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Physical review. B, Condensed matter

Equations of state of the Potts ferromagnet in an anisotropic square lattice: Renormalization-group approach

Anna Chame, Constantino Tsallis, Uriel M. S. Costa

Published May 1, 1988• 5 citations
Publisher

Abstract

We extend to the anisotropic case (not necessarily equal coupling constants along the various crystalline directions) the real-space renormalization-group procedure recently introduced by Caride and Tsallis, which directly yields the order parameter for arbitrary temperatures. The method does not demand the calculation of the thermodynamic energy; it is as simple as a mean-field calculation although it provides nontrivial results, which can be systematically improved. We apply the procedure to the q-state Potts ferromagnet in an anisotropic square lattice, whose exact equation of state is still unknown. The results compare satisfactorily with available exact results.

DOI
10.1103/physrevb.37.7549

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We extend to the anisotropic case (not necessarily equal coupling constants along the various crystalline directions) the real-space renormalization-group procedure recently introduced by Caride and Tsallis, which directly yields the order parameter for arbitrary temperatures. The method does not demand the calculation of the thermodynamic energy; it is as simple as a mean-field calculation although it provides nontrivial results, which can be systematically improved. We apply the procedure to the q-state Potts ferromagnet in an anisotropic square lattice, whose exact equation of state is still unknown. The results compare satisfactorily with available exact results.
Renormalisation group (RG) techniques are currently used to derive relevant quantities in the vicinity of the critical point. The authors present a real space RG procedure which directly yields the order parameter for all values of the external parameters (e.g. temperature). It is as simple as a mean-field calculation, although it provides non-trivial results, which can be systematically improved. The method is successfully illustrated on the square lattice Potts ferromagnet. The whole approach suggests that the order parameter on a hierarchical lattice is, on every site, proportional to its coordination number.
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.
Within a real-space renormalization-group framework, we study the qqq-state Potts model in a square lattice, assuming a positive second-neighbor coupling constant and an arbitrary sign for the first-neighbor coupling constant. The qqq evolution of the full phase diagram as well as the thermal and magnetic critical exponents are obtained. Whenever comparison with available results is possible, the agreement is either exact or very satisfactory.
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 (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.