Spin-spin correlations in central rows of Ising models with holes

TL;DR

This paper investigates spin-spin correlations in the central rows of layered Ising models with holes, expressing them as Toeplitz determinants and analyzing their asymptotic behavior near criticality.

Contribution

It introduces a new method to compute correlations in layered Ising models with holes using Toeplitz determinants and explores their asymptotic properties.

Findings

Correlations can be expressed as Toeplitz determinants.

Near critical temperature, behaviors are two-dimensional Ising-like.

Different layer sizes affect non-critical correlation behaviors.

Abstract

In our previous works on infinite horizontal Ising strips of width $m$ alternating with layers of strings of Ising chains of length $n$, we found the surprising result that the specific heats are not much different for different values of $N$, the separation of the strings. For this reason, we study here for $N=1$ the spin-spin correlation in the central row of each strip, and also the central row of a strings layer. We show that these can be written as a Toeplitz determinants. Their generating functions are ratios of two polynomials, which in the limit of infinite vertical size become square roots of polynomials whose degrees are $m+1$ where $m$ is the size of the strips. We find the asymptotic behaviors near the critical temperature to be two-dimensional Ising-like. But in regions not very close to criticality the behavior may be different for different $m$ and $n$. Finally, in the appendix we shall present results for generating functions in more general models.