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Journal of the Physical Society of Japan

Exact Ground-State Energies of the Random-Field Ising Chain and Ladder

Toshiyuki Hamasaki, Hidetoshi Nishimori

Published Jun 15, 2004• 3 citations
Publisher

Abstract

We derive the exact ground-state energy of the one-dimensional Ising model in random fields taking values h , 0 and - h with general probabilities. The random-field Ising model on a ladder is also analyzed by showing its equivalence to the random-field Ising chain with field values ± h and 0 for h < J . The zero-temperature transfer matrix is used to obtain the results.

DOI
10.1143/jpsj.73.1490

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We derive the exact ground-state energy of the one-dimensional Ising model in random fields taking values h , 0 and - h with general probabilities. The random-field Ising model on a ladder is also analyzed by showing its equivalence to the random-field Ising chain with field values ± h and 0 for h < J . The zero-temperature transfer matrix is used to obtain the results.
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.
We propose a new method for exact analytical calculation of the ground-state energy of the Ising spin glass on strips. An outstanding advantage of this method over the numerical transfer matrix technique is that the energy is obtained for complex values of the probability describing quenched randomness. We study the ±J\pm J±J and the site-random models using this method for strips of various sizes up to 5×∞5\times\infty5×∞. The ground-state energy of these models is found to have singular points in the complex-probability plane, reminiscent of Lee-Yang zeros in the complex-field plane for the Ising ferromagnet. The ±J\pm J±J Ising model has a series of singularities which may approach a limiting point around p∼0.9p \sim 0.9p∼0.9 on the real axis in the limit of infinite width.
We propose a new method for exact analytical calculation of the ground-state energy of the Ising spin glass on strips. An outstanding advantage of this method over the numerical transfer matrix technique is that the energy is obtained for complex values of the probability describing quenched randomness. We study the ±J\pm J±J and the site-random models using this method for strips of various sizes up to 5×∞5\times\infty5×∞. The ground-state energy of these models is found to have singular points in the complex-probability plane, reminiscent of Lee-Yang zeros in the complex-field plane for the Ising ferromagnet. The ±J\pm J±J Ising model has a series of singularities which may approach a limiting point around p∼0.9p \sim 0.9p∼0.9 on the real axis in the limit of infinite width.
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 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.