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Physical review. B, Condensed matter

Tricritical points in systems with random fields

Amnon Aharony A

Published Oct 1, 1978• 402 citations
Publisher

Abstract

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 $d>4$ 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 $d=5$, and not at $d=3$. Although quantitative results are calculated only at $d=5\ensuremath{-}\ensuremath{\epsilon}$ dimensions, the qualitative results are expected to apply at $d=3$.

DOI
10.1103/physrevb.18.3318

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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−2d\ensuremath{-}2d\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−\ensuremathϵd=5\ensuremath{-}\ensuremath{\epsilon}d=5\ensuremath−\ensuremathϵ dimensions, the qualitative results are expected to apply at d=3d=3d=3.
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 effect of bond randomness on tricritical and critical end-point phenomena is studied by renormalization-group theory. In three dimensions, the pure-system tricritical point is replaced by a line segment of second-order transitions dominated by randomness and bounded by a multicritical point and a random-bond tricritical point, which reaches zero temperature at threshold randomness. This topology indicates a violation of the empirical universality principle. The random-bond tricritical point renormalizes onto the fixed distribution of random-field Ising criticality.
A formalism is given whereby high-temperature series for the random-field Ising model are obtained by partitioning the vertices of the pure Ising series' diagrams; for any given order in the high-temperature variable, the random fields are thereby treated exactly. Series for the susceptibility up to seventh order in the high-temperature variable have been obtained for an arbitrary distribution of the random fields. Series for a bimodal and a Gaussian distribution were analyzed by Dlog-Pad\'e methods. The analysis predicts that, for a bimodal distribution of random fields, the tricritical point predicted by mean-field theory persists in all dimensions above the lower critical dimension, but moves to weaker fields as the dimension is reduced. For the Gaussian distribution, the transition is predicted to be second order over the whole phase boundary for dimension d\ensuremath{\ge}4; however, for d<4 the analysis suggests that fluctuations drive the transition first order for sufficiently strong fields implying that d=4 is a ``critical'' dimension for this model. Above four dimensions the value of the susceptibility exponent determined from the series is consistent with the predictions of the d\ensuremath{\rightarrow}d-2 rule for both distributions.
Mean-field theory, presented in the preceding chapter, correctly describes the qualitative features of most phase transitions and, in some cases, the quantitative features. Since mean-field theory replaces the actual configurations of the local variables (e.g. spins) by their average value, it neglects the effects of fluctuations about this mean. These fluctuations may or may not be important. The more spins that interact with a particular test spin, the more the test spin sees an effective average or mean field. If the test spin interacts with two neighbors, the averaging is minimal and the fluctuations are large and important. The number of spins producing the effective field increases with the range of the interaction and with the dimension. Thus we will find that mean-field theory is a good approximation in high dimensions but fails to provide a quantitatively correct description of second-order critical points in low dimensions. This chapter will be devoted to the study of second-order phase transitions when mean-field theory is not a good approximation. We will begin by considering fluctuations of the order parameter about its spatially uniform mean-field value. We will find that for spatial dimensions below an upper critical dimension d c (typically d c = 4), fluctuations always become important and invalidate mean-field theory for temperatures sufficiently close to T c . This will motivate a generalization of Landau theory that incorporates spatial fluctuations about a mean-field free energy minimum.
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.