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diff --git a/ising_scaling.tex b/ising_scaling.tex index 2055b9b..909d99f 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -55,31 +55,34 @@ universality class. The continuous phase transition in the two-dimensional Ising model is the most well studied, and its universal thermodynamic functions have likewise received -the most attention. Without a field, an exact solution is known for some lattice models \cite{Onsager_1944_Crystal}. Precision numeric work both on lattice models and on the -``Ising'' conformal field theory (related by universality) have yielded -high-order polynomial expansions of those functions, along with a comprehensive -understanding of their analytic properties \cite{Fonseca_2003_Ising, -Mangazeev_2008_Variational, Mangazeev_2010_Scaling}. In parallel, smooth -approximations of the Ising equation of state produce convenient, evaluable, -differentiable empirical functions \cite{Caselle_2001_The}. Despite being -differentiable, these approximations become increasingly poor when derivatives -are taken due to the neglect of subtle singularities. +the most attention. Without a field, an exact solution is known for some +lattice models \cite{Onsager_1944_Crystal}. Precision numeric work both on +lattice models and on the ``Ising'' conformal field theory (related by +universality) have yielded high-order polynomial expansions of those functions, +along with a comprehensive understanding of their analytic properties +\cite{Fonseca_2003_Ising, Mangazeev_2008_Variational, Mangazeev_2010_Scaling}. +In parallel, smooth approximations of the Ising equation of state produce +convenient, evaluable, differentiable empirical functions +\cite{Caselle_2001_The}. Despite being differentiable, these approximations +become increasingly poor when derivatives are taken due to the neglect of +subtle singularities. This paper attempts to find the best of both worlds: a smooth approximate -universal thermodynamic function that respects the global analytic properties of the -Ising free energy. By constructing approximate functions with the correct -singularities, corrections converge exponentially to the true function. To make -the construction, we review the analytic properties of the Ising scaling -function. Parametric coordinates are introduced that remove unnecessary -singularities that are a remnant of the coordinate choice. The singularities known to be present in the scaling function are incorporated in their simplest form. Then, the arbitrary -analytic functions that compose those coordinates are approximated by truncated -polynomials whose coefficients are fixed by matching the series expansions of -the universal function. +universal thermodynamic function that respects the global analytic properties +of the Ising free energy. By constructing approximate functions with the +correct singularities, corrections converge \emph{exponentially} to the true +function. To make the construction, we review the analytic properties of the +Ising scaling function. Parametric coordinates are introduced that remove +unnecessary singularities that are a remnant of the coordinate choice. The +singularities known to be present in the scaling function are incorporated in +their simplest form. Then, the arbitrary analytic functions that compose those +coordinates are approximated by truncated polynomials whose coefficients are +fixed by matching the series expansions of the universal function. \section{Universal scaling functions} A renormalization group analysis predicts that certain thermodynamic functions -will be universal in the vicinity of any critical point in the Ising +will be universal in the vicinity of \emph{any} critical point in the Ising universality class, from perturbed conformal fields to the end of the liquid--gas coexistence line. Here we will review precisely what is meant by universal. @@ -131,15 +134,17 @@ Solving these equations for $u_f$ yields &=|u_h|^{D\nu/\Delta}\mathcal F_0(u_t|u_h|^{-1/\Delta})+\frac{|u_t|^{D\nu}}{8\pi}\log u_h^{2/\Delta} \\ \end{aligned} \end{equation} -where $\mathcal F_\pm$ and $\mathcal F_0$ are undetermined scaling functions related by a change of coordinates \footnote{To connect the -results of this paper with Mangazeev and Fonseca, one can write $\mathcal -F_0(\eta)=\tilde\Phi(-\eta)=\Phi(-\eta)+(\eta^2/8\pi) \log \eta^2$ and -$\mathcal F_\pm(\xi)=G_{\mathrm{high}/\mathrm{low}}(\xi)$.