})$ and $C_2({\\bf k})$,\nare given by Eqs.~(2.22) and (2.23) in Ref.~\\onlinecite{Igarashi92-1}.\nNote that $C_1({\\bf k})$ and $C_2({\\bf k})$ diverge as $1\/\\epsilon_{\\bf k}$ \nwith $|{\\bf k}|\\to 0$. \n       \n\\section{\\label{sect.3}Green's Function} \n\nFor systematically carrying out the 1\/S-expansion,\nit is convenient to introduce the Green's functions for spin-waves,\n\\begin{eqnarray}\n G_{\\alpha\\alpha}({\\bf k},t) &=& -i \\langle T(\\alpha_{\\bf k}(t)\n \\alpha_{\\bf k}^\\dagger (0)) \\rangle, \\\\\n G_{\\alpha\\beta}({\\bf k},t) &=& -i \\langle T(\\alpha_{\\bf k}(t)\n \\beta_{-{\\bf k}}(0)) \\rangle, \\\\\n G_{\\beta\\alpha}({\\bf k},t) &=& -i \\langle T(\\beta_{-{\\bf k}}^\\dagger(t)\n \\alpha_{\\bf k}^\\dagger (0)) \\rangle, \\\\\n G_{\\beta\\beta}({\\bf k},t) &=& -i \\langle T(\\beta_{-{\\bf k}}^\\dagger(t)\n \\beta_{-{\\bf k}}(0)) \\rangle,\n\\end{eqnarray}\nwhere $ \\langle \\cdots \\rangle$ denotes the average over the ground state, and \n{\\textit T} is the time-ordering operator.\n\nIn this paper, we measure energies in units of $2JS(1+\\zeta)$.\nThe unperturbed propagators corresponding to $H_0$ are given by\n\\begin{eqnarray}\n G_{\\alpha\\alpha}^0({\\bf k},\\omega) &=& [\\omega \n - \\epsilon_{\\bf k} + i\\delta]^{-1}, \\\\\n G_{\\alpha\\beta}^0({\\bf k},\\omega) &=& G_{\\beta\\alpha}^0({\\bf k},\\omega) = 0, \n \\\\\n G_{\\beta\\beta}^0({\\bf k},\\omega) &=& [-\\omega \n - \\epsilon_{\\bf k} + i\\delta]^{-1}.\n\\end{eqnarray}\nThe self-energy is defined by\n\\begin{equation}\n G_{\\mu\\nu}({\\bf k},\\omega) = G_{\\mu\\nu}^0({\\bf k},\\omega)\n + \\sum_{\\mu'\\nu'} G_{\\mu\\mu'}^0({\\bf k},\\omega)\n \\Sigma_{\\mu'\\nu'}({\\bf k},\\omega)G_{\\nu'\\nu}({\\bf k},\\omega). \n\\end{equation}\nIt is expanded in powers of $1\/(2S)$,\n\\begin{equation}\n \\Sigma_{\\mu\\nu}({\\bf k},\\omega) = \\frac{1}{2S}\n    \\Sigma_{\\mu\\nu}^{(1)}({\\bf k},\\omega)\n + \\frac{1}{(2S)^2}\\Sigma_{\\mu\\nu}^{(2)}({\\bf k},\\omega) + \\cdots. \n\\end{equation}\n>From $H_1$ we have the first-order correction as\n\\begin{equation}\n \\Sigma_{\\alpha\\alpha}^{(1)}({\\bf k},\\omega)\n  = \\Sigma_{\\beta\\beta}^{(1)}({\\bf k},\\omega) = A\\epsilon_{\\bf k}, \\quad\n \\Sigma_{\\alpha\\beta}^{(1)}({\\bf k},\\omega)\n   = \\Sigma_{\\beta\\alpha}^{(1)}({\\bf k},\\omega)= 0.\n\\end{equation}\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.1.eps}%\n\\caption{\\label{fig.diagram1}\nSecond-order diagrams for the self-energy, \n(a) $\\Sigma_{\\alpha\\alpha}({\\bf k},\\omega)$ \nand (b) $\\Sigma_{\\alpha\\beta}({\\bf k},\\omega)$.\nSolid lines represent the unperturbed Green's functions.\n}\n\\end{figure}\n\nThe second-order term $\\Sigma_{\\mu\\nu}^{(2)}({\\bf k},\\omega)$ is obtained from\nthe second-order perturbation, whose diagrams are shown \nin Fig.~\\ref{fig.diagram1}.\nWe obtain formally the same expression for the self-energy as \nin our previous paper:\\cite{Igarashi92-1}\n\\begin{eqnarray}\n \\Sigma_{\\alpha\\alpha}^{(2)}({\\bf k},\\omega)\n  &=& \\Sigma_{\\beta\\beta}^{(2)}(-{\\bf k},-\\omega) \\nonumber \\\\\n  &=& C_1({\\bf k}) + \\left(\\frac{2}{N}\\right)^2\n  \\sum_{{\\bf p}{\\bf q}}2\\ell_{\\bf k}^2\\ell_{\\bf p}^2\n  \\ell_{\\bf q}^2\\ell_{{\\bf k}+{\\bf p}-{\\bf q}}^2  \\nonumber \\\\\n &\\times& \\left[\\frac{\\mid B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(4)}\n \\mid^2}{\\omega-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q}+i\\delta}\n -\\frac{\\mid B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}\\mid^2}\n{\\omega+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}-i\\delta}\\right], \n\\label{eq.self1}\\\\\n \\Sigma_{\\alpha\\beta}^{(2)}({\\bf k},\\omega)\n  &=& \\Sigma_{\\beta\\alpha}^{(2)}(-{\\bf k},-\\omega) \\nonumber \\\\\n  &=& C_2({\\bf k}) + \\left(\\frac{2}{N}\\right)^2\n  \\sum_{{\\bf p}{\\bf q}}2\n  \\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{{\\bf k}+\n {\\bf p}-{\\bf q}}^2 {\\rm sgn}(\\gamma_{\\bf G}) \\nonumber \\\\\n &\\times& B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(4)}\n          B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}\n \\frac{2(\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{{\\bf k}+{\\bf p}-{\\bf q}})}\n {\\omega^2-(\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q})^2+i\\delta},\n\\label{eq.self2}\n\\end{eqnarray}\nwhere $\\delta\\to 0$, and $[{\\bf k+p-q}]$ is the vector ${\\bf k+p-q}$ reduced to\nthe 1st BZ by a reciprocal vector ${\\bf G}$.