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Journal of Physics A Mathematical and General

Metastable states of the random-field Ising chain

Sayuri Masui, Allan E. Jacobs, Cameron Wicentowich, B. W. Southern

Published Jan 7, 1993• 4 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/26/1/008

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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 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 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,….
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 ground state critical properties of the Random Field Ising Model (RFIM) on the diamond hierarchical lattice are investigated via a combining method encompassing real space renormalization group and an exact recurrence procedure. The local magnetization and the nearest neighbors pair correlation function are exactly calculated. The fixed-point joint probability distribution of couplings and local fields are numerically obtained and analyzed, indicating that the critical behavior of the model is governed by the zero temperature disorder fixed point. The critical exponents associated with the order parameter and correlation length are estimated showing an universal behavior regarding the choice of the initial probability distribution for initial local fields being the symmetric continuous Gaussian or discrete delta-bimodal.