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arXiv:1907.04748

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

Helen Au-Yang, Jacques H. H. Perk

Published Jul 10, 2019• 0 citations
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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.

DOI
10.48550/arxiv.1907.04748

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In our previous works on infinite horizontal Ising strips of width mmm alternating with layers of strings of Ising chains of length nnn, we found the surprising result that the specific heats are not much different for different values of NNN, the separation of the strings. For this reason, we study here for N=1N=1N=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+1m+1m+1 where mmm 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 mmm and nnn. Finally, in the appendix we shall present results for generating functions in more general models.
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In a previous paper of this series, we studied $\mathfrak{P}(\overline{\ensuremath{\sigma}})$, the probability that the average boundary spin for a half-plane of Ising spins is $\overline{\ensuremath{\sigma}}$. We extend our study of such functions by generalizing the previous derivation to any magnetic system and using them to relate the magnetic behavior at the critical isotherm to the spin-spin correlations at the critical temperature. We also clarify the meaning of the secondary maxima previously found in $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ by a more accurate calculation and by examining the magnetization on interior rows of the half-plane. These considerations show that $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ is more properly to be considered as the sum of two separate spin probability functions. The various ways in which the thermodynamic limit may be taken and the effects that these different limiting procedures have on $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ are discussed in detail. The influence of boundary conditions on spin probability functions is studied by reversing the sign of one column of horizontal bonds. We compute the additional free energy resulting from the misfit bonds and use this to show that $\mathfrak{P}(\overline{\ensuremath{\sigma}})$ is no longer bimodal but is rectangular for $|\overline{\ensuremath{\sigma}}|$ less than the boundary spontaneous magnetization. Finally, we present in graphical form numerical integrations of the boundary magnetization and susceptibility.
In this paper, we use the dimer method to obtain the free energy of Ising models consisting of repeated horizontal strips of width mmm connected by sequences of vertical strings of length nnn mutually separated by distance NNN, with NNN arbitrary, to investigate the effects of connectivity and proximity on the specific heat. The decoration method is used to transform the strings of n+1n+1n+1 spins interacting with their nearest neighbors with coupling JJJ into a pair with coupling Jˉ\bar JJˉ between the two spins. The free energy per site is given as a single integral and some results for critical temperatures are derived.