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Physical review. B, Solid state

Magnetization upper bound for quenched Ising models with random ferromagnetic interactions

Harold Falk

Published May 1, 1977• 1 citation
Publisher

Abstract

For the quenched spin-1/2 Ising model in a uniform magnetic field and having only nearest-neighbor interactions which are statistically independent non-negative (ferromagnetic) identically distributed random variables, an exact upper bound is obtained for the magnetization. The bound can be identified with the value computed from a generalized Bethe approximation and thereby provides an extension to random interactions of some recent work of Krinsky.

DOI
10.1103/physrevb.15.4288

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For the quenched spin-1/2 Ising model in a uniform magnetic field and having only nearest-neighbor interactions which are statistically independent non-negative (ferromagnetic) identically distributed random variables, an exact upper bound is obtained for the magnetization. The bound can be identified with the value computed from a generalized Bethe approximation and thereby provides an extension to random interactions of some recent work of Krinsky.
Ising systems are approximated by a generalized mean-field approximation in which only some of the interactions are replaced by their expectation values. If all the interactions are positive then it is shown that a self-consistent solution exists for the magnetization in this model and that it gives an upper bound for the true magnetization. Using this mean-field approximation, a generalization of the random-phase approximation is obtained. This is shown to give an upper bound for two-spin correlation functions.
In this work, we simulated a quenched random mixed-spin Ising model on the square lattice. The model system consists of two different particles with spins σ = 1/2 (states ±1/2) and S = 1 (states ±1, 0). These particles are randomly distributed on the lattice, and we considered only nearest-neighbor interactions. This model can represent a random magnetic binary alloy AxB1-x, obtained from the high-temperature quenching of a liquid mixture. We performed Monte Carlo simulations for several lattice sizes and temperatures, and we found its critical temperature through the reduced fourth-order cumulant. We also determined the magnetization, the susceptibility, and the specific heat as a function of temperature. We used finite-size scaling arguments to estimate the critical exponents β, γ, and ν of the model. We showed that the quenched model is in the same universality class of the two-dimensional pure Ising model. We also investigated the sample to sample fluctuations that occur in the values of the thermodynamic quantities in order to describe the self-averaging properties at the critical point.
We studied a new and simple method which gives successive approximate expressions for the magnetization (or any order parameter) of a random Ising model as a function of temperature, site, or bond dilution, frustration, magnetic field,... To obtain the various approximations a trace is performed on finite clusters increasing in size. In particular, we investigated a honeycomb Ising model with random nearest-neighbor interactions and calculated sequences of critical temperatures, percolation thresholds,... which converge towards exact known results.
Studies a correlation function which is given by the canonical average of a product of one or more spin variables, for the random-bond Ising model, in which the exchange integrals are +J(J>O) and -J with probabilities p and 1-p, respectively. The authors show that an upper bound to the configurational average of the correlation functions, calculated in the thermodynamic limit in the zero external field limit, is the product of the same correlation function for the corresponding ferromagnetic Ising model at the temperature under consideration and the same quantity at the temperature T 1 which is determined by the condition T 1 =2J/ mod k B 1n(p/(1-p)) mod , where k B is the Boltzmann constant. The results are given for the random-bond Ising model of an arbitrary spin S, and also for the diluted random-bond Ising models of an arbitrary spin S, with the pair interaction and with general interaction, and for the diluted and undiluted random-bond n-vector model.
The Random Field Ising Model (RFIM) represents one of the simplest models of cooperative behaviour with quenched disorder, and it is, in a way, complementary to the Ising Spin Glass which will be extensively treated later in this book. It accounts for the presence of a random external magnetic field which antagonizes the ordering induced by the ferromagnetic spin–spin interactions. From an experimental point of view, on the other hand, as shown by Fishman and Aharony (1979) and Cardy (1984), it is equivalent to a dilute anti-ferromagnet in a uniform field (see Belanger, 1998 for a recent review on experimental results). Despite twenty-five years of active and continuous research the RFIM is not yet completely understood. The problem seems related to the presence of bound states in the ferromagnetic phase, which make the standard theoretical approaches not adequate to analyze the critical behaviour. Here we discuss the RFIM in the context of perturbative field theory. The chapter is organized as follows: in Section 2.1 we define the model and outline the main expectations for its qualitative behaviour. In Section 2.2 we introduce an effective replicated ϕ 4 field model where the disorder has been integrated out. Then we perform a perturbative analysis on this model (Section 2.3) and illustrate how the so-called dimensional reduction arises (Section 2.4). Finally, in Section 2.5 we introduce some generalized couplings which need to be taken into account to properly describe the system; we perform a perturbative Renormalization Group (RG) close to the upper critical dimension (Sections 2.6 and 2.7) and discuss the occurrence of a vitrous transition (Section 2.8).