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arXiv:cond-mat/0004199 · Journal of Physics A Mathematical and General

Partition function zeros and leading-order scaling correction of the 3D Ising model from multicanonical simulations

Nelson A. Alves, J. R. Drugowich de Felício, Ulrich H. E. Hansmann

Published Oct 11, 2000• 13 citations
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Abstract

The density of states for the three-dimensional Ising model is calculated with high precision by means of multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L = 32. We have evaluated the critical exponent ν and the correction to scaling through an analysis of a multi-parameter fit and of the Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known.

DOI
10.1088/0305-4470/33/42/302

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The density of states for the three-dimensional Ising model is calculated with high precision by means of multicanonical simulations. This allows us to estimate the leading partition function zeros for lattice sizes up to L = 32. We have evaluated the critical exponent ν and the correction to scaling through an analysis of a multi-parameter fit and of the Bulirsch-Stoer (BST) extrapolation algorithm. The performance of the BST algorithm is also explored in case of the 2D Ising model, where the exact partition function zeros are known.
We used a Monte Carlo simulation of the structurally disordered three dimensional Ising model. For the systems with spin concentrations p = 0.95 ,0.8, 0.6 and 0.5 we calculated the correlation-length critical exponent ν by finite-size scaling. Extrapolations to the thermodynamic limit yield ν(0.95) = 0.705(5) ,ν(0.8) = 0.711(6),ν(0.6) = 0.736(6) and ν(0.5) = 0.744(6). These results are compatible with some previous estimates from a variety of sources. The analysis of the results demonstrates the nonuniversality of the critical behavior in the disordered Ising model.
We have performed Monte Carlo simulations for the three-dimensional Ising model. Using histogram techniques, we calculate the density of states on L3{L}^{3}L3 block lattices up to size L=14. Statistical jackknife methods are employed to perform a thorough error analysis. We obtain high-precision estimates for the leading zeros of the partition function, which, using finite-size scaling, translate into \ensuremath{\nu}=0.6285\ifmmode\pm\else\textpm\fi{}0.0019. Along a different line of approach following recent work in lattice-gauge theories, we accurately determine the mass gap m=1/\ensuremath{\xi} (\ensuremath{\xi} correlation length) for cylindrical L2{L}^{2}L2Lz{L}_{z}Lz​ lattices (with Lz{L}_{z}Lz​=256 and L up to 12). The finite-size-scaling analysis of the mass-gap data leads to \ensuremath{\nu}=0.6321\ifmmode\pm\else\textpm\fi{}0.0019.
We give results from an accurate determination of the partition function of the three-dimensional Ising model on lattices of size up to 103{10}^{3}103. We compute the two complex zeros of Z closest to the real temperature axis. From a finite-size scaling analysis of these, we get the estimates \ensuremath{\nu}=0.6295(10) for the correlation-length exponent and \ensuremath{\varphi}=52.2(7) for the angle between the first two zeros and the real u=${e}^{\mathrm{\ensuremath{-}}4\ensuremath{\beta}}$ axis.
Corrections to scaling in the 3D Ising model are studied based on Monte Carlo (MC) simulation results for very large lattices with linear lattice sizes up to L=3456. Our estimated values of the correction-to-scaling exponent omega tend to decrease below the usually accepted value about 0.83 when the smallest lattice sizes are discarded from the fits. This behavior apparently confirms some of the known estimates of the Monte Carlo renormalization group (MCRG) method, i.e., omega about 0.7 and omega = 0.75(5). We discuss the possibilities that omega is either really smaller than usually expected or these values of omega describe some transient behavior. We propose refining MCRG simulations and analysis to resolve this issue. In distinction from omega, our actual MC estimations of the critical exponents eta and nu provide stable values eta=0.03632(13) and nu=0.63017(31), which well agree with those of the conformal bootstrap method, i.e., eta=0.0362978(20) and nu=0.6299709(40).
We perform high-statistics Monte Carlo simulations of three three-dimensional Ising spin glass models: the ± J Ising model for two values of the disorder parameter p , p = 1/2 and 0.7, and the bond-diluted ± J model for bond-occupation probability p b = 0.45. A finite-size scaling analysis of the quartic cumulants at the critical point shows conclusively that these models belong to the same universality class and allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator, ω = 1.0(1). We also determine the critical exponents ν and η. Taking into account the scaling corrections, we obtain ν = 2.53(8) and η = −0.384(9).