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Physical review. B, Condensed matter

Universality of surface critical behavior of the three-dimensional Ising model in a random surface field

K. K. Mon, M. P. Nightingale

Published Mar 1, 1988• 14 citations
Publisher

Abstract

The Harris criterion is generalized to determine the relevance at bulk criticality (in the scaling theory sense) of a random ordering field acting at a surface only. For surface critical behavior of the three-dimensional Ising model the criterion predicts the random field to be irrelevant for the ordinary transition, and relevant for the surface-bulk multicritical point. Monte Carlo results for the ordinary transition verify the prediction. Moreover, surface critical behavior near substrates chemically altered in part, as used in a series of experiments by Franck and co-workers, is in the universality class mentioned above, and the value of ${\ensuremath{\beta}}_{1}$ found in a random surface experiment [D. J. Durian, and C. Franck, Phys. Rev. Lett. 59, 555 (1987)] agrees with the scaling prediction.

DOI
10.1103/physrevb.37.3815

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The Harris criterion is generalized to determine the relevance at bulk criticality (in the scaling theory sense) of a random ordering field acting at a surface only. For surface critical behavior of the three-dimensional Ising model the criterion predicts the random field to be irrelevant for the ordinary transition, and relevant for the surface-bulk multicritical point. Monte Carlo results for the ordinary transition verify the prediction. Moreover, surface critical behavior near substrates chemically altered in part, as used in a series of experiments by Franck and co-workers, is in the universality class mentioned above, and the value of \ensuremathβ1{\ensuremath{\beta}}_{1}\ensuremathβ1​ found in a random surface experiment [D. J. Durian, and C. Franck, Phys. Rev. Lett. 59, 555 (1987)] agrees with the scaling prediction.
With use of a novel Monte Carlo method suited for the study of surface phenomena, the authors have investigated a three-dimensional Ising model with modified surface exchange. The results verify the scaling behavior of the magnetization profile and confirm and extend renormalization-group predictions. The surface-layer magnetization exponent is ${\ensuremath{\beta}}_{1}=0.78\ifmmode\pm\else\textpm\fi{}0.02$ except at the "special" (multicritical) transition where ${{\ensuremath{\beta}}_{1}}^{m}=0.175\ifmmode\pm\else\textpm\fi{}0.025$ and the crossover exponent is $\ensuremath{\phi}=0.56\ifmmode\pm\else\textpm\fi{}0.04$.
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.
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We study the critical behavior at the ordinary surface universality class of the three-dimensional O(N)\mathrm{O}(N)O(N) model, bounded by a two-dimensional surface. Using high-precision Monte Carlo simulations of an improved lattice model, where the leading bulk scaling correction is suppressed, and finite-size scaling analysis of the fourth cumulant of the surface magnetization, we obtain precise estimates of the scaling dimension of the surface field operator for N=2,3,4N=2,3,4N=2,3,4. We also determine the fixed-point values of two renormalization-group invariant observables, which characterize the finite-size scaling behavior at the ordinary transition.
Recent advances in conformal field theory and critical phenomena have focused on the characterization of boundary or defects in a conformally invariant system. In this Letter, we study the critical behavior of the three-dimensional Ising universality class in the presence of a surface, realizing the ordinary, the special, and the normal universality classes. By combining high-precision Monte Carlo simulations of an improved model, where leading scaling corrections are suppressed, with a finite-size scaling analysis informed by conformal field theory, we determine unbiased, accurate estimates of universal boundary operator product expansion coefficients of experimental relevance. Furthermore, we improve the value of the scaling dimension of the surface field at the special transition by the estimate Δ ̂ σ = 0.3531 ( 3 ) .