\nWe have used the relations\n\\begin{equation}\n\\begin{array}{l}\nB_{[{\\bf k+p-q}],{\\bf q},{\\bf p},{\\bf k}}^{(5)} =\n{\\rm sgn}(\\gamma_{\\bf G})B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(4)},\\\\\nB_{{\\bf q},[{\\bf k+p-q}],{\\bf k},{\\bf p}}^{(6)} =\n{\\rm sgn}(\\gamma_{\\bf G})B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}.  \\\\\n\\end{array}\n\\end{equation}\nThe terms divergent with ${\\bf k}\\to 0$ in $C_1({\\bf k})$ and $C_2{\\bf k})$\nare canceled by the second-order perturbation terms in Eqs.~(\\ref{eq.self1}) \nand (\\ref{eq.self2}).\nOne can prove $\\Sigma_{\\mu\\nu}^{(2)}({\\bf k}\\to 0,\\omega=0)\\to 0$\nfrom these equations.\n\n\\section{\\label{sect.4}Sublattice Magnetization} \n\nOnce the Green's function is known, the sublattice magnetization is \ncalculated from the relation,\n\\begin{eqnarray}\n M &\\equiv& S-\\langle a_i^\\dagger a_i\\rangle \\nonumber \\\\\n   &=& S - \\frac{2}{N}\\sum_{\\bf k}\\lim_{\\eta\\to 0^+}\\int_{-\\infty}^{+\\infty}\n\\frac{{\\rm d}\\omega}{2\\pi}i{\\rm e}^{i\\omega\\eta}\n\\Bigl\\{\\ell_{\\bf k}^2G_{\\alpha\\alpha}({\\bf k},\\omega) \\nonumber\\\\\n&+& \\ell_{\\bf k}m_{\\bf k}\\bigl[G_{\\alpha\\beta}({\\bf k},\\omega)\n+G_{\\beta\\alpha}({\\bf k},\\omega)\\bigr]\n+ m_{\\bf k}^2G_{\\beta\\beta}({\\bf k},\\omega)\\Bigr\\},\n\\end{eqnarray}\nwith $\\eta\\to 0^+$.\nAfter carrying out the integration with respect to $\\omega$,\nwe obtains\n\\begin{equation}\n M = S -\\Delta S + \\frac{M_2}{(2S)^2},\n\\end{equation}\nwith\n\\begin{eqnarray}\n \\Delta S &=& \\frac{2}{N}\\sum_{\\bf k}\\frac{1}{2}(\\epsilon_{\\bf k}^{-1}-1), \\\\\n M_2 &=& \\frac{2}{N}\\sum_{\\bf k}\\Biggl\\{\n \\frac{\\ell_{\\bf k}m_{\\bf k}}{\\epsilon_{\\bf k}}\\Sigma_{\\alpha\\beta}^{(2)}\n({\\bf k},-\\epsilon_{\\bf k}) \\nonumber\\\\\n &-& \\left(\\frac{2}{N}\\right)^2\\sum_{{\\bf pq}}\n 2\\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{\\bf k+p-q}^2\n \\biggl[\\frac{(\\ell_{\\bf k}^2+m_{\\bf k}^2)\\mid\n B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}\\mid^2}\n {(\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q})^2} \n \\nonumber\\\\\n &+&\\frac{2\\ell_{\\bf k} m_{\\bf k}{\\rm sgn}(\\gamma_{\\bf G})\n B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(4)}\n B_{{\\bf k},{\\bf p},{\\bf q},[{\\bf k+p-q}]}^{(6)}}\n {\\epsilon_{\\bf k}^2-(\\epsilon_{\\bf p}+\\epsilon_{\\bf q}\n +\\epsilon_{\\bf k+p-q})^2}\\biggr]\\Biggr\\}.\n\\label{eq.zero}\n\\end{eqnarray}\nHere $[{\\bf k+p-q}]$ was defined before, and \n${\\bf G}={\\bf k+p-q}-[{\\bf k+p-q}]$. The zeroth-order correction\n$\\Delta S$ represents the well-known ``zero-point\" reduction in the LSW\ntheory. \nTo evaluate Eq.~(\\ref{eq.zero}), we sum up the values of $N_L^2\/4$ points\nof ${\\bf k}$ in the 1\/4 part of the first BZ and $N_L^2$ points of\n${\\bf p}$ and ${\\bf q}$ in the first BZ, with $N_L=20,48$.\nFor $J'\/J=1$, the convergence is very good; $M_2=0.0035059$ for $N_L=20$,\nand $M_2=0.0035065$ for $N_L=48$.\n\n\\begin{table}\n\\caption{\\label{table2}\nSublattice magnetization}\n\\begin{ruledtabular}\n\\begin{tabular}{llllll}\n $J'\/J$    & $1$ & $0.5$ & $0.1$ & $0.075$ & $0.05$ \\\\\n\\hline\n $\\Delta S$ & $0.196$  & $0.213$ & $0.355$ & $0.391$ & $0.445$ \\\\ \n $M_2$      & $0.0035$ & $0.024$ & $0.323$ & $0.481$ & $0.818$ \\\\ \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nTable \\ref{table2} lists the values of $\\Delta S$ and $M_2$ for several \nvalues of $J'\/J$. These values are evaluated for $N_L=48$.\nFor the isotropic coupling, we reproduce the values obtained previously.\n\\cite{Igarashi92-1} \nThe zero-point reduction $\\Delta S$ increases with decreasing values of $J'\/J$.\nOn the other hand, $M_2$ is found always positive, tending to cancel\nthe zero-point reduction. The value increases with decreasing values of\n$J'\/J$, and becomes comparable to $\\Delta S$ around \n$J'\/J=0.1$, suggesting the applicability limit of the expansion for \nthe case of $S=1\/2$. \n\n\\section{\\label{sect.5}Spin-Wave Dispersion} \n\nWithin the second order in $1\/S$,\nthe renormalized spin-wave energy\n${\\tilde \\epsilon_{\\bf k}}$ in units of $2JS(1+\\zeta)$\nis obtained from \n\\begin{equation}\n {\\tilde\\epsilon}_{\\bf k} = \\epsilon_{\\bf k} + \\frac{1}{2S}A\\epsilon_{\\bf k} \n + \\frac{1}{(2S)^2}\\Sigma_{\\alpha\\alpha}^{(2)}({\\bf k},\\epsilon_{\\bf k}).