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Physical review. B, Condensed matter

Harris criterion for direct and orthogonal quenched randomness

A. Nihat Berker

Published Nov 1, 1990• 19 citations
Publisher

Abstract

Critical-behavior stability is studied, with respect to the introduction of quenched randomness that is direct (e.g., field randomness at field approach to criticality) or orthogonal (e.g., field randomness at temperature approach to criticality). Rather than resorting to renormalization-group theory, an argument is conducted by considering the effective uniform field per spin within a correlated region. By comparing the correlation lengths limited by this effective uniform field and by the actual uniform deviation from criticality, it is recovered that nonrandom critical behavior is not maintained under random fields when its susceptibility critical exponent \ensuremath{\gamma} is positive, and conversely.

DOI
10.1103/physrevb.42.8640

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Critical-behavior stability is studied, with respect to the introduction of quenched randomness that is direct (e.g., field randomness at field approach to criticality) or orthogonal (e.g., field randomness at temperature approach to criticality). Rather than resorting to renormalization-group theory, an argument is conducted by considering the effective uniform field per spin within a correlated region. By comparing the correlation lengths limited by this effective uniform field and by the actual uniform deviation from criticality, it is recovered that nonrandom critical behavior is not maintained under random fields when its susceptibility critical exponent \ensuremath{\gamma} is positive, and conversely.
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.
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Critical properties are studied in systems with quenched bond disorder that is correlated along d1{d}_{1}d1​ of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion \ensuremath{\varphi}=\ensuremath{\alpha}+d1{d}_{1}d1​\ensuremath{\nu} is used. For d1{d}_{1}d1​, we find fixed distributions at finite temperatures, yielding new ``random'' exponents for various q-state Potts models. For d1{d}_{1}d1​=d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for d1{d}_{1}d1​=1 are relevant to random quantum systems.
The critical properties of systems with quenched bond disorder are determined from a fixed distribution, under renormalization group, of the random bonds. Full fixed distributions with all moments are obtained numerically by histograms and, to a good approximation, in terms of $\ensuremath{\Gamma}$ distributions. For such systems, the specific-heat exponent $\ensuremath{\alpha}$ does not equal the crossover exponent $\ensuremath{\varphi}$ at random criticality. We derive a new relation between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$, which invokes characteristics of the fixed distribution. The difference between $\ensuremath{\alpha}$ and $\ensuremath{\varphi}$ is noted for nnn-vector models in $4\ensuremath{-}\ensuremath{\epsilon}$ dimensions and for Potts models on hierarchical lattices solved exactly. In general, stable random critical behavior with positive $\ensuremath{\alpha}$ appears to be possible. We develop a general treatment of quenched disorder and illustrate it by calculating specific-heat curves. It is suggested that the critical exponents of the three- and four-state random-bond Potts models in two dimensions are $\ensuremath{\nu}\ensuremath{\simeq}1.06 \mathrm{and} 1.19$.
We employ scaling arguments and optimal fluctuation theory to establish a general relation between quantum Griffiths singularities and the Harris criterion for quantum phase transitions in disordered systems. If a clean critical point violates the Harris criterion, it is destabilized by weak disorder. At the same time, the Griffiths dynamical exponent z' diverges upon approaching the transition, suggesting unconventional critical behavior. In contrast, if the Harris criterion is fulfilled, power-law Griffiths singularities can coexist with clean critical behavior, but z' saturates at a finite value. We present applications of our theory to a variety of systems including quantum spin chains, classical reaction-diffusion systems and metallic magnets, and we discuss modifications for transitions above the upper critical dimension. Based on these results we propose a unified classification of phase transitions in disordered systems.
We investigate and contrast, via the Wang-Landau (WL) algorithm, the effects of quenched bond randomness on the self-averaging properties of two Ising spin models in 2d. The random bond version of the superantiferromagnetic (SAF) square model with nearest- and next-nearest-neighbor competing interactions and the corresponding version of the simple ferromagnetic Ising model are studied. We find that, the random bond SAF model shows a strong violation of self-averaging, much stronger than that observed in the case of the random bond Ising model. Our analysis of the asymptotic scaling behavior of the variance of the distribution of the sample-dependent pseudocritical temperatures is found to be consistent with the renormalization group prediction of Aharony and Harris. Using this alternative approach, we find estimates of the correlation length exponent ννν in agreement with results obtained from the usual finite-size scaling (FSS) methodology.