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Journal of Physics C Solid State Physics

A generalized coherent-potential approximation for site-disordered spin systems

Eduardo Lage, Robin B. Stinchcombe

Published Jan 28, 1977• 18 citations
Publisher

Abstract

An extension of the usual diagrammatic coherent-potential approximation is considered to account for the presence of disorder both in off-diagonal and inhomogeneous terms. The method is applied to the quenched site-disordered Ising model (S= 1 / 2 ) above the transition, where the fundamental equation for the correlation function is a generalized random-phase approximation. The results for the initial slope of the critical temperature and the critical concentration are presented for the simple cubic lattice with nearest-neighbour interactions, and they are in reasonable agreement with the values derived by other methods. The differences are studied and a discussion is presented to show that, near the critical concentration, terms not considered in the coherent-potential approximation are as important as the terms included. The application of this method to other models is also briefly discussed and, in the particular case of the Heisenberg model, it is shown that the method reproduces the basic results derived by Harris et al. (1974) using an effective-medium approach.

DOI
10.1088/0022-3719/10/2/013

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Journal of Physics C Solid State Physics•Mar 14, 1977•12 citations
An extension of the usual diagrammatic coherent-potential approximation is considered to account for the presence of disorder both in off-diagonal and inhomogeneous terms. The method is applied to the quenched site-disordered Ising model (S= 1 / 2 ) above the transition, where the fundamental equation for the correlation function is a generalized random-phase approximation. The results for the initial slope of the critical temperature and the critical concentration are presented for the simple cubic lattice with nearest-neighbour interactions, and they are in reasonable agreement with the values derived by other methods. The differences are studied and a discussion is presented to show that, near the critical concentration, terms not considered in the coherent-potential approximation are as important as the terms included. The application of this method to other models is also briefly discussed and, in the particular case of the Heisenberg model, it is shown that the method reproduces the basic results derived by Harris et al. (1974) using an effective-medium approach.
The thermal and magnetic properties of the simple cubic Ising model with random, quenched, site impurities have been determined by Monte Carlo calculations. The transition appears to remain sharp, and the critical temperature decreases monotonically towards zero as the impurity concentration is increased towards the percolation limit. No clear evidence is found for any change in the critical exponents away from the pure lattice values. These results are discussed in terms of Harris's predictions for the critical behavior of impure systems.
Changes in magnetic critical behaviour of quenched structurally-disordered magnets are usually exemplified in experiments and in MC simulations by diluted systems consisting of magnetic and non-magnetic components. In our study we aim to show that similar effects can be observed not only for diluted magnets with non-magnetic impurities but may be implemented, e.g., by the presence of two (and more) chemically different magnetic components as well. Therefore we consider a model of the structurally-disordered quenched magnet where all lattice sites are occupied by Ising-like spins of different lengths L. In such a random spin length Ising model, the length L of each spin is a random variable governed by the distribution function p(L). We demonstrate that this model belongs to the universality class of the site-diluted Ising model. This proves that both models are described by the same values of asymptotic critical exponents. However, their effective critical behaviour differs. As a case study, we consider a quenched mixture of two different magnets with values of elementary magnetic moments L1=1 and L2=s, and of concentration c and 1−c, correspondingly. We apply field-theoretical renormalization group approach to analyse the renormalization group flow for different initial conditions, triggered by s and c, and to calculate effective critical exponents further away from the fixed points of the renormalization group transformation. We show how the effective exponents are governed by difference in properties of the magnetic components.
We study the correlated-disorder-driven zero-temperature equilibrium phase transition of the random-field Ising magnet (RFIM) using exact numerical ground-state calculations for cubic lattices. We consider correlations of the quenched disorder decaying proportionally to ra{r}^{a}ra, where rrr is the distance between two lattice sites and a<0a<0a<0. To obtain exact ground states, we use a well-established mapping to the graph-theoretical maximum-flow problem, which allows us to study large system sizes of more than $2\ifmmode\times\else\texttimes\fi{}{10}^{6}$ spins. We use finite-size-scaling analyses for values $a={\ensuremath{-}1,\ensuremath{-}2,\ensuremath{-}3,\ensuremath{-}7}$ to calculate the critical point and the critical exponents characterizing the behavior of the specific heat, magnetization, susceptibility, and the correlation length close to the critical point. We find basically the same critical behavior as for the RFIM with $\ensuremath{\delta}$-correlated disorder, except for the finite-size exponent of the susceptibility and for the case $a\ensuremath{\ge}\ensuremath{-}2$, where the results are also compatible with a phase transition at infinitesimal disorder strength. We numerically confirm earlier predictions.
We study the diluted site-bond-correlated (SBC) Ising model within the scheme of the mean-field renormalization group. The SBC Ising model has been proposed to describe the phase diagram for KNip{\mathrm{KNi}}_{\mathit{p}}KNip​${\mathrm{Mg}}_{1\mathrm{\ensuremath{-}}\mathit{p}}$F3{\mathrm{F}}_{3}F3​. One treats the square (Z=4) and simple cubic (Z=6) lattices by using two different exchange interactions (denoted by models A and B). In the present paper, the phase diagram for some values of the correlation parameter \ensuremath{\alpha}, which range from zero to unity, is studied. We obtain two critical concentrations pc{\mathit{p}}_{\mathit{c}}pc​(\ensuremath{\alpha}=0) and pc{\mathit{p}}_{\mathit{c}}pc​(\ensuremath{\alpha}\ensuremath{\ne}0) in agreement with results from Monte Carlo methods. In addition, an interesting universal relation between the initial slope ${\mathrm{\ensuremath{\Delta}}}^{(\mathrm{\ensuremath{\mu}})}$(\ensuremath{\alpha}=1) and ${\mathrm{\ensuremath{\Delta}}}^{(\mathrm{\ensuremath{\mu}})}$(\ensuremath{\alpha}=0) is obtained (\ensuremath{\mu} denotes model A or B). From this relation, we can estimate the possible exact values to the initial slope on the completely correlated case. Our approach is valid for lattices with any coordination number Z.
The techniques developed by Edwards and Anderson (1975) for the spin-glass problem are here applied to the bond-diluted quenched Ising model (S= 1 / 2 ). The n to 0 trick is used to express exactly the free energy in terms of a generalized Hamiltonian representing n different but coupled replicas. The only approximation of the theory consists of decoupling these replicas in a way that reduces the free energy of the diluted system to that of a pure system with renormalized, concentration-dependent parameters. To first order in the concentration of missing bonds the author obtains all the results previously derived by Harris (1974) using a cumulant expansion and which are exact in this limit. More generally, the theory allows a discussion of the concentration dependence of the critical temperature; in particular, the values deduced for the critical concentration agree rather well with those obtained exactly or through series expansions. The behaviour of the system near the transition is also studied and the critical exponents are found to be Fisher-renormalized.