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Journal of Physics C Solid State Physics

Mean-field renormalisation group study of the anisotropic Ising model

J. A. Plascak, N P Silva

Published Aug 20, 1986• 8 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/19/23/012

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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.
The recently proposed multi-interaction mean-field renormalization technique [C. N. Likos and A. Maritan, Phys. Rev. E 53, 3303 (1996)] is applied to the Ising model on a triangular lattice with equal couplings in two directions and a different coupling in the third, a model equivalent to a square Ising model with additional second-neighbor interactions along a single diagonal direction. Three different clusters are considered and the possible mappings between them are discussed. The estimates for critical couplings and exponents are in satisfactory agreement with exact results. An explanation of the way in which the finite-size features of the method bring about a systematic overestimation of the critical temperature is also given.
The critical behaviour of the spin- Ising model with a transverse field at zero temperature in triangular and square lattices is investigated through a finite-size rescaling transformation. This technique enables the critical properties of infinite systems to be calculated from the scaling properties of systems of finite size, in a generalisation of the so-called phenomenological renormalisation group. The critical field and correlation length exponent calculated using systems of fairly small sizes have reasonable accuracy, but calculations on larger clusters are limited by computer time and storage.