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Journal of Physics Conference Series

Boundary magnetization of the generalized McCoy-Wu type random Ising models

Kazuyuki Matsumoto, Yuko Akutsu

Published Jan 1, 2010• 0 citations
Publisher

Abstract

This paper investigates the generalized McCoy-Wu type random Ising model. It is found that boundary magnetization behaves as (Tc − T)β', β' = 1 just below the critical temperature. From numerical analyses it is revealed that this type of critical behavior, β' = 1, is universal for any type of randomness. This universal behavior is explained in terms of the destruction of the devil's staircase structure.

DOI
10.1088/1742-6596/200/2/022033

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This paper investigates the generalized McCoy-Wu type random Ising model. It is found that boundary magnetization behaves as (Tc − T)β', β' = 1 just below the critical temperature. From numerical analyses it is revealed that this type of critical behavior, β' = 1, is universal for any type of randomness. This universal behavior is explained in terms of the destruction of the devil's staircase structure.
(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.
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.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
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.
We study the critical and off-critical (Griffiths-McCoy) regions of the random transverse-field Ising spin chain by analytical and numerical methods and by phenomenological scaling considerations. Here we extend previous investigations to surface quantities and to the ferromagnetic phase. The surface magnetization of the model is shown to be related to the surviving probability of an adsorbing walk and several critical exponents are exactly calculated. Analyzing the structure of low-energy excitations we present a phenomenological theory which explains both the scaling behavior at the critical point and the nature of Griffiths-McCoy singularities in the off-critical regions. In the numerical part of the work we used the free-fermion representation of the model and calculated the critical magnetization profiles, which are found to follow very accurately the conformal predictions for different boundary conditions. In the off-critical regions we demonstrated that the Griffiths-McCoy singularities are characterized by a single, varying exponent, the value of which is related through duality in the paramagnetic and ferromagnetic phases.