\n\\label{eq.spin-wave}\n\\end{equation}\nFrom this equation, we define the renormalized spin-wave velocity $V_x$ \nalong the $x$ direction by $V_x \\equiv \\lim_{k_x\\to 0}\n2JS(1+\\zeta)\\tilde \\epsilon_{\\bf k}\/(\\hbar k_x)$ with $k_y=0$.\nThus the renormalization factor is expressed as\n\\begin{equation}\n Z_v\\equiv\\frac{V_x}{2JS(1+\\zeta)^{1\/2}}= 1+\\frac{v_1}{2S}+\\frac{v_2}{(2S)^2}, \n\\end{equation}\nwith $v_1=A$ given by Eq.~(\\ref{eq.A}).\nIn the following numerical evaluation, we divide\nthe first BZ into $N_L^2$ meshes with $N_L=64$.\n\n\\subsection{Isotropic case}\n\nFigure \\ref{fig.iso.disp} shows the spin-wave energy \n$2JS(1+\\zeta)\\tilde\\epsilon_{\\bf k}$\nas a function of momentum for the isotropic coupling ($J'\/J=1$)\nwith $S=1\/2$, in comparison with the experimental data\ntaken from the INS for \nCu(DCOO)$_2\\cdot$4D$_2$O (CFTD).\\cite{Ronnow01,Christensen04}\nMomentum is measured in units of (nearest neighbor distance)$^{-1}$.\nIn the whole BZ, both the first and second order corrections make\nthe spin-wave energy larger.\nThe curve along the line $(0,0)-(\\pi,0)$ has already been reported\nin our previous paper.\\cite{Igarashi92-1}\nThe dispersion along $(\\pi\/2,\\pi\/2)-(\\pi,0)$ is completely \nflat within the first-order correction.\nThe second-order correction makes the excitation energy \nat $(\\pi,0)$ about $2\\%$ smaller than the energy at $(\\pi\/2,\\pi\/2)$.\nExplicitly they are \n$\\tilde\\epsilon_{(\\pi\/2,\\pi\/2)}=1.196$ and $\\tilde\\epsilon_{(\\pi,0)}=1.179$. \nA previous series expansion study predicted the energy difference \nabout $7\\%$,\\cite{Singh95} and a recent study gave about $9\\%$ difference,\nthat is, $\\tilde\\epsilon_{(\\pi\/2,\\pi\/2)}=1.192$ and \n$\\tilde\\epsilon_{(\\pi,0)}=1.09$.\\cite{Zheng04}\nA Monte Carlo simulation has given \n$\\tilde\\epsilon_{(\\pi\/2,\\pi\/2)}=1.195$ and \n$\\tilde\\epsilon_{(\\pi,0)}=1.08$.\\cite{Sandvik01}\nThese values at $(\\pi\/2,\\pi\/2)$ agree well with our value, while\nthe values at $(\\pi,0)$ is rather different from our estimate.\nThe experimental data indicate that the excitation energy at $(\\pi,0)$ is \n$6\\%$ smaller than that at $(\\pi\/2,\\pi\/2)$.\\cite{Ronnow01,Christensen04}\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.2.eps}%\n\\caption{\\label{fig.iso.disp}\nSpin-wave energy as a function of momentum for the isotropic coupling\n($J'\/J=1$). $S=1\/2$.\nThe dotted, broken and solid lines represent the values calculated \nwithin the LSW theory, up to the first-order correction, \nand up to the second-order correction, respectively. \nExperimental data are taken from Ref.~\\onlinecite{Christensen04}.\nInset indicates high symmetry lines which momentum varies along.\n}\n\\end{figure}\n\n\\subsection{Anisotropic case}\n\nFigure \\ref{fig.aniso.disp} shows the renormalized spin-wave energy \nas a function of momentum along $(0,0)-(\\pi,0)$ for $J'\/J<1$.\nAs the same as the isotropic coupling,\nboth the first-order and the second-order \ncorrections are found to be positive, making the energy larger. \nBoth corrections increase with decreasing values of $J'\/J$.\nIn quite small interchain couplings ($J'\/J < 0.1$),\nthe excitation energy seems approaching \nthe {\\em des Cloizeaux-Pearson boundary} \nin one dimension.\\cite{desCloizeaux62}\nAs shown in Table \\ref{table1}, the renormalization constant\n$Z_v$ seems approaching $\\pi\/2$, corresponding to the value of\nthe boundary.\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.3.eps}%\n\\caption{\\label{fig.aniso.disp}\nSpin-wave energy as a function of momentum for anisotropic couplings,\n(a) $\\zeta=0.5$, (b) $\\zeta=0.1$, (c) $\\zeta=0.075$, and (d) $\\zeta=0.05$.\nThe dotted, broken and solid lines represent the values calculated \nwithin the LSW theory, up to the first-order correction, and up to\nthe second-order correction, respectively. \nThe thin solid line labelled ``1D Exact\" represents the des Cloizeaux-Pearson \nboundary.}\n\\end{figure}\n\n\\begin{table}\n\\caption{\\label{table1}\nRenormalization of spin-wave velocity}\n\\begin{ruledtabular}\n\\begin{tabular}{llllll}\n $J'\/J$& $1$     & $0.5$   & $0.1$   & $0.075$ & $0.05$ \\\\\n\\hline\n $v_1$ & $0.158$ & $0.174$ & $0.272$ & $0.287$ & $0.306$ \\\\ \n $v_2$ & $0.021$ & $0.053$ & $0.130$ & $0.141$ & $0.155$ \\\\ \n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\section{\\label{sect.6}Dynamical Structure Factor} \n\nThe dynamical structure factor is an important quantity, \nsince it is directly related to the INS spectra.