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Journal of Physics A Mathematical and General

Finite size studies of the Ashkin-Teller model

Francisco Castilho Alcaraz, J. R. Drugowich de Felício

Published Aug 21, 1984• 21 citations
Publisher

Abstract

The authors study the Ashkin-Teller model in the time-continuous Hamiltonian version. Finite-size scaling is used to calculate the magnetic ( gamma M ), electric ( gamma P ) and thermal ( alpha and nu ) critical exponents for several values of the coupling constant ( lambda ). The results confirm the believed extended scaling relations and suggest a conjecture relating the mass-gap amplitudes and critical indices in the Hamiltonian context.

DOI
10.1088/0305-4470/17/12/005

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The authors study the Ashkin-Teller model in the time-continuous Hamiltonian version. Finite-size scaling is used to calculate the magnetic ( gamma M ), electric ( gamma P ) and thermal ( alpha and nu ) critical exponents for several values of the coupling constant ( lambda ). The results confirm the believed extended scaling relations and suggest a conjecture relating the mass-gap amplitudes and critical indices in the Hamiltonian context.
Monte Carlo simulations in the three-dimensional (3D) Ashkin-Teller model on a cubic lattice are performed in the regions of the two-parameter space diagram where Ising-type phase transitions are expected. The scaling behavior of the Binder cumulant Q and the magnetic susceptibility in the critical region are exploited. In simulations the periodic boundary conditions and the Metropolis algorithm are used. Starting from Ising critical exponents and applying the finite-size-scaling analysis of the cumulant Q with nonlinear corrections, the accurate positions of the critical couplings on the continuous phase transition lines are calculated. For these couplings the critical exponent y(h) is calculated analyzing the magnetic susceptibility in the framework of the finite-size scaling with corrections. The values of y(h) agree with the 3D Ising model value along two lines determined by the order parameter [ssigma] .
The critical behavior of a bond-disordered Ashkin-Teller model on a square lattice is investigated by intensive Monte Carlo simulations. A duality transformation is used to locate a critical plane of the disordered model. This critical plane corresponds to the line of critical points of the pure model, along which critical exponents vary continuously. Along this line the scaling exponent corresponding to randomness \ensuremath{\varphi}=(\ensuremath{\alpha}/\ensuremath{\nu}) varies continuously and is positive so that the randomness is relevant, and different critical behavior is expected for the disordered model. We use a cluster algorithm for the Monte Carlo simulations based on the Wolff embedding idea, and perform a finite size scaling study of several critical models, extrapolating between the critical bond-disordered Ising and bond-disordered four-state Potts models. The critical behavior of the disordered model is compared with the critical behavior of an anisotropic Ashkin-Teller model, which is used as a reference pure model. We find no essential change in the order parameters' critical exponents with respect to those of the pure model. The divergence of the specific heat C is changed dramatically. Our results favor a logarithmic type divergence at Tc{\mathit{T}}_{\mathit{c}}Tc​, C\ensuremath{\sim}lnL for the random-bond Ashkin-Teller and four-state Potts models and C\ensuremath{\sim}ln lnL for the random-bond Ising model.