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diff --git a/ising_scaling.tex b/ising_scaling.tex index 2b1fb62..7fa1ae6 100644 --- a/ising_scaling.tex +++ b/ising_scaling.tex @@ -26,10 +26,10 @@ linkcolor=purple \begin{document} -\title{Smooth and global Ising universal scaling functions} +\title{Precision approximation of the universal scaling functions for the 2D Ising model in an external field} \author{Jaron Kent-Dobias} -\affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} +\affiliation{\textsc{DynSysMath}, Istituto Nazionale di Fisica Nucleare, Sezione di Roma} \author{James P.~Sethna} \affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} @@ -46,8 +46,8 @@ linkcolor=purple the low- and high-temperature zero-field limits fixes the parametric coordinate transformation. For the two-dimensional Ising model, we show that this procedure converges exponentially with the order to which the series are - matched, up to seven digits of accuracy. - To facilitate use, we provide Python and Mathematica implementations of the code at both lowest order (three digit) and high accuracy. + matched, up to seven digits of accuracy. + To facilitate use, we provide a Mathematica implementation of the code at both lowest order (three digit) and high accuracy. %We speculate that with appropriately modified parametric coordinates, the method may converge even deep into the metastable phase. \end{abstract} @@ -67,10 +67,11 @@ 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 +the most attention. Onsager provided an exact solution in the absence of an external field \cite{Onsager_1944_Crystal}. Here we provide a high-precision, rapidly converging calculation of the universal scaling function for the 2D Ising model in a field. Our solution is not an exact formula in terms of well-known special functions (as is Onsager's result). Indeed, it seems likely that there is no such formula. The critical exponents for the 3D Ising model have recently been determined to high-precision calculations using conformal bootstrap methods, which should be viewed as a solution to that outstanding problem. The universal scaling function for the 2D Ising model in a field is a well-defined function with known singularities; in analogy, we tentatively suggest that our convergent, high-precision approximation for the function can be viewed as the complete solution to the universal part of the 2D Ising free energy in an external field. + +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, +universality) have yielded high-order polynomial expansions of the free energy and other universal thermodynamic 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 @@ -83,7 +84,7 @@ 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 \emph{exponentially} to the true -function. To make the construction, we review the analytic properties of the +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 @@ -99,9 +100,10 @@ With six derivatives, it is accurate to about $10^{-7}$. We hope that with some refinement, this idea might be used to establish accurate scaling functions for critical behavior in other universality classes, doing for scaling functions what advances in conformal bootstrap did for critical exponents -\cite{Gliozzi_2014_Critical}. Mathematica and Python implementations will be provided in the supplemental material. +\cite{Gliozzi_2014_Critical}. A Mathematica implementation will be provided in the supplemental material. \section{Universal scaling functions} +\label{sec:UniversalScalingFunctions} A renormalization group analysis predicts that certain thermodynamic functions will be universal in the vicinity of \emph{any} critical point in the Ising @@ -130,8 +132,16 @@ $\delta=15$ are dimensionless constants. The combination $\Delta=\beta\delta=\frac{15}8$ will appear often. The flow equations are truncated here, but in general all terms allowed by the symmetries of the parameters are present on their righthand side. By making a near-identity -transformation to the coordinates and the free energy of the form $u_t(t, -h)=t+\cdots$, $u_h(t, h)=h+\cdots$, and $u_f(f,u_t,u_h)\propto f(t,h)-f_a(t,h)$, one can bring +transformation to the coordinates and the free energy of the form +\begin{align} + \label{eq:AnalyticCOV} + u_t(t,h)=t+\cdots + && + u_h(t, h)=h+\cdots + && + u_f(f,u_t,u_h)\propto f(t,h)-f_a(t,h), +\end{align} +one can bring the flow equations into the agreed upon simplest normal form \begin{align} \label{eq:flow} \frac{du_t}{d\ell}=\frac1\nu u_t @@ -149,6 +159,7 @@ matter of convention, fixing the scale of $u_t$. Here the free energy $f=u_f+f_a Solving these equations for $u_f$ yields \begin{equation} +\label{eq:FpmF0eqns} \begin{aligned} u_f(u_t, u_h) &=|u_t|^{D\nu}\mathcal F_\pm(u_h|u_t|^{-\Delta})+\frac{|u_t|^{D\nu}}{8\pi}\log u_t^2 \\ @@ -163,7 +174,7 @@ $\mathcal F_\pm(\xi)=G_{\mathrm{high}/\mathrm{low}}(\xi)$.