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@@ -4,4 +4,8 @@ *.blg *Notes.bib *.dvi +*.fdb_latexmk +*.synctex.gz +*.fls +*.out /*.pdf diff --git a/essential-ising.tex b/essential-ising.tex deleted file mode 100644 index e125229..0000000 --- a/essential-ising.tex +++ /dev/null @@ -1,478 +0,0 @@ - -% -% Created by Jaron Kent-Dobias on Thu Apr 20 12:50:56 EDT 2017. -% Copyright (c) 2017 Jaron Kent-Dobias. All rights reserved. -% -\documentclass[aps,prl,reprint]{revtex4-1} - -\usepackage[utf8]{inputenc} -\usepackage{amsmath,amssymb,latexsym,mathtools,xifthen} - -% uncomment to label only equations that are referenced in the text -%\mathtoolsset{showonlyrefs=true} - -% I want labels but don't want to type out ``equation'' -\def\[{\begin{equation}} -\def\]{\end{equation}} - -% math not built-in -\def\arcsinh{\mathop{\mathrm{arcsinh}}\nolimits} -\def\arccosh{\mathop{\mathrm{arccosh}}\nolimits} -\def\ei{\mathop{\mathrm{Ei}}\nolimits} % exponential integral Ei -\def\re{\mathop{\mathrm{Re}}\nolimits} -\def\im{\mathop{\mathrm{Im}}\nolimits} -\def\sgn{\mathop{\mathrm{sgn}}\nolimits} -\def\dd{d} % differential -\def\O{O} % big O -\def\o{o} % little O - -% subscript for ``critical'' values, e.g., T_\c -\def\c{\mathrm c} - -% scaling functions -\def\fM{\mathcal M} % magnetization -\def\fX{\mathcal Y} % susceptibility -\def\fXt{\check{\mathcal Y}} % susceptibility -\def\fF{\mathcal F} % free energy -\def\fiF{\mathcal H} % imaginary free energy -\def\fS{\mathcal S} % surface tension -\def\fG{\mathcal G} % exponential factor - -% lattice types -\def\sq{\mathrm{sq}} -\def\tri{\mathrm{tri}} -\def\hex{\mathrm{hex}} - -% dimensions -\def\dim{d} -\def\twodee{\textsc{2d}} -\def\threedee{\textsc{3d}} -\def\fourdee{\textsc{4d}} - -% fancy partial derivative -\newcommand\pd[3][]{ - \ifthenelse{\isempty{#1}} - {\def\tmp{}} - {\def\tmp{^#1}} - \frac{\partial\tmp#2}{\partial#3\tmp} -} - -% used to reformat display math to fit in two-column ``reprint'' mode -\makeatletter -\newif\ifreprint -\@ifclasswith{revtex4-1}{reprint}{\reprinttrue}{\reprintfalse} -\makeatother - -\begin{document} - -\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} -\author{Jaron Kent-Dobias} -\author{James P.~Sethna} -\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} - -\date\today - -\begin{abstract} - Renormalization group ideas and results from critical droplet theory are - used to construct a scaling ansatz for the imaginary component of the free - energy of an Ising model in its metastable state close to the critical - point. The analytic properties of the free energy are used to determine - scaling functions for the free energy in the vicinity of the critical point - and the abrupt transition line. These functions have essential singularities - at the abrupt transition. Analogous forms for the magnetization and susceptibility in - two dimensions are fit to numeric data and show good agreement, especially - when their nonsingular behavior is modified to match existing numeric results. -\end{abstract} - -\maketitle - -The Ising model is the canonical example of a system with a continuous phase -transition, and the study of its singular properties marked the first success -of the renormalization group (\textsc{rg}) method in statistical physics -\cite{wilson.1971.renormalization}. Its status makes sense: it's a simple -model whose continuous phase transition contains all the essential features of -more complex ones, but admits \textsc{rg} methods in a straightforward way and -has exact solutions in certain dimensions and for certain parameter -restrictions. However, the Ising critical point is not simply a continuous -transition: it also ends a line of abrupt phase transitions extending from it -at zero field below the critical temperature. Though