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Effective field methods in lattice systems

Raf Dekeyser, G. Kamieniarz

Published Jan 1, 1992• 0 citations
OpenAlex

Abstract

It is possible to generalise in several ways the sefconsistency conditions inherent in the mean field approximation. One can then obtain a substantial improvement on the estimates of critical parameters through rather simple calculations; also critical exponents can be obtained through a combination with the mean-field renormalization group method. These techniques can also be applied to quantummechanical systems.

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It is possible to generalise in several ways the sefconsistency conditions inherent in the mean field approximation. One can then obtain a substantial improvement on the estimates of critical parameters through rather simple calculations; also critical exponents can be obtained through a combination with the mean-field renormalization group method. These techniques can also be applied to quantummechanical systems.
The selfconsistency conditions inherent in the mean field approximation are generalized in several ways according to the symmetry of the Hamiltonian. A substantial improvement on the estimates of thermodynamic properties and critical parameters is obtained through rather simple calculations. Non-classical values of critical exponents are found, implementing the mean-field renormalization group method. These techniques can be applied both to classical and to quantum-mechanical systems, especially in low dimensions.
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.
The large scale behavior of the simplest non-mean-field spin-glass system is analyzed, and the critical exponent related to the divergence of the correlation length is computed at two loops within the ε-expansion technique with two independent methods. The techniques presented show how the underlying ideas of the renormalization group apply also in this disordered model, in such a way that an ε-expansion can be consistently set up. By pushing such calculation to high orders in ε, a consistent non-mean-field theory for such disordered system could be established, giving a substantial contribution the development of a predictive theory for real spin glasses.