Harris-Luck criterion in the plateau transition of the Integer Quantum Hall Effect

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 $\nu$. It states that the new disorder can be relevant only if $d \nu < 2$. We analyze the applicability of the Harris criterion to the GKNS network disorder formulated in the paper [I. A. Gruzberg, A. Kl\"umper, W. Nuding, and A. Sedrakyan, Phys. Rev. B 95, 125414 (2017)] and show that the fluctuations of the geometry are relevant despite $d \nu> 2$, implying that 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.