}. The scaling functions are universal in the sense that any system in the same universality class will share the free energy \eqref{eq:flow}, for suitable analytic functions $u_t$, $u_h$, and analytic background $f_a$ -- the singular behavior is universal up to an analytic coordinate change. %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 $u_h$. +%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 $\eta=u_t|u_h|^{-1/\Delta}$. @@ -196,7 +207,7 @@ literature \cite{Mangazeev_2010_Scaling, Clement_2019_Respect}. In the low temperature phase, the free energy has an essential singularity at zero field, which becomes a branch cut along the negative-$h$ axis when -analytically continued to negative $h$ \cite{Langer_1967_Theory}. The origin +analytically continued to negative $h$ \cite{Langer_1967_Theory, Isakov_1984_Nonanalytic}. The origin can be schematically understood to arise from a singularity that exists in the imaginary free energy of the metastable phase of the model. When the equilibrium Ising model with positive magnetization is subjected to a small @@ -235,7 +246,7 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant \begin{figure} - \includegraphics{figs/F_lower_singularities.pdf} + \includegraphics{figs/F_lower_singularities} \caption{ Analytic structure of the low-temperature scaling function $\mathcal F_-$ in the complex $\xi=u_h|u_t|^{-\Delta}\propto H$ plane. The circle @@ -282,7 +293,7 @@ branch cuts beginning at $\pm i\xi_{\mathrm{YL}}$ for a universal constant $\xi_{\mathrm{YL}}$. \begin{figure} - \includegraphics{figs/F_higher_singularities.pdf} + \includegraphics{figs/F_higher_singularities} \caption{ Analytic structure of the high-temperature scaling function $\mathcal F_+$ in the complex $\xi=u_h|u_t|^{-\Delta}\propto H$ plane. The squares @@ -417,7 +428,7 @@ entirely fixed, and it will be truncated at finite order. [0:20:0.1] '+' u ($1*f(12*-t0/16)):($1**del*g(12*-t0/16)) dt 2 lc black lw 2 , \ [0:20:0.1] '+' u ($1*f(13*-t0/16)):($1**del*g(13*-t0/16)) dt 2 lc black lw 2 , \ [0:20:0.1] '+' u ($1*f(14*-t0/16)):($1**del*g(14*-t0/16)) dt 2 lc black lw 2 , \ - [0:20:0.1] '+' u ($1*f(15*-t0/16)):($1**del*g(15*-t0/16)) dt 2 lc black lw 2 + [0:20:0.1] '+' u ($1*f(15*-t0/16)):($1**del*g(15*-t0/16)) dt 2 lc black lw 2 \end{gnuplot} \caption{ Example of the parametric coordinates. Solid lines are of constant @@ -484,7 +495,7 @@ $\theta$. Therefore, The location $\theta_0$ is not fixed by any principle. \begin{figure} - \includegraphics{figs/F_theta_singularities.pdf} + \includegraphics{figs/F_theta_singularities} \caption{ Analytic structure of the global scaling function $\mathcal F$ in the complex $\theta$ plane. The circles depict essential singularities of the @@ -541,7 +552,7 @@ Fixing these requirements for the imaginary part of $\mathcal F(\theta)$ fixes its real part up to an analytic even function $G(\theta)$, real for real $\theta$. \begin{figure} - \includegraphics{figs/contour_path.pdf} + \includegraphics{figs/contour_path} \caption{ Integration contour over the global scaling function $\mathcal F$ in the complex $\theta$ plane used to produce the dispersion relation. The @@ -594,6 +605,7 @@ Because the real part of $\mathcal F$ is even, the imaginary part must be odd. T \end{equation} Evaluating these ordinary integrals, we find for $\theta\in\mathbb R$ \begin{equation} +\label{eq:FfromFoFYLG} \operatorname{Re}\mathcal F(\theta)=\operatorname{Re}\mathcal F_0(\theta)+\mathcal F_\mathrm{YL}(\theta)+G(\theta) \end{equation} where @@ -611,6 +623,7 @@ where $\mathcal R$ is given by the function \end{equation} and \begin{equation} +\label{eq:FYL} \mathcal F_{\mathrm{YL}}(\theta)=2C_\mathrm{YL}\left[2(\theta^2+\theta_\mathrm{YL}^2)^{(1+\sigma)/2}\cos\left((1+\sigma)\tan^{-1}\frac\theta{\theta_\mathrm{YL}}\right)-\theta_\mathrm{YL}^{1+\sigma}\right] \end{equation} We have also included the analytic part $G$, which we assume has a simple @@ -1101,7 +1114,7 @@ Notice that this infelicity does not appear to cause significant errors in the f function of polynomial order $m$, rescaled by their asymptotic limit $\mathcal F_-^\infty(m)$ from \eqref{eq:low.asymptotic}. The numeric values are from Table \ref{tab:data}, and those of Caselle \textit{et al.