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Physical review. B, Solid state

Theory of layered Ising models. II. Spin correlation functions parallel to the layering

Helen Au-Yang, Barry M. McCoy

Published Nov 1, 1974• 31 citations
Publisher

Abstract

We study the spin correlation $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{0N}〉$, which is expressable as a block Toeplitz determinant, of a nth-order layered Ising model, in the direction parallel to the layering. The generating function of this block Toeplitz determinant can be written as a scalar function times a 2\ifmmode\times\else\texttimes\fi{}2 matrix whose elements are nth-order trigonometric polynomials. We show that it is possible to calculate asymptotically for large $N$ the block Toeplitz determinant generated by such a matrix function. As an example we compute, for large $N$, $〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{0N}〉$ for the second-order layered Ising model whose horizontal bonds are all equal and whose vertical bonds between the jth row and (j + 1)th row is ${E}_{2}$, if $j$ even and ${E}_{2}^{\ensuremath{'}}$, if $j$ odd.

DOI
10.1103/physrevb.10.3885

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We study the spin correlation 〈\ensuremathσ00\ensuremathσ0N〉〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{0N}〉〈\ensuremathσ00​\ensuremathσ0N​〉, which is expressable as a block Toeplitz determinant, of a nth-order layered Ising model, in the direction parallel to the layering. The generating function of this block Toeplitz determinant can be written as a scalar function times a 2\ifmmode\times\else\texttimes\fi{}2 matrix whose elements are nth-order trigonometric polynomials. We show that it is possible to calculate asymptotically for large NNN the block Toeplitz determinant generated by such a matrix function. As an example we compute, for large NNN, 〈\ensuremathσ00\ensuremathσ0N〉〈{\ensuremath{\sigma}}_{00}{\ensuremath{\sigma}}_{0N}〉〈\ensuremathσ00​\ensuremathσ0N​〉 for the second-order layered Ising model whose horizontal bonds are all equal and whose vertical bonds between the jth row and (j + 1)th row is E2{E}_{2}E2​, if jjj even and E2\ensuremath′{E}_{2}^{\ensuremath{'}}E2\ensuremath′​, if jjj odd.
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​.
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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.
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 in detail the asymptotic behavior, for large separations, of the correlation between two spins in the case of the two-dimensional Ising model without magnetic field. In the limit of infinite separation, this correlation is equal to the square of the spontaneous magnetization per spin. This paper is devoted to answering the question how, for fixed temperature, the correlation approaches this limiting value. The investigation is restricted to the situation where the two spins under consideration lie on the same row of the two-dimensional lattice. There are three distinct cases where the fixed temperature is above, below, or equal to the critical temperature or the Curie temperature. In the first two cases, the limiting value is approached exponentially as a function of the separation, while at the critical temperature the correlation behaves asymptotically as the inverse fourth root of the separation. In this paper, we evaluate exactly the first four terms of the asymptotic expansion in the first case, the first three terms in the second case, and the coefficients of the terms proportional to the -\textonequarter{} and the $\ensuremath{-}\frac{9}{4}$ powers of the separation in the third case. All these results are obtained by first expressing the correlation as a Toeplitz determinant, and the method used is capable of giving in principle the entire asymptotic expansion for large classes of such determinants, when the size of the determinant approaches infinity. The explicit results for the two-dimensional Ising model also serve as an example where the prescription of summing the leading terms, or the most divergent terms, fails.