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Physica A Statistical Mechanics and its Applications

Correlated cluster mean-field theory for Ising-like spin systems

M. Schmidt, P.F. Dias

Published Mar 5, 2021• 6 citations
Publisher
DOI
10.1016/j.physa.2021.125884

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Mean-field approaches provide a qualitative understanding of the behavior of interacting many-body systems. Weiss's molecular-field theory, in particular, has been successfully applied to the studies of various phenomena. However, since local fluctuations from the average are ignored, this simple mean-field theory does not work for systems whose distinctive behavior is caused by fluctuation effects. In the present work, we introduce a new cluster mean-field approach, which we refer to as the correlated cluster mean-field approach, and the applications to classical and quantum spin systems are demonstrated. We divide the lattice sites into clusters consisting of several sites and include the effect of cluster-cluster correlations into the effective fields acting on the spins. In doing so, the effect of spin fluctuations can be taken into account, and our method gives qualitatively and even semiquantitatively correct results for both classical and quantum systems.

Journal of Physics Conference Series•Jan 1, 2010•5 citations
A cluster mean-field method is introduced and the applications to the Ising and Heisenberg models are demonstrated. We divide the lattice sites into clusters whose size and shape are selected so that the equivalence of all sites in a cluster is preserved. Since the strength of interactions of a cluster with its surrounding clusters is strongly dependent on the spin configuration of the central cluster itself, we include this contribution in the effective fields acting on the spins. The effects of ``correlations'' between clusters can be taken into account beyond the standard mean-field level and as a result our cluster-based method gives qualitatively (and even quantitatively) correct results for the both Ising and Heisenberg models. Especially, for the Ising model on the honeycomb and square lattices, the calculated results of the critical temperature are very close (overestimated by only less than 5%) to the exact values.
Mean-field theory (MFT) is one of the main available tools for analytical calculations entailed in investigations regarding many-body systems. Recently, there has been a surge of interest in ameliorating this kind of method, mainly with the aim of incorporating geometric and correlation properties of these systems. The correlated cluster MFT (CCMFT) is an improvement that succeeded quite well in doing that for classical spin systems. Nevertheless, even the CCMFT presents some deficiencies when applied to quantum systems. In this article, we address this issue by proposing the quantum CCMFT (QCCMFT), which, in contrast to its former approach, uses general quantum states in its self-consistent mean-field equations. We apply the introduced QCCMFT to the transverse Ising model in honeycomb, square, and simple cubic lattices and obtain fairly good results both for the Curie temperature of thermal phase transition and for the critical field of quantum phase transition. Actually, our results match those obtained via exact solutions, series expansions or Monte Carlo simulations.
Mean-field approaches provide a qualitative understanding of the behavior of interacting many-body systems. Weiss's molecular-field theory, in particular, has been successfully applied to the studies of various phenomena. However, since local fluctuations from the average are ignored, this simple mean-field theory does not work for systems whose distinctive behavior is caused by fluctuation effects. In the present work, we introduce a new cluster mean-field approach, which we refer to as the correlated cluster mean-field approach, and the applications to classical and quantum spin systems are demonstrated. We divide the lattice sites into clusters consisting of several sites and include the effect of cluster-cluster correlations into the effective fields acting on the spins. In doing so, the effect of spin fluctuations can be taken into account, and our method gives qualitatively and even semiquantitatively correct results for both classical and quantum systems.