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Journal of Modern Physics

Modified Percolation Theory and Its Relevance to Quantum Critical Phenomena

Tom Heitmann, John Gaddy, Wouter Montfrooij

Published Jan 1, 2014• 6 citations
Publisher

Abstract

We present a new family of percolation models. We show, using theory and computer simulations, that these represent a new universality class. Interestingly, systems in this class appear to violate the Harris criterion, making model systems within these class ideal systems for studying the influence of disorder on critical behavior. We argue that such percolative systems have already been realized in practice in strongly correlated electron systems that have been driven to the quantum critical point by means of chemical substitution.

DOI
10.4236/jmp.2014.58076

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We present a new family of percolation models. We show, using theory and computer simulations, that these represent a new universality class. Interestingly, systems in this class appear to violate the Harris criterion, making model systems within these class ideal systems for studying the influence of disorder on critical behavior. We argue that such percolative systems have already been realized in practice in strongly correlated electron systems that have been driven to the quantum critical point by means of chemical substitution.
We describe a new universality class - dubbed protected percolation - that we show to be relevant to quantum critical systems. Percolation theory describes phase transitions where long-range order is lost when parts of a system become disconnected from other parts; in the vicinity of the transition, critical behavior is observed, captured by universal power laws. Protected percolation has the added restriction that only sites from the system spanning connection can be removed. We developed a new technique to simulate protected percolation, and we used it to determine the critical exponents of this new universality class in 2, 3, and 4 dimensions. We relate the exponents analytically to those of standard percolation. The Harris criterion predicts whether a phase transition is stable against impurities. We prove that protected percolation violates this criterion in 3 dimensions and higher, implying that impurities result in the loss of universal behavior in systems governed by protected percolation. We investigated the change in critical exponents for various types of impurities, focusing on the case for three dimensions where protected percolation models quantum critical systems. We detail how our simulations can be used for direct comparison to experimental results on such quantum critical systems.
We present the results of a percolation-like model that has been restricted compared to standard percolation models in the sense that we do not allow finite sized clusters to break up once they have formed. We calculate the critical exponents for this model and derive relationships between these exponents and those of standard percolation models. We argue that this restricted model represents a new universality class that is directly relevant to the critical physics as observed in quantum critical systems, and we describe under what conditions our percolation results can be applied to the observed temperature and field dependencies of the specific heat and susceptibility in such systems.
Abstract We present computer simulations on a class of percolative systems that forms a new universality class. We determine the universal critical exponents for this new class from simulations on lattices consisting of up to one billion sites. These new percolative systems differ from standard systems in that once a cluster breaks off the lattice spanning cluster, its sites become protected and cannot be removed. We demonstrate that despite this restriction on the evolution of isolated clusters, the scaling relationships between the critical exponents remain valid. Protected percolation closely mimics the situation in heavily-doped quantum critical systems where isolated magnetic clusters are protected from Kondo screening. We show that protected percolation in three dimensions violates the Harris criterion, explaining why universal exponents for quantum phase transitions have been elusive.
This paper is a study of the percolation problem with long-range correlations in the site or bond occupations. An extension of the Harris criterion for the relevance of the correlations is derived for the case that the correlations decay as ${x}^{\ensuremath{-}a}$ for large distances xxx.. For a<da<da<d the correlations are relevant if $a\ensuremath{\nu}\ensuremath{-}2<0$ (where $\ensuremath{\nu}$ is the percolation correlation-length exponent), while for a>da>da>d the correlations are relevant if $d\ensuremath{\nu}\ensuremath{-}2<0$.. Applying this criterion to the behavior that results when the correlations are relevant, we argue that the new behavior will have ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$.. It is shown that the correlated bond percolation problem is equivalent to a qqq-state Potts model with quenched disorder in the limit $q\ensuremath{\rightarrow}1$.. With the use of this result, a renormalization-group study of the problem is presented, expanding in $\ensuremath{\epsilon}=6\ensuremath{-}d$ and in $\ensuremath{\delta}=4\ensuremath{-}a$.. In addition to the normal percolation fixed point, we find a new long-range fixed point. The crossover to this new fixed point follows the extended Harris criterion, and the fixed point has exponents ${\ensuremath{\nu}}_{\mathrm{long}}=\frac{2}{a}$ (as predicted) and ${\ensuremath{\eta}}_{\mathrm{long}}=\frac{1}{11}(\ensuremath{\delta}\ensuremath{-}\ensuremath{\epsilon})$. Finally, several results on the percolation properties of the Ising model at its critical point are shown to be in agreement with the predictions of this paper.
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.
Quantum critical points in quasiperiodic magnets can realize new universality classes, with critical properties distinct from those of clean or disordered systems. Here, we study quantum phase transitions separating ferromagnetic and paramagnetic phases in the quasiperiodic q-state Potts model in 2+1D. Using a controlled real-space renormalization group approach, we find that the critical behavior is largely independent of q, and is controlled by an infinite-quasiperiodicity fixed point. The correlation length exponent is found to be ν=1, saturating a modified version of the Harris-Luck criterion.