\nWe have already reported the expression within the second order \nof $1\/S$ in the isotropic coupling situation.\\cite{Igarashi92-2}\nAs is evident from the forms of $H_0$ and $H_1$, the\nformulas for the anisotropic coupling are\nformally the same as those for\nthe isotropic coupling.\n\nWe consider only the transverse component, \nwhich is defined by\n\\begin{equation}\n S^{+-}_{u(s)}({\\bf k}, \\omega) = \\frac{1}{2\\pi}\n \\int {\\rm d}t {\\rm e}^{i\\omega t}\n   \\langle Q_{u(s)}({\\bf k},t)Q_{u(s)}({\\bf k},0)^{\\dagger}\\rangle,\n\\end{equation}\nwhere\n\\begin{equation}\n Q_{u(s)}({\\bf k})=S^{+}_a({\\bf k})\\pm S^{+}_b({\\bf k}),\n \\label{eq.Q}\n\\end{equation}\nwith \n\\begin{eqnarray}\n S^{+}_a({\\bf k})&=&[S^{-}_a({\\bf k})]^{\\dagger}=\n  \\left(\\frac{2}{N}\\right)^{1\/2}\\sum_i S^{+}_i \\exp(-i{\\bf k}\\cdot{\\bf r}_i),\n  \\label{eq.sa}\\\\\n S^{+}_b({\\bf k})&=&[S^{-}_b({\\bf k})]^{\\dagger}=\n  \\left(\\frac{2}{N}\\right)^{1\/2}\\sum_j S^{+}_j \\exp(-i{\\bf k}\\cdot{\\bf r}_j).\n  \\label{eq.sb}\n\\end{eqnarray}\nWe need the ``uniform\" and ``staggered\" parts because the momentum \nis defined inside the first BZ.\nThey are labelled as the suffix ``u\" and ``s\", \nand correspond to upper and lower signs in Eq.~(\\ref{eq.Q}), respectively.\n\nWe start by introducing the operators,\n\\begin{eqnarray}\n Y^{+}_{\\alpha}({\\bf k})&=&[Y^{-}_{\\alpha}({\\bf k})]^{\\dagger}\n   = [\\ell_{\\bf k}S^{+}_{a}({\\bf k})-m_{\\bf k}S^{+}_{b}({\\bf k})]\/(2S)^{1\/2},\\\\\n Y^{+}_{\\beta}({\\bf k})&=&[Y^{-}_{\\beta}({\\bf k})]^{\\dagger}\n   = [-m_{\\bf k}S^{+}_{a}({\\bf k})+\\ell_{\\bf k}S^{+}_{b}({\\bf k})]\/(2S)^{1\/2},\n\\end{eqnarray}\nand the associated Green's functions,\n\\begin{equation}\n F_{\\mu\\nu}({\\bf k},\\omega)=-i\\int_{-\\infty}^{\\infty}dt {\\rm e}^{i\\omega t}\n   \\langle T[Y^{+}_{\\mu}({\\bf k},t)Y^{-}_{\\nu}({\\bf k},0)]\\rangle.\n\\end{equation}\nThen, with the help of the fluctuation-dissipation theorem,\nwe have\n\\begin{eqnarray}\n& & S^{+-}_{u(s)}({\\bf k},\\omega) = \n2S(\\ell_{\\bf k}\\pm m_{\\bf k})^2 \\left(- \\frac{1}{\\pi} \\right) \\nonumber\\\\\n&  &\\times {\\rm Im}[F_{\\alpha\\alpha}({\\bf k},\\omega) \n                \\pm F_{\\alpha\\beta}({\\bf k},\\omega)\n                \\pm F_{\\beta\\alpha}({\\bf k},\\omega)\n                   +F_{\\beta\\beta}({\\bf k},\\omega)],\n\\nonumber \\\\\n \\label{eq.dyna}\n\\end{eqnarray}\nwhere the upper (lower) signs correspond to the uniform (staggered) part.\n\nFor calculating $F_{\\mu\\nu}({\\bf k},\\omega)$, we expand the operator \n$Y^{+}_{\\mu}({\\bf k})$ in terms of spin-wave operators with the help of\nthe HP transformation and the Bogoliubov transformation.\nAfter lengthy calculations, we have\n\\begin{eqnarray}\n Y^{+}_{\\alpha}({\\bf k}) &=&\n  D\\alpha_{\\bf k}-\\frac{1}{2S}\\frac{2}{N}\\sum_{234}\n    \\delta_{\\bf G}({\\bf k}+2-3-4)\\frac{1}{2}\\ell_{\\bf k}\\ell_2\\ell_3\\ell_4\n \\nonumber\\\\\n &\\times& (M^{(1)}_{{\\bf k}234}\\beta_{-2}\\alpha_3\\alpha_4\n          +M^{(2)}_{{\\bf k}234}\\alpha^{\\dagger}_{2}\\beta^{\\dagger}_{-3}\n           \\beta^{\\dagger}_{-4}+ \\cdots),\n  \\label{eq.y1}\\\\\n Y^{+}_{\\beta}({\\bf k}) &=&\n  D\\beta^{\\dagger}_{-\\bf k} \\nonumber \\\\\n&-&\\frac{1}{2S}\\frac{2}{N}\\sum_{234}\n    \\delta_{\\bf G}({\\bf k}+2-3-4)\\frac{1}{2}\\ell_{\\bf k}\\ell_2\\ell_3\\ell_4\n    {\\rm sgn}(\\gamma_{\\bf G})\\nonumber\\\\\n &\\times& (M^{(2)}_{{\\bf k}234}\\beta_{-2}\\alpha_3\\alpha_4\n          +M^{(1)}_{{\\bf k}234}\\alpha^{\\dagger}_{2}\\beta^{\\dagger}_{-3}\n           \\beta^{\\dagger}_{-4}+ \\cdots),\n  \\label{eq.y2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n D&=&1-\\frac{\\Delta S}{2S}-\\frac{1}{4}\\frac{\\Delta S(1+3\\Delta S)}{(2S)^2},\n \\label{eq.D}\\\\\n M^{(1)}_{{\\bf k}234} &=& -x_2+ {\\rm sgn}(\\gamma_{\\bf G})x_{\\bf k}x_3x_4, \\\\\n M^{(2)}_{{\\bf k}234} &=& x_3x_4 - {\\rm sgn}(\\gamma_{\\bf G})x_{\\bf k}x_2.\n\\end{eqnarray}\nwith ${\\bf G}={\\bf k}+2-3-4$. \nThe first-order and second-order corrections in Eq.~(\\ref{eq.D}) arises from \nsetting four and six boson operators in the HP transformation \ninto the normal product forms with spin-wave operators, respectively.\nThereby, the second terms in Eqs. (\\ref{eq.y1}) and (\\ref{eq.y2})\nare normally ordered.\nNote that ${\\rm sgn}(\\gamma_{\\bf G})$ arises from the phase difference\nin the definitions, Eqs.~(\\ref{eq.sa}) and (\\ref{eq.sb}).