typically neglected in -\textsc{rg} scaling analyses of the critical point, we demonstrate that there -are numerically measurable contributions to scaling due to the abrupt -transition line that cannot be accounted for by analytic changes of control or -thermodynamic variables. - -\textsc{Rg} analysis predicts that the singular part of the free energy per -site $F$ as a function of reduced temperature $t=1-T_\c/T$ and field $h=H/T$ -in the vicinity of the critical point takes the scaling form -$F(t,h)=|t|^{2-\alpha}\fF(h|t|^{-\beta\delta})$ for the low temperature phase -$t<0$ \cite{cardy.1996.scaling}. When studying the properties of the Ising -critical point, it is nearly always assumed that the universal scaling -function $\fF$ is analytic, i.e., has a convergent Taylor series. However, it -has long been known that there exists an essential singularity in $\fF$ at -zero argument, though its effects have long been believed to be unobservable -\cite{fisher.1967.condensation}, or simply just neglected -\cite{guida.1997.3dising,schofield.1969.parametric,schofield.1969.correlation,caselle.2001.critical,josephson.1969.equation,fisher.1999.trigonometric}. -With careful analysis, we have found that assuming the presence of the -essential singularity is predictive of the scaling form of, for instance, the -susceptibility and magnetization. - -The provenance of the essential singularity can be understood using the -methods of critical droplet theory for the decay of an Ising system in a -metastable state, i.e., an equilibrium Ising state for $T<T_\c$, $H>0$ -subjected to a small negative external field $H<0$. The existence of an -essential singularity has also been suggested by transfer matrix -\cite{mccraw.1978.metastability,enting.1980.investigation,mangazeev.2008.variational,mangazeev.2010.scaling} and \textsc{rg} -methods \cite{klein.1976.essential}, and a different kind of essential -singularity is known to exist in the zero-temperature susceptibility -\cite{orrick.2001.susceptibility,chan.2011.ising,guttmann.1996.solvability,nickel.1999.singularity,nickel.2000.addendum,assis.2017.analyticity}. It has long been known that the decay -rate $\Gamma$ of metastable states in statistical mechanics is often related -to the metastable free energy $F$ by $\Gamma\propto\im F$ -\cite{langer.1969.metastable,penrose.1987.rigorous,gaveau.1989.analytic,privman.1982.analytic}. -`Metastable free energy' can be thought of as either an analytic continuation -of the free energy through the abrupt phase transition, or restriction of the -partition function trace to states in the vicinity of the local free energy -minimum that characterizes the metastable state. In any case, the free energy -develops a nonzero imaginary part in the metastable region. Heuristically, -this can be thought of as similar to what happens in quantum mechanics with a -non-unitary Hamiltonian: the imaginary part describes loss rate of probability -that the system occupies any `accessible' state, which corresponds to decay. - -In critical droplet theory, the metastable state decays when a domain of the -equilibrium state forms whose surface-energy cost for growth is outweighed by -bulk-energy gains. There is numerical evidence that, near the critical point, -these droplets are spherical \cite{gunther.1993.transfer-matrix}. The free energy -cost of the surface of a droplet of radius $R$ is $\Sigma S_\dim R^{\dim-1}$ -and that of its bulk is $-M|H|V_\dim R^\dim$, where $S_\dim$ and $V_\dim$ are -the surface area and volume of a $(\dim-1)$-sphere, respectively, and $\Sigma$ -is the surface tension of the equilibrium--metastable interface. The critical -droplet size is then $R_\c=(\dim-1)\Sigma/M|H|$ and the free energy of the -critical droplet is $\Delta