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Journal of Physics C Solid State Physics

One-dimensional Ising model with random bonds

J M Kowalski, Andrzej Pękalski

Published Jul 7, 1975• 3 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3719/8/13/018

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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

•Jan 1, 2008•4 citations
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.
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.
We calculate the thermodynamics of the one-dimensional spin-1/2 Ising model in the presence of a constant skew magnetic field. We obtain the high-temperature expansion of its Helmholtz free energy (HFE), for the ferromagnetic and antiferromagnetic cases, up to order beta7. This expansion permits us to obtain the behaviour of the model for $ |J| \beta \stackrel {<}{_\sim}} 1 $, when it cannot be described by its classical version. Among the calculated thermodynamical functions of the model, we have the diagonal elements of the magnetic susceptibility tensor for the transverse and logitudinal Ising models, obtained by taking the limits h z -> 0 and h y -> 0, respectively, of the beta-expansion of the HFE. The y-component of the magnetization and the chiyy component of the magnetic susceptibility tensor are almost the same for the antiferro- and ferromagnetic models, at least for $ |J| \beta \stackrel {<}{_\sim}} 1 $; and, c yy is practically independent of the direction of the external magnetic. We also show that, in this region of temperature, the thermodynamics of the Ising model with skew magnetic field and that of an XXZ model with longitudinal magnetic field are not similar.
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 two-dimensional Ising model with two kinds of bonds –ferro and antiferromagnetic– is investigated in both quenched and annealed approximations. High temperature series expansion technique is used to get the respective free energies and susceptibilities. Form the analysis of the ratio method results, the phase diagrams are drawn and critical concentrations are estimated. It is shown that in both approximations the system behaves similarly, although in the annealed case the magnetically ordered phases are more stable against increasing impurity concentration. It is also proved that for Bethe lattices initial susceptibilities for quenced and annealed systems coincide.
Using the integral equation method, we have got the analytic expressions of the internal energy and specific heat for the one-dimensional Ising model with nearest-neighbour antiferromagnetic repulsion Js and infinitely long-range exponential (Kae-Helfand-type) ferromagnetic attraction J1 acting only on even-numbered neighbours. Under the conditions of J1=1 and Js being 0.05, 0.1 and 0.3 respectively, we have drawn the curves of the internal energy and specific heat varying with temperature in zero external magnetic field, and pointed out its critical phase transition point. Under the conditions of J1=1,Js=0.1, we have drawn the curves of influence of the external magnetic field upon the internal energy and specific heat. Because there exist double solutions for the system under low temperatures, zero or weak external magnetic field and weaker antiferromagnetic interactions, it is inferred that there should exist another quasi-stable state of the model under the given conditions.
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