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Journal of Physics A Mathematical and General

Stochastic truncation for the (2+1)D Ising model

P. F. Price, Chris J. Hamer, D O'Shaughnessy

Published Jun 21, 1993• 11 citations
Publisher

Abstract

The method of 'symmetrized' stochastic truncation is applied to the (2+1)-dimensional Ising model, to calculate the ground-state energy, mass gap, magnetization and susceptibility on lattices of sizes up to 6*6 sites. The method is shown to be a powerful and accurate Monte Carlo technique, provided the lattice size is not too large, and the model is suitably 'convergent'. Finite-size scaling estimates of the critical indices are nu =0.627(2), alpha =0.12(2), beta =0.324(3), and delta =1.23(1). An estimate of the combination gamma / nu +2 beta / nu =3.000(1) is obtained, in outstanding agreement with the expected hyperscaling relation.

DOI
10.1088/0305-4470/26/12/023

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The method of 'symmetrized' stochastic truncation is applied to the (2+1)-dimensional Ising model, to calculate the ground-state energy, mass gap, magnetization and susceptibility on lattices of sizes up to 6*6 sites. The method is shown to be a powerful and accurate Monte Carlo technique, provided the lattice size is not too large, and the model is suitably 'convergent'. Finite-size scaling estimates of the critical indices are nu =0.627(2), alpha =0.12(2), beta =0.324(3), and delta =1.23(1). An estimate of the combination gamma / nu +2 beta / nu =3.000(1) is obtained, in outstanding agreement with the expected hyperscaling relation.
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.
The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J ', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J '/ J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.
Abstract. The critical properties of the randomly site-diluted two-dimensional Ising model were studied using the Wang-Landau algorithm. The concentration of nonmagnetic sites was q = 0.1; the remaining sites were occupied by magnetic particles. The study was carried out in the appro-priate restricted but dominant energy subspaces. The main effort was focused on the specific heat and magnetic susceptibility by using the density of states for the model for lattices with linear size L = 20-120. The finite size scaling behavior of the specific heat and susceptibility was studied. The ratio of the critical exponents (α/ν) is negative (consistent with the Harris criterion), while the ratio (γ/ν) appears to assume its pure Ising model value (weak universality). 1.
We use a new updated algorithm scheme to investigate the critical behaviour of the two-dimensional ferromagnetic Ising model on a triangular lattice with the nearest neighbour interactions. The transition is examined by generating accurate data for lattices with L = 8, 10, 12, 15, 20, 25, 30, 40 and 50. The updated spin algorithm we employ has the advantages of both a Metropolis algorithm and a single-update method. Our study indicates that the transition is continuous at T c = 3.6403(2). A convincing finite-size scaling analysis of the model yields ν = 0.9995(21), β/ν = 0.12400(17), γ/ν = 1.75223(22), γ'/ν = 1.7555(22), α/ν = 0.00077(420) (scaling) and α/ν = 0.0010(42) (hyperscaling). The present scheme yields more accurate estimates for all the critical exponents than the Monte Carlo method, and our estimates are shown to be in excellent agreement with their predicted values.