Disordered Electrons in a Strong Magnetic Field: Transfer Matrix Approaches to the Statistics of the Local Density of States

Abstract

We present two novel approaches to establish the local density of states as an order parameter field for the Anderson transition problem. We first demonstrate for 2D quantum Hall systems the validity of conformal scaling relations which are characteristic of order parameter fields. Second we show the equivalence between the critical statistics of eigenvectors of the Hamiltonian and of the transfer matrix, respectively. Based on this equivalence we obtain the order parameter exponent α0 ≈ 3.4 for 3D quantum Hall systems.The absence of diffusion in coherent disordered electron systems is known as Anderson localization [1]. In dimensions d > 2 a disorder induced localizationdelocalization (LD) transition occurs quite generally at some value of the Fermi energy [2]. In d = 2 all states are localized unless a certain amount of spin-orbit scattering [3] or a strong magnetic field is present [4,5]. The LD transitions of independent (spinless) 2D electrons subject to a strong perpendicular magnetic field are located at the Landau energies. These transitions are generally believed to be responsible for the integer quantum Hall effect [6,5].In general, LD transitions are characterized by the critical exponent ν of the localization length [7] and by the multifractal f (α) spectrum of the local amplitudes of critical eigenstates [8,9]. The f (α) spectrum describes the statistics and scaling behavior of the local density of states (LDOS). Although the average density of states does not reflect the LD transition, the typical value (i.e. geometrically averaged) of the LDOS does: it vanishes with an exponent β typ = (α 0 − d)ν on approaching the LD transition point, where α 0 > d is the maximum position of f (α). It is thus tempting to interpret the LDOS as an order parameter field of the LD problem [8] (see also [10,11]). Our aim is to support this interpretation by establishing two characteristic features of order parameter fields for the LDOS: First, scaling exponents of the order parameter field are related to the critical exponents of the corresponding spatial correlation functions. These correlation functions show conformal invariance. Second, the scaling exponents are universal in the sense of (one-parameter) scaling theory, i.e. any local quantity containing contributions from the relevant scaling field shows asymptotically the same spectrum of scaling exponents.In this article we demonstrate that the f (α) spectrum of critical eigenstates is related to correlation functions in different geometries by conformal invariance. We derive the conformal mapping relations appropriate for a multifractal situation and check them numerically in 2D quantum Hall systems (QHS). Furthermore, we have calculated numerically f (α) for the local components of transfer matrix eigenvectors in 2D QHS and show that it coincides with f (α) of the Hamiltonian eigenstates. Thus, these two local quantities share the same spectrum of scaling exponents although their microscopic construction is quite different. Our findings support the identificati...

Statistics and scaling in disordered mesoscopic electron systems

“…Here Λ * appears since it is just the typical localization length at criticality divided by the width L t . Equation (294) has been confirmed in numerical calculations by Dohmen et al [40], and Eq. (295) is, so far, in accordance with all known numerical results.…”

Conformal invariance, multifractality, and finite-size scaling at Anderson localization transitions in two dimensions

“…, where x ρ = 0, in previous work. 9,13,[17][18][19][20] ] We finally verify numerically Eqs. (6) and (7) for the spin quantum Hall transition in symmetry class C of Ref.…”

Finite-Size Effects and Irrelevant Corrections to Scaling Near the Integer Quantum Hall Transition

“…Concomitant with this was an estimate for the ratio Γ c of the quasi-one dimensional localization length and the system width at the IQHT. Conformal invariance, which is expected to hold at the IQHT, predicts the relation Γ c /π = α 0 − 2, where α 0 = d∆ q /dq q=0 [25][26][27]. The values α 0 − 2 = 0.2596 and 0.2617 obtained in Refs.…”