# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1907.04748