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Journal of Physics A Mathematical and General

Mean-field-like variational and renormalisation group studies of (1+1)-dimensional spin models

Ernst Niebur, Jenő Sólyom

Published Jul 11, 1987• 1 citation
Publisher

Abstract

Two recently proposed methods, the variational cluster wavefunction approximation and the mean-field-like renormalisation group transformation, are used to describe the phase transitions in the Hamiltonian version of the two-dimensional Z(q) symmetric spin models. The convergence of the two schemes to the known results is analysed. It is shown that in their usual form neither of these models can reproduce fine details of the phase diagram, although the renormalisation group method gives in general better convergence.

DOI
10.1088/0305-4470/20/10/037

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Two recently proposed methods, the variational cluster wavefunction approximation and the mean-field-like renormalisation group transformation, are used to describe the phase transitions in the Hamiltonian version of the two-dimensional Z(q) symmetric spin models. The convergence of the two schemes to the known results is analysed. It is shown that in their usual form neither of these models can reproduce fine details of the phase diagram, although the renormalisation group method gives in general better convergence.
The phase diagrams of general qqq-state planar spin models on a square lattice are studied for $q=4\ensuremath{-}7$. The phase diagrams are calculated by a variational method and the infinitesimal Migdal renormalization-group (RG) transformations. Since the Migdal RG commutes with the self-dual transformations of Z(q)Z(q)Z(q) models, one expects reliable phase diagrams (but not accurate critical exponents). In fact all exactly known critical points are reproduced by our calculations together with a fair numerical simulation of lines of critical points. Our results clearly indicate the existence of massless phases for q=6and7q=6 \mathrm{and} 7q=6and7 but are inconclusive for q=5q=5q=5. More accurate methods are needed to determine the phase structure of the Z(5)Z(5)Z(5) model; see, for example, the finite-lattice approximation of Roomany and Wyld (preceding paper).