} are - from the most accurate scaling function listed in \cite{Caselle_2001_The}. Note that our $n=6$ fit generates significant deviations in polynomial coefficients $m$ above around 10. + from the most accurate scaling function listed in \cite{Caselle_2001_The}. Note that our $n=6$ fit generates significant deviations in polynomial coefficients $m$ above around 10. } \label{fig:glow.series.scaled} \end{figure} @@ -1197,14 +1210,21 @@ the ratio. We have introduced explicit approximate functions forms for the two-dimensional Ising universal scaling function in the relevant variables. These functions are smooth to all orders, include the correct singularities, and appear to converge -exponentially to the function as they are fixed to larger polynomial order. +exponentially to the function as they are fixed to larger polynomial order. The universal scaling function will be available in Mathematica in the supplemental material. It is implicitly defined by $\mathcal{F}_0$ and $\mathcal{F}_\pm$ in Eq.~\eqref{eq:scaling.function.equivalences.2d}, where $g(\theta)$ is defined in Eq.~\eqref{eq:schofield.funcs}, $\mathcal{F}$ in Eqs.~\eqref{eq:FfromFoFYLG}--\eqref{eq:FYL}, and the fit constants at various levels of approximation are given in Table~\ref{tab:fits}. This method, although spectacularly successful, could be improved. It becomes difficult to fit the unknown functions at progressively higher order due to the complexity of the chain-rule derivatives, and we find an inflation of predicted coefficients in our higher-precision fits. These problems may be related to the precise form and method of truncation for the unknown functions. -The successful smooth description of the Ising free energy produced in part by +It would be natural to extend our approach to the 3D Ising model, where enough high-precision information is available to provide the first few levels of approximation. In 3D, there is an important singular correction to scaling, which could be incorporated as a third invariant scaling variable in the universal scaling function. Indeed, it is believed that there are singular corrections to scaling also in 2D, which happen to vanish for the exactly solvable models~\cite{BarmaFisherPRB}. + +Derivatives of our Ising free energy provides most bulk thermodynamic properties, but not the correlation functions. The 2D Ising correlation function has been estimated~\cite{ChenPMSnn}, but without incorporating the effects of the essential singularity as one crosses the abrupt transition line. This correlation function would be experimentally useful, for example, in analyzing FRET data for two-dimensional membranes. + +It is interesting to note the close analogy between our analysis and the incorporation of analytic corrections to scaling discussed in section~\ref{sec:UniversalScalingFunctions}. Here the added function $G(\theta)$ corresponds to the analytic part of the free energy $f_a(t,h)$, and the coordinate change $g(\theta)$ corresponds to the scaling field change of variables $u_t(t,h)$ and $u_h(t,h)$ +(Eqs.~\ref{eq:AnalyticCOV} and~\ref{eq:FpmF0eqns}). One might view the universal scaling form for the Ising free energy as a scaling function describing the crossover scaling between the universal essential singularities at the two abrupt, `first-order' transition at $\pm H$, $T<T_c$. + +Finally, the successful smooth description of the Ising free energy produced in part by analytically continuing the singular imaginary part of the metastable free energy inspires an extension of this work: a smooth function that captures the universal scaling \emph{through the coexistence line and into the metastable |