\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.4.eps}%\n\\caption{\\label{fig.struc.diagram}\nDiagrams for $F_{\\mu\\nu}({\\bf k},\\omega)$.\nSolid lines represent the unperturbed Green's functions \n$G_{\\mu\\nu}^0({\\bf k},\\omega)$. \n}\n\\end{figure}\n\nWith the use of Eqs.~(\\ref{eq.y1}) and (\\ref{eq.y2}),\n$F_{\\mu\\nu}({\\bf k},\\omega)$ is expanded up to the second order by\nthe diagrams shown in Fig.~\\ref{fig.struc.diagram}.\nExplicitly, it is given by\n\\begin{eqnarray}\n F_{\\mu\\nu}({\\bf k},\\omega) &=& D^2G_{\\mu\\nu}^0({\\bf k},\\omega)\\delta_{\\mu\\nu}\n   + G_{\\mu\\mu}^0({\\bf k},\\omega)\\frac{1}{(2S)^2}\n   \\Sigma_{\\mu\\nu}^{(2)}({\\bf k},\\omega)G_{\\nu\\nu}^0({\\bf k},\\omega)\\nonumber\\\\\n  &+& I_{\\mu\\nu}({\\bf k},\\omega)G_{\\nu\\nu}^0({\\bf k},\\omega) \n   + G_{\\mu\\mu}^0({\\bf k},\\omega)\\tilde I_{\\mu\\nu}({\\bf k},\\omega)\n   + J_{\\mu\\nu}({\\bf k},\\omega),\n \\label{eq.f}\n\\end{eqnarray}\nwhere each term in Eq.~(\\ref{eq.f}) corresponds to the diagrams \n(a), (b), (c), (d), and (e), respectively.\nExplicit expressions for $I_{\\mu\\nu}({\\bf k},\\omega)$,\n$\\tilde I_{\\mu\\nu}({\\bf k},\\omega)$, and $J_{\\mu\\nu}({\\bf k},\\omega)$ \nare given by Eqs.~(4.5)-(4.22) in Ref.~\\onlinecite{Igarashi92-1}.\n\nThe dynamical structure factor is obtained by substituting Eq.~(\\ref{eq.f}) \ninto Eq.~(\\ref{eq.dyna}). \nIt consists of the $\\delta$-function-like peak of the one spin-wave excitation\nand the continuum of three spin-wave excitations:\n\\begin{equation}\n S_{u(s)}^{+-}({\\bf k},\\omega) = \n   \\rho_{u(s)}^{(1)}({\\bf k})\\delta(\\omega-\\epsilon_{\\bf k})\n                        + \\rho_{u(s)}^{(2)}({\\bf k},\\omega).\n \\label{eq.s+-}\n\\end{equation}\nFor the term of one spin-wave excitation, the bare energy $\\epsilon_{\\bf k}$\nis to be replaced by the renormalized value $\\tilde\\epsilon_{\\bf k}$\ngiven by Eq.~(\\ref{eq.spin-wave}).\nHowever, the spectral weight $\\rho^{(1)}({\\bf k})$ \nwithin the second order of $1\/S$ is safely evaluated \nby putting $\\omega=\\epsilon_{\\bf k}$ in Eq.~(\\ref{eq.f}).\nIt is expressed as\n\\begin{equation}\n \\rho_{u(s)}^{(1)}({\\bf k}) = 2S(\\ell_{\\bf k}\\pm m_{\\bf k})^2\n \\left(1+\\frac{d_{u(s),1}}{2S}+\\frac{d_{u(s),2}}{(2S)^2}\\right),\n\\end{equation}\nwith\n\\begin{eqnarray}\n d_{u(s),1} &=& -2\\Delta S, \n \\label{eq.dyna1}\\\\\n d_{u(s),2} &=& -\\frac{1}{2}\\Delta S(1+\\Delta S) \n             \\mp\\frac{1}{\\epsilon_{\\bf k}} \n             \\Sigma_{\\alpha\\beta}^{(2)}({\\bf k},\\epsilon_{\\bf k}) \\nonumber\\\\\n  &+& \\left(\\frac{2}{N}\\right)^2\\sum_{\\bf pq}\n     2\\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{\\bf k+p-q}^2\\nonumber\\\\\n  &\\times&\\left[\\frac{-|B^{(4)}_{\\bf k,p,q,[k+p-q]}|^2}\n  {(\\epsilon_{\\bf k}-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q})^2}\n  +\\frac{|B^{(6)}_{\\bf k,p,q,[k+p-q]}|^2}\n  {(\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q})^2}\n \\right] \\nonumber\\\\\n  &+& \\left(\\frac{2}{N}\\right)^2\\sum_{\\bf pq}\n     2\\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{\\bf k+p-q}^2\n (M_{\\bf k,p,q,[k+p-q]}^{(1)}\n     \\pm{\\rm sgn}(\\gamma_{\\bf G})M_{\\bf k,p,q,[k+p-q]}^{(2)})\\nonumber\\\\\n &\\times& \\left[\\frac{B^{(4)}_{\\bf k,p,q,[k+p-q]}}\n   {(\\epsilon_{\\bf k}-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q})}\n   \\mp\\frac{{\\rm sgn}(\\gamma_{\\bf G})B^{(6)}_{\\bf k,p,q,[k+p-q]}}\n   {\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}}\n  \\right].\n \\label{eq.dyna2}\n\\end{eqnarray}\nThe upper (lower) signs correspond to the uniform (staggered) part.\nThe first-order correction Eq.~(\\ref{eq.dyna1}) arises from the first term of\nEq.~(\\ref{eq.f}).\nIn the second-order correction given by Eq.~(\\ref{eq.dyna2}), \nthe first term arises\nfrom the first term of Eq.~(\\ref{eq.f}), and the second term arises from \nthe second term of Eq.~(\\ref{eq.f}),\n\\[ G_{\\alpha\\alpha}^0({\\bf k},\\omega)\\frac{1}{(2S)^2}\n   \\Sigma_{\\alpha\\beta}^{(2)}({\\bf k},\\omega)G_{\\beta\\beta}^0({\\bf k},\\omega)\n  +G_{\\beta\\beta}^0({\\bf k},\\omega)\\frac{1}{(2S)^2}\n \\Sigma_{\\beta\\alpha}^{(2)}({\\bf k},\\omega)G_{\\alpha\\alpha}^0({\\bf k},\\omega).\n\\]\nThe third term of Eq.~(\\ref{eq.dyna2}) is equivalent to\n\\[ \\left.\n  \\frac{1}{(2S)^2}\\frac{\\partial\\Sigma^{(2)}_{\\alpha\\alpha}({\\bf k},\\omega)}\n {\\partial\\omega}\\right|_{\\omega=\\epsilon_{\\bf k}}, \\]\nand arises from the second term of Eq.