Henk W. J. Blöte

Ising universality in three dimensions: a Monte Carlo study

We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as yt=1.587 (2), yh=2.4815 (15) and yi=-0.82 (6). The universal ratio Q=(m2)2/(m4) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is Kc=0.2216546 (10).

THE MAGNETIZATION OF THE 3D ISING MODEL

We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to spins. These results were obtained by means of the Cluster Processor, a newly built special-purpose computer for the Wolff cluster simulation of the 3D Ising model. We find that the spontaneous magnetization M(t) is accurately described by , where , in a wide temperature range 0.0005 < t < 0.26. Any corrections to scaling with higher powers of t could not be resolved from our data, which implies that they are very small. The magnetization exponent is determined as . An analysis of the magnetization distribution near criticality yields a new determination of the critical point: , with a standard deviation of .

Universal ratio of magnetization moments in two-dimensional Ising models

The authors calculate the universal critical-point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic Ising models on the square and on the triangular lattices. Periodic boundary conditions are used in accordance with the four-fold and six-fold rotational symmetries of the respective lattices. These results, which are obtained by means of an analysis of finite-size data computed with a transfer-matrix technique, have an accuracy of the order of one millionth. This analysis is also applied to rectangular systems with arbitrary aspect ratios.

Classical critical behavior of spin models with long-range interactions

We present the results of extensive Monte Carlo simulations of Ising models with algebraically decaying ferromagnetic interactions in the regime where classical critical behavior is expected for these systems. We corroborate the values for the exponents predicted by renormalization theory for systems in one, two, and three dimensions and accurately observe the predicted logarithmic corrections at the upper critical dimension. We give both theoretical and numerical evidence that above the upper critical dimension the decay of the critical spin-spin correlation function in finite systems consists of two different regimes. For one-dimensional systems our estimates for the critical couplings are more than two orders of magnitude more accurate than existing estimates. In two and three dimensions we are unaware of any other results for the critical couplings.

Susceptibility calculations for alternating antiferromagnetic chains

Earlier work of Duffy and Barr consisting of exact calculations on alternating antiferromagnetic Heisenberg spin‐1/2 chains is extended to longer chains of up to 12 spins, and subsequent extrapolations of thermodynamic properties, particularly the susceptibility, are extended to the weak alternation region close to the uniform limit. This is the region of interest in connection with the recent experimental discovery of spin‐Peierls systems. The extrapolated susceptibility curves are compared with corresponding curves calculated from the model of Bulaevskii, which has been used extensively in approximate theoretical treatments of a variety of phenomena. Qualitative agreement is observed in the uniform limit and persists for all degrees of alternation, but quantitative differences of about 10% are present over the whole range, including the isolated dimer limit. Potential application of the new susceptibility calculations to experiment is discussed.

Probability distribution of the order parameter for the three-dimensional Ising-model universality class: A high-precision Monte Carlo study

We study the probability distribution P(M) of the order parameter (average magnetization) M, for the finite-size systems at the critical point. The systems under consideration are the 3-dimensional Ising model on a simple cubic lattice, and its 3-state generalization known to have remarkably small corrections to scaling. Both models are studied in a cubic box with periodic boundary conditions. The model with reduced corrections to scaling makes it possible to determine P(M) with unprecedented precision. We also obtain a simple, but remarkably accurate, approximate formula describing the universal shape of P(M).

MONTE CARLO METHOD FOR SPIN MODELS WITH LONG-RANGE INTERACTIONS

We introduce a Monte Carlo method for the simulation of spin models with ferromagnetic long-range interactions in which the amount of time per spin-flip operation is independent of the system size, in spite of the fact that the interactions between each spin and all other spins are taken into account. We work out two algorithms for the q-state Potts model and discuss the generalization to systems with other interactions and to O(n) models. We illustrate the method with a simulation of the mean-field Ising model, for which we have also analytically calculated the leading finite-size correction to the dimensionless amplitude ratio 2/ at the critical temperature.

Dynamic exponent of the two-dimensional ising model and Monte Carlo computation of the subdominant eigenvalue of the stochastic matrix

We introduce a novel variance-reducing Monte Carlo algorithm for accurate determination of correlation times. We apply this method to two-dimensional Ising systems with sizes up to 15 3 15, using single-spin flip dynamics, random site selection, and transition probabilities according to the heat-bath method. From a finite-size scaling analysis of these correlation times, the dynamic critical exponent z is determined as z › 2.1665s12d. [S0031-9007(96)00379-1]

Triangular SOS models and cubic-crystal shapes

A solid-on-solid (SOS) model in a field h conjugate to the orientation of the surface is exactly solved with the aid of Pfaffians. The free energy (h) directly gives the equilibrium shape of a finite crystal. The phase diagram exhibits rough and smooth phases, corresponding to rounded and flat portions of the crystal surface. The solid-on-solid model undergoes transitions of the Pokrovsky-Talapov type (1979) characterised by a specific heat exponent alpha =1/2. One special point of the phase diagram corresponds to the appearance of a facet via an alpha =0 transition. Height-height correlations are derived along a special line in the phase diagram. With the aid of the known equivalence of this SOS model with an Ising model, several exponents can be translated from one model to the other. This enables one to derive the topology of the phase diagram of the antiferromagnetic triangular Ising model with first- and second-neighbour couplings in a field.

THE CLUSTER PROCESSOR: NEW RESULTS

We present a progress report on the Cluster Processor, a special-purpose computer system for the Wolff simulation of the three-dimensional Ising model, including an analysis of simulation results obtained thus far. These results allow, within narrow error margins, a determination of the parameters describing the phase transition of the simple-cubic Ising model and its universality class. For an improved determination of the correction-to-scaling exponent, we include Monte Carlo data for systems with nearest-neighbor and third-neighbor interactions in the analysis.