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Journal of Physics A Mathematical and General

Low-temperature thermodynamics of random-field Ising chains: exact results

J. M. Luck, M Funke, Theo M. Nieuwenhuizen

Published Sep 7, 1991• 19 citations
Publisher

Abstract

The authors consider the random-field ferromagnetic Ising chain, for a class of continuous symmetric probability distributions of the random magnetic fields, of a diluted power-times-exponential type. This class of distribution is 'exactly solvable', in the sense that the disorder can be integrated explicitly, at any temperature. A detailed analysis of the low-temperature thermodynamics is presented. Exact expressions are obtained for the ground-state energy, the zero-temperature entropy, and the amplitude of the specific heat, which vanishes linearly at low temperature. The diluted symmetric binary distribution, where the magnetic fields can only assume the values +or-H B or zero, can be viewed as a limiting case of the class of exactly solvable distributions. The low-temperature physics of this discrete model is investigated in detail, including the exponential fall-off of the specific heat.

DOI
10.1088/0305-4470/24/17/030

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The authors consider the random-field ferromagnetic Ising chain, for a class of continuous symmetric probability distributions of the random magnetic fields, of a diluted power-times-exponential type. This class of distribution is 'exactly solvable', in the sense that the disorder can be integrated explicitly, at any temperature. A detailed analysis of the low-temperature thermodynamics is presented. Exact expressions are obtained for the ground-state energy, the zero-temperature entropy, and the amplitude of the specific heat, which vanishes linearly at low temperature. The diluted symmetric binary distribution, where the magnetic fields can only assume the values +or-H B or zero, can be viewed as a limiting case of the class of exactly solvable distributions. The low-temperature physics of this discrete model is investigated in detail, including the exponential fall-off of the specific heat.
The authors present an exact solution for the one-dimensional Ising model in a random field. The distribution of field strengths is site independent, symmetric and has a three-peak structure. The free energy is obtained for all finite temperatures. The low-temperature behaviour is studied in detail. They find an expansion for the free energy in integer powers of temperature. The ground-state energy and the zero-point entropy are calculated explicitly. The specific heat is linear for low temperatures, in contradiction with mean-field theory. The origin of an additional exponential parameter is discussed.
The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.
Using a numerical transfer-matrix method, the authors have determined the number of zero-temperature metastable states and their distribution in energy and magnetization for random-field Ising chains. Both discrete and continuous distributions of the random fields are considered. The degeneracy and structure of the metastable states and ground states for the discrete case may be understood in terms of a domain picture. Algorithms for generating exact ground-state configurations for both discrete and continuous distributions are described, and these demonstrate that the ground states of the random-field Ising chain have a hierarchical structure.
The low-temperature behavior ($T\ensuremath{\ll}h, J$) of a random-bond Ising chain in a magnetic field is considered [$H=\ensuremath{-}(\frac{1}{2})J\ensuremath{\Sigma}{T}_{i}{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{i+1}\ensuremath{-}h\ensuremath{\Sigma}{\ensuremath{\sigma}}_{i}, {\ensuremath{\sigma}}_{i}=\ifmmode\pm\else\textpm\fi{}1$, Ti{{T}_{i}}Ti​ is a fixed random sequence of numbers +1 and -1 with concentrations c1{c}_{1}c1​ and ${c}_{2}=1\ensuremath{-}{c}_{1}$, respectively]. The ground-state energy E0{E}_{0}E0​, magnetization ${\ensuremath{\mu}}_{0}$ and zero-point entropy S0{S}_{0}S0​ are calculated exactly. It is shown that ${\ensuremath{\mu}}_{0}$ and S0{S}_{0}S0​ are discontinuous functions of magnetic field having jumps at h=Jnh=\frac{J}{n}h=nJ​, n=1,2,…n=1, 2, \dots{}n=1,2,….
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.