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Journal of Physics A Mathematical and General

Corner critical exponents from the mean-field renormalisation group

Joseph O. Indekeu, G. Menu

Published May 7, 1988• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/21/9/008

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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.
Critical exponents for the O(n) spin model local field near a surface, an edge and a corner are calculated by means of the renormalisation group method for the special geometry where the wedge or corner is formed by mutually perpendicular (d-1)-dimensional walls, each having von Neumann (special transition) or Dirichlet (ordinary transitions) boundary conditions. The combined effect of two or more interacting surfaces leads to new local susceptibility exponents which are calculated to order in ( in =4-d). The case of two or more von Neumann surfaces presents some difficulty in that a double divergence arises in the first order perturbation calculation of the local susceptibilities.