The nearest-neighbor-interaction ferromagnetic Ashkin-Teller model is investigated on a square lattice through a powerful computational method for dealing with correlation functions in magnetic systems. This technique, which is based on damage-spreading numerical simulations, makes use of exact relations involving special kinds of damage and correlation functions, as well as the corresponding order parameters of the model. The computation of correlation functions, which represents usually a hard task in standard Monte Carlo simulations, due to large fluctuations, turns out to be much simpler within the present approach. We concentrate our analysis along the Baxter line, well known for its continuously varying critical exponents; seven different points along this line are investigated. The critical exponents associated with correlation functions along the Baxter line are successfully evaluated, by means of numerical methods, within damage-spreading simulations. The efficiency of this method is confirmed through precise estimates of the critical exponents associated with the order parameters (magnetization and polarization), as well as with their corresponding correlation functions, in spite of the small lattice sizes considered.
A powerful computational method for dealing with correlation functions in magnetic systems, based on damage-spreading simulations, is reviewed and tested, by investigating the q-state Potts ferromagnet, on a square lattice, at criticality. Exact relations involving special kinds of damage and the spin-spin correlation function, as well as the magnetization, are used. The efficiency of the method arises with a significant reduction of the finite-size effects, with respect to conventional Monte Carlo simulations. Correlation functions, which represent usually a hard task within this latter procedure, appear to be much more easily estimated through the present damage-spreading simulations. The effectiveness of the technique is illustrated by an accurate estimate of the exponent eta, of the spin-spin correlation function, for q=2, 3, and 4, with rather small lattice sizes. In the cases q > or = 5, an analysis of the magnetization is consistent with the well-known first-order phase transition.
The nearest-neighbor-interaction spin-1 Ising model is investigated within the damage-spreading approach. Exact relations involving quantities computable through damage-spreading simulations and thermodynamic properties are derived for such a model, defined in terms of a very general Hamiltonian that covers several spin-1 models of interest in the literature. Such relations presuppose translational invariance and hold for any ergodic dynamical procedure, leading to an efficient tool for obtaining thermodynamic properties. The implementation of the method is illustrated through damage-spreading simulations for the ferromagnetic spin-1 Ising model on a square lattice. The two-spin correlation function and the magnetization are obtained, with precise estimates of their associated critical exponents and of the critical temperature of the model, in spite of the small lattice sizes considered. These results are in good agreement with the universality hypothesis, with critical exponents in the same universality class of the spin- 12 Ising model. The advantage of the present method is shown through a significant reduction of finite-size effects by comparing its results with those obtained from standard Monte Carlo simulations.
Some exact results are derived relating quantities computable by the so-called damage spreading method and thermodynamic functions for the N -color Ashkin–Teller model. The results are valid for any ergodic dynamics. Since we restrict our analysis to the ferromagnetic case, the results are also valid for any translationally invariant lattice. The relations derived can be used in order to determine the N -color Ashkin–Teller critical exponents numerically, providing better accuracy and requiring less computational effort than standard Monte Carlo simulations.
We investigate the short-time universal behavior of the two-dimensional Ashkin-Teller model at the Baxter line by performing time-dependent Monte Carlo simulations. First, as preparatory results, we obtain the critical parameters by searching the optimal power-law decay of the magnetization. Thus, the dynamic critical exponents θ_{m} and θ_{p}, related to the magnetic and electric order parameters, as well as the persistence exponent θ_{g}, are estimated using heat-bath Monte Carlo simulations. In addition, we estimate the dynamic exponent z and the static critical exponents β and ν for both order parameters. We propose a refined method to estimate the static exponents that considers two different averages: one that combines an internal average using several seeds with another, which is taken over temporal variations in the power laws. Moreover, we also performed the bootstrapping method for a complementary analysis. Our results show that the ratio β/ν exhibits universal behavior along the critical line corroborating the conjecture for both magnetization and polarization.
We study magnetic critical behavior in the Ashkin-Teller model with an asymmetric defect line. This system is represented by two Ising lattices of spins
σ and
τ interacting through a four-spin coupling
ε. In addition, the couplings between
σ-spins are modified along a particular line, whereas couplings between
τ-spins are kept unaltered. This problem has been previously considered by means of analytical field-theoretical methods and by numerical techniques, with contradictory results. For
ε>0 field-theoretical calculations give a magnetic critical exponent corresponding to
σ-spins which depends on the defect strength only (it is independent of
ε), while
τ-spins magnetization decay with the universal Ising value
1/8. On the contrary, numerical computations based on density matrix renormalization (DMRG) give, for
ε>0 similar scaling behaviors for
σ and
τ spins, which depend on both
ε and defect intensity. In this paper we revisit the problem by performing a direct Montecarlo simulation. Our results are in well agreement with DMRG computations. We also discuss some possible sources for the disagreement between numerical and analytical results.