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Progress of Theoretical Physics

Monte Carlo Study of the Surface Critical Phenomena of the Ising Model

Macoto Kikuchi, Yutaka Okabe

Published Jan 1, 1985• 19 citations
Publisher

Abstract

Monte Carlo technique is applied to the three-dimensional Ising model with free surface. The critical behavior of the layer magnetization, the layer susceptibility and the local susceptibility near the ordinary transition point is investigated. The universal properties of the surface critical exponents and the surface critical amplitude ratios are confirmed. The scaling of magnetization profile is studied, and the bulk amplitude ratio for the correlation length ξ0h / ξ0- is calculated.

DOI
10.1143/ptp.73.32

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Monte Carlo technique is applied to the three-dimensional Ising model with free surface. The critical behavior of the layer magnetization, the layer susceptibility and the local susceptibility near the ordinary transition point is investigated. The universal properties of the surface critical exponents and the surface critical amplitude ratios are confirmed. The scaling of magnetization profile is studied, and the bulk amplitude ratio for the correlation length ξ0h / ξ0- is calculated.
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We present the results of extensive Monte Carlo simulations of phase transitions and critical behavior at the surface of a simple cubic Ising model. Profiles of the magnetization and internal energy are determined as a function of the distance from the surface, and we extract surface and bulk properties as a function of temperature and surface coupling Js{\mathit{J}}_{\mathit{s}}Js​. The surface-bulk multicritical point is located with improved precision, Js{\mathit{J}}_{\mathit{s}}Js​/J=1.52\ifmmode\pm\else\textpm\fi{}0.02, and crossover behavior is studied. New estimates for critical exponents are extracted, ${\ensuremath{\gamma}}_{1}$=0.78\ifmmode\pm\else\textpm\fi{}0.06, ${\ensuremath{\gamma}}_{1}^{\mathit{m}}$=1.41\ifmmode\pm\else\textpm\fi{}0.14, ${\ensuremath{\gamma}}_{11}^{\mathit{m}}$=0.96\ifmmode\pm\else\textpm\fi{}0.09, and along with critical amplitude ratios these are compared with theoretically predicted values.
We simulated the critical behavior of the d=3 Ising model with surfaces with a cluster Monte Carlo method at the multicritical point and at the ordinary transition. The surface exponents ${\mathrm{\ensuremath{\beta}}}_{1}$ have been determined using finite-size scaling. The errors of the resulting values ${\mathrm{\ensuremath{\beta}}}_{1}^{\mathit{m}}$=0.2375(15) and ${\mathrm{\ensuremath{\beta}}}_{1}^{0}$=0.807(4) are an order of magnitude smaller than in previous determinations. Whereas the values of the exponents agree with the prediction of the \ensuremath{\epsilon} expansion, the universal ratios of susceptibility and magnetization amplitudes at the special transition are at variance with those predictions.