On General-n Coefficients in Series Expansions for Row Spin-Spin Correlation Functions in the Two-Dimensional Ising Model

TL;DR

This paper develops a method to compute general coefficients in series expansions of row spin-spin correlation functions in the 2D Ising model, aiding the search for differential equations governing these correlations.

Contribution

It introduces a new approach for calculating high-order coefficients in series expansions of row correlations in the 2D Ising model, extending understanding of their mathematical structure.

Findings

Derived expressions for higher-order coefficients in series expansions.

Provided insights that may help find differential equations for correlation functions.

Enhanced computational methods for analyzing spin-spin correlations.

Abstract

We consider spin-spin correlation functions for spins along a row, $R_n = \langle \sigma_{0,0}\sigma_{n,0}\rangle$, in the two-dimensional Ising model. We discuss a method for calculating general-$n$ expressions for coefficients in high-temperature and low-temperature series expansions of $R_n$ and apply it to obtain such expressions for several higher-order coefficients. In addition to their intrinsic interest, these results could be useful in the continuing quest for an ordinary differential equation whose solution would determine $R_n$, analogous to the known ordinary differential equation whose solution determines the diagonal correlation function $\langle \sigma_{0,0}\sigma_{n,n}\rangle$ in this model.