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Physical review. B, Solid state

Thermodynamics of an Ising model with random exchange interactions

Michael F. Thorpe, David Beeman

Published Jul 1, 1976• 92 citations
Publisher

Abstract

An Ising model with an arbitrary distribution $P(J)$ of exchange interactions $J$ is solved exactly for the annealed case in which the system is allowed to reach complete thermal equilibrium at each temperature. The solution is expressed in terms of an Ising model on the same lattice with a single exchange parameter, allowing an exact solution to be obtained in one and two dimensions. Some feeling for the effect of annealing can be found by examining correlations between exchange interactions on neighboring bonds. Some special distributions $P(J)$ are examined in detail.

DOI
10.1103/physrevb.14.188

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An Ising model with an arbitrary distribution P(J)P(J)P(J) of exchange interactions JJJ is solved exactly for the annealed case in which the system is allowed to reach complete thermal equilibrium at each temperature. The solution is expressed in terms of an Ising model on the same lattice with a single exchange parameter, allowing an exact solution to be obtained in one and two dimensions. Some feeling for the effect of annealing can be found by examining correlations between exchange interactions on neighboring bonds. Some special distributions P(J)P(J)P(J) are examined in detail.
Thermodynamic properties of one-dimensional Ising chains with random nearest neighbor exchange interaction J are studied exactly in the case of distributions of the type P(J) = A|J|ν for Jmin ≤J ≤Jmax and P(J) = 0, otherwise. The results are compared with those for a periodic chain. We find that the specific heat C ∼Tν+1 for Jmin = 0 and kT ≪Jmax. The dependence of spin-spin correlation function on distance and temperature is discussed. The absence of the peak in the zero field χ⊥ as a function of temperature when ν = 0 is noted in contrast to the case of the periodic chain. The effect of randomness on the low temperature behaviour of these thermodynamic quantities is different. For instance whereas C becomes analytic at T = 0, χ|| either remains singular (ferromagnetic) or becomes analytic (antiferromagnetic). The effect of random xy interaction (∑iJi⊥(SixSi + 1x + SiySi + 1y)) on the high temperature behaviour of χ|| and χ⊥ is discussed.
The effect of a discrete distribution of annealed random fields on the properties of the two-dimensional Ising model is studied. The random field is allowed to assume the values 0,\ifmmode\pm\else\textpm\fi{}h with the probabilities p0{p}_{0}p0​,${p}_{\ifmmode\pm\else\textpm\fi{}}$, and a parameter \ensuremath{\mu} is introduced which is determined by specifying p0{p}_{0}p0​. Exact results are obtained by mapping the model onto an effective Ising model on the rectangular lattice without field, and the probabilities p0{p}_{0}p0​,${p}_{\ifmmode\pm\else\textpm\fi{}}$ are expressed in terms of the magnetization and two-spin correlations of the effective model. It is shown that the long-range order is not destroyed by the random field and the transition temperature increases as p0{p}_{0}p0​ decreases. For a given p0{p}_{0}p0​ the critical temperature increases with the field strength and these combined effects show that the annealed random field favors the ordering of the system, therefore presenting behavior opposite to that expected for the system in a quenched random field.
The critical behavior of the two-dimensional Ising model (square lattice, exchange constant J) in an uniform field, and in an annealed random field is considered. The random field is generated by decorating the horizontal and vertical bonds of the lattice, and it satisfies an arbitrary distribution which is imposed by introducing a pseudo-chemical potential. By decimating the decorating variables the model can be mapped onto a homogeneous Ising model with effective exchange constant J′ and effective external field h′, dependent on the temperature. These parameters, which satisfy a set of coupled equations, depend on the spin average and nearest-neighbor two-spin correlation, and are obtained numerically. For the symmetric field distribution [Formula: see text] the mapping of the critical frontier on the (K′=βJ′,H′=βh′) plane onto the (K=β J,H=βh) plane is determined and, as in the model introduced by Essam and Place, there is a region on the (K, H) plane which cannot be reached from any real values of (K′, H′). The critical exponents are determined numerically, and it is shown that they do not satisfy renormalization relations obtained for their model.
We study the equilibrium properties of an Ising model on a disordered random network where the disorder can be quenched or annealed. The network consists of fourfold coordinated sites connected via variable length one-dimensional chains. Our emphasis is on nonuniversal properties and we consider the transition temperature and other equilibrium thermodynamic properties, including those associated with one-dimensional fluctuations arising from the chains. We use analytic methods in the annealed case, and a Monte Carlo simulation for the quenched disorder. Our objective is to study the difference between quenched and annealed results with a broad random distribution of interaction parameters. The former represents a situation where the time scale associated with the randomness is very long and the corresponding degrees of freedom can be viewed as frozen, while the annealed case models the situation where this is not so. We find that the transition temperature and the entropy associated with one-dimensional fluctuations are always higher for quenched disorder than in the annealed case. These differences increase with the strength of the disorder up to a saturating value. We discuss our results in connection to physical systems where a broad distribution of interaction strengths is present.
The one-dimensional Ising model in an external magnetic field with uniform long-range interactions and random short-range interactions satisfying bimodal annealed distributions is studied. This generalizes the random model discussed by Paladin et al. (J. Phys. I France 4, 1994, p. 1597). Exact results are obtained for the thermodynamic functions at arbitrary temperatures, and special attention is given to the induced and spontaneous magnetization. At low temperatures the system can exist in a “ferrimagnetic” phase with magnetization 0 < σ < 1, in addition to the usual paramagnetic, ferromagnetic and antiferromagnetic phases. For a fixed distribution of the random variables the system presents up to three tricritical points for different intensities of the long-range interactions. Field-temperature diagrams can present up to four critical points. 1