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Physical review. B, Solid state

Tricritical points in random systems

Michael J. Stephen

Published Mar 1, 1976• 5 citations
Publisher

Abstract

The critical behavior of the thermodynamic functions near a tricritical point in a quenched random system is investigated. It is shown that a single stable fixed point exists corresponding to the tricritical point in the random system. The logarithmic corrections to the mean-field theory of the tricritical point three dimensions are obtained.

DOI
10.1103/physrevb.13.2007

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The critical behavior of the thermodynamic functions near a tricritical point in a quenched random system is investigated. It is shown that a single stable fixed point exists corresponding to the tricritical point in the random system. The logarithmic corrections to the mean-field theory of the tricritical point three dimensions are obtained.
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.
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.
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.
Critical behaviors in quenched random-spin systems with N-spin component are studied in the limit M → 0 of the non-random MN-component models by means of the renormalization group theory. As the static critical phenomena the stability of the fixed points is investigated and the critical exponents η[~ O(ε3); ε ≡ 4 – d], γ, α, and crossover index [Formula: see text] and the equation of state [~ O(ε)] are obtained. Within the approximation up to the order ε2, even the random-spin systems with N = 2 or 3 are unstable in the three dimensions and the pure systems are stable there.
The asymptotic critical form of thermodynamic functions is analyzed by means of renormalization-group techniques. If certain exponent relations are satisfied, then the critical behavior is not described by a simple power law, but a power law multiplied by a fractional power of a logarithm. The approach is applied to two special systems whose critical exponents are molecular-field-like. (i) For ordinary critical transitions in four dimensions we find the same logarithmic factors previously computed by Larkin and Khmel'nitskii. (ii) For tricritical transitions in three dimensions we compute the logarithmic corrections to the molecularfield tricritical behavior discussed in an earlier publication.