The Ising correlation $C(M,N)$ for $\nu=-k$

TL;DR

This paper derives Painlevé VI sigma form equations for the Ising model's two-point correlation functions at special temperature conditions, revealing new nonlinear ODEs and a remarkable sum property of sigma functions.

Contribution

It introduces four new nonlinear ODEs for the Ising correlation functions at $ u=-k$, and uncovers a sum property of Painlevé VI sigma functions at this special case.

Findings

Derivation of four nonlinear ODEs depending on $M$ and $N$

Identification of a sum of four Painlevé VI sigma functions for $C(0,N)$ with odd $N$

Representation of $C(M,N)$ as an $N imes N$ Toeplitz determinant for $ u=-k$

Abstract

We present Painlev{\'e} VI sigma form equations for the general Ising low and high temperature two-point correlation functions $ C(M,N)$ with $M \leq N $ in the special case $\nu = -k$ where $\nu = \, \sinh 2E_h/k_BT/\sinh 2E_v/k_BT$. More specifically four different non-linear ODEs depending explicitly on the two integers $M $ and $N$ emerge: these four non-linear ODEs correspond to distinguish respectively low and high temperature, together with $ M+N$ even or odd. These four different non-linear ODEs are also valid for $M \ge N$ when $ \nu = -1/k$. For the low-temperature row correlation functions $ C(0,N)$ with $ N$ odd, we exhibit again for this selected $\nu = \, -k$ condition, a remarkable phenomenon of a Painlev\'e VI sigma function being the sum of four Painlev\'e VI sigma functions having the same Okamoto parameters. We show in this $\nu = \, -k$ case for $ T < T_c $ and also $ T > T_c$, that $ C(M,N)$ with $ M \leq N $ is given as an $ N \times N$ Toeplitz determinant.