}. -The scaling functions are universal in the sense that if another system whose -critical point belongs to the same universality class has its parameters -brought to the form \eqref{eq:flow}, one will see the same functional form, up to the units of $u_t$ and $u_h$. The invariant scaling combinations that appear -as the arguments to the universal scaling functions will come up often, and we -will use $\xi=u_h|u_t|^{-\Delta}$ and $\eta=u_t|u_h|^{-1/\Delta}$. +where $\mathcal F_\pm$ and $\mathcal F_0$ are undetermined scaling functions +related by a change of coordinates \footnote{To connect the results of this + paper with Mangazeev and Fonseca, one can write $\mathcal + F_0(\eta)=\tilde\Phi(-\eta)=\Phi(-\eta)+(\eta^2/8\pi) \log \eta^2$ and +$\mathcal F_\pm(\xi)=G_{\mathrm{high}/\mathrm{low}}(\xi)$.}. The scaling +functions are universal in the sense that if another system whose critical +point belongs to the same universality class has its parameters brought to the +form \eqref{eq:flow}, one will see the same functional form, up to the units of +$u_t$ and $u_h$. The invariant scaling combinations that appear as the +arguments to the universal scaling functions will come up often, and we will +use $\xi=u_h|u_t|^{-\Delta}$ and $\eta=u_t|u_h|^{-1/\Delta}$. The analyticity of the free energy at places away from the critical point implies that the functions $\mathcal F_\pm$ and $\mathcal F_0$ have power-law @@ -152,10 +157,10 @@ the case at infinity: since \mathcal F_\pm(\xi) =\xi^{D\nu/\Delta}\mathcal F_0(\pm \xi^{-1/\Delta})+\frac1{8\pi}\log\xi^{2/\Delta} \end{equation} -and $\mathcal F_0$ has a power-law expansion about zero, $\mathcal -F_\pm$ has a series like $\xi^{D\nu/\Delta-j/\Delta}$ for $j\in\mathbb N$ at -large $\xi$, along with logarithms. The nonanalyticity of these functions at -infinite argument can be understood as an artifact of the chosen coordinates. +and $\mathcal F_0$ has a power-law expansion about zero, $\mathcal F_\pm$ has a +series like $\xi^{D\nu/\Delta-j/\Delta}$ for $j\in\mathbb N$ at large $\xi$, +along with logarithms. The nonanalyticity of these functions at infinite +argument can be understood as an artifact of the chosen coordinates. For the scale of $u_t$ and $u_h$, we adopt the same convention as used by \cite{Fonseca_2003_Ising}. The dependence of the nonlinear scaling variables on @@ -189,18 +194,22 @@ This critical bubble occurs with free energy cost \simeq\frac{\pi\sigma^2}{2M|H|} \simeq T\left(\frac{2M_0}{\pi\sigma_0^2}|\xi|\right)^{-1} \end{equation} -where $\sigma_0=\lim_{t\to0}t^{-\mu}\sigma$ and $M_0=\lim_{t\to0}t^{-\beta}M$ are the critical amplitudes for the surface tension -and magnetization at zero field in the low-temperature phase \cite{Kent-Dobias_2020_Novel}. In the context -of statistical mechanics, Langer demonstrated that the decay rate is +where $\sigma_0=\lim_{t\to0}t^{-\mu}\sigma$ and $M_0=\lim_{t\to0}t^{-\beta}M$ +are the critical amplitudes for the surface tension and magnetization at zero +field in the low-temperature phase \cite{Kent-Dobias_2020_Novel}. In the +context of statistical mechanics, Langer demonstrated that the decay rate is asymptotically proportional to the imaginary part of the free energy in the metastable phase, with \begin{equation} \operatorname{Im}F\propto\Gamma\sim e^{-\beta\Delta F_c}\simeq e^{-1/b|\xi|} \end{equation} which can be more rigorously related in the context of quantum field theory -\cite{Voloshin_1985_Decay}. The constant $b=2M_0/\pi\sigma_0^2$ is predicted by known properties, e.g., for the square lattice $M_0$ and $\sigma_0$ are both predicted by Onsager's solution \cite{Onsager_1944_Crystal}, -but for our conventions for $u_t$ and $u_h$, $M_0/\sigma_0^2=\bar -s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant \cite{Fonseca_2003_Ising}. +\cite{Voloshin_1985_Decay}. The constant $b=2M_0/\pi\sigma_0^2$ is predicted by +known properties, e.g., for the square lattice $M_0$ and $\sigma_0$ are both +predicted by Onsager's solution \cite{Onsager_1944_Crystal}, but for our +conventions for $u_t$ and $u_h$, $M_0/\sigma_0^2=\bar +s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant +\cite{Fonseca_2003_Ising}. \begin{figure} @@ -216,14 +225,17 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant \cite{Fonseca_2003_ To lowest order, this singularity is a function of the scaling invariant $\xi$ alone. It is therefore suggestive that this should be considered a part