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Physical review. B, Condensed matter

Mean-field renormalization group: Unified approach to bulk and surface critical behavior

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

Published Jan 1, 1987• 66 citations
Publisher

Abstract

The previously introduced method of mean-field renormalization is reexamined in a finite-size-scaling perspective. The method is shown to provide simultaneously estimates for bulk and surface critical exponents. These are obtained within a scheme which guarantees convergence to exact results assuming the validity of standard finite-size scaling. The unifying scheme leads to rather accurate computations as illustrated in Ising and Potts models in two (and three) dimensions.

DOI
10.1103/physrevb.35.305

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Journal of Physics A Mathematical and General•Mar 7, 1994•1 citation
The previously introduced method of mean-field renormalization is reexamined in a finite-size-scaling perspective. The method is shown to provide simultaneously estimates for bulk and surface critical exponents. These are obtained within a scheme which guarantees convergence to exact results assuming the validity of standard finite-size scaling. The unifying scheme leads to rather accurate computations as illustrated in Ising and Potts models in two (and three) dimensions.
The previously introduced method of mean-field renormalisation has provided a unified approach to bulk and surface critical behaviour. The method is used for the computation of the critical exponent associated with the order parameter at corners. Applications to the square and triangular Ising model are presented. The angle dependence of the corner critical exponent is qualitatively reproduced. The results for the bulk critical exponents are more accurate than in previously employed mean-field renormalisation schemes.
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.
With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\ensuremath{\beta}}_{1}=0.78\ifmmode\pm\else\textpm\fi{}0.02$ except at the "special" (multicritical) transition where ${{\ensuremath{\beta}}_{1}}^{m}=0.175\ifmmode\pm\else\textpm\fi{}0.025$ and the crossover exponent is $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.
A modification of Kadanoff's lower-bound renormalization transformation is used to analyze the critical behavior of the semi-infinite Ising model with surface interactions which may differ from the bulk interactions. The d=2d=2d=2 square lattice and the d=3d=3d=3 bcc lattice are considered. Surface critical exponents, various critical couplings, and a phase diagram for d=3d=3d=3 are calculated. The surface critical exponents are compared with the scaling laws relating surface and bulk exponents and the $\ensuremath{\epsilon}$-expansion results due to Bray and Moore. The eigenvalues determining the surface exponents agree within 10% with the predictions of Bray and Moore except in the case of the eigenvalue determining the surface-bulk crossover exponent, where the discrepancy is much larger.
A new real space renormalization group transformation combining ideas from mean-field and finite-size scaling theories is presented. Application to the two-dimensional site directed percolation problem gives better values for the percolation threshold and critical exponent of longitudinal correlation length than those obtained previously with other real space renormalization group approaches. When applied to one-, two- and three-dimensional) Ising models, the results are comparable to the ones from previous mean-field-like renormalization group transformations. This method can easily be applied to other systems having second-order phase transitions.