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Physica A Statistical Mechanics and its Applications

Multi-effective-field theory: Applications to the CAM analysis of the two-dimensional Ising model

Kazuhiko Minami, Yoshihiko Nonomura, Makoto Katori, Masuo Suzuki

Published Jun 1, 1991• 18 citations
Publisher
DOI
10.1016/0378-4371(91)90344-c

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Journal of the Physical Society of Japan•Jul 15, 1994•3 citations
The double-cluster approximation (DCA) and the multi-effective-field approximation (MEFA) are formulated for the two-dimensional S=1/2 transverse Ising model on a square lattice in the ground state. These two cluster-effective-field approximations expand the sizes of clusters effectively. Combining some suitable series of these approximations with the coherent-anomaly method (CAM), the authors have analysed the critical phenomenon of this model. They have obtained g * c =1.528+or-0.016 and gamma =1.25+or-0.08 using the DCA, and gamma =1.26+or-0.10 using the MEFA. These estimates are consistent with the other results, g * c approximately=1.52 and gamma approximately=1.24. Their scheme is expected to be useful in the study of frustrated systems, because various shapes of clusters can be used in their calculations.
Two systematic series of multi-body effective field approximations are constructed and the CAM analyses are performed for the Ising mode on the two-dimensional square lattice. In the first series a two-body effective field is introduced in addition to the usual one-body one and a set of strips are used. The critical temperature and the critical index of the susceptibility are estimated as T* c approximately=2.2697 J/k B and gamma approximately=1.749. In the second series only a three-line strip is used and applied with increasing numbers of multi-body effective fields. The expected CAM scaling behaviour has also been observed.
Layered square-lattice ferromagnetic Ising models (∞×∞× n lattices) are studied by the Monte Carlo simulation for n =3 to 7. The results show the dimensionality crossover from the two-dimensional square lattice Ising model to the three-dimensional simple cubic Ising model. Furthermore, the CAM analysis is performed to evaluate the critical exponent γ of the three-dimensional simple cubic ferromagnetic Ising model, which shows that γ =1.21(4). The present analysis is also applicable to the study of the phase transition of quantum spin systems.