of the singular free energy and moreover part of the scaling function that composes -it. There is substantial numeric evidence for this as well \cite{Enting_1980_An, Fonseca_2003_Ising}. We will therefore -make the ansatz that +it. There is substantial numeric evidence for this as well +\cite{Enting_1980_An, Fonseca_2003_Ising}. We will therefore make the ansatz +that \begin{equation} \label{eq:essential.singularity} \operatorname{Im}\mathcal F_-(\xi+i0)=A_0\Theta(-\xi)\xi e^{-1/b|\xi|}\left[1+O(\xi)\right] \end{equation} The linear prefactor can be found through a more careful accounting of the entropy of long-wavelength fluctuations in the droplet surface -\cite{Gunther_1980_Goldstone, Houghton_1980_The}. In the Ising conformal field theory, the prefactor is known to be $A_0=\bar s/2\pi$ \cite{Voloshin_1985_Decay, Fonseca_2003_Ising}. +\cite{Gunther_1980_Goldstone, Houghton_1980_The}. In the Ising conformal field +theory, the prefactor is known to be $A_0=\bar s/2\pi$ +\cite{Voloshin_1985_Decay, Fonseca_2003_Ising}. \subsection{Yang--Lee edge singularity} @@ -252,10 +264,10 @@ $\xi_{\mathrm{YL}}$. } \label{fig:higher.singularities} \end{figure} -The Yang--Lee singularities, although only accessible with complex fields, are critical points in their own right, with their -own universality class different from that of the Ising model -\cite{Fisher_1978_Yang-Lee}. Asymptotically close to this point, the scaling -function $\mathcal F_+$ takes the form +The Yang--Lee singularities, although only accessible with complex fields, are +critical points in their own right, with their own universality class different +from that of the Ising model \cite{Fisher_1978_Yang-Lee}. Asymptotically close +to this point, the scaling function $\mathcal F_+$ takes the form \begin{equation} \label{eq:yang.lee.sing} \mathcal F_+(\xi) =A(\xi) +B(\xi)[1+(\xi/\xi_{\mathrm{YL}})^2]^{1+\sigma}+\cdots @@ -288,7 +300,8 @@ The Schofield coordinates $R$ and $\theta$ are implicitly defined by && u_h(R, \theta) = R^{\Delta}g(\theta) \end{align} -where $g$ is an odd function whose first zero lies at $\theta_0>1$ \cite{Schofield_1969_Parametric}. We take +where $g$ is an odd function whose first zero lies at $\theta_0>1$ +\cite{Schofield_1969_Parametric}. We take \begin{align} \label{eq:schofield.funcs} g(\theta)=\left(1-\frac{\theta^2}{\theta_0^2}\right)\sum_{i=0}^\infty g_i\theta^{2i+1}. \end{align} @@ -304,7 +317,8 @@ combinations depend only on $\theta$, as \xi=u_h|u_t|^{-\Delta}=\frac{g(\theta)}{|1-\theta^2|^{\Delta}} && \eta=u_t|u_h|^{-1/\Delta}=\frac{1-\theta^2}{|g(\theta)|^{1/\Delta}} \end{align} -Moreover, both scaling variables have polynomial expansions in $\theta$ near zero, with +Moreover, both scaling variables have polynomial expansions in $\theta$ near +zero, with \begin{align} &\xi= g'(0)\theta+\cdots && \text{for $\theta\simeq0$}\\ &\xi=g'(\theta_0)(\theta_0^2-1)^{-\Delta}(\theta-\theta_0)+\cdots && \text{for $\theta\simeq\theta_0$} @@ -478,11 +492,14 @@ and \begin{equation} \mathcal F_{\mathrm{YL}}(\theta)=C_{\mathrm{YL}}\left[(\theta^2+\theta_{\mathrm{YL}}^2)^{1+\sigma}-\theta_{\mathrm{YL}}^{2(1+\sigma)}\right] \end{equation} -We have also included the analytic part $G$, which we assume has a simple series expansion +We have also included the analytic part $G$, which we assume has a simple +series expansion \begin{equation} G(\theta)=\sum_{i=1}^\infty G_i\theta^{2i} \end{equation} -From the form of the real part, we can infer the form of $\mathcal F$ that is analytic for the whole complex plane except at the singularities and branch cuts previously discussed. +From the form of the real part, we can infer the form of $\mathcal F$ that is +analytic for the whole complex plane except at the singularities and branch +cuts previously discussed. For $\theta\in\mathbb C$, we take \begin{equation} \mathcal F(\theta)=\mathcal F_0(\theta)+\mathcal F_{\mathrm{YL}}(\theta)+G(\theta), @@ -531,7 +548,9 @@ and \right)\right\} \end{aligned} \end{equation} -fixing $B$ and $C_0$. Similarly, \eqref{eq:yang-lee.theta} puts a constraint on the value of $\theta_\mathrm{YL}$, while the known amplitude of the Yang--Lee branch cut fixes the value of $C_\mathrm{YL}$ by +fixing $B$ and $C_0$. Similarly, \eqref{eq:yang-lee.theta} puts a constraint on +the value of $\theta_\mathrm{YL}$, while the known amplitude of the Yang--Lee +branch cut fixes the value of $C_\mathrm{YL}$ by \begin{equation} \begin{aligned} u_f |