Nonuniversal Delocalization in a Strong Magnetic Field

Abstract

A method is described for a proper numerical simulation of various random potentials and is used to compare the distributions of the potentials, projected onto the fist two Landau bands. In the case of a slowly varying potential (correlation length >> cyclotron radius) we find a universal distribution, proving an expected universality for the delocalization transition, whereas for a short-range potential the distributions in the higher Landau bands become nonuniversal. Numerical simulations in the first Landau level support nonuniversal localization properties for very short-range correlations. * * *I would like to thank H. A. WEIDENMULLER for valuable discussions and advice and the Heidelberger Akademie der Wissenschaften for its financial support.

Mesoscopic effects in the quantum Hall regime

“…Calculations for the second lowest Landau level suggest that the value of the localization length exponentν depends on the correlation length l of the random potential [26,30]. Mieck showed that ν decreases from 6.2 for l 0 to 2.3 for l 4l c , where l c hc eBµ 1 2 is the magnetic length [29,30]. Huckestein also obtained for a correlation length l l c the same scaling behavior as in the lowest Landau level [31].…”

“…The localization length in higher Landau levels grows rapidly with increasing Landau level number (exponentially as its square in simple approximations), raising great difficulties in the numerical calculation of the critical exponent in higher Landau levels. This difficulty has been reflected, in particular from the observation of non-universal (larger-than-expected) exponent, in early works [23,[26][27][28][29][30]. The universal scaling behavior can, therefore, only be restored in much larger systems, unless the localization length be reduced.…”

The instanton vacuum of generalized CPN−1 models

“…The reduced dimensionality of the quantum Hall system offers a rare opportunity to perform numerical work on the mobility edge problem and extract accurate results on quantum critical behavior. By now there exists an impressive stock of numerical data on the critical indices of the plateau transitions, including the correlation or localization length exponent (m) [32][33][34][35][36][37], the multifractal f (a) spectrum [38][39][40][41][42][43] and even the leading irrelevant exponent (y r ) in the problem [44,45].…”

“…(23) and (24), in general violate the invariance under Eq. (34). This is so because we have expressed S 0 [q] to lowest order in a series expansion in powers of the derivatives acting on the q field.…”

θ renormalization, electron–electron interactions and super universality in the quantum Hall regime

“…2 The initial success of the free electron theory has primarily led to a widely spread believe in Fermi liquid type of ideas 3,4,5,6,7,8 as well as an extended literature on scaling and critical exponent phenomenology. 9,10,11,12,13,14,15,16,17,18,19,20,21,22 Except for experimental considerations, however, there exists absolutely no valid (microscopic) argument that would even remotely justify any of the different kinds of free (or nearly free) electron scenarios that have frequently been proposed over the years. In fact, Fermi liquid principles are fundamentally in conflict with the novel insights that have more recently emerged from the development of a microscopic theory on interaction effects.…”