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Journal of Physics C Solid State Physics

Critical temperature for the 2D Heisenberg model

E. Praveczki

Published Apr 20, 1980• 3 citations
Publisher

Abstract

Series, in powers of inverse coordination number, are derived for the critical temperature of 2D classical and quantum Heisenberg models from high-temperature series-expansion results. The results show convincingly that a phase transition occurs at a non-zero temperature.
DOI
10.1088/0022-3719/13/11/015

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The anisotropic Heisenberg model has been investigated by interpolating anisotropy between the Ising and isotropic Heisenberg limits via general spin high temperature series (HTS) for the zero-field susceptibility of the ferromagnets for the square lattice. This is done by locating the critical temperature ( T c (2) ) and the susceptibility exponent (γ) from the analysis of the series by Pade approximant and ratio methods. The critical temperature is found to be spin dependent. It decreases as anisotropy changes from the Ising limit towards the isotropic limit and remains finite (lower than that of Stanley and Kaplan) for all the spins except for S =1/2 at the isotropic limit. Exponent γ is almost Ising like for the anisotropic system and anomalous jump in it occurs for the isotropic case. In addition, expressions for the magnetization are derived by using spin-wave theory. The limit of complete isotropy has been found not to be a special case for the susceptibility, in contrast to magnetization.
The phase transition for the spin systems described by the Hamiltonian H =-2 J ∑ < i , j > (ξ S i x S j x +η S i y S j y + S i z S j z )- g µ H ∑ i S i z is studied by means of the high-temperature series expansion of the zero field susceptibility. The first five terms in the susceptibility series are explicitly obtained for the two-dimensional square lattice in the cases of (i) 0≤ξ≤1, η=0 and (ii) 0≤ξ=η≤1. The critical temperature T c and the critical exponent γ (susceptibility) are obtained as functions of the spin S and the anisotropy parameter ξ. The critical temperature has a finite value in the XY model (ξ=1, η=0) as well as the Heisenberg model (ξ=η=1). The simple formulae expressing T c and γ valid for arbitrary S and ξ are proposed.
Using the vertex renormalized form of the technique developed by Englert,1 high-temperature series expansions for the spin-spin correlation function of the classical anisotropic Heisenberg model are calculated for various lattices and anisotropies through order T−8 (close-packed lattices) and T−9 (loose-packed lattices). These series are combined and then extrapolated to give the high-temperature critical indices γ (zero-field susceptibility), ν (correlation range), and α (specific heat) as functions of anisotropy. The results are consistent with the hypothesis that the critical indices only change when there is a change in the symmetry of the system, e.g., in interpolating between the Ising and isotropic Heisenberg models, indices remain Ising-like until the system becomes isotropic, at which point they appear to change discontinuously. Previous results for the limiting cases are confirmed and extended. Series for the limiting cases (spin-infinity Ising, XY and isotropic Heisenberg models) as well as for the spin-½ Ising model2 have been derived for the B-site spinel lattice. Analysis of these series shows that the anomalies in the critical indices, reported on the basis of six-term expansions,3 disappear with the use of longer series.
We suggest that renormalization-group recursion relations can profitably be combined with high-temperature series expansions in systems with critical points at zero temperature. Low-temperature recursion relations are integrated out to high temperatures, where conventional high-temperature series should accurately describe quantities such as the susceptibility. A "variational" principle is suggested which gives the optimal temperature at which to splice together the low- and high-temperature results. In this way, we obtain a sequence of approximants to the susceptibility of the classical two-dimensional Heisenberg model which apparently converges exponentially fast at all temperatures. Although further investigation reveals that these approximants are not as accurate as their exponential convergence suggests, they do appear to give a good quantitative description at all temperatures of the susceptibilities we have studied.
The system described by the Hamiltonian shown is investigated on the basis of exact high-temperature series expansions and the Green function technique. Expansions for the zero-field ( B = 0) free energy and susceptibility (valid for arbitrary lattice structure) are developed to fifth order in reciprocal temperature for the S = ½ system with nearest-neighbour forces. The critical behaviour is discussed in two and three dimensions using the random phase approximation and Padé approximants. In particular the variation of the critical temperature with η is accurately determined for common two- and three-dimensional lattices. It is conjectured that the susceptibility index γ remains constant for 0 ⩽ η < 1 but changes discontinuously at η = 1.