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Computational Methods in Science and Technology

VERIFICATION OF THE THREE-CLUSTER MOLECULAR-FIELD RENORMALIZATION GROUP APPROACH FOR THE ISING MODEL

G. Kamieniarz, Grzegorz Musiał

Published Jan 1, 2002• 0 citations
Publisher

Abstract

A transfer matrix approach has been worked out to test the predictions of the three-step molecular-field renormalization group (MFRG) for square Ising clusters with linear size up to L = 11.The convergence of the finite-size critical couplings and the critical exponents towards the exact values is shown.
DOI
10.12921/cmst.2002.08.01.63-67

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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.