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Journal of Physics A Mathematical and General

Renormalisation group recursions by mean-field approximations

Joseph O. Indekeu, Amos Maritan, Attilio L. Stella

Published Jun 1, 1982• 160 citations
Publisher

Abstract

Within a renormalisation group strategy, different 'rescaled' mean-field approximations for the magnetisation are combined. Good qualitative estimates are obtained for the critical properties of classical and quantum spin systems. A particularly simple realisation of the method yields recursions for arbitrary dimensionality d. The critical couplings are accurate for all d, and have the correct asymptotic behaviour for d approaching infinity.

DOI
10.1088/0305-4470/15/6/008

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Texts and readings in physical sciences•Jan 1, 2001•0 citations
Within a renormalisation group strategy, different 'rescaled' mean-field approximations for the magnetisation are combined. Good qualitative estimates are obtained for the critical properties of classical and quantum spin systems. A particularly simple realisation of the method yields recursions for arbitrary dimensionality d. The critical couplings are accurate for all d, and have the correct asymptotic behaviour for d approaching infinity.
Renormalization group recursions based on mean-field approximations [J. O. Indekeu, A. Maritan, and A. L. Stella, J. Phys.A15, L291 (1982)], commonly referred to as mean-field renormalization group methods (MFRG), have proven to be efficient and easily applicable for computing non-classical critical properties of lattice models. We give a fairly complete bibliography of applications to date, and extend previous test calculations of bulk, surface, and corner critical exponents in the two-dimensional Ising model to larger cluster sizes on triangular, square (including crossing bonds), and honeycomb lattices. Without much effort the exact value of the critical exponent ratioyH/yT is reproduced systematically with a precision of 2%. This ratio turns out to be the most accurate probe of non-classical critical behaviour that is available in the MFRG method.
These lectures describe the physics of phase transitions and critical phenomena. After introducing the thermodynamics of phase transitions, these general considerations are made explicit by presenting statistical mechanical calculations on a lattice gas model in the mean-field approximation. Then the Landau theory of phase transitions is presented. The form of free energy in the Landau theory is motivated by an explicit mean field calculation for the Ising model. This is followed by a discussion of spatial correlations and the breakdown of the mean field theory due to increase of spatial fluctuations near the critical point. This calculation also brings out the role of dimension of the system for the critical behaviour. The basic ideas of Renormalization Group (RG) are then introduced and illustrated by a simple RG calculation for the one-dimensional Ising model. Real-space RG calculations are then applied to the two-dimensional Ising model on the triangular lattice, which, in fact, brings forth several key features of the RG technique. A discussion of more general aspects of the RG formalism, such as scaling for spatial correlation functions, relevant and irrelevant couplings, universality, upper critical dimension, etc. are introduced.