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Journal of Physics A Mathematical and General

Mean field renormalisation group for the spin-1/2anisotropic Heisenberg model

J. A. Plascak

Published Aug 1, 1984• 32 citations
Publisher

Abstract

The mean field renormalisation group approach is applied to the spin- 1 / 2 anisotropic Heisenberg model. The critical temperature and the critical exponent are obtained as a function of the anisotropy for the two-dimensional model on square, triangular, and hexagonal lattices. The three-dimensional model on a simple cubic lattice is also analysed.

DOI
10.1088/0305-4470/17/11/008

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The mean field renormalisation group approach is applied to the spin- 1 / 2 anisotropic Heisenberg model. The critical temperature and the critical exponent are obtained as a function of the anisotropy for the two-dimensional model on square, triangular, and hexagonal lattices. The three-dimensional model on a simple cubic lattice is also analysed.
The mean-field renormalisation group approach is applied to the anisotropic Ising model by considering clusters consisting of finite blocks of spins and clusters consisting of infinite strips of spins with the finite widths. The critical temperature and estimates of the critical exponent nu are obtained as functions of the anisotropy. A two-step renormalisation group procedure using a phenomenological renormalisation transformation is also presented. In general, the results for nu are consistent with universality arguments.
A real-space effective-field renormalization-group method (ERFG) recently derived for computing critical properties of Ising spins is extended to treat the quantum spin-1/2 anisotropic Heisenberg model. The formalism is based on a generalized but approximate Callen–Suzuki spin relation and utilizes a convenient differential operator expansion technique. The method is illustrated in several lattice structures by employing its simplest approximation version in which clusters with one (N′=1) and two (N=2) spins are used. The results are compared with those obtained from the standard mean-field (MFRG) and Migdal–Kadanoff (MKRG) renormalization-group treatments and it is shown that this technique leads to rather accurate results. It is shown that, in contrast with the MFRG and MKRG predictions, the EFRG, besides correctly distinguishing the geometries of different lattice structures, also provides a vanishing critical temperature for all two-dimensional lattices in the isotropic Heisenberg limit. For the simple cubic lattice, the dependence of the transition temperature Tc with the exchange anisotropy parameter Δ [i.e., Tc(Δ)], and the resulting value for the critical thermal crossover exponent φ [i.e., Tc≂Tc(0)+AΔ1/φ ] are in quite good agreement with results available in the literature in which more sophisticated treatments are used.
The anisotropic nearest-neighbour Heisenberg model for the simple cubic lattice has been investigated by interpolating the anisotropy between the Ising and isotropic Heisenberg limits via general spin high-temperature series expansions of the zero-field suspectibility. This is done by estimating the critical temperature (Tc(3)) and the susceptibility exponent γ from the analysis of the series by the Ratio and Pade approximants methods. It is noted that Tc(3) varies with anisotropy while γ is almost the same for the anisotropic system, and a jump in it occurs for the isotropic case in agreement with the universality hypothesis. The effect of anisotropy on the susceptibility is also shown. Further, it is seen that estimates of γ for the two extreme limits agree well with those of previous theoretical as well as experimental investigations. In addition, critical temperatures have been summarised in a relation, and expressions for the magnetisation have been derived.
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.