F_\c=\pi^{\dim/2}\Sigma^\dim((\dim-1)/M|H|)^{\dim-1}/\Gamma(1+\dim/2)$. -Assuming the typical singular scaling forms -$\Sigma/T=|t|^\mu\fS(h|t|^{-\beta\delta})$ and $M=|t|^\beta\mathcal -M(h|t|^{-\beta\delta})$ and using known hyperscaling relations -\cite{widom.1981.interface}, this implies a scaling form -\def\eqcritformone{ - T\frac{\pi^{\dim/2}(\dim-1)^{\dim-1}}{\Gamma(1+\dim/2)} - \frac{\fS^\dim(h|t|^{-\beta\delta})}{(-h|t|^{-\beta\delta} - \fM(h|t|^{-\beta\delta}))^{\dim-1}} -} -\def\eqcritformtwo{ - T\fG^{-(\dim-1)}(h|t|^{-\beta\delta}) -} -\ifreprint -\[ - \begin{aligned} - \Delta F_\c - &=\eqcritformone\\ - &\sim\eqcritformtwo - \end{aligned} -\] -\else -\[ - \Delta F_\c=\eqcritformone\sim\eqcritformtwo -\] -\fi -for the free energy change due to the critical droplet. -Since both surface tension and magnetization are finite and nonzero for $H=0$ -at $T<T_\c$, $\fG(X)=-BX+\O(X^2)$ for small negative $X$ with -\[ - B=\frac{\fM(0)}{\dim-1}\bigg(\frac{\Gamma(1+\dim/2)} - {\pi^{\dim/2}\fS(0)^\dim}\bigg)^{1/(\dim-1)}. -\] -This first term in the scaling function $\fG$ is related to the ratio between -the correlation length $\xi$ and the critical domain radius $R_\c$, with -\[ - Bh|t|^{-\beta\delta} - =\frac\xi{R_\c}\bigg(\frac{\Gamma(1+\dim/2)} - {\pi^{\dim/2}\fS(0)(\xi_0^-)^{\dim-1}}\bigg)^{1/(\dim-1)} -\] -where the critical amplitude for the correlation length $\xi_0^-$ is defined -by $\xi=\xi_0^-|t|^{-\nu}$ for $t<T_\c$ and $H=0$. Since $\fS(0)(\xi_0^-)^{\dim-1}$ is a -universal amplitude ratio \cite{zinn.1996.universal}, -$(Bh|t|^{-\beta\delta})/(\xi/R_\c)$ is a universal quantity. The decay rate -of the metastable state is proportional to the Boltzmann factor corresponding -to the -creation of a critical droplet, yielding -\[ - \im F\sim\Gamma\propto e^{-\Delta F_\c/T} - =e^{-\fG(h|t|^{-\beta\delta})^{-(\dim-1)}}. -\] -For $d>1$ this function has an essential singularity in the invariant -combination $h|t|^{-\beta\delta}$. - -This form of $\im F$ for small $h$ is well known -\cite{langer.1967.condensation,harris.1984.metastability}. We make the -scaling ansatz that the imaginary part of the metastable free energy has the -same singular behavior as the real part of the equilibrium free energy, and -that for small $t$, $h$, $\im F(t,h)=|t|^{2-\alpha}\fiF(h|t|^{-\beta\delta})$, -where -\[ - \fiF(X)=A\Theta(-X)(-BX)^be^{-1/(-BX)^{\dim-1}} - \label{eq:im.scaling} -\] -and $\Theta$ is the Heaviside function. Results from combining an analysis of -fluctuations on the surface of critical droplets with \textsc{rg} recursion -relations suggest that $b=-(d-3)d/2$ for $d=2,4$ and $b=-7/3$ for $d=3$ -\cite{houghton.1980.metastable,rudnick.1976.equations,gunther.1980.goldstone}. -Assuming that $F$ is analytic in the upper complex-$h$ plane, the real part of -$F$ in the equilibrium state can be extracted from this imaginary metastable -free energy using the Kramers--Kronig relation -\[ - \re F(t,h)=\frac1\pi\int_{-\infty}^\infty\frac{\im F(t,h')}{h'-h}\,\dd h'. - \label{eq:kram-kron} -\] -This relationship has been used to compute high-order moments of the free -energy with $H$ in good agreement with transfer matrix expansions -\cite{lowe.1980.instantons}. Here, we evaluate the integral explicitly to come -to -functional forms. In \threedee\ and \fourdee\ this can be done directly -given our scaling ansatz, yielding -\def\eqthreedeeone{ - \fF^\threedee(Y/B)&= - \frac{A}{12}\frac{e^{-1/Y^2}}{Y^2} - \bigg[\Gamma(\tfrac16)E_{7/6}(-Y^{-2}) -} -\def\eqthreedeetwo{ - -4Y\Gamma(\tfrac23)E_{5/3}(-Y^{-2})\bigg] -} -\def\eqfourdeeone{ - \fF^\fourdee(Y/B)&= - -\frac{A}{9\pi}\frac{e^{1/Y^3}}{Y^2} - \Big[3\ei(-Y^{-3}) -} -\def\eqfourdeetwo{ - +3\Gamma(\tfrac23)\Gamma(\tfrac13,Y^{-3}) - +\Gamma(\tfrac13)\Gamma(-\tfrac13,Y^{-3})\Big] -} -\ifreprint -\begin{align} - &\begin{aligned} - \eqthreedeeone\\ - &\hspace{7em} - \eqthreedeetwo - \end{aligned} - \\ - &\begin{aligned} - \eqfourdeeone - \\ - &\hspace{-0.5em} - \eqfourdeetwo, - \end{aligned} -\end{align} -\else -\begin{align} - \eqthreedeeone\eqthreedeetwo - \\ - \eqfourdeeone\eqfourdeetwo, -\end{align} -\fi -where $E_n$ is the generalized exponential integral and $\Gamma(x,y)$ is the -incomplete gamma function. -At the level of truncation of \eqref{eq:im.scaling} at which we are working -the Kramers--Kronig relation does not converge in \twodee. However, higher -moments can still be extracted, e.g., the susceptibility, by taking -\[ - \chi=\pd MH=-\frac1{T}\pd[2]Fh - =-\frac2{\pi T}\int_{-\infty}^\infty\frac{\im F(t,h')}{(h'-h)^3}\,\dd h'. -\] -With a scaling form defined by $T\chi=|t|^{-\gamma}\fX(h|t|^{-\beta\delta})$, -this yields -\[ - \fX^\twodee(Y/B)=\frac{AB^2}{\pi Y^3}\big[Y(Y-1)-e^{1/Y}\ei(-1/Y)\big] - \label{eq:sus_scaling} -\] -Scaling forms for the free energy can then be extracted by direct integration -and their constants of integration fixed by known zero-field values, yielding -\begin{align} - \label{eq:mag_scaling} - \fM^\twodee(Y/B) - &=\fM(0)+\frac{AB}{\pi}\bigg(1-\frac{Y-1}Ye^{1/Y}\ei(-1/Y)\bigg)\\ - \fF^\twodee(Y/B) - &=-Y\bigg(\frac{\fM(0)}B-\frac{A}\pi e^{1/Y}\ei(-1/Y)\bigg) - \label{eq:2d_free_scaling} -\end{align} -with $F(t,h)=t^2\fF(h|t|^{-15/8})+t^2\log t^2$ in -two dimensions, as $\alpha=0$ and $\beta\delta=\frac{15}8$. - -How are these functional forms to be interpreted? Though the scaling function -\eqref{eq:im.scaling} for the imaginary free energy of the metastable state is -asymptotically correct sufficiently close to the critical point, the results -of the integral relation \eqref{eq:kram-kron} are not, since there is no limit -of $t$ or $h$ in which it becomes arbitrarily correct for a given truncation -of \eqref{eq:im.scaling}. It is well established that this method of using -unphysical or metastable elements of a theory to extract properties of the -stable or equilibrium theory is only accurate for high moments of those -predictions \cite{parisi.1977.asymptotic,bogomolny.1977.dispersion}. The -functions above should be understood as possessing exactly the correct -singularity at the coexistence line, but requiring polynomial corrections, -especially for smaller integer powers. Using these forms in conjunction with -existing methods of describing the critical equation of state or critical -properties with analytic functions in $h$ will incorporate these low-order -corrections while preserving the correct singular structure. In other words, -the scaling functions can be \emph{exactly} described by -$\tilde\fF(X)=\fF(X)+f(X)$ for some analytic function $f$. Higher order terms -in the expansion of $\tilde\fF$ become asymptotically equal to those of $\fF$ -because, as an analytic function, progressively higher order terms of $f$ must -eventually become arbitrarily small \cite{flanigan.1972.complex}. - -How predictive are these scaling forms in the proximity of the critical point -and the abrupt transition line? We simulated the \twodee\ Ising model on square -lattice using a form of the Wolff algorithm modified to remain efficient in -the presence of an external field. Briefly, the external field $H$ is applied -by adding an extra spin $s_0$ with coupling $|H|$ to all others -\cite{dimitrovic.1991.finite}. A quickly converging estimate for the -magnetization in the finite-size system was then made by taking -$M=\sgn(H)s_0\sum s_i$, i.e., the magnetization relative to the external spin -\cite{kent-dobias.2018.wolff}. -Data was then taken for susceptibility and magnetization for -$T_\c-T,H\leq0.1$. This data, rescaled as appropriate to collapse onto a -single curve, is plotted in Fig.