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Journal of Physics A Mathematical and General

Anisotropic cubic lattice Potts ferromagnet: renormalisation group treatment

L. R. da Silva, Constantino Tsallis, Georges Schwachheim

Published Nov 11, 1984• 9 citations
Publisher

Abstract

Within a real space renormalisation group framework, the authors discuss the criticality of the full anisotropic (arbitrary J x , J y and J z )q-state Potts ferromagnet in the simple cubic lattice. Several previously known exact results for the d=1 and d=2 particular cases are recovered. Furthermore they obtain: (i) the q-dependence of the d=3 correlation length critical exponent nu 3 (in particular, if q to 0, nu 3 (q) approximately nu 3 (0)+ nu 3 '(0)q where the present approximate values are nu 3 (0) approximately=1.105 and nu 3 '(0) approximately=-0.66); (ii) the q-dependence of the d=2 to or from d=3 crossover critical exponent phi 23 (in particular, phi 23 varies as 1/ square root q if q to 0); (iii) through a convenient numerical extrapolation, a quite accurate proposal for the critical temperatures corresponding to arbitrary ratios, J y /J x and J z /J x and values of q.

DOI
10.1088/0305-4470/17/16/022

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Within a real space renormalisation group framework, the authors discuss the criticality of the full anisotropic (arbitrary J x , J y and J z )q-state Potts ferromagnet in the simple cubic lattice. Several previously known exact results for the d=1 and d=2 particular cases are recovered. Furthermore they obtain: (i) the q-dependence of the d=3 correlation length critical exponent nu 3 (in particular, if q to 0, nu 3 (q) approximately nu 3 (0)+ nu 3 '(0)q where the present approximate values are nu 3 (0) approximately=1.105 and nu 3 '(0) approximately=-0.66); (ii) the q-dependence of the d=2 to or from d=3 crossover critical exponent phi 23 (in particular, phi 23 varies as 1/ square root q if q to 0); (iii) through a convenient numerical extrapolation, a quite accurate proposal for the critical temperatures corresponding to arbitrary ratios, J y /J x and J z /J x and values of q.
The choice of a convenient self-dual cell within a real space renormalisation group framework enables a satisfactory treatment of the anisotropic square lattice q-state Potts ferromagnet criticality. The exact critical frontier and dimensionality crossover exponent Hf as well as the expected universality behaviour (renormalisation flow sense) are recovered for any linear scaling factor b and all values of q(q<or=4). The b=2 and b=3 approximate correlation length critical exponent nu is calculated for all values of q and compared with the den Nijs conjecture. The same calculation is performed, for all values of b for the exponent nu (d=1) associated with the one-dimensional limit and the exact result nu (d=1)=1 is recovered in the limit b to infinity .
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 .).
Within a real-space renormalisation group framework, the authors treat the q-state Potts ferromagnet in a simple cubic lattice consisting of two different semi-infinite bulks (respectively characterised by the coupling constants J1and J2) separated by a (1,0,0) interface (characterised by Js). The use of a rather sophisticated two-terminal cluster allows a quite reliable discussion of the phase diagram and its universality classes. Four physically different phases are obtained, namely the paramagnetic and the double-bulk, single-bulk and surface ferromagnetic ones. The critical surface contains a multicritical line (associated with J1not=J2) which in turn contains a special point associated with J1=J2.
The one-dimensional q-state Potts model with ferromagnetic pair interactions which decay with the distance r as is considered. We calculate, through a real-space renormalization group technique using Kadanoff blocks of length b, the critical temperature and the correlation length critical exponent as a function of for different values of q. Some of the very few known rigorous results for general q are reproduced by our approach. Several asymptotic behaviours are derived analytically for q = 2, 3 in the limit. We also obtain extrapolated critical temperatures for arbitrary values of and for q = 2, 3, 4, which we believe approximate the exact ones well, except in the region near . Furthermore, the use of another extrapolation procedure suitable only in the vicinity of led us to conjecture that the exact critical temperature is the same for any value of q . We also verify that , which is consistent with a recent conjecture of Tsallis.
Within a real space renormalisation group (RG) scheme, the authors study the criticality of the ferromagnetic Z(4) model on an anisotropic square lattice. They use an RG cluster which has already proved to be very efficient for the Potts model on the same lattice. The establishment of the RG recurrence relations is greatly simplified through the break-collapse method. The phase diagram (exhibiting ferromagnetic, paramagnetic and nematic-like phases) recovers all the available exact results, and is believed to be of high precision everywhere. If the model is alternatively thought of as being associated with a particular hierarchical lattice rather than with the square lattice, then it is exact everywhere.
The free and internal energies and specific heat of the q-state Potts ferromagnet are discussed. A real-space renormalisation-group approach is presented, which reproduces a considerable amount of exact particular results for all dimensionalities (hypercubic lattices). The case of the square lattice is investigated in detail using self-dual clusters (which provide the exact critical point for all q). Comparison with Onsager results (q=2) is satisfactory; the general tendencies for q not=2 (1<or=q<or=4) are exhibited, and the bound percolation limit (q to 1) is worked out in some detail.