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Journal of Physics C Solid State Physics

Tricritical point in systems with randomly oriented local fields

V. K. Saxena

Published Sep 10, 1981• 14 citations
Publisher

Abstract

A spin 1 / 2 Heisenberg model with randomly oriented quenched local magnetic fields is studied in the mean field theory. The system is found to show a tricritical behaviour and existence of two phase transitions, one of second order and another of first order at a temperature lower than the second-order transition temperature, for certain values of the strength of the field.

DOI
10.1088/0022-3719/14/25/001

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A spin 1 / 2 Heisenberg model with randomly oriented quenched local magnetic fields is studied in the mean field theory. The system is found to show a tricritical behaviour and existence of two phase transitions, one of second order and another of first order at a temperature lower than the second-order transition temperature, for certain values of the strength of the field.
Mean-field theory and renormalization-group arguments are used to show that the phase transition in a system with a random ordering field becomes first order at sufficiently low transition temperature, provided the (symmetric) random-field distribution function has a minimum at zero field. The first-order region is separated from the second-order region by a tricritical point. Both the critical and the tricritical exponents at d>4d>4d>4 dimensions are shown to be the same as for the pure system at $d\ensuremath{-}2$ dimensions. The relevance to spin glasses and other systems is discussed. The new tricritical point is very different from all previously studied tricritical points, as it deviates from mean-field theory at d=5d=5d=5, and not at d=3d=3d=3. Although quantitative results are calculated only at $d=5\ensuremath{-}\ensuremath{\epsilon}$ dimensions, the qualitative results are expected to apply at d=3d=3d=3.
In view of the recently seen dramatic effect of quenched random bonds on tricritical systems, we have conducted a renormalization-group study on the effect of quenched random magnetic fields on the tricritical phase diagram of the spin-1 Ising model in d=3d\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3d=3. We find that random fields convert first-order phase transitions into second-order phase transitions more effectively than random bonds. The coexistence region is extremely flat, attesting to an unusually small tricritical exponent ${\ensuremath{\beta}}_{u}$. The random-field phase diagram is strikingly asymmetric. To accommodate this asymmetry, the second-order boundary exhibits reentrance.
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.
The dynamical phase transitions of the kinetic Ising model in the presence of a random magnetic field with a bimodal probability distribution is studied by using effective-field theory (EFT) with correlations. We have used a Glauber-type stochastic dynamic to describe the time evolution of the system, where the system strongly depends on the H≡√<H(i)(2)>(c) root mean square deviation of the magnetic field. The EFT dynamic equation is given for the simple cubic lattice (z=6), and the dynamic order parameter is calculated. The system presents ferromagnetic and paramagnetic states for low and high temperatures, respectively. Our results predict first-order transitions at low temperatures and large disorder strengths, which corresponds to the existence of a nonequilibrium tricritical point (TCP) in a phase diagram in the T-H plane. We compare the results with the equilibrium phase diagram, where only the first-order line is different. Our qualitative results are compatible with recent Monte Carlo simulations.