Mesoscopic effects in the quantum Hall regime

Abstract

We report results of a study of (integer) quantum Hall transitions in a single or multiple Landau levels for non-interacting electrons in disordered two-dimensional systems, obtained by projecting a tight-binding Hamiltonian to the corresponding magnetic subbands. In finite-size systems, we find that mesoscopic effects often dominate, leading to apparent non-universal scaling behavior in higher Landau levels. This is because localization length, which grows exponentially with Landau level index, exceeds the system sizes amenable to the numerical study at present. When band mixing between multiple Landau levels is present, mesoscopic effects cause a crossover from a sequence of quantum Hall transitions for weak disorder to classical behavior for strong disorder. This behavior may be of relevance to experimentally observed transitions between quantum Hall states and the insulating phase at low magnetic fields.

Localization-length exponent in two models of quantum Hall plateau transitions

“…22 However, given the fact that earlier studies had large error bars of 0.1, the new result is still consistent with the earlier estimates. 4,22,23 Our analysis of both models, using fully two-dimensional scaling, suggests that corrections to scaling due to irrelevant operators are small, unlike what is found in strip-geometry methods using the crossover to one-dimension to extract critical exponents. Even in the lattice model where the corrections are apparently larger, the leading irrelevant length exponent is found to be y = 4.3 ± 0.2, which is substantially larger than the value found in the methods using the crossover to one dimension using the strip geometry on the CCN model.…”

“…4 The same result was also found in the disordered Hofstadter model. [22][23][24] No attempt, however, has been made for the two models including corrections due to irrelevant length scales, as done in the CCN model.…”

Mobility gap in fractional quantum Hall liquids: Effects of disorder and layer thickness

“…In the noninteracting IQHE, the calculation of topologically invariant Chern numbers has been established [16][17][18][19][20][21][22][23] as a reliable way to obtain the Hall conductance, to measure the localization length critical exponent, and to determine whether a single-particle state is localized or conductingthus where the mobility edge is. Physically, the Chern number of a state is the ͑dimensionless͒ Hall conductance, which can be derived from the Kubo formular, averaged over boundary conditions of a finite system on a torus.…”

Numerical study of spin quantum Hall transitions in superconductors with broken time-reversal symmetry

“…One of the well-developed numerical methods in the study of the quantum Hall effect is the calculation of Chern numbers of either single-or many-electron states, which allows one to distinguish between current carrying and insulating states unambiguously, even in a finite-size system. The Chern number method has been very successful in the studies of quantum Hall transitions, for both integral 27,28,29,30 and fractional effects, 31 as well as in bilayer systems, 32 and also in other contexts. 33,34,35 For example, by finite-size scaling, the localization length exponent ν ≈ 2.3 has been obtained, 28,29,30 consistent with other estimates.…”

“…The Chern number method has been very successful in the studies of quantum Hall transitions, for both integral 27,28,29,30 and fractional effects, 31 as well as in bilayer systems, 32 and also in other contexts. 33,34,35 For example, by finite-size scaling, the localization length exponent ν ≈ 2.3 has been obtained, 28,29,30 consistent with other estimates. 36 More recent application of this method to fractional quantum Hall states (where one inevitably has to deal with interacting electrons) allows one to determine the transport gap numerically, 31 which is not possible from other known numerical methods.…”