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Physical Review Letters

Accurate estimate ofνfor the three-dimensional Ising model from a numerical measurement of its partition function

Gyan V Bhanot, Román Salvador, Steve Black, Paul M. Carter, Raúl Toral

Published Aug 17, 1987• 77 citations
Publisher

Abstract

We give results from an accurate determination of the partition function of the three-dimensional Ising model on lattices of size up to ${10}^{3}$. 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.

DOI
10.1103/physrevlett.59.803

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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 use a high-precision Monte Carlo simulation to determine the universal specific-heat amplitude ratio A + / A − in the three-dimensional Ising model via the impact angle ϕ of complex temperature zeros. We also measure the correlation-length critical exponent ν from finite-size scaling and the specific-heat exponent α through hyperscaling. Extrapolations to the thermodynamic limit yield ϕ = 59.2(1.0)°, A + / A − = 0.56(3), ν = 0.630 48(32) and α = 0.1086(10). These results are compatible with some previous estimates from a variety of sources and rule out recently conjectured exact values.