The Three-State Square-Lattice Potts Antiferromagnet at Zero Temperature

We study the 3-state square-lattice Potts antiferromagnet at zero temperature by a Monte Carlo simulation using the Wang–Swendsen–Kotecký cluster algorithm, on lattices up to 1024 × 1024. We confirm the critical exponents predicted by Burton and Henley based on the height representation of this model.     

Corrections to finite-size-scaling in two dimensional Potts models

High resolution Monte Carlo simulations are used to examine the finite size behavior of Q-state Potts models in two dimensions. For Q = 3 we find that at the critical point bulk properties are subject to much larger corrections to finite size scaling than were previously realized. For Q = 4 we find that corrections to finite size scaling are subtle and that the multiplicative logarithmic correction is insufficient to correct the dominant terms.     

Static and dynamic critical phenomena of the two-dimensionalq-state Potts model

Theq-state Potts model on the square lattice is studied by Monte Carlo simulation forq=3, 4, 5, 6. Very good agreement is obtained with exact results of Kiharaet al. and Baxter for energy and free energy at the critical point. Critical exponent estimates forq=3 are0.4,0.1,1.45, in rough agreement with high-temperature series extrapolation and real space renormalization-group methods. The transition forq=5, 6 is found to be a very weakly first-order transition, i.e., pronounced pseudocritical phenomena occur, specific heat, susceptibility, etc. (nearly) diverge at the first-order transition temperature. Dynamics is associated to the model in the same way as for the kinetic Ising model, and the nonlinear slowing down of the order parameter and of the energy is studied. The dynamic exponent is estimated to be (=zv)1.9. Within our accuracy noq dependence is detected. The relaxation is found to be consistent with dynamic scaling predictions, and dynamic scaling functions associated with the nonlinear relaxation are estimated.     

Critical behavior of a three-state Potts model on a Voronoi lattice

We use the single-histogram technique to study the critical behavior of the three-state Potts model on a (random) Voronoi-Delaunay lattice with size ranging from 250 to 8 000 sites. We consider the effect of an exponential decay of the interactions with the distance, , with a>0, and observe that this system seems to have critical exponents and which are different from the respective exponents of the three-state Potts model on a regular square lattice. However, the ratio remains essentially the same. We find numerical evidences (although not conclusive, due to the small range of system size) that the specific heat on this random system behaves as a power-law for a=0 and as a logarithmic divergence for a=0.5 and a=1.0 Received 5 April 2000     

Critical exponents for the self-avoiding random walk in three dimensions

We compute by direct Monte Carlo simulation the main critical exponents, , ₄, andv and the effective coordination number for the self-avoiding random walk in three dimensions on a cubic lattice. We find both hyperscaling relationsdv=2– anddv– 2 ₄+=0 satisfied ind = 3.     

Critical exponents of Manhattan Hamiltonian walks in two dimensions,from Potts andO(n) models

We consider a set of Hamiltonian circuits filling a Manhattan lattice, i.e., a square lattice with alternating traffic regulation. We show that the generating function (with fugacityz) of this set is identical to the critical partition function of aq-state Potts model on an unoriented square lattice withq ^1/2 =z. The set of critical exponents governing correlations of Hamiltonian circuits is derived using a Coulomb gas technique. These exponents are also found to be those of an O(n) vector model in the low-temperature phase withn =q ^1/2 =z. The critical exponents in the limitz = 0 are then those of spanning trees (q= 0) and of dense polymers (n=0,T < T_c), corresponding to a conformal theory with central chargeC = –2. This shows that the Manhattan orientation and the Hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of Hamiltonian walks.     

Investigating the nonequilibrium aspects of long-range Potts model: Refinement of critical temperatures and raw exponents

In this work, we analyze the q-state Potts model with long-range interactions through nonequilibrium scaling relations commonly used when studying short-range systems. We determine the critical temperature via an optimization method for short-time Monte Carlo simulations. The study takes into consideration two different boundary conditions and three different values of range parameters of the couplings. We also present estimates of some critical exponents, named as raw exponents for systems with long-range interactions, which confirm the non-universal character of the model. Finally, we provide some preliminary results addressing the relations between the raw exponents and the exponents obtained for systems with short-range interactions. The results assert that the methods employed in this work are suitable to study the considered model and can easily be adapted to other systems with long-range interactions.     

