A. Hoogland

Monte Carlo renormalization of the three-dimensional Ising model

Abstract We have investigated the simple cubic Ising model by means of the Monte Carlo renormalization technique. The emphasis of our study concerns the influence of truncation, i.e. the dimensionality of the coupling subspace in which the analysis of the correlation functions generated by the Monte Carlo and spin blocking algorithms is performed. To this purpose we have included up to 36 even and 21 odd couplings in our analysis. We find that the increase in the number of couplings has a significant influence on the largest eigenvalues of the linearized renormalization transformation matrices. These eigenvalues serve to estimate the renormalization exponents yI and yH. Remarkably, we find no significant finite-size effect on these eigenvalues when the maximum number of couplings is included in the analysis, except for the smallest system sizes (83 → 43). After a suitable extrapolation to the fixed point, we find good agreement with existing results for the critical exponents. We have determined the critical point of the simple cubic Ising model as K = 0.221652(6), also in agreement with existing results.

Monte Carlo renormalization of the 3D ising model: Analyticity and convergence

We review the assumptions on which the Monte Carlo renormalization technique is based, in particular the analyticity of the block spin transformations. On this basis, we select an optimized Kadanoff blocking rule in combination with the simulation of a d=3 Ising model with reduced corrections to scaling. This is achieved by including interactions with second and third neighbors. As a consequence of the improved analyticity properties, this Monte Carlo renormalization method yields a fast convergence and a high accuracy. The results for the critical exponents are y_H=2.481(1) and y_T=1.585(3).

The simple quadratic Ising model with crossing bonds

We investigate the ferromagnetic critical line of the simple quadratic Ising model with nearest-neighbour and next-nearest-neighbour interactions, by means of Monte Carlo, renormalisation group and analytic methods. A comparison of the Monte Carlo results with the renormalisation group trajectories yields the asymptotic behaviour of the critical line in the strong-coupling limit. A number of figures of spin configurations is included to illustrate the physical behaviour of the model.

Smooth finite-size behaviour of the three-dimensional Ising model

We have obtained Monte Carlo results for finite, simple cubic Ising models at criticality. The susceptibility data are consistent with a smooth approach to the asymptotic behaviour for large system sizes.

Monte Carlo renormalization of the three-dimensional Ising model: the influence of truncation

We have applied the Monte Carlo renormalization technique to the simple cubic Ising model. In particular we have investigated the influence of truncation, i.e. the number of coupling used in the analysis of the conjugate correlation functions generated by the Monte Carlo and spin blocking algorithms. To this purpose we have included up to 36 even and 21 odd couplings in our analysis, which is considerably more than used so far. We find that the addition of extra couplings does significantly influence the largest eigenvalues of the stability matrices. However, after correction for the finite-size effect and extrapolation to the fixed point, the effects on the estimated critical exponents are no longer significant.

Critical behaviour of two Ising models with multispin interactions

We have performed extensive Monte Carlo simulations on the two-dimensional Ising models with n-spin interactions described recently by Debierre and Turban. Results for n = 3 models with sizes up to 128 × 128 are analyzed by means of finite-size scaling. This yields a value of the magnetic exponent yh close to 158. Direct estimates of the temperature exponent yT do not converge convincingly. However, assuming the presence of logarithmic corrections such as in the 4-state Potts model, we obtain an improved estimate of the temperature exponent close to 32, in agreement with 4-state Potts universality. This result is further supported by an exact mapping between the n = 3 model and the 4-state Potts model in an anisotropic limit. For the n = 4 model, we confirm that the phase transition is first order, and we estimate the discontinuities in the energy and the magnetization.

The Delft Ising system processor

Publisher Summary This chapter focuses on the Delft Ising system processor (DISP). Numerical simulation by means of general-purpose computers is a standard technique in statistical mechanics. In principle, the method is straightforward, but in practice—particularly in the field of phase transitions and critical phenomena—it may be difficult to produce sufficiently accurate results within reasonable limits of time and money. The use of supercomputers alleviates the first of these restraints but is detrimental for the other. The stochastic algorithm used to simulate Ising spin systems with short-range interactions is a perfect test-case for the construction of a special-purpose computer, because of the binary character of the main variables involved, that is, the spin values, and because of the extremely long relaxation times and correlation distances in the critical region, which necessitate very long calculation times. The chapter also discusses the performance of the DISP in comparison with general-purpose computers.

CRITICAL PROPERTIES OF 3D ISING SYSTEMS WITH NON-HAMILTONIAN DYNAMICS

We investigate two three-dimensional Ising models with non-Hamiltonian Glauber dynamics. The transition probabilities of these models can, just as in the case of equilibrium models, be expressed in terms of Boltzmann factors depending only on the interacting spins and the bond strengths. However, the bond strength associated with each lattice edge assumes different values for the two spins involved. The first model has cubic symmetry and consists of two sublattices at different temperatures. In the second model a preferred direction is present. These two models are investigated by Monte Carlo simulations on the Delft Ising System Processor. Both models undergo a phase transition between an ordered and a disordered state. Their critical properties agree with Ising universality. The second model displays magnetization bistability.

Monte Carlo Renormalization of the 3-D-Ising Model and the Analyticity of Block-Spin Transformations

We present a new analysis on Monte Carlo Renormalization Group (MCRG) results obtained earlier by means of the Delft Ising System Processor (DISP). The MCRG data involve a total of 57 coupling constants, 36 even and 21 odd. Simulations were carried out for simple cubic lattices with 643, 323 and 163 spins. The RG transformation is assumed to be analytic. A number of relations exist between correlation functions at different renormalization levels. Some of these involve the derivatives of the stability matrix. These correlation functions enable an analysis of the so-called regular part of the RG transformation. If the Hamiltonian of the original lattice only contains nearest-neighbour couplings then the regular contributions to the specific heat and the magnetic susceptibility can be easily determined. These contributions must depend only weakly on the initial lattice size, at least if the RG transformation is analytic. We investigated whether this is indeed true when the majority-rule is applied. New simulations involving higher-order correlations will enable us to study the analytic contributions in more detail.