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

The Harris criterion imposes a constraint on the critical behavior of a system upon introduction of new disorder, based on its dimension <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>d</a:mi></a:math> and localization length exponent <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>ν</b:mi></b:math>. It states that the new disorder can be relevant only if <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>d</c:mi><c:mi>ν</c:mi><c:mo>&lt;</c:mo><c:mn>2</c:mn></c:mrow></c:math>. 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:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>d</d:mi><d:mi>ν</d:mi><d:mo>&gt;</d:mo><d:mn>2</d:mn></d:mrow></d:math>, 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msup><e:mi>L</e:mi><e:mn>0</e:mn></e:msup></e:math>, i.e., it does not depend on block size <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>L</f:mi></f:math> in contrast to the expectation based on the Harris criterion that they should decrease as <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msup><g:mi>L</g:mi><g:mrow><g:mo>−</g:mo><g:mi>d</g:mi><g:mo>/</g:mo><g:mn>2</g:mn></g:mrow></g:msup></g:math> according to the central limit theorem. Since <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:msup><h:mi>L</h:mi><h:mn>0</h:mn></h:msup><h:mo>&gt;</h:mo><h:mrow><h:mo>(</h:mo><h:mi>x</h:mi><h:mo>−</h:mo><h:msub><h:mi>x</h:mi><h:mi>c</h:mi></h:msub><h:mo>)</h:mo></h:mrow></h:mrow></h:math> 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.