Continuous phase transition in a disordered eight-states Potts model

We investigate the two-dimensional eight-states ferromagnetic Potts model in the Voronoi-Delaunay tessellation. In this study, we assume that the coupling factor J varies with the distance r between the first neighbors as , with . The disordered system is simulated applying the single-cluster Monte-Carlo update algorithm and the reweighting technique. We find that this model displays a first-order phase transition if , in agreement with previous recent studies. For and 1.0, a typical second order transition is observed and the critical exponents for magnetization and susceptibility are calculated. Received 19 May 1999 and Received in final form 2 June 1999     

TheXY model and the three-state antiferromagnetic Potts model in three dimensions: Critical properties from fluctuating boundary conditions

We present the results of a Monte Carlo study of the three-dimensionalXY model and the three-dimensional antiferromagnetic three-state Potts model. In both cases we compute the difference of the free energies of a system with periodic and a system with antiperiodic boundary conditions in a neighborhood of the critical coupling. From the finite-size scaling behaviour of this quantity we extract values for the critical temperature and the critical exponentv that are compatible with recent high-statistics Monte Carlo studies of the models. The results for the free energy difference at the critical temperature and for the exponentv confirm that both models belong to the same universality class.     

Finite-size effects and phase transition in the three-dimensional three-state Potts model

The three-state Potts model in three dimensions is studied by Monte Carlo and finite-size scaling techniques. Using a histogram method recently proposed by Ferrenberg and Swendsen, the finite-size dependence for the maximum of the specific heat is found to scale with the volume of the system, indicating that the phase transition is of first order. The value of the latent heat per spin and the correlation length at the transition are estimated.     

Monte Carlo studies of the square Ising model with next-nearest-neighbor interactions

We apply a new entropic scheme to study the critical behavior of the square-lattice Ising model with nearest- and next-nearest-neighbor antiferromagnetic interactions. Estimates of the present scheme are compared with those of the Metropolis algorithm. We consider interactions in the range where superantiferromagnetic (SAF) order appears at low temperatures. A recent prediction of a first-order transition along a certain range (0.5–1.2) of the interaction ratio (R=J_nnn/J_nn) is examined by generating accurate data for large lattices at a particular value of the ratio (R=1). Our study does not support a first-order transition and a convincing finite-size scaling analysis of the model is presented, yielding accurate estimates for all critical exponents for R=1. The magnetic exponents are found to obey “weak universality” in accordance with a previous conjecture.     

Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models

At the critical point in two dimensions, the number of percolation clusters of enclosed area greater than A is proportional to A ^–1, with a proportionality constant C that is universal. We show theoretically (based upon Coulomb gas methods), and verify numerically to high precision, that . We also derive, and verify to varying precision, the corresponding constant for Ising spin clusters, and for Fortuin–Kasteleyn clusters of the Q = 2, 3 and 4-state Potts models.     

Monte Carlo test of the Goldstone mode singularity in 3D XY model

Monte Carlo simulations of magnetization and susceptibility in the 3D XY model are performed for system sizes up to L=384 (significantly exceeding the largest size L=160 considered in work published previously), and fields h ≥ 0.0003125 at two different coupling constants β=0.5, and β=0.55 in the ordered phase. We examine the prediction of the standard theory that the longitudinal susceptibility χ_∥ has a Goldstone mode singularity such that χ_∥ ∝h^-1/2 holds when h↦0. Most of our results, however, support another theoretical prediction that the singularity is of a more general form χ_∥ ∝h^ρ-1, where 1/2<ρ<1 is a universal exponent related to the ∼h^ρ variation of the magnetization.     

Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas

Many two-dimensional spin models can be transformed into Coulomb-gas systems in which charges interact via logarithmic potentials. For some models, such as the eight-vertex model and the Ashkin-Teller model, the Coulomb-gas representation has added significantly to the insight in the phase transitions. For other models, notably theXY model and the clock models, the equivalence has been instrumental for almost our entire understanding of the critical behavior. Recently it was shown that theq-state Potts model and then-vector model are equivalent to a Coulomb gas with an asymmetry between positive and negative charges. Fieldlike operators in these spin models transform noninteger charges and magnetic monopoles. With the aid of exactly solved models the Coulombgas representation allows analytic calculation of some critical indices.     

Quenching of the Haldane gap in LiVSi<Subscript>2</Subscript>O<Subscript>6</Subscript> and related compounds

We report results of susceptibility χ and ⁷Li NMR measurements on LiVSi₂O₆. The temperature dependence of the magnetic susceptibility χ(T) exhibits a broad maximum, typical for low-dimensional magnetic systems. Quantitatively it is in agreement with the expectation for an S=1 spin chain, represented by the structural arrangement of V ions. The NMR results indicate antiferromagnetic ordering below T_N=24 K. The intra- and interchain coupling J and J_p for LiVSi₂O₆, and also for its sister compounds LiVGe₂O₆, NaVSi₂O₆ and NaVGe₂O₆, are obtained via a modified random phase approximation which takes into account results of quantum Monte Carlo calculations. While J_p is almost constant across the series, J varies by a factor of 5, decreasing with increasing lattice constant along the chain direction. The comparison between experimental and theoretical susceptibility data suggests the presence of an easy-axis magnetic anisotropy, which explains the formation of an energy gap in the magnetic excitation spectrum below T_N, indicated by the variation of the NMR spin-lattice relaxation rate at T≪T_N.