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Progress of Theoretical Physics

Finite Size Effects and Pseudocritical Behavior. II: Potts Spin Model

Shingo Ohya, Hiroshi Katsumori

Published May 1, 1996• 1 citation

Abstract

Using the transfer matrices constructed by Kramers and Wannier's successive screw method for the 2-d (l × 蜴) and 3-d (l ×m ×蜴) q state Potts model (2 蠆q 蠆7), we examine the extent to which the presently available supercomputers can provide us with finite size effects and pseudocritical properties of the systems. We estimate the bulk coupling parameters Kc from the specific heat anomaly in two and three dimensions and also from the correlation length anomaly in two dimensions. In two dimensions, our results Kc are in good agreement with those given by the Potts duality formula. In three dimensions, however, we need extended precision computations to obtain the numerical results with sufficient accuracy. We regard the behavior of the exponent µ, which characterizes the relation between the specific heat peak Cv0(l) and the lattice size l, as an indicator to discriminate the order of the phase transition by investigating Cv0(l) for several small values of l.

DOI: 10.1143/ptp.95.851

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Using the transfer matrices constructed by Kramers and Wannier's successive screw method for the 2-d (l × 蜴) and 3-d (l ×m ×蜴) q state Potts model (2 蠆q 蠆7), we examine the extent to which the presently available supercomputers can provide us with finite size effects and pseudocritical properties of the systems. We estimate the bulk coupling parameters Kc from the specific heat anomaly in two and three dimensions and also from the correlation length anomaly in two dimensions. In two dimensions, our results Kc are in good agreement with those given by the Potts duality formula. In three dimensions, however, we need extended precision computations to obtain the numerical results with sufficient accuracy. We regard the behavior of the exponent µ, which characterizes the relation between the specific heat peak Cv0(l) and the lattice size l, as an indicator to discriminate the order of the phase transition by investigating Cv0(l) for several small values of l.

The transfer matrices for the ferromagnetic Ising model on the two- and three-dimensional lattices with finite size are formulated using the screw method introduced by Kramers and Wannier, and the largest and next largest eigenvalues of matrices are obtained by a computer. Then the specific heats and magnetic susceptibilities are calculated for each system with various sizes. By use of the finite-size scaling and Roomany-Wyld approximations, the critical exponents of the infinite-size systems are evaluated. They are in quite good agreement with the reliable ones.

Monte Carlo simulations are performed to analyze phase transitions in three-dimensional systems described by the 3-state Potts model with nonmagnetic impurities. Numerical results are presented for systems with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 on lattices of size L varying between 20 and 44. Binder’s cumulant analysis shows that the introduction of quenched disorder in the form of non-magnetic impurities induces a crossover from first-order to second-order phase transition. The finite-size scaling method is used to calculate the static critical exponents α, γ, β, and ν for specific heat, susceptibility, magnetization, and correlation length, respectively.

The results of computer simulations of the critical behavior of three-dimensional structures described by disordered Potts models with q = 3 and q = 4 spin states are discussed. Systems with linear dimensions L = 20–60 with spin concentrations p = 1.00, 0.95, 0.90, 0.80, 0.70, and 0.65 are examined. It is found that in a three-dimensional Potts model with q = 3 spin states, introducing nonmagnetic impurities induces a second order phase transition. In a Potts model with q = 4, on the other hand, a shift from a first order phase transition to a second order phase transition is observed at high dilution (p = 0.65). Numerical values of the critical parameters are found using finite-size scaling theory and it is shown that they form a universality class corresponding to disordered systems.

We present a large- q expansion of the 2d q -states Potts model free energies up to order 9 in 1/√ q . Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q = 4, and predicts, for q > 4, the n -th energy cumulant scales as the power (3 n /2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q = 10 with lattices of linear dimensions up to L = 50. The pure-phase specific heats are predicted to be much larger, at q ⩽ 10, than the values extracted from the current finite-size scaling analysis of the extrema. Implications for safe numerical determinations of interface tensions are discussed.

The static critical behavior of the three-dimensional weakly diluted Potts model with the state q = 3 on a simple cubic lattice has been investigated by the Monte Carlo method using the Wolff single-cluster algorithm. It is shown that at the spin concentrations p = 0.9 and 0.8 a second-order phase transition is observed in the three-dimensional weakly diluted Potts model with the state q = 3. On the basis of the finite-size scaling theory, we calculated the static critical exponents of the specific heat α, susceptibility γ, magnetization β, and the correlation-length exponent v .

By introducing a small amount of non-magnetic impurities into an antiferromagnetic (AF) two-dimensional (2D) Potts model on a triangular lattice it is that the impurities in spin systems described by this model result in the change of a first order to a second-order phase transition. The systems with linear sizes L × L = N, L = 9-144 are considered. Investigations are performed using the standard Metropolis algorithm along with Monte-Carlo single-cluster Wolff algorithm. On the basis of the theory of finite-size scaling, critical exponents (CE) are calculated: the heat capacity α, the susceptibility γ, the order parameter β, and the CE of the correlation radius ν.