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Journal of Physics A Mathematical and General

Mean-field renormalisation group study of the semi-infinite Potts model

M A Santos, M C Marques

Published Dec 1, 1987• 2 citations
Publisher

Abstract

The MFRG method is used to study the critical behaviour of the semi-infinite three-dimensional q-state Potts model. Estimates of Delta c , the critical enhancement at the special transition, and the phase diagram are obtained for different values of q and compared with results from other techniques.
DOI
10.1088/0305-4470/20/17/048

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Within a real space renormalisation group framework which uses rather sophisticated clusters, the authors discuss the phase diagram and the universality classes of a semi-infinite cubic-lattice q-state Potts ferromagnet. In particular, they study the influence, on the surface magnetism, of q and Delta identical to J S /J B -1 (where J S and J B are respectively the surface and bulk coupling constants). The exact d=2 critical temperature T c 2D is recovered for all values of q. The q-dependence of the value Delta c above which surface magnetic order can exist even if the bulk is disordered, is calculated and, through a convenient extrapolation, reliable results are obtained (for the Ising particular case, i.e. q=2, they obtain the extrapolated value Delta c =0.569, which compares satisfactorily with the series result 0.6+or-0.1 and the Monte Carlo one, 0.50+or-0.03). At the surface-bulk (SB) multicritical point they calculate the q-dependences of the critical amplitude A and the crossover exponent phi (defined in the Delta to Delta c +0 limit through (T c S ( Delta )/T c 3D -1) approximately A( Delta / Delta c -1) 1 phi /, where T c S ( Delta ) and T c 3D identical to T c S ( Delta c ) respectively are the surface and the bulk critical temperatures), as well as the critical exponent y t1 SB . For the Ising particular case they obtain phi equivalent to 0.641 (which compares satisfactorily with the epsilon -expansion (0.68) and the Monte Carlo (0.56+or-0.04) results), A equivalent to 0.4 and 1/y t1 SB equivalent to 1.623. (As far as we know, no series or Monte Carlo results are available in the literature for A or 1/y t1 SB .).