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Journal of Experimental and Theoretical Physics

Critical behavior of disordered systems with a free surface

С. В. Белим

Published Oct 1, 2006• 11 citations
Publisher

Abstract

The behavior of homogeneous and disordered systems with a free boundary is described on the basis of group theory in the two-loop approximation directly in three-dimensional space. The effect of the free boundary on the regime of the bulk critical behavior is revealed. It is shown that the boundedness of the system slightly affects the regime of the bulk critical behavior in the case of the ordinary transition, whereas this effect is more noticeable in the case of the special transition. Surface critical phenomena are described for homogeneous and disordered systems, and the critical exponents are calculated in the two-loop approximation. It is shown that the effect of impurities is insignificant in the special phase transition, whereas it is more noticeable in the ordinary phase transition. The derived critical exponents are compared with the computer-simulation results.

DOI
10.1134/s106377610610013x

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The behavior of homogeneous and disordered systems with a free boundary is described on the basis of group theory in the two-loop approximation directly in three-dimensional space. The effect of the free boundary on the regime of the bulk critical behavior is revealed. It is shown that the boundedness of the system slightly affects the regime of the bulk critical behavior in the case of the ordinary transition, whereas this effect is more noticeable in the case of the special transition. Surface critical phenomena are described for homogeneous and disordered systems, and the critical exponents are calculated in the two-loop approximation. It is shown that the effect of impurities is insignificant in the special phase transition, whereas it is more noticeable in the ordinary phase transition. The derived critical exponents are compared with the computer-simulation results.
We calculate the surface critical exponents of the ordinary transition occurring in semi-infinite, quenched dilute Ising-like systems. This is done by applying the field theoretic approach directly in d=3 dimensions up to the two-loop approximation as well as in 4-epsilon dimensions. At d=4-epsilon we extend, up to the next-to-leading order, the previous first-order results of the square root of epsilon expansion by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. In both cases numerical estimates for surface exponents are computed using Padé approximants extrapolating the perturbation theory expansions. The obtained results indicate that the critical behavior of semi-infinite systems with quenched bulk disorder is characterized by the new set of surface critical exponents.
We use a three-dimensional massive field theory up to the two-loop approximation to study the critical behavior of semi-infinite quenched random Ising-like systems at the special surface transition. Besides, we extend up to the next-to leading order, the previous first-order results of the sqrt[epsilon] expansion obtained by Ohno and Okabe [Phys. Rev. B 46, 5917 (1992)]. The numerical estimates for surface critical exponents in both cases are computed by means of the Padé analysis. Moreover, in the case of the massive field theory we perform Padé-Borel resummation of the resulting two-loop series expansions for surface critical exponents. The most reliable estimates for critical exponents of semi-infinite systems with quenched bulk randomness at the special surface transition, which we can obtain in the frames of the present approximation scheme, are eta(//)=-0.238, Delta(1)=1.098, eta( perpendicular )=-0.104, beta(1)=0.258, gamma(11)=0.839, gamma(1)=1.426, delta(1)=6.521, and delta(11)=4.249. These values are different from critical exponents for pure semi-infinite Ising-like systems and show that in a system with quenched bulk randomness the plane boundary is characterized by a new set of critical exponents at the special surface transition.
We consider the crossover behavior between special and ordinary surface transitions in three-dimensional semi-infinite Ising-like systems with random quenched bulk disorder. We calculate the surface crossover critical exponent Phi, the critical exponents of the layer alpha(1), and local specific heats alpha(11) by applying the field theoretic approach directly in three spatial dimensions (d=3) up to the two-loop approximation. The numerical estimates of the resulting two-loop series expansions for the surface critical exponents are computed by means of Padé and Padé-Borel resummation techniques. We find that Phi, alpha(1), alpha(11) obtained in the present paper are different from their counterparts of pure Ising systems. The obtained results support the idea that in a system with random quenched bulk disorder the plane boundary is characterized by a new set of critical exponents.
This paper is a field theoretic description of the critical dynamics of spin systems with frozen nonmagnetic impurities. For three-dimensional systems the dynamical critical exponent is found directly by employing the three-loop approximation with the Padé-Borel summation technique. The results are compared with those obtained by calculating the dynamical exponent for homogeneous systems in the four-loop approximation, and with the values obtained by computer simulation of the critical dynamics by Monte Carlo methods. Calculations of the dynamical exponent for the two-dimensional Ising model in the four-loop approximation are also presented.
We discuss the universality classes of the local critical behavior of a bulk system near a free surface and near an internal plane of defects. A phase diagram and plausible renormalization-group flow properties for a general model which encompasses both cases are described qualitatively. Several of the crossover exponents are related to bulk critical exponents or semi-infinite surface critical exponents. The picture which is proposed accomodates systems with nonuniversal local critical exponents near a plane of defects, such as the two-dimensional Ising model near a defect line, in a natural way. Necessary conditions for nonuniversal local critical exponents and for logarithmic corrections are derived. The singularities of the energy densities near a free surface and near an internal defect plane on approaching the bulk critical temperature are analyzed.