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Physical Review

One-Dimensional Ising Model with Random Exchange Energy

Chungpeng Fan, Barry M. McCoy

Published Jun 10, 1969• 45 citations
Publisher

Abstract

The one-dimensional Ising model with random exchange energy is formulated in terms of a homogeneous integral equation. Assuming the input distribution has a narrow width proportional to ${N}^{\ensuremath{-}1}$, the integral equation is solved by perturbation in ${N}^{\ensuremath{-}1}$. The shift in the free energy of the system up to the order of ${N}^{\ensuremath{-}2}$ is given. It is found that for a symmetrical distribution, the shift due to the randomness is second order in ${N}^{\ensuremath{-}1}$ and negative, depending only upon the variance of the input distribution. The first-order shift for the asymmetric distribution comes entirely from the asymmetry. After a shift in energy is made to account for this asymmetry, the effective shift is identical to that for the symmetrical case. The shifts of all the thermodynamic properties of the system are also given. The randomness is found to decrease the magnetization for all temperature and applied field. However, shifts in magnetic susceptibility and specific heat are oscillatory in sign.

DOI
10.1103/physrev.182.614

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The one-dimensional Ising model with random exchange energy is formulated in terms of a homogeneous integral equation. Assuming the input distribution has a narrow width proportional to N\ensuremath−1{N}^{\ensuremath{-}1}N\ensuremath−1, the integral equation is solved by perturbation in N\ensuremath−1{N}^{\ensuremath{-}1}N\ensuremath−1. The shift in the free energy of the system up to the order of N\ensuremath−2{N}^{\ensuremath{-}2}N\ensuremath−2 is given. It is found that for a symmetrical distribution, the shift due to the randomness is second order in N\ensuremath−1{N}^{\ensuremath{-}1}N\ensuremath−1 and negative, depending only upon the variance of the input distribution. The first-order shift for the asymmetric distribution comes entirely from the asymmetry. After a shift in energy is made to account for this asymmetry, the effective shift is identical to that for the symmetrical case. The shifts of all the thermodynamic properties of the system are also given. The randomness is found to decrease the magnetization for all temperature and applied field. However, shifts in magnetic susceptibility and specific heat are oscillatory in sign.
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 thermodynamics of the 1D Ising model with arbitrarily varying exchange integrals is investigated. Using the 'levelling-exponential' method and a simple diagrammatic technique, the asymptotically exact expansion of the free energy with respect to the parameter tanh( beta H) (where H is the external magnetic field) is obtained. General results are specified and discussed for the case of the so-called exclusive double-bond model when one of the exchange integrals is ferro- and the other antiferromagnetic.
Abstract We reconsider the problem of a one‐dimensional Ising model with an arbitrary nearest‐neighbor random exchange integral, temperature, and random magnetic field in each site. A convenient formalism is developed that reduces the partition function to a recurrence equation, which is convenient both for numerical as well as for analytical approaches. We have calculated asymptotic expressions for an ensemble averaged free energy and the averaged magnetization in the case of strong and weak couplings in external constant magnetic field. With a random magnetic field at each site in addition to nearest‐neighbor random exchange integrals we also evaluated the free energy. We show that the zeros of the partition function for the Ising model in the complex external magnetic field plane formally coincide with the singularities of the real part of electron's transmission amplitude through the chain of δ‐function potentials.
The one-dimensional Ising model is investigated by generalizing the Bethe approximation, which, in this case, gives exact solutions. The energy, specific heat and the zero field susceptibility for S =1, 3/2 and 2 except the susceptibility for S =2 are calculated exactly and compared with the results of Suzuki et al. . As a direct application of this method, the special lattice called the Bethe lattice is treated, and the Curie temperature of the lattice for S =1 is obtained. This method is also applied to solve the one-dimensional Ising model with the second neighbor interactions as well as the first neighbor ones, and the energy and specific heat for S =1/2 are calculated.
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.
We determine the critical equation of state of three-dimensional randomly dilute Ising systems--- i.e., of the random-exchange Ising universality class. We first consider the small-magnetization expansion of the Helmholtz free energy in the high-temperature phase. Then, we apply a systematic approximation scheme of the equation of state in the whole critical regime, which is based on polynomial parametric representations matching the small-magnetization expansion of the Helmholtz free energy and satisfying a global stationarity condition. These results allow us to estimate several universal amplitude ratios, such as the ratio ${A}^{+}{/A}^{\ensuremath{-}}$ of the specific-heat amplitudes. Our best estimate ${A}^{+}{/A}^{\ensuremath{-}}=1.6(3)$ is in good agreement with experimental results on dilute uniaxial antiferromagnets.