~\ref{fig:scaling_fits}. - -For the \twodee\ Ising model on a square lattice, exact results at zero -temperature have $\fS(0)=4/T_\c$, $\fM(0)=(2^{5/2}\arcsinh1)^\beta$ -\cite{onsager.1944.crystal}, and $\fX(0)=C_0^-=0.025\,536\,971\,9$ \cite{barouch.1973.susceptibility}, so that -$B=T_\c^2\fM(0)/\pi\fS(0)^2=(2^{27/16}\pi(\arcsinh1)^{15/8})^{-1}$. If we -assume incorrectly that \eqref{eq:sus_scaling} is the true asymptotic form of -the susceptibility scaling function, then -$T\chi(t,0)|t|^\gamma=\lim_{X\to0}\fX^\twodee(X)=2AB^2/\pi$ and the constant -$A$ is fixed to $A=\pi\fX(0)/2B^2=2^{19/8}\pi^3(\arcsinh1)^{15/4}C_0^-$. The -resulting scaling functions $\fX$ and $\fM$ are plotted as solid blue lines in -Fig.~\ref{fig:scaling_fits}. Though there is good agreement -between our functional forms and what is measured, there -are systematic differences that can be seen most clearly in the -magnetization. This is to be expected based on our earlier discussion: these -scaling forms should only be expected to well-describe the singularity at the -abrupt transition. Our forms both exhibit incorrect low-order -coefficients at the transition (Fig.~\ref{fig:series}) and incorrect -asymptotics as $h|t|^{-\beta\delta}$ becomes very large. - -In forthcoming work, -we develop a method to incorporate the essential singularity in the scaling -functions into a form that also incorporates known properties of the scaling -functions in the rest of the configuration space using a Schofield-like -parameterization \cite{schofield.1969.parametric,caselle.2001.critical,kent-dobias.2018.parametric}. Briefly, we define -parameters $R$ and $\theta$ by -\begin{align} - t=R(1-\theta^2) - && - h=h_0R^{\beta\delta}g(\theta) -\end{align} -where $h_0$ is a constant and $f$ is an arbitrary odd function whose first -finite zero $\theta_\c>1$ corresponds to the abrupt transition. In these -coordinates the invariant combination $h|t|^{-\beta\delta}$ is given by -\[ - h|t|^{-\beta\delta}=\frac{h_0g(\theta)}{|1-\theta^2|^{\beta\delta}}=\frac{h_0(-g'(\theta_\c))}{(\theta_c^2-1)^{\beta\delta}}(\theta_\c-\theta) - +\O\big( (\theta_\c-\theta)^2\big), -\] -an analytic function of $\theta$ about $\theta_\c$. -The simplest -function of the coordinate $\theta$ that exhibits the correct singularity at -the abrupt transitions at $h=\pm0$, $t<0$ is -\[ - \fX(\theta)=\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c-\theta)\bigg)+\fX\bigg(\frac{h_0(-g'(\theta_\c))}{(\theta_\c^2-1)^{\beta\delta}}(\theta_\c+\theta)\bigg) -\] -This function is analytic in the range $-\theta_\c<\theta<\theta_\c$. -In order to correct its low-order behavior to match that expected, we -make use of both the freedom of the coordinate transformation $g$ and an -arbitrary analytic additive function $Y$, -\begin{align} - g(\theta)=\bigg(1-\frac{\theta^2}{\theta_\c^2}\bigg)\sum_{n=0}^\infty - g_n\theta^{2n+1} - && - Y(\theta)=\sum_{n=0}^\infty Y_n\theta^{2n} -\end{align} -so that $\tilde\fX(\theta)=\fX(\theta)+Y(\theta)$. By manipulating these -coefficients, we can attempt to give the resulting scaling form a series -expansion consistent with known values. One such prediction---made by fixing -the first four terms in the low-temperature, critical isotherm, and -high-temperature expansions -of $\tilde\fX$---is shown as a dashed yellow line in -Fig.~\ref{fig:scaling_fits}. -As shown in Fig.