Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region

TLDR

The authors aim to derive exact expressions for the scaling functions of the two‑dimensional Ising spin‑spin correlation function, expressed via a Painlevé III transcendent, and to analyze their small‑ and large‑t behavior. They compute the exact spin‑spin correlation functions for the square‑lattice Ising model in zero field, then study these functions in the scaling limit \(T\to T_c\) with \(M^2+N^2\to\infty\) at fixed \((T-T_c)\), deriving the scaling functions in terms of Painlevé III and examining their asymptotics. In this scaling regime the correlation function admits the universal expansion \(\langle\sigma_{0,0}\sigma_{M,N}\rangle=R^{-1/4}F_{\pm}(t)+R^{-5/4}F_{1\pm}(t)+\dots\); the authors supply ten‑digit tables for \(F_{\pm}(t)\) and use the results to determine the leading coefficients \(C_{0\pm}\) and \(C_{1\pm}\) in the zero‑field susceptibility expansion near \(T_c\).

Abstract

We compute exactly the spin-spin correlation functions $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ for the two-dimensional Ising model on a square lattice in zero magnetic field for $T>{T}_{c}$ and $T<{T}_{c}$. We then analyze the correlation functions in the scaling limit $T\ensuremath{\rightarrow}{T}_{c},{M}^{2}+{N}^{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that $(T\ensuremath{-}{T}_{c})$ is fixed. In this scaling limit $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉={R}^{\ensuremath{-}\frac{1}{4}}{F}_{\ifmmode\pm\else\textpm\fi{}}(t)+{R}^{\ensuremath{-}\frac{5}{4}}{F}_{1\ifmmode\pm\else\textpm\fi{}}(t)+o({R}^{\ensuremath{-}\frac{5}{4}})$, where $t$ is the scaling variable $\frac{R}{\ensuremath{\xi}}$ and ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ and ${F}_{1\ifmmode\pm\else\textpm\fi{}}(t)$ are the scaling functions ($\ensuremath{\xi}$ is the correlation length). We derive exact expressions for these scaling functions, in terms of a Painlev\'e function of the third kind and analyze both the small- and large-$t$ behavior. A table of values for ${F}_{\ifmmode\pm\else\textpm\fi{}}(t)$ (good to ten significant digits) is also given. As an application we computer the coefficients ${C}_{0\ifmmode\pm\else\textpm\fi{}}$ and ${C}_{1\ifmmode\pm\else\textpm\fi{}}$ in the expansion ${k}_{B}T\ensuremath{\chi}(T)={C}_{0\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{7}{4}}+{C}_{1\ifmmode\pm\else\textpm\fi{}}{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{3}{4}}+O(1)$ of the zero-field susceptibility $\ensuremath{\chi}(T)$ as $T\ensuremath{\rightarrow}{T}_{c}^{\ifmmode\pm\else\textpm\fi{}}$.