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arXiv:1806.00873

Ising Models with Holes: Crossover Behavior

Helen Au-Yang, Jacques H. H. Perk

Published Jun 3, 2018• 1 citation
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Abstract

In order to investigate the effects of connectivity and proximity in the specific heat, a special class of exactly solvable planar layered Ising models has been studied in the thermodynamic limit. The Ising models consist of repeated uniform horizontal strips of width $m$ connected by sequences of vertical strings of length $n$ mutually separated by distance $N$, with $N=1,2$ and $3$. We find that the critical temperature $T_c(N,m,n)$, arising from the collective effects, decreases as $n$ and $N$ increase, and increases as $m$ increases, as it should be. The amplitude $A(N,m,n)$ of the logarithmic divergence at the bulk critical temperature $T_c(N,m,n)$ becomes smaller as $n$ and $m$ increase. A rounded peak, with size of order $\ln m$ and signifying the one-dimensional behavior of strips of finite width $m$, appears when $n$ is large enough. The appearance of these rounded peaks does not depend on $m$ as much, but depends rather more on $N$ and $n$, which is rather perplexing. Moreover, for fixed $m$ and $n$, the specific heats are not much different for different $N$. This is a most surprising result. For $N=1$, the spin-spin correlation in the center row of each strip can be written as a Toeplitz determinant with a generating function which is much more complicated than in Onsager's Ising model. The spontaneous magnetization in that row can be calculated numerically and the spin-spin correlation is shown to have two-dimensional Ising behavior.

DOI
10.48550/arxiv.1806.00873

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In order to investigate the effects of connectivity and proximity in the specific heat, a special class of exactly solvable planar layered Ising models has been studied in the thermodynamic limit. The Ising models consist of repeated uniform horizontal strips of width mmm connected by sequences of vertical strings of length nnn mutually separated by distance NNN, with N=1,2N=1,2N=1,2 and 333. We find that the critical temperature Tc(N,m,n)T_c(N,m,n)Tc​(N,m,n), arising from the collective effects, decreases as nnn and NNN increase, and increases as mmm increases, as it should be. The amplitude A(N,m,n)A(N,m,n)A(N,m,n) of the logarithmic divergence at the bulk critical temperature Tc(N,m,n)T_c(N,m,n)Tc​(N,m,n) becomes smaller as nnn and mmm increase. A rounded peak, with size of order ln⁡m\ln mlnm and signifying the one-dimensional behavior of strips of finite width mmm, appears when nnn is large enough. The appearance of these rounded peaks does not depend on mmm as much, but depends rather more on NNN and nnn, which is rather perplexing. Moreover, for fixed mmm and nnn, the specific heats are not much different for different NNN. This is a most surprising result. For N=1N=1N=1, the spin-spin correlation in the center row of each strip can be written as a Toeplitz determinant with a generating function which is much more complicated than in Onsager's Ising model. The spontaneous magnetization in that row can be calculated numerically and the spin-spin correlation is shown to have two-dimensional Ising behavior.
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.
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.
The specific heats of exactly solvable alternating layered planar Ising models with strips of width m_{1} lattice spacings and "strong" couplings J_{1} sandwiched between strips of width m_{2} and "weak" coupling J_{2}, have been studied numerically to investigate the effects of connectivity and proximity. We find that the enhancements of the specific heats of the strong layers and of the overall or "bulk" critical temperature T_{c}(J_{1},J_{2};m_{1},m_{2}) arising from the collective effects reflect the observations of Gasparini and co-workers in experiments on confined superfluid helium. Explicitly, we demonstrate that finite-size scaling holds in the vicinity of the upper limiting critical point T_{1c} (∝J_{1}/k_{B}) and close to the corresponding lower critical limit T_{2c} (∝J_{2}/k_{B}) when m_{1} and m_{2} increase. However, the residual enhancement, defined via appropriate subtractions of leading contributions from the total specific heat, is dominated (away from T_{1c} and T_{2c}) by a decay factor 1/(m_{1}+m_{2}) arising from the seams (or boundaries) separating the strips; close to T_{1c} and T_{2c} the decay is slower by a factor lnm_{1} and lnm_{2}, respectively.
Part I of this article studied the specific heats of planar alternating layered Ising models with strips of strong coupling J_{1} sandwiched between strips of weak coupling J_{2}, to illustrate qualitatively the effects of connectivity, proximity, and enhancement in analogy to those seen in extensive experiments on superfluid helium by Gasparini and co-workers. It was demonstrated graphically that finite-size scaling descriptions hold in a variety of temperature regions including in the vicinity of the two specific heat maxima. Here we provide exact theoretical analyses and asymptotics of the specific heat that support and confirm the graphical findings. Specifically, at the overall or bulk critical point, the anticipated (and always present) logarithmic singularity is shown to vanish exponentially fast as the width of the stronger strips increases.
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.
We study the asymptotic behavior, for large separations, of the spin-spin correlation function $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{M,N}〉$ in the two-dimensional Ising model, where the two spins are not necessarily on the same row. Besides the limiting value for infinite separation, which is the square of the spontaneous magnetization, we evaluate the two leading terms in the asymptotic expression in each of the two cases T<TcT<{T}_{c}T<Tc​ and T>TcT>{T}_{c}T>Tc​. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case T>TcT>{T}_{c}T>Tc​, but much more complicated for T<TcT<{T}_{c}T<Tc​. In an Appendix, we also give exactly in a very simple form the correlation $〈{\ensuremath{\sigma}}_{0,0}{\ensuremath{\sigma}}_{N,N}〉$ for symmetrical Ising lattice at the critical temperature Tc{T}_{c}Tc​.