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Modern Physics Letters B

MEAN FIELD RENORMALIZATION GROUP FOR ISING MODELS WITH S≥1

D. Peña Lara, J. A. Plascak

Published Sep 20, 1996• 11 citations
Publisher

Abstract

The mean field renormalization group is extended in order to study spin-S Ising models (S≥1) by introducing additional parameters in the Hamiltonians of the clusters, in the same spirit as the mean field approach. These new parameters are then consistently obtained according to finite size scaling ideas and quite good results are obtained, even for the smallest choice of the clusters. Moreover, the mean value of the square of the spin along the second-order transition line can also be obtained from the present formalism.

DOI
10.1142/s0217984996001218

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The mean field renormalization group is extended in order to study spin-S Ising models (S≥1) by introducing additional parameters in the Hamiltonians of the clusters, in the same spirit as the mean field approach. These new parameters are then consistently obtained according to finite size scaling ideas and quite good results are obtained, even for the smallest choice of the clusters. Moreover, the mean value of the square of the spin along the second-order transition line can also be obtained from the present formalism.
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.
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.
We analyse in some detail a recently proposed transfer matrix mean-field approximation which yields the exact critical point for several two-dimensional nearest-neighbour Ising models. For the square lattice model we show explicitly that this approximation yields not only the exact critical point, but also the exact boundary magnetization of a semi-infinite Ising model, independent of the size of the strips used. Then we develop a new mean-field renormalization-group strategy based on this approximation and make connections with finite size scaling. Applying our strategy to the quadratic Ising and three-state Potts models we obtain results for the critical exponents which are in very good agreement with the exact ones. In this way we also clarify some advantages and limitations of the mean field renormalization group approach.