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arXiv:2411.01651 · Physical review. B./Physical review. B

Harris-Luck criterion in the plateau transition of the integer quantum Hall effect

Hrant Topchyan, W. Nuding, Andreas Klümper, A. Sedrakyan

Published Mar 12, 2025• 1 citation

Abstract

The Harris criterion imposes a constraint on the critical behavior of a system upon introduction of new disorder, based on its dimension d and localization length exponent ν . It states that the new disorder can be relevant only if d ν < 2 . We analyze the applicability of the Harris criterion to the GKNS network disorder formulated in the paper [I. A. Gruzberg, A. Klümper, W. Nuding, and A. Sedrakyan, ] and show that the fluctuations of the geometry are relevant despite d ν > 2 , implying that the Harris criterion should be modified. We have observed that the fluctuations of the critical point in different quenched configurations of disordered network blocks is of order L 0 , i.e., it does not depend on block size L in contrast to the expectation based on the Harris criterion that they should decrease as L − d / 2 according to the central limit theorem. Since L 0 > ( x − x c ) is always satisfied near the critical point, the mentioned network disorder is relevant and the critical indices of the system can be changed. We have also shown that the GKNS disordered network is fundamentally different from Voronoi-Delaunay and dynamically triangulated random lattices: The probability of higher connectivity in the GKNS network decreases in a power law as opposed to an exponential, indicating that we are dealing with a “scale-free” network, such as the internet, protein-protein interactions, etc.

DOI: 10.1103/physrevb.111.l100201

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The Harris criterion imposes a constraint on the critical behavior of a system upon introduction of new disorder, based on its dimension d and localization length exponent ν . It states that the new disorder can be relevant only if d ν < 2 . We analyze the applicability of the Harris criterion to the GKNS network disorder formulated in the paper [I. A. Gruzberg, A. Klümper, W. Nuding, and A. Sedrakyan, ] and show that the fluctuations of the geometry are relevant despite d ν > 2 , implying that the Harris criterion should be modified. We have observed that the fluctuations of the critical point in different quenched configurations of disordered network blocks is of order L 0 , i.e., it does not depend on block size L in contrast to the expectation based on the Harris criterion that they should decrease as L − d / 2 according to the central limit theorem. Since L 0 > ( x − x c ) is always satisfied near the critical point, the mentioned network disorder is relevant and the critical indices of the system can be changed. We have also shown that the GKNS disordered network is fundamentally different from Voronoi-Delaunay and dynamically triangulated random lattices: The probability of higher connectivity in the GKNS network decreases in a power law as opposed to an exponential, indicating that we are dealing with a “scale-free” network, such as the internet, protein-protein interactions, etc.

We study the effects of topological (connectivity) disorder on phase transitions. We identify a broad class of random lattices whose disorder fluctuations decay much faster with increasing length scale than those of generic random systems, yielding a wandering exponent of ω=(d-1)/(2d) in d dimensions. The stability of clean critical points is thus governed by the criterion (d+1)ν>2 rather than the usual Harris criterion dν>2, making topological disorder less relevant than generic randomness. The Imry-Ma criterion is also modified, allowing first-order transitions to survive in all dimensions d>1. These results explain a host of puzzling violations of the original criteria for equilibrium and nonequilibrium phase transitions on random lattices. We discuss applications, and we illustrate our theory by computer simulations of random Voronoi and other lattices.

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.

Motivated by the Harris criterion, the absorbing phase transitions of the conserved lattice gas (CLG) model were studied on lattices with a quenched disorder, i.e., on infinite percolation networks, in two and three dimensions. The Harris criterion suggests that, in a magnetic system, the pure fixed point will be unstable if the specific-heat exponent is positive. For the CLG model, the specific-heat exponent α calculated by the hyperscaling relation α=2-dν, where ν is the spatial correlation length exponent in d dimensions, will be positive in two dimensions if the value of ν obtained earlier by Lübeck and Heger is employed. On the other hand, it will be close to 0 if the more recent value by Lee and Lee is used and it is positive in three dimensions with the available value of ν. Extensive numerical simulations showed that, when the concentration of disordered sites is less than the critical concentration, the critical exponents were similar to those on a regular lattice both in two and three dimensions. When the concentration of disordered sites becomes critical, the density of active particles showed nonuniversal power-law behavior for all particle densities considered in both dimensions. These results were in contrast to the results for the diluted contact process. The cause of such a nonuniversal behavior was addressed.

We describe the critical behavior of the Anderson-like transition predicted to occur in the 2D tight-binding model with isotropic scale-free long-range correlated disorder characterized by a power-law spectral density. We explore the scale invariance of the participation function relative fluctuation at the critical point to locate the mobility edge as a function of the power-law spectral density exponent α 2D . The states near the band center, which exhibit power-law localization for uncorrelated disorder, become delocalized for α 2D >2. In addition, we consider the finite-size scaling hypothesis to estimate the correlation length critical exponent. We find that the critical exponent ν depends on α 2D , thus indicating that correlations in the disorder distribution are indeed relevant in this regime, in agreement with the extended Harris criterion.

Quenched disorder--in the sense of the Harris criterion--is generally a relevant perturbation at an absorbing state phase transition point. Here using a strong disorder renormalization group framework and effective numerical methods we study the properties of random fixed points for systems in the directed percolation universality class. For strong enough disorder the critical behavior is found to be controlled by a strong disorder fixed point, which is isomorph with the fixed point of random quantum Ising systems. In this fixed point dynamical correlations are logarithmically slow and the static critical exponents are conjecturedly exact for one-dimensional systems. The renormalization group scenario is confronted with numerical results on the random contact process in one and two dimensions and satisfactory agreement is found. For weaker disorder the numerical results indicate static critical exponents which vary with the strength of disorder, whereas the dynamical correlations are compatible with two possible scenarios. Either they follow a power-law decay with a varying dynamical exponent, like in random quantum systems, or the dynamical correlations are logarithmically slow even for a weak disorder. For models in the parity conserving universality class there is no strong disorder fixed point according to our renormalization group analysis.

We consider the Ising model on the square lattice with biaxially correlated random ferromagnetic couplings, the critical point of which is fixed by self-duality. The disorder, which has a correlator, $G(r)\ensuremath{\sim}{r}^{\ensuremath{-}1}$, represents a relevant perturbation according to the extended Harris criterion. Critical properties of the system are studied by large scale Monte Carlo simulations. The correlation length critical exponent $\ensuremath{\nu}=2.005(5)$ corresponds to that expected in a system with isotropic correlated long-range disorder, whereas the scaling dimension of the magnetization density ${x}_{m}=\ensuremath{\beta}∕\ensuremath{\nu}=0.1294(7)$ is somewhat larger than in the pure system. Conformal properties of the magnetization and energy density profiles are also examined numerically.