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Physical review. B, Condensed matter

Thermodynamics of Ising models with layered randomness: Exact solutions on square and triangular lattices

Theo M. Nieuwenhuizen, Henri Orland

Published Sep 1, 1989• 7 citations
Publisher

Abstract

Two-dimensional Ising models are studied on square and triangular lattices with layered disordered bonds (McCoy-Wu-type models). It is shown that exact solutions exist for diluted exponential distributions of couplings. In the ferromagnetic case the free energy exhibits an essential singularity (slightly different from the result of McCoy and Wu) at the transition point. If frustration is present the transitions occurs at T=0, and the specific heat is linear.

DOI
10.1103/physrevb.40.5094

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(Received ) This paper investigates the generalized McCoy-Wu type random Ising model. It is found that boundary magnetization behaves as (TciT) fl 0 , fl 0 = 1:0 just below the ’critical temperature’. From numerical analyses it is revealed that this type of critical behavior, fl 0 = 1:0, would be universal for any type of randomness. This universal behavior is explained in terms of the destruction of the devil’s staircase structure. As an observable result of this model, the integrated distribution function of boundary magnetization is suggested.

•Mar 20, 1988•0 citations
Two-dimensional Ising models are studied on square and triangular lattices with layered disordered bonds (McCoy-Wu-type models). It is shown that exact solutions exist for diluted exponential distributions of couplings. In the ferromagnetic case the free energy exhibits an essential singularity (slightly different from the result of McCoy and Wu) at the transition point. If frustration is present the transitions occurs at T=0, and the specific heat is linear.
Exactly solvable two-dimensional Ising models with McCoy-Wu type layered quenched bond randomness are studied both analytically and numerically. For binary type bond distributions, it is proved that the distribution function of a random variable, which appears in the exact solution, shows the devil's staircase structure for a certain range of temperatures. By using a direct iternation method, the free energy and the specific heat are calculated in such a way as to give the numerically exact solution for a finite system. The specific heat does not show divergent behavior around the ‘critical’ temperature. The behavior is explained in terms of the distribution function.
We study a d=2 Ising model where the veritcal bonds are fixed and ferromagnetic and the horizontal bonds can vary randomly in sign and in magnitude (within some limits) but are same within each now. The model therefore generalizes that of McCoy and Wu since it allows for the interesting case of frustration. We use the transfer matrix to map our problem to a collection of random field d=1 problems about which a lot is known. We find generally three transitions: a Griffiths transition, its dual version, and one with infinite correlation length and index \ensuremath{\nu}=1. In all cases the free energy has infinitely differentiable singularities. In addition there are some zero-temperature transitions.
We consider a system with randomly layered ferromagnetic bonds (McCoy-Wu model) and study its critical properties in the frame of mean-field theory. In the low-temperature phase there is an average spontaneous magnetization in the system, which vanishes as a power law at the critical point with the critical exponents and in the bulk and at the surface of the system, respectively. The singularity of the specific heat is characterized by an exponent . The samples reduced critical temperature has a power law distribution and we show that the difference between the values of the critical exponents in the pure and in the random system is just . Above the critical temperature the thermodynamic quantities behave analytically, thus the system does not exhibit Griffiths singularities.
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.
Ising models, with quenched bond disorder along one direction on a square lattice, are studied by the transfer-matrix method. The exact criticality condition is rederived, and extended to an arbitrary variation, in magnitude and/or sign, of the interactions from layer to layer. With competing interactions, a phase diagram is obtained, with a zero-temperature spin-glass point. We consider 2n2n2n different nearest-neighbor couplings layered and repeated periodically along one direction. The phase boundaries, including zero-temperature bicritical points, the order parameters, and the ground-state degeneracies are illustrated by a detailed discussion for n=2n=2n=2. Closed-form free energies are given for n=2and3n=2 \mathrm{and} 3n=2and3. The unique criticality condition is derived for any nnn, the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ corresponding to quenched disorder. Our extended result is made possible by a similarity transformation in the free-fermion representation. A conjecture emerges for the criticality condition of randomly layered ferromagnetic qqq-state Potts models.
(Received ) This paper investigates the generalized McCoy-Wu type random Ising model. It is found that boundary magnetization behaves as (TciT) fl 0 , fl 0 = 1:0 just below the ’critical temperature’. From numerical analyses it is revealed that this type of critical behavior, fl 0 = 1:0, would be universal for any type of randomness. This universal behavior is explained in terms of the destruction of the devil’s staircase structure. As an observable result of this model, the integrated distribution function of boundary magnetization is suggested.