~(\\ref{eq.f}),\n\\[ G_{\\alpha\\alpha}^0({\\bf k},\\omega)\\frac{1}{(2S)^2}\n  \\Sigma_{\\alpha\\alpha}^{(2)}({\\bf k},\\omega)\n  G_{\\alpha\\alpha}^0({\\bf k},\\omega).\n\\]\nThis is related to the second-order correction to the residue of \nthe spin-wave pole in $G_{\\alpha\\alpha}({\\bf k},\\omega)$,\n\\[ \\frac{1}{1- \\left.\n  \\frac{1}{(2S)^2}\\frac{\\partial\\Sigma^{(2)}_{\\alpha\\alpha}\n ({\\bf k},\\omega)}\n {\\partial\\omega}\\right|_{\\omega=\\epsilon_{\\bf k}}}\n \\approx \n 1+ \\left.\n \\frac{1}{(2S)^2}\\frac{\\partial\\Sigma^{(2)}_{\\alpha\\alpha}({\\bf k},\\omega)}\n {\\partial\\omega}\n \\right|_{\\omega=\\epsilon_{\\bf k}} .  \\]\nThe fourth term arises from the third and fourth terms of Eq.~(\\ref{eq.f}),\n\\[ I_{\\mu\\nu}({\\bf k},\\omega)G_{\\nu\\nu}^0({\\bf k},\\omega) \n + G_{\\mu\\mu}^0({\\bf k},\\omega)\\tilde I_{\\mu\\nu}({\\bf k},\\omega).\n\\]\nNo contribution arises from the last term of Eq.~(\\ref{eq.f}),\n$J_{\\mu\\nu}({\\bf k},\\omega)$.\nNote that  the main momentum dependence around ${\\bf k}=(0,0)$ arises from\nthe prefactors, $(\\ell_{\\bf k}+m_{\\bf k})^2\\propto \\epsilon_{\\bf k}$\nand $(\\ell_{\\bf k}-m_{\\bf k})^2\\propto 1\/\\epsilon_{\\bf k}$.\n\nThe three-spin-wave continuum arises only from the second-order corrections.\nNote that the first term of Eq.~(\\ref{eq.f}) has no contribution. \nAfter careful evaluation of other terms in Eq.~(\\ref{eq.f}), we obtain\n\\begin{eqnarray}\n \\rho_{u(s)}^{(2)}({\\bf k},\\omega) &=& 2S(\\ell_{\\bf k}\\pm m_{\\bf k})^2\n  \\frac{1}{(2S)^2}\\left(\\frac{2}{N}\\right)^2\\sum_{\\bf pq}\n  \\delta(\\omega-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q})\n  \\nonumber\\\\\n &\\times& \\frac{1}{2}\n \\ell_{\\bf k}^2\\ell_{\\bf p}^2\\ell_{\\bf q}^2\\ell_{\\bf k+p-q}^2\n \\Biggl[M_{\\bf k,p,q,[k+p-q]}^{(1)}\n     \\pm{\\rm sgn}(\\gamma_{\\bf G})M_{\\bf k,p,q,[k+p-q]}^{(2)}\\nonumber\\\\\n &-& \\frac{2B^{(4)}_{\\bf k,p,q,[k+p-q]}}\n   {\\epsilon_{\\bf k}-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q}}\n   \\mp\\frac{2{\\rm sgn}(\\gamma_{\\bf G})B^{(6)}_{\\bf k,p,q,[k+p-q]}}\n   {\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}}\n  \\Biggr]^2.\n \\label{eq.conti}\n\\end{eqnarray}\nThe spectral shape may be modified by the renormalization of spin-wave\nenergies and by taking account of scattering spin waves due to mutual\ninteraction, which terms are present in $H_1$.\nTherefore, it may be difficult to determine the spectral shape\nin a consistent way with the $1\/S$ expansion.\nHowever, the total intensity, which is given by\n\\begin{equation}\n I_{u(s)}^{(2)}({\\bf k})=\\int_0^{\\infty}{\\rm d}\\omega\n             \\rho_{u(s)}^{(2)}({\\bf k},\\omega),\n\\end{equation}\nmay be safely evaluated from Eq.~(\\ref{eq.conti}).\nNote that around ${\\bf k}=(0,0)$ \n\\begin{eqnarray}\n M_{\\bf k,p,q,[k+p-q]}^{(1)}\n     &\\pm&{\\rm sgn}(\\gamma_{\\bf G})M_{\\bf k,p,q,[k+p-q]}^{(2)} \\propto\n     \\epsilon_{\\bf k}, \\\\\n -\\frac{B^{(4)}_{\\bf k,p,q,[k+p-q]}}\n   {\\epsilon_{\\bf k}-\\epsilon_{\\bf p}-\\epsilon_{\\bf q}-\\epsilon_{\\bf k+p-q}}\n   &\\mp&\\frac{{\\rm sgn}(\\gamma_{\\bf G})B^{(6)}_{\\bf k,p,q,[k+p-q]}}\n   {\\epsilon_{\\bf k}+\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}}\n \\nonumber \\\\\n &\\approx&\n \\frac{B^{(4)}_{\\bf k,p,q,[k+p-q]} \n   \\mp{\\rm sgn}(\\gamma_{\\bf G})B^{(6)}_{\\bf k,p,q,[k+p-q]}}\n   {\\epsilon_{\\bf p}+\\epsilon_{\\bf q}+\\epsilon_{\\bf k+p-q}} \n  \\propto\\epsilon_{\\bf k},\n\\end{eqnarray}\nand $(\\ell_{\\bf k}+m_{\\bf k})^2\\ell_{\\bf k}^2\\propto\nconst.$, $(\\ell_{\\bf k}-m_{\\bf k})^2\\ell_{\\bf k}^2\n\\propto 1\/\\epsilon_{\\bf k}^2$, we notice the dependences\naround ${\\bf k}=(0,0)$ as\n$I^{(2)}_u({\\bf k})\\propto \\epsilon_{\\bf k}^2$,\n$I^{(2)}_s({\\bf k})\\propto const.$ \n\nIn the numerical evaluation of Eqs.~(\\ref{eq.dyna2}) and (\\ref{eq.conti}),\nwe sum up the values on $N_L^2$ points of ${\\bf p}$ and ${\\bf q}$ in the\nfirst BZ, with $N_L=64$.\n\n\\subsection{Isotropic case}\n\nFigure \\ref{fig.iso.struc} shows the spin-wave-peak intensity \nand the intensity of three-spin-wave continuum as a function of momentum \nalong high-symmetry lines for $S=1\/2$. \nUsing the extended zone scheme, we assign the staggered part for line\n$(0,0)-(\\pi\/2,\\pi\/2)$ to the values for line $(\\pi,\\pi)-(\\pi\/2,\\pi\/2)$ \nand also the staggered part for line $(0,0)-(\\pi,0)$ to\nthe vales for line $(\\pi,\\pi)-(\\pi,0)$.