~\ref{fig:series}, the low-order free energy coefficients of this prediction match known values -exactly up to $n=5$, and improve the agreement with higher-order coefficients. -Unlike scaling forms which treat $\fX$ as analytic at the coexistence line, -the series coefficients of the scaling form developed here increase without -bound at high order. - - -\begin{figure} - \input{fig-susmag} - \caption{ - Scaling functions for (top) the susceptibility and (bottom) the - magnetization plotted in terms of the invariant combination - $h|t|^{-\beta\delta}$. Points with error bars show data with sampling - error taken from simulations of a $4096\times4096$ square-lattice Ising - model with periodic boundary conditions and $T_\c-T=0.01,0.02,\ldots,0.1$ - and $H=0.1\times(1,2^{-1/4},\ldots,2^{-50/4})$. The solid blue lines - show our analytic results \eqref{eq:sus_scaling} and - \eqref{eq:mag_scaling}, the dashed yellow lines show - a scaling function modified to match known series expansions of the - susceptibility - to third order, and the - dotted green lines show the - polynomial resulting from truncating the known series expansion after the eight terms - reported by \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. - } - \label{fig:scaling_fits} -\end{figure} - -\begin{figure} - \input{fig-series} - \caption{ - The series coefficients defined by $\tilde\fF(X)=\sum_nf_nX^n$. The blue - pluses correspond to the scaling form \eqref{eq:2d_free_scaling}, the - yellow saltires correspond to a scaling function modified to match known - series expansions of the susceptibility to third order---and therefore - the free energy to fifth order---and the green - stars - correspond to the first eight coefficients from - \cite{mangazeev.2008.variational,mangazeev.2010.scaling}. The modified - scaling function and the known coefficients match exactly up to $n=5$. - } - \label{fig:series} -\end{figure} - -Abrupt phase transitions, such as the jump in magnetization in the Ising -model below $T_\c$, are known to imply essential singularities in the free -energy that are usually thought to be unobservable in practice. We have -argued that this essential singularity controls the universal scaling -behavior near continuous phase transitions, and have derived an explicit -analytical form for the singularity in the free energy, magnetization, -and susceptibility for the Ising model. We have developed a Wolff algorithm -for the Ising model in a field, and showed that incorporating our singularity -into the scaling function gives good convergence to the simulations in \twodee. - -Our results should allow improved high-precision functional forms for the free -energy~\cite{caselle.2001.critical}, and should have implications for the scaling -of correlation functions~\cite{chen.2013.universal,wu.1976.spin}. Our methods might be generalized -to predict similar singularities in systems where nucleation and metastability -are proximate to continuous phase transitions, such as 2D superfluid -transitions~\cite{ambegaokar.1978.dissipation,ambegaokar.1980.dynamics}, the melting of 2D crystals~\cite{dahm.1989.dynamics}, and -freezing transitions in glasses, spin glasses, and other disordered systems. - - -%We have used results from the properties of the metastable Ising ferromagnet -%and the analytic nature of the free energy to derive universal scaling -%functions for the free energy, and in \twodee the magnetization and -%susceptibility, in the limit of small $t<0$ and $h$. Because of an essential -%singularity in these functions at $h=0$---the abrupt transition line---their -%form cannot be brought into that of any regular function by analytic -%redefinition of control or thermodynamic variables. These predictions match -%the results of simulations well. Having demonstrated that the essential -%singularity in thermodynamic functions at the abrupt transition leads to -%observable scaling effects, we hope that these functional forms will be used in -%conjunction with traditional perturbation methods to better express the -%equation of state of the Ising model in the whole of its parameter