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Journal of Physics A Mathematical and General

Quantum spin systems: dynamical mean field renormalisation group approach

J. A. Plascak

Published Sep 11, 1984• 15 citations
Publisher

Abstract

The mean field renormalisation group approach is applied to the study of the static and dynamical critical properties of the spin- 1 / 2 anisotropic Heisenberg model in a transverse field. The critical surface and estimates of static and dynamical critical exponents are obtained for the one-, two-, and three-dimensional models.
DOI
10.1088/0305-4470/17/13/004

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The phenomenological renormalisation group transformation, introduced by Nightingale (1976), is extended to a form suitable for the study of a quantum mechanical system at T=0. The spin- 1 / 2 Ising model in a transverse field Gamma , in one and two dimensions, is considered. The new scaling form, called phenomenological time scaling, is derived by exploiting the connection between a d dimensional quantum mechanical system at T=0 and a (d+1) dimensional classical system in the anisotropic limit of weakly coupled chains. Estimates are made of the critical field Gamma c and the exponent nu associated with the vanishing of the first excitation energy at T=0. These results also provide some information about the thermal critical behaviour of the anisotropic three-dimensional Ising model.
The renormalization group approach to critical phenomena is developed for quantum mechanical problems with non-commuting operators. Applying the theory to the spin 1 / 2 Ising model with a transverse field Gamma it is found that the critical exponents are those of the Ising model with Gamma =0 if the transition occurs at T>0. However, for transitions at T=0 the critical behaviour of the d-dimensional transverse system corresponds to that of the (d+1)-dimensional Ising model with Gamma =0, in agreement with series expansion predictions. At T=0 the dynamic, as well as static, critical behaviour is given by mean field theory for d>3.
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.