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Physical review. B, Condensed matter

Phase diagram of the transverse Ising model in a random field

F. S. Milman, Paulo R. Hauser, W. Figueiredo

Published Jun 1, 1991• 25 citations
Publisher

Abstract

We determine the phase diagram of the transverse Ising model with a trimodal distribution (sum of three \ensuremath{\delta} functions) for a longitudinal random field at T=0, using a mean-field approximation. The phase diagram includes tricritical points, ordered critical points, a fourth-order point, critical end points, and a double critical end point. Our T=0 phase diagram is completely equivalent to the one obtained by Kaufman, Klunzinger, and Khurana for the random-field Ising model. We show that the temperature and the magnitude of the transverse field play a similar role.

DOI
10.1103/physrevb.43.13641

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We determine the phase diagram of the transverse Ising model with a trimodal distribution (sum of three \ensuremath{\delta} functions) for a longitudinal random field at T=0, using a mean-field approximation. The phase diagram includes tricritical points, ordered critical points, a fourth-order point, critical end points, and a double critical end point. Our T=0 phase diagram is completely equivalent to the one obtained by Kaufman, Klunzinger, and Khurana for the random-field Ising model. We show that the temperature and the magnitude of the transverse field play a similar role.
The transverse-random-field mixed Ising model consisting of arbitrary spin values has been studied by combining the pair model approximation with discretized path-integral representation. The phase diagrams of systems with mixed spins: \ensuremath{\sigma}=1/2, S=1; \ensuremath{\sigma}=1/2, S=3/2, and \ensuremath{\sigma}=1, S=2 are plotted. Not only the discontinuity at T=0 K is found between the trimodal and bimodal distributions of transverse fields, which is similar to the single-spin counterpart, but also the tricritical behavior is observed in these phase diagrams when both transverse fields are trimodal distributed, which is different from the single-spin one. The appearance of tricritical point is independent of the coordination number and spin values. \textcopyright{} 1996 The American Physical Society.
The form of the field distribution, in a random-field Ising model, is known to play an important role in determining the order of the phase transition and the existence of a tricritical point. We have studied a random-field Ising model with a trimodal distribution (sum of three \ensuremath{\delta} functions), which simulates a Gaussian distribution for a certain range of its parameters, within the mean-field renormalization-group scheme. We have considered clusters of one, two, and four spins, and the recursion relations for the determination of the fixed points are numerically analyzed. Using this analysis we have constructed the phase diagram of the system in the space of coupling (K), field (h), and zero-field probability (p). It is shown that the line of tricritical points ends for values of p near (2/3, very different from that obtained in the usual mean-field approximation. Mean-field renormalization-group (p\ensuremath{\simeq}(2/3) results are found to be consistent with the equivalence of the trimodal and Gaussian (absence of tricritical points) distributions until the fourth-order moments for p=(2/3.