\nThe uniform part is assigned inside the first BZ.\nAt the zone boundary of the reduced BZ, the uniform and staggered parts\ncoincide with each other. \nThe second-order corrections to the spin-wave-peak intensity becomes\none order of magnitude smaller than the first-order correction,\ndue to a cancellation among contributions of four terms \nin Eq.~(\\ref{eq.dyna2}).\nThe intensity is almost determined within the first-order correction.\nThus the correction relative to the zero-th order value is independent \nof momentum.  We obtain around ${\\bf k}=(0,0)$, \n\\begin{equation}\n \\rho^{(1)}_u({\\bf k})=0.215 |{\\bf k}|, \\quad\n \\rho^{(1)}_s({\\bf k})= 1.72\/|{\\bf k}| \\quad (S=1\/2).\n\\end{equation}\nThese values should be compared with the $1\/S$-expansion analysis\nbased on the Dyson-Maleev transformation,\\cite{Canali93} 0.202 and 1.86.\nA series expansion analysis by Singh\\cite{Singh93} \ngives the values, 0.246 and 2.10, while a recent analysis by Zheng \n\\textit{et al}.\\cite{Zheng04} gives the values, 0.216 and 1.86.\n\nThe small second-order correction to $\\rho^{(1)}_{u(s)}({\\bf k})$\ndoes not necessarily mean small three-spin-wave continuum.\nAt the zone boundary $(\\pi,0)$, for example, we have \n$I^{(2)}(\\pi,0)=0.143$ in addition to $\\rho^{(1)}(\\pi,0)=0.618$.\nSuch a considerable intensity of three spin-wave continuum has been\npredicted in our previous paper\\cite{Igarashi92-2} and \nothers.\\cite{Canali93,Zheng04}\nIt varies as proportional to $\\epsilon_{\\bf k}^2$ at the zone center,\nand converges to a constant value at $(\\pi,\\pi)$.\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.5.eps}%\n\\caption{\\label{fig.iso.struc}\nTransverse dynamical structure factor as a function of momentum \nfor the isotropic coupling. $S=1\/2$.\nThe dotted, broken and solid lines represent the intensity of one spin-wave\nexcitation calculated within the LSW theory, the first-order correction, \nand the second-order correction, respectively. \nThin solid line represents the intensity of three spin-wave excitations.\nExperimental data are taken from Ref.~\\onlinecite{Christensen04}.\nInset indicates high symmetry lines which momentum varies along.} \n\\end{figure}\n\n\\subsection{Anisotropic case}\n\nFigure \\ref{fig.aniso.struc} shows the spin-wave-peak intensity \nand the intensity of three-spin-wave continuum along the symmetry line\n$(0,0)-(\\pi,0)$ for several anisotropic couplings. $S=1\/2$.\n\nThe spin-wave-peak intensity is reduced by the first-order correction,\nbut is increased by the second-order correction.\nNote that, although the residue of the spin-wave pole in the Green's function\nis reduced by the self-energy, the other terms of the second-order\ncorrection work to increase the intensity. The second-order correction \nincreases with decreasing values of $J'\/J$, but\nthe reduction due to the first-order correction is much larger than\nthe gain due to the second-order correction. As a result, the spin-wave-peak\nintensity is strongly reduced. \n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig.6.eps}%\n\\caption{\\label{fig.aniso.struc}\nTransverse dynamical structure factor \nas a function of momentum along the symmetric line $(0,0)-(\\pi,0)$\nfor anisotropic couplings;\n(a)$J'\/J=0.5$, (b)$J'\/J=0.1$, (c)$J'\/J=0.075$, and (d)$J'\/J=0.05$. $S=1\/2$.\nThe dotted, broken and solid lines represent the spin-wave-peak intensity\ncalculated within the LSW theory, up to the first-order correction, \nand up to the second-order correction, respectively. \nThe thin solid line represents the three-spin-wave continuum intensity.}\n\\end{figure}\n\nOn the other hand, the intensity of three-spin-wave continuum increases\nwith decreasing values of $J'\/J$.\nIt exceeds the spin-wave-peak intensity in the quasi-one dimensional situation.\nSuch large intensities of continuum spectra have been observed \nin the recent INS experiments on the quasi-one dimensional QHAF\nsuch as KCuF$_3$\\cite{Lake00} and BaCu$_2$Si$_2$O$_7$.\\cite{Zheludev02}\n\n\\section{\\label{sect.7} Concluding remarks}\n\nWe have systematically carried out the $1\/S$ expansion up to the second order\non the basis of the HP transformation in the two-dimensional QHAF.\nWe have calculated the spin-wave energy in the whole BZ,\nin comparison with the recent INS experiment \nfor CFTD.