Bond-diluted Ising Model Research Articles

Dynamical linked cluster expansions with applications to disordered systems

Dynamical linked cluster expansions are linked cluster expansions with hopping parameter terms endowed with their own dynamics. We discuss physical applications to systems with annealed and quenched disorder. Examples are the bond-diluted Ising model and the Sherrington-Kirkpatrick spin glass. We derive the rules and identify the full set of graphs that contribute to the series in the quenched case. This way it becomes possible to avoid the vague extrapolation from positive integer n to n = 0, that usually goes along with an application of the replica trick.

Crossover effects in the bond-diluted Ising model in three dimensions

We investigate by Monte Carlo simulations the critical properties of the three-dimensional bond-diluted Ising model. The phase diagram is determined by locating the maxima of the magnetic susceptibility and is compared to mean-field and effective-medium approximations. The calculation of the size-dependent effective critical exponents shows the competition between the different fixed points of the model as a function of the bond dilution.

High-temperature series expansions for random-bond Potts models on

Abstract We use a star-graph expansion technique to compute high-temperature series for the free energy and susceptibility of random-bond q -state Potts models on hypercubic lattices. This method allows us to calculate quenched disorder averages for arbitrary uncorrelated coupling distributions. Moreover, we can keep the disorder strength p as well as the dimension d as symbolic parameters. This enables scans over large regions of the ( p , d ) parameter space for any value of q . For the bond-diluted Ising model ( q =2) in three dimensions we present first results for the critical temperature and exponent γ obtained from the analysis of susceptibility series up to order 19.

Effect of disorder on critical short-time dynamics

Critical short-time dynamics in a bond-diluted Ising model is investigated in this paper using numerical simulations. The effective static and dynamic critical exponents determined by the power-law scaling are found to depend strongly on bond concentration and initial state. For weak disorder, the short-time scaling relations for the system quenched from high temperature are observed to hold. In the strong dilution limit, multiscaling relations for the system starting from the ordered state are found. Corrections to the short-time scaling are proposed. The effect of disorder on critical short-time dynamics is discussed.

Diffusive scaling of the spectral gap for the dilute Ising lattice-gas dynamics below the percolation threshold

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on ℤd at inverse temperature β. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any β, with probability one, the spectral gap of the generator of the dyamics in a box of side L centered at the origin scales like L−2. Such an estimate is then used to prove a decay to equilibrium for local functions of the form \(\) where e is positive and arbitrarily small and α = ½ for d = 1, α=1 for d≥2. In particular our result shows that, contrary to what happes for the Glauber dynamics, there is no dynamical phase transition when β crosses the critical value βc of the pure system.

Spin Glass and Percolation Behaviour on Chromium Spinels

We compute the phase diagram of the non-metallic compounds ZnCr2pAl2—2pS(Se)4 and Zn1—pCdpCr2S(Se)4, for which we consider the bond-diluted Ising model on the spinel B-site (S.B.S.) lattice with competitive exchange interactions, and show that quenched randomness may cause fundamental changes in their low-temperature magnetic behaviour. Dilution and competition are found to be responsible of the spin glass phase and the percolation behaviour. The possible appearance of a reentrant spin glass phase is also mentioned.

Non-Equilibrium Dynamics of the Disordered Ising Model

The non-equilibrium dynamics of the bond-diluted Ising model in two-dimensions (2d) is investigated numerically for a range of bond concentration 0 ≤ p ≤ 1. The fraction of spins which never flip, P (∞), is found to increase monotonically from zero for p = 0 with increasing bond concentration to a maximum value of about 0.46 for p = 0.5 and then decreases to zero for p = 1. For strong bond-dilution (0 ≤ p ≤ 0.5) we find that r(t) = P (t)−P (∞) decays exponentially to zero at large times. For weak dilution (0.975 ≤ p < 1.0), three distinct regimes are identified: a short time regime where the behaviour is purelike; an intermediate regime where the persistence probability decays nonalgebraically with time; and a long time ‘frozen’ regime where the domains cease to grow. Our results for strong dilution are consistent with recent work of Newman and Stein, and suggest that persistence in diluted and pure systems falls into different classes. Furthermore, its behaviour would also appear to depend crucially on the strength of the dilution present.

Phase Diagram of ZnCr2pAl2—2pS4 and Zn1—pCdpCr2Se4

We compute the phase diagram of the non-metallic compounds ZnCr 2p Al 2-2p S 4 (I) and Zn 1-p Cd p Cr 2 Se 4 (II). We consider the bond-diluted Ising model on the spinel B site (S.B.S.) lattice with competitive exchange interactions, i.e. the ferromagnetic exchange interaction J 1 between nearest neighbours (n.n.) and the antiferromagnetic superexchange interaction J 2 between next-nearest neighbours (n.n.n.) (or and the more distant superexchange interactions J i (i > 1)). Dilution and competition are found to be responsible of the spin glass phase and the percolation behaviour.

