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Journal of Physics A Mathematical and General

On the Z(4) spin model

Vera Lúcia Vieira Baltar, Maria Elena Pol

Published Apr 1, 1986• 0 citations
Publisher

Abstract

The authors study the Z(4) spin model in two dimensions in the region of parameters where a critical phase may occur. The authors results, obtained by Monte Carlo simulations, Monte Carlo renormalisation group technique and a study of the elementary excitations near the SOS model, indicate that there is no critical phase in the Z(4) model.

DOI
10.1088/0305-4470/19/5/027

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The authors study the Z(4) spin model in two dimensions in the region of parameters where a critical phase may occur. The authors results, obtained by Monte Carlo simulations, Monte Carlo renormalisation group technique and a study of the elementary excitations near the SOS model, indicate that there is no critical phase in the Z(4) model.
Monte Carlo studies of the Z(4)Z(4)Z(4) (or Ashkin-Teller) model on the triangular lattice were carried out. In particular, we investigated whether the massless phase, which was shown to exist on the boundary of the physical parameter space, may extend into a region of finite width. The existence of such a phase would imply that Nc{N}_{c}Nc​, the critical value above which Z(N)Z(N)Z(N) models have intermediate phases, is nonuniversal, i.e., lattice and interaction dependent. Although, on the basis of our Monte Carlo results we cannot completely rule out this possibility, most aspects of our findings indicate consistency with existing theories of the Z(4)Z(4)Z(4) model which exclude the possibility of such an intermediate massless phase.
Within a real-space renormalization-group (RG) framework, we study the criticality of the Z(4) ferromagnet on the square lattice. The phase diagram (exhibiting ferromagnetic, paramagnetic, and nematiclike phase) recovers all the available exact results, and possibly is of high precision everywhere. In particular, we establish the main asymptotic behaviors (bifurcation and Ising regions). In addition, we develop an operational procedure (break-collapse method) which considerably simplifies the exact calculation of arbitrary Z(4) two-terminal clusters (commonly appearing in RG approaches).
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.