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Physical review. B, Condensed matter

Magnetic susceptibility of the two-dimensional Ising antiferromagnet

Yoshiaki Tanaka, Norikiyo Uryû

Published Mar 1, 1980• 16 citations
Publisher

Abstract

With the use of a rigorous relation of the correlations in the Ising spin system, an approach to the exact magnetic susceptibility of the two-dimensional Ising antiferromagnet has been attempted. The zero-field magnetic susceptibilities for the rectangular as well as the square antiferromagnets are shown to be expressed in terms of finite near-spin correlations. With some decoupling approximations for many-spin correlations, the result for the square antiferromagnet by the present method is proved to be within 0.35% from that of the series expansion.

DOI
10.1103/physrevb.21.1994

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Journal of the Physical Society of Japan•Sep 15, 1997•0 citations
With the use of a rigorous relation of the correlations in the Ising spin system, an approach to the exact magnetic susceptibility of the two-dimensional Ising antiferromagnet has been attempted. The zero-field magnetic susceptibilities for the rectangular as well as the square antiferromagnets are shown to be expressed in terms of finite near-spin correlations. With some decoupling approximations for many-spin correlations, the result for the square antiferromagnet by the present method is proved to be within 0.35% from that of the series expansion.
In this part labelled II we examine the spin correlations and the susceptibility. We use a similar method which has allowed to derive a closed-form expression of the zero-field partition function ZN(0), for 2D square lattices composed of (2N+1)2classical spins isotropically coupled [1]. We rigorously show that the spin correlation vanishes in the zero-field limit, except at T=0 K. Thus, the critical temperature is TC=0 K, in agreement with Mermin-Wagner's theorem. For calculating the spin-spin correlation, we show that it is necessary to distinguish a correlation domain in which the correlation path is confined and a wing domain (Theorem 1). In the thermodynamic limit (N→+∞), we give a general closed-form expression for the spin-spin correlation between any two lattice sites. We prove that all the possible paths have the same analytic expression and correspond to the shortest ones in agreement with the classical principle of least action and its quantum version (Theorem 2). As a result and for the first time, we derive the closed-form expression for the susceptibility, without any approximation. We finally test previous experimental fits and we show that the use of a truncated expansion for the susceptibility was totally justified.
Extended series expansions for the high temperature zero-field susceptibility of the Ising model are given in powers of the usual high temperature counting variable v=tanh K; for the triangular lattice to v 16 , for the square lattice to v 21 and for the honeycomb lattice to v 32 , inclusive. The asymptotic behaviour of the ferromagnetic and antiferromagnetic susceptibility is studied. It is concluded that the ferromagnetic singularity is not exactly factorizible. The antiferromagnetic susceptibility of the square and honeycomb lattices has a singularity of the same type as the energy at the antiferromagnetic critical temperature.
The first two terms in the expansion near TC{\mathit{T}}_{\mathit{C}}TC​ of the zero-field susceptibility of the antiferromagnetic square Ising lattice are known exactly. We have isolated the next four terms in this expansion by carrying out a differential-approximant analysis of Nickel's 55-term high-temperature ferromagnetic series with the variable v=tanhJ/kB{\mathit{k}}_{\mathit{B}}kB​T replaced by -v. The four new terms are found to vary as t, t2{\mathit{t}}^{2}t2ln\ensuremath{\Vert}t\ensuremath{\Vert}, t2{\mathit{t}}^{2}t2, and t3{\mathit{t}}^{3}t3ln\ensuremath{\Vert}t\ensuremath{\Vert} where t=1-TC{\mathit{T}}_{\mathit{C}}TC​/T and values for their respective coefficients are obtained. It is established that to order t3{\mathit{t}}^{3}t3ln\ensuremath{\Vert}t\ensuremath{\Vert}, there are no confluent singularities associated with irrelevant variables for this system. A direct extrapolation of Nickel's series coefficients is also carried out and found to be consistent with the present results.