Kinetic lattice models of disorder

We study a class of stochastic Ising (or interacting particle) systems that exhibit a spatial distribution of impurities that change with time. It may model, for instance, steady nonequilibrium conditions of the kind that may be induced by diffusion in some disordered materials. Different assumptions for the degree of coupling between the spin and the impurity configurations are considered. Two interesting well-defined limits for impurities that behave autonomously are (i) the standard (i.e., quenched) bond-diluted, random-field, random-exchange, and spin-glass Ising models, and (ii) kinetic variations of these standard cases in which conflicting kinetics simulate fast and random diffusion of impurities. A generalization of the Mattis model with disorder that describes a crossover from the equilibrium case (i) to the nonequilibrium case (ii) and the microscopic structure of a generalized heat bath are explicitly worked out as specific realizations of our class of models. We sketch a simple classification of transition rates for the time evolution of the spin configuration based on the critical behavior that is exhibited by the models in case (ii). The latter are shown to have an exact solution for any lattice dimension for some special choice of rates.

Absence of a percolation threshold in a quantum diluted Ising model.

A quantum bond-diluted Ising model is introduced. The model is reducible to its classical corresponding pure Ising model through the standard iteration-decoration transformation. The model considers a finite probability amplitude t for the bond hopping to and from an infinite bond reservoir. The exact phase diagram of the model on a square lattice is obtained and displays no percolation threshold for any finite t. Possible relevance of the present result for systems with hole or electron doping is discussed

One-dimensional Ising model: dilute antiferromagnet versus ferromagnet in a random field

The one-dimensional bond and site dilute antiferromagnetic Ising model in a uniform field and the ferromagnetic model in a random field are studied. Exact results are obtained for annealed distributions of the random parameters. For the random field problem the magnetization is explicitly evaluated for the discrete distribution p( h i ) = pδ( h i − h + (1 − p) δ( h i + h) and it is shown that it presents a critical behaviour for p = 0.5 only. On the other hand, it is shown that the bond dilute model presents no critical behaviour, and the site dilute model presents critical behaviour for values of the uniform field less than the exchange interaction only.

Dynamics in a dilute ferromagnet at the percolation threshold

The dynamics of the spin autocorrelation function and the relaxation of the magnetization in the Griffiths phase of the two-dimensional bond-diluted Ising model at the percolation threshold are studied using Monte Carlo techniques. The results resolve a previous ambiguity about the decay law.

A bond-diluted Ising model on a generalized Fibonacci lattice

A bond-diluted Ising model on a class of one-dimensional quasiperiodic lattices is studied. The free energy of this model is obtained. It is shown that the transition from a self-similar structure to a translationally invariant structure is smooth.

Remarks on a RG approach to the critical behaviour of 2D dilute systems

A continuous expression for the spin-spin correlator of the 2D bond-dilute Ising model is considered. It is shown that its evaluation through straightforward use of RG techniques leads to inconsistent results. Connections with related results in the literature are discussed.

Exact Decimation Approach with Mean-Field Approximation for Randomly Diluted Ising Models

A new renormalization approach developed by authors previously is generalized to disordered systems. Calculations of the critical temperature, concentration, exponents, and of limiting slope of critical curve for the random bond-diluted Ising model on the square lattice, even in the lowest approximation, are in very satisfactory agreement with all known exact or series results.

Bond-diluted Ising model with alternated interaction along one direction

Within the framework of an effective — field theory, the critical temperature and critical percolation values are investigated for a bond diluted two-dimensional Ising system with alternated interactions along one direction.

Self-similar and translationally invariant structures in a bond-diluted Ising model

Self-similar and translationally invariant structures in a bond-diluted Ising model

The breakdown of dynamic scaling of the bond-diluted Ising model at the percolation threshold for hypercubic lattices

The authors use real-space renormalisation group techniques to study the kinetic Ising model (with Glauber dynamics) on bond-diluted d-dimensional hypercubic lattices, at the percolation threshold. The results confirm and generalise the recently proposed breakdown of dynamic scaling and they predict a dependence on dimensionality of the factors A and B on the temperature-dependent dynamic exponent z=A log xi +B (where xi is the thermal correlation length). These factors also depend on the lattice scaling factor (b); the authors suggest that the limit b=1 may be taken as the best choice for the important coefficient A, leading to A=d(d-1)/2 nu p.

Self-similar and translationally invariant structures in a bond-diluted Ising model

A bond-diluted Ising model is used to simulate a process from a Cantor set to a translationally invariant lattice. The free energy function shows that there is no singularity, which seem to imply that no transition exists in this process.

Continuum limit on a 2-dimensional random lattice

We investigate the continuum limit of the Ising model on a 2-dimensional random lattice using Monte Carlo and strong coupling techniques. No evidence is found for deviation from the behaviour of the standard Ising model on a regular lattice; we do not see the modified critical properties found in the bond-diluted Ising model. Our results suggest that the continuum limit is the usual non-interacting Majorana fermion theory.