\\cite{Ronnow01,Christensen04}\nWe have found that the spin-wave energy at $(\\pi,0)$ is about $2\\%$\nsmaller than that at $(\\pi\/2,\\pi\/2)$ due to the second-order correction.\nThis is a correct tendency, but the value is somewhat smaller than\nthe experimental value $6\\%$ and other theoretical estimates \n$7-9\\%$.\\cite{Singh95,Sandvik01,Zheng04}\nWe have so far not been able to find the reason for the difference, \nsince the corrections higher than the second order of $1\/S$ is expected\nto be quite small. We have also calculated the transverse dynamical\nstructure factor. The second-order correction is found extremely small\nin the one-spin-wave-peak intensity due to the cancellation\nin the second-order terms, while it gives rise to substantial intensities \nof three-spin-wave continuum. This is consistent with previous \nstudies.\\cite{Igarashi92-2,Canali93,Singh95,Zheng04}\n\nCanali and Wallin\\cite{Canali93} reported that the value of the perpendicular \nsusceptibility $\\chi_{\\perp}$ is different from the value \nin our previous paper\\cite{Igarashi92-1} (for $J'\/J=1$),\nalthough they confirmed the same values for the second-order correction\nto the sublattice magnetization $M_2$ and for the spin-wave velocity $V_x$.\nThe difference is not due to numerical errors, since the convergence \nwith respect to $N_L$ has been carefully checked with changing \n$N_L=160, 320, 480$ (see Ref.~\\onlinecite{Igarashi92-1}). \nAs already discussed there, the value in Ref.~\\onlinecite{Igarashi92-1}\nsatisfies the hydrodynamic relation, $V_x=(\\rho_s\/\\chi)^{1\/2}$ in an \nappropriate unit, with independently-evaluated spin-stiffness constant \n$\\rho_s$.\nAs regards the dynamical structure factor, the difference\nbetween the values by Canali and Wallin and the present values\nmight have the same origin as the difference in the perpendicular \nsusceptibility, because the diagrams for the perpendicular susceptibility \nare closely related to those for the transverse dynamical structure factor.\n\nThe second-order correction is expected to become more important in \nquasi-one dimensional systems.\nWe have studied the crossover from two dimensions to one dimension\nby weakening the exchange coupling in one direction.\nAll formulas are found formally the same as those for the isotropic coupling\nwith replacing $JSz$ by $2JS(1+\\zeta)$.\nIt is found that the excitation energy is pushed up by the first-order and \nsecond-order corrections. With approaching the quasi-one dimensional situation,\nthe corrections make the excitation energy close to the des Cloizeaux-Pearson \nboundary in the one-dimensional QHAF for $S=1\/2$. \nIn the transverse dynamical structure factor,\nthe spin-wave-peak intensity is reduced by the first-order\ncorrection, but is increased by the second-order correction.\nThe former exceeds the latter in the quasi-one dimensional situation, \nand thereby the peak intensity is strongly reduced from the LSW value.\nOn the other hand, the intensity of three-spin-wave continuum is found to \nbecome larger and exceeds the spin-wave-peak intensity.\n\nIn purely one-dimension, spin-one excitations are considered excitations\nof two spinons of a spin-one-half excitation.\nIn this respect, the spin wave might be considered \nas a bound state of two spinons.  Our finding that the weight \n$\\rho_1({\\bf k})$ of the spin-wave peak decreases with $J'\/J\\to 0$\nis consistent with this picture. Large intensities\nof three-spin-wave continuum might be replaced by two-spinon continuum\nin the one-dimensional limit. \nRecently, INS experiments have been carried out at low temperatures\nin the quasi-one dimensional systems such as KCuF$_3$\\cite{Lake00} \nand BaCu$_2$Si$_2$O$_7$,\\cite{Zheludev02} and large broad spectra \nhave been observed in addition to a peak in the transverse dynamical structure \nfactors. This behavior as well as the behavior of the longitudinal component\nhave been analyzed by the chain-mean-field and random phase \napproximation.\\cite{Essler97,Zheludev03}\nIt may be interesting to analyze these data in terms of the $1\/S$ expansion\nby starting from a detailed three-dimensional model \nwith directional anisotropy.\n\n\\begin{acknowledgments}\nJ.I. thanks MPI-PKS at Dresden for hospitality during his stay, \nwhere this work started.\nThis work was partially supported by a Grant-in-Aid for Scientific Research \nfrom the Ministry of Education, Science, Sports and Culture, Japan.\n\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}