Simultaneous quantization of edge and bulk Hall conductivity

Abstract

The edge Hall conductivity is shown to be an integer multiple of e 2 /h which is almost surely independent of the choice of the disordered configuration. Its equality to the bulk Hall conductivity given by the Kubo-Chern formula follows from K-theoretic arguments. This leads to quantization of the Hall conductance for any redistribution of the current in the sample. It is argued that in experiments at most a few percent of the total current can be carried by edge states.Soon after the discovery of the integer quantum Hall effect (QHE) [19], several geometric interpretations of the observed quantization of the Hall conductance of a two dimensional electron gas were put forward in the framework of non-relativistic quantum mechanics. Laughlin proposed an adiabatic Gedankenexperiment in order to calculate the Hall conductance [15], Halperin and later on Büttiker studied the conduction by edge channels [13,10], while Thouless, Kohmoto, Nightingale and den Nijs investigated the Hall conductivity as given by the Kubo formula [20]. Laughlin's argument was rigorously analyzed by Avron, Seiler and Simon even for multiparticle Hamiltonians and in presence of a disordered potential [3,4,5]. Bellissard, recently joint by van Elst and Schulz-Baldes, generalized the TKN 2 -work in order to show quantization of the Hall conductivity also in presence of a disordered potential as long as the Fermi level lies in a region of dynamically localized states [6,7], a result that was also obtained by Aizenman and Graf [1]. All these beautiful mathematical approaches exhibit the Hall conductance and conductivity respectively to have a deep geometrical meaning and allow to calculate them as an index of a certain Fredholm operator. In [20,5,7,1], the edges of the sample play no particular rôle.Recently there has been a revived interest in edge states of magnetic Schrödinger operators. Hatsugai linked an edge state winding number to the Chern numbers for Harper's equation with rational flux [14]. Akkermans, Avron, Narevich and Seiler introduced spectral boundary conditions giving rise to a linear dispersion relation for edge states and a natural setting for the Laughlin wave function as a many body bulk state [2]. The stability of the absolutely continuous spectrum associated to edge states under the perturbation with a random potential was studied by several authors with Mourre's positive commutator estimates [16,8,12].Our first main result is a rigorous proof of the edge current quantization in the sense of Halperin for a discrete magnetic half-plane operator containing a disordered potential, notably we show quantization of what we call the edge Hall conductivity. Our second mathematical result is its equality to the bulk Hall conductivity as calculated by the Kubo-Chern formula [20,6,7]. The proof of this equality unveals a deep connection between the plane and edge geometry as it is based on Bott periodicity, the heart of K-theory [9]. We still need a gap in the spectrum of the plane operator, but a generalization to a region of...

Equality of the Bulk and Edge Hall Conductances in a Mobility Gap

“…In this case, the conductance σ (1) E = σ (2) E defined here coincides with σ E defined in (1.7). This statement follows from Theorem 1 and the known equality σ E = σ B [32,12], but can also be seen directly. For completeness, we include a proof of this fact in Section 2 below.…”

“…Since σ E is independent of ρ as long as it conforms with (1.8), see [32] and Theorem 1 below, it is indeed the conductance σ E = I/δ for sufficiently small δ.…”

“…The equality σ B = σ E , suggested by Halperin's analysis [17] of the Laughlin argument [21], has been established in the context of an effective field theory description [14]. It was later derived in a microscopic treatment of the integral QHE [32,12,24] for the case that the Fermi energy lies in a spectral gap ∆ of the single-particle Hamiltonian H B . We prove this equality, by quite different means, in the more general setting that H B exhibits Anderson localization in ∆ -more precisely, dynamical localization (see (1.2) below).…”

“…The current operator across the line x 1 = 0 is −i [H a , Λ 1 ]. Matters are simpler if we temporarily assume that ∆ is a gap for H B , i.e., if σ(H B ) ∩ ∆ = ∅, in which case one may set [32] σ E := −i tr…”

Boundary Maps for C*-Crossed Products with with an Application to the Quantum Hall Effect

“…In the Landau model and its restriction to the half-space, a proof can be given by explicitly calculating both pairings and then seeing that both numbers are the same [SKR00]. This calculation makes use of the translation invariance in the direction along the boundary.…”