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Journal of Physics A Mathematical and General

Critical phenomena at surfaces in a model of nonequilibrium phase transitions

Michel Droz, Laurent Frachebourg, M C Marques

Published Jun 21, 1991• 4 citations
Publisher

Abstract

Critical phenomena at surfaces are studied in a simple two-dimensional model of non-equilibrium phase transitions belonging to the class of interacting particle systems. The mean field renormalization group approach and numerical simulations are used to determine the phase diagram for this system and some of the critical exponents associated with ordinary, special, extraordinary and surface phase transitions.

DOI
10.1088/0305-4470/24/12/025

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Critical phenomena at surfaces are studied in a simple two-dimensional model of non-equilibrium phase transitions belonging to the class of interacting particle systems. The mean field renormalization group approach and numerical simulations are used to determine the phase diagram for this system and some of the critical exponents associated with ordinary, special, extraordinary and surface phase transitions.
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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 use simple position-space renormalization-group procedures involving only nearest-neighbor interactions to discuss the critical behavior of the ferromagnetically coupled semi-infinite Ising model with a free surface in two and three dimensions. In three dimensions for suitably enhanced surface couplings there are surface transitions and simultaneous surface-bulk transitions as well as the bulk transition. A phase diagram, critical exponents for the various transitions, and some representative magnetization profiles are calculated. Despite their simplicity, the renormalization transformations exhibit complete sets of fixed points for describing the various phase transitions and yield phase diagrams which are a significant improvement over mean-field theory. However, the calculated magnetization profiles show some unphysical features due to the finite cluster size on which the transformations are based.
We study the special point in the phase diagram of a semi-infinite system where the bulk transition is in the three-dimensional Ising universality class. To this end we performed a finite-size-scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interaction. In order to check for the effect of leading bulk corrections we also simulated the spin-1/2 Ising model on the simple cubic lattice. We accurately estimated the surface enhancement coupling at the special point of these models. We find yt1=0.718(2){y}_{{t}_{1}}=0.718(2)yt1​​=0.718(2) and yh1=1.6465(6){y}_{{h}_{1}}=1.6465(6)yh1​​=1.6465(6) for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.