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.Comment: 14 pages, latex2

Critical Point Correlation Function for the 2D Random Bond Ising Model

High accuracy Monte Carlo simulation results for 1024*1024 Ising system with ferromagnetic impurity bonds are presented. Spin-spin correlation function at a critical point is found to be numerically very close to that of a pure system. This is not trivial since a critical temperature for the system with impurities is almost two times lower than pure Ising $T_c$. Finite corrections to the correlation function due to combined action of impurities and finite lattice size are described.Comment: 7 pages, 2 figures after LaTeX fil

Critical region of the random bond Ising model

We describe results of the cluster algorithm Special Purpose Processor simulations of the 2D Ising model with impurity bonds. Use of large lattices, with the number of spins up to $10^6$, permitted to define critical region of temperatures, where both finite size corrections and corrections to scaling are small. High accuracy data unambiguously show increase of magnetization and magnetic susceptibility effective exponents $\beta$ and $\gamma$, caused by impurities. The $M$ and $\chi$ singularities became more sharp, while the specific heat singularity is smoothed. The specific heat is found to be in a good agreement with Dotsenko-Dotsenko theoretical predictions in the whole critical range of temperatures.Comment: 11 pages, 16 figures (674 KB) by request to authors: [email protected] or [email protected], LITP-94/CP-0

The Magnetization of the 3D Ising Model

We present highly accurate Monte Carlo results for simple cubic Ising lattices containing up to $256^3$ 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 magnetization $M(t)$ is perfectly described by $M(t)=(a_0-a_1 t^{\theta} - a_2 t) t^{\beta}$, where $t=(T_{\rm c}-T)/T_{\rm c}$, in a wide temperature range $0.0005 < t < 0.26$. If there exist corrections to scaling with higher powers of $t$, they are very small. The magnetization exponent is determined as $\beta=0.3269$ (6). An analysis of the magnetization distribution near criticality yields a new determination of the critical point: $K_{\rm c}=J/k_B T_{\rm c}=0.2216544$, with a standard deviation of $3\cdot 10^{-7}$.Comment: 7 pages, 5 Postscript figure

Higher moments of spin-spin correlation functions for the ferromagnetic random bond Potts model

Using CFT techniques, we compute the disorder-averaged p-th power of the spin-spin correlation function for the ferromagnetic random bonds Potts model. We thus generalize the calculation of Dotsenko, Dotsenko and Picco, where the case p=2 was considered. Perturbative calculations are made up to the second order in epsilon (epsilon being proportional to the central charge deviation of the pure model from the Ising model value). The explicit dependence of the correlation function on $p$ gives an upper bound for the validity of the expansion, which seems to be valid, in the three-states case, only if p-alpha in final formula

The RANLUX generator: resonances in a random walk test

Using a recently proposed directed random walk test, we systematically investigate the popular random number generator RANLUX developed by Luescher and implemented by James. We confirm the good quality of this generator with the recommended luxury level. At a smaller luxury level (for instance equal to 1) resonances are observed in the random walk test. We also find that the lagged Fibonacci and Subtract-with-Carry recipes exhibit similar failures in the random walk test. A revised analysis of the corresponding dynamical systems leads to the observation of resonances in the eigenvalues of Jacobi matrix.Comment: 18 pages with 14 figures, Essential addings in the Abstract onl

Critical behavior of the pure and random-bond two-dimensional triangular Ising ferromagnet

We investigate the effects of quenched bond randomness on the critical properties of the two-dimensional ferromagnetic Ising model embedded in a triangular lattice. The system is studied in both the pure and disordered versions by the same efficient two-stage Wang-Landau method. In the first part of our study we present the finite-size scaling behavior of the pure model, for which we calculate the critical amplitude of the specific heat's logarithmic expansion. For the disordered system, the numerical data and the relevant detailed finite-size scaling analysis along the lines of the two well-known scenarios - logarithmic corrections versus weak universality - strongly support the field-theoretically predicted scenario of logarithmic corrections. A particular interest is paid to the sample-to-sample fluctuations of the random model and their scaling behavior that are used as a successful alternative approach to criticality.Comment: 10 pages, 8 figures, slightly revised version as accepted for publication in Phys. Rev.

Universal amplitude ratios from numerical studies of the three-dimensional O(2) model

We investigate the three-dimensional O(2) model near the critical point by Monte Carlo simulations and calculate the major universal amplitude ratios of the model. The ratio U_0=A+/A- is determined directly from the specific heat data at zero magnetic field. The data do not, however, allow to extract an accurate estimate for alpha. Instead, we establish a strong correlation of U_0 with the value of alpha used in the fit. This numerical alpha-dependence is given by A+/A- = 1 -4.20(5) alpha + O(alpha^2). For the special alpha-values used in other calculations we find full agreement with the corresponding ratio values, e. g. that of the shuttle experiment with liquid helium. On the critical isochore we obtain the ratio xi+/xi-_T=0.293(9), and on the critical line the ratio xi_T^c/xi_L^c=1.957(10) for the amplitudes of the transverse and longitudinal correlation lengths. These two ratios are independent of the used alpha or nu-values.Comment: 34 pages, 19 Ps-figures, Latex2e, revised version, to be published in J. Phys.

The Critical Finite Size Scaling Relation of the Order-Parameter Probability Distribution for the Three-Dimensional Ising Model on the Creutz Cellular Automaton

We study the order parameter probability distribution at the critical point for the three-dimensional spin-1/2 and spin-1 Ising models on the simple cubic lattice with periodic boundary conditions. The finite size scaling relation for the order parameter probability distribution is tested and verified numerically by microcanonical Creutz cellular automata simulations. The state critical exponent \delta, which characteries the far tail regime of the scaling order parameter probability distribution, is estimated for 3-d Ising models using the cellular automaton simulations at the critical temperature. The results are in good agreement with the monte carlo calculations.Comment: 8 pages 5 figure

Logarithmic corrections to gap scaling in random-bond Ising strips

Numerical results for the first gap of the Lyapunov spectrum of the self-dual random-bond Ising model on strips are analysed. It is shown that finite-width corrections can be fitted very well by an inverse logarithmic form, predicted to hold when the Hamiltonian contains a marginal operator.Comment: LaTeX code with Institute of Physics macros for 7 pages, plus 2 Postscript figures; to appear in Journal of Physics A (Letter to the Editor