space. - -\begin{acknowledgments} - The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien - for helpful conversations. The authors would also like to think Jacques Perk - for pointing us to several insightful studies. JPS thanks Jim Langer for past inspiration, - guidance, and encouragement. This work was supported by NSF grants - DMR-1312160 and DMR-1719490. -\end{acknowledgments} - -\bibliography{essential-ising} - -\end{document} - diff --git a/ising_scaling.bib b/ising_scaling.bib new file mode 100644 index 0000000..36fdbe9 --- /dev/null +++ b/ising_scaling.bib @@ -0,0 +1,57 @@ +@article{Butera_2011_Free, + author = {Butera, P. and Pernici, M.}, + title = {Free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model}, + journal = {Phys. Rev. B}, + publisher = {American Physical Society}, + year = {2011}, + month = {2}, + volume = {83}, + pages = {054433}, + url = {https://link.aps.org/doi/10.1103/PhysRevB.83.054433}, + doi = {10.1103/PhysRevB.83.054433}, + issue = {5}, + numpages = {15} +} + +@article{Campostrini_2000_Critical, + author = {Campostrini, Massimo and Pelissetto, Andrea and Rossi, Paolo and Vicari, Ettore}, + title = {Critical equation of state of three-dimensional {$XY$} systems}, + journal = {Physical Review B}, + publisher = {American Physical Society ({APS})}, + year = {2000}, + month = {9}, + number = {9}, + volume = {62}, + pages = {5843--5854}, + url = {https://doi.org/10.1103%2Fphysrevb.62.5843}, + doi = {10.1103/physrevb.62.5843} +} + +@article{Mangazeev_2008_Variational, + author = {Mangazeev, Vladimir V and Batchelor, Murray T and Bazhanov, Vladimir V and Dudalev, Michael Yu}, + title = {Variational approach to the scaling function of the 2D Ising model in a magnetic field}, + journal = {Journal of Physics A: Mathematical and Theoretical}, + publisher = {IOP Publishing}, + year = {2008}, + month = {12}, + number = {4}, + volume = {42}, + pages = {042005}, + url = {https://doi.org/10.1088%2F1751-8113%2F42%2F4%2F042005}, + doi = {10.1088/1751-8113/42/4/042005} +} + +@article{Mangazeev_2010_Scaling, + author = {Mangazeev, Vladimir V. and Dudalev, Michael Yu. and Bazhanov, Vladimir V. and Batchelor, Murray T.}, + title = {Scaling and universality in the two-dimensional Ising model with a magnetic field}, + journal = {Physical Review E}, + publisher = {American Physical Society (APS)}, + year = {2010}, + month = {6}, + number = {6}, + volume = {81}, + url = {https://doi.org/10.1103%2Fphysreve.81.060103}, + doi = {10.1103/physreve.81.060103} +} + + diff --git a/ising_scaling.tex b/ising_scaling.tex new file mode 100644 index 0000000..b7b76b2 --- /dev/null +++ b/ising_scaling.tex @@ -0,0 +1,42 @@ +\documentclass[aps,prb,reprint,longbibliography,floatfix]{revtex4-2} + +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{newtxtext, newtxmath} +\usepackage[ + colorlinks=true, + urlcolor=purple, + citecolor=purple, + filecolor=purple, + linkcolor=purple +]{hyperref} % ref and cite links with pretty colors + +\usepackage{amsmath, graphicx, xcolor} + +\begin{document} + +\title{Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line} +\author{Jaron Kent-Dobias} +\author{James P.~Sethna} +\affiliation{Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA} + +\date\today + +\begin{abstract} +\end{abstract} + +\maketitle + +\cite{Campostrini_2000_Critical} + +\begin{acknowledgments} + The authors would like to thank Tom Lubensky, Andrea Liu, and Randy Kamien + for helpful conversations. The authors would also like to think Jacques Perk + for pointing us to several insightful studies. JPS thanks Jim Langer for past + inspiration, guidance, and encouragement. This work was supported by NSF + grants DMR-1312160 and DMR-1719490. +\end{acknowledgments} + +\bibliography{ising_scaling} + +\end{document} |