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Journal of Physics A Mathematical and General

Renormalized coupling constant in the Ising model

George Allen Baker Jr., Naoki Kawashima

Published Nov 21, 1996• 20 citations
Publisher

Abstract

The three-dimensional Ising model is studied by means of the Swendsen - Wang, cluster, Monte Carlo method. The simulations are performed on finite sized systems up to the 96-cube, and the renormalized coupling constant is estimated to be . We make further arguments and computations and say that and so that which implies the validity of hyperscaling . The Josephson hyperscaling relation also appears to hold. We give estimates for some of the critical indices.

DOI
10.1088/0305-4470/29/22/018

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The three-dimensional Ising model is studied by means of the Swendsen - Wang, cluster, Monte Carlo method. The simulations are performed on finite sized systems up to the 96-cube, and the renormalized coupling constant is estimated to be . We make further arguments and computations and say that and so that which implies the validity of hyperscaling . The Josephson hyperscaling relation also appears to hold. We give estimates for some of the critical indices.
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Monte Carlo calculations of the renormalized coupling constant for the three-dimensional Ising model are reported. Improved estimators both for the second partial of the susceptibility with respect to the magnetic field and for the correlation length are developed. These estimators show greatly improved convergence properties, and allow us to compute for system sizes and temperature rather more relevant to the thermodynamic limit than previously. Our results indicate strongly that hyperscaling holds for this model.
We report extensive Monte Carlo simulations and finite-size scaling analysis of the correlation length (${\ensuremath{\xi}}_{\mathrm{L}}$) for the spin-spin correlation function and the renormalized coupling (gL{\mathrm{g}}_{\mathrm{L}}gL​) of the simple-cubic Ising model with a wide range of lattice sizes (L\ensuremath{\le}96). New methods of analysis which avoid assumptions related to the correlation length were introduced. The exponents \ensuremath{\nu} and 2-\ensuremath{\alpha} were estimated from separate analysis of ${\ensuremath{\xi}}_{\mathrm{L}}$ and gL{\mathrm{g}}_{\mathrm{L}}gL​, respectively. A value of ${\mathrm{\ensuremath{\omega}}}^{\mathrm{*}}$\ensuremath{\approxeq}0.001\ifmmode\pm\else\textpm\fi{}0.07 for the Fisher exponent is obtained. This is consistent with a very small or no violations of hyperscalings.