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Journal of Physics C Solid State Physics

Dilute semi-infinite Ising model

A. Benyoussef, N. Boccara, M. Saber

Published Aug 10, 1985• 50 citations
Publisher

Abstract

The influence of bond and site dilution on the phase transitions of a semi-infinite three-dimensional Ising model is investigated. To determine, in particular, phase diagrams two different techniques have been used: an extension of Migdal's approach to disordered systems and a more tractable approximation introduced recently. Temperature-concentration phase diagrams for fixed values of the ratio of surface and bulk interactions can exhibit five different types of phase transitions.

DOI
10.1088/0022-3719/18/22/011

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The influence of bond and site dilution on the phase transitions of a semi-infinite three-dimensional Ising model is investigated. To determine, in particular, phase diagrams two different techniques have been used: an extension of Migdal's approach to disordered systems and a more tractable approximation introduced recently. Temperature-concentration phase diagrams for fixed values of the ratio of surface and bulk interactions can exhibit five different types of phase transitions.
Abstract The Migdal–Kadanoff renormalization group technique is applied to study a three-dimensional semi-infinite Ising system consisting of spin-1/2 sites in the bulk and spin-1 sites at the surface. According to the competition ratio between bulk and surface exchange interactions ( R = J B / J S ), two types of surface phase diagrams have been identified in the (surface crystal field, temperature) plane. These diagrams exhibit a rich variety of phase transition phenomena: surface, ordinary, extraordinary, and special second-order transitions, first-order transitions at the surface, as well as tricritical and multicritical points. Additionally, we list in a table the fixed points that govern the observed phases and phase transitions.
We study by Monte Carlo simulations the influence of bond dilution on the three-dimensional Ising model. This paradigmatic model in its pure version displays a second-order phase transition with a positive specific heat critical exponent α\alphaα . According to the Harris criterion disorder should hence lead to a new fixed point characterized by new critical exponents. We have determined the phase diagram of the diluted model, starting from the pure model limit down to the neighbourhood of the percolation threshold. For the estimation of critical exponents, we have first performed a finite-size scaling study, where we concentrated on three different dilutions to check the stability of the disorder fixed point. We emphasize in this work the great influence of the cross-over phenomena between the pure, disorder and percolation fixed points which lead to effective critical exponents dependent on the concentration. In a second set of simulations, the temperature behaviour of physical quantities has been studied in order to characterize the disorder fixed point more accurately. In particular this allowed us to estimate ratios of some critical amplitudes. In accord with previous observations for other models this provides stronger evidence for the existence of the disorder fixed point since the amplitude ratios are more sensitive to the universality class than the critical exponents. Moreover, the question of non-self-averaging at the disorder fixed point is investigated and compared with recent results for the bond-diluted q = 4 Potts model. Overall our numerical results provide evidence that, as expected on theoretical grounds, the critical behaviour of the bond-diluted model is indeed governed by the same universality class as the site-diluted model.
The effect of random bond (site) dilution on the phase transition of a three-dimensional transverse Ising model is studied by a cluster variation, along with perturbation expansion method. Complete phase diagrams in the temperature-concentration-transverse-field space are constructed for both the cases. The value of critical transverse field as a function of concentration, at zero temperature, exhibits a discontinuity at the critical value of concentration for the existence of phase transition, as conjectured by Harris. As is expected for the existence of a second-order phase transition, the static susceptibility diverges on the critical surface.
2D spin-1 Ising models with nearest-neighbour and crystal-field interactions exhibit tricritical behaviour. A new method that applies to a wide class of random systems is used to study the influence of site and bond dilution on this behaviour. The authors have calculated temperature-crystal-field-concentration phase diagrams and determined, in particular, the influence of dilution on the zero-temperature critical field and on the tricritical temperature. The results are in agreement with those obtained recently for bond dilution by a real-space renormalisation-group technique. The method appears more tractable.
As a function of the ratio R of bulk and surface interactions and the ratio D of bulk and surface crystal field we have determined, within the framework both of the mean-field approximation and by renormalization-group techniques, various types of phase diagram of a three-dimensional semi-infinite ferromagnetic spin-3/2 Blume-Capel model. We found that there exist four main types of phase diagram showing a variety of phase transitions associated with the surface and unusual multicritical topologies, including ordinary, extraordinary, surface and special phase transitions. Moreover, in the ( R , D ) plane, we determined the domains in which the system exhibits a particular type of phase diagram.