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arXiv:cond-mat/0312462 · International Journal of Modern Physics B

SOME EXACT FORMULAE ON LONG-RANGE CORRELATION FUNCTIONS OF THE RECTANGULAR ISING LATTICE

SHU-CHIUAN CHANG, Masuo Suzuki

Published Jan 20, 2004• 1 citation

Abstract

We study long-range correlation functions of the rectangular Ising lattice with cyclic boundary conditions. Specifically, we consider the situation in which two spins are on the same column, and at least one spin is on or near free boundaries. The low-temperature series expansions of the correlation functions are presented when the spin–spin couplings are the same in both directions. The exact correlation functions can be obtained by D log Padé for the cases with simple algebraic resultant expressions. The present results show that when the two spins are infinitely far from each other, the correlation function is equal to the product of the row magnetizations of the corresponding spins, as expected. In terms of low-temperature series expansions, the approach of this m th row correlation function to the bulk correlation function for increasing m can be understood from the observation that the dominant terms of their series expansions are the same successively in the above two correlation functions. The number of these dominant terms increases monotonically as m increases.

DOI: 10.1142/s0217979204023842

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Abstract We consider spin–spin correlation functions for spins along a row, R n = ⟨ σ 0,0 σ n ,0 ⟩, in the two-dimensional Ising model. We discuss a method for calculating general- n expressions for coefficients in high-temperature and low-temperature series expansions of R n and apply it to obtain such expressions for several higher-order coefficients. In addition to their intrinsic interest, these results could be useful in the continuing quest for a nonlinear ordinary differential equation whose solution would determine R n , analogous to the known nonlinear ordinary differential equation whose solution determines the diagonal correlation function ⟨ σ 0,0 σ n , n ⟩ in this model.

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<{T}_{c}

and

T>{T}_{c}

. It is found that the nearest singularity of the generating function for the correlation is quite simple in the case

T>{T}_{c}

, but much more complicated for

T<{T}_{c}

. 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

{T}_{c}

.