Propagating Edge States for a Magnetic Hamiltonian

Abstract

We study the quantum mechanical motion of a charged particle moving in a half plane (x > 0) subject to a uniform constant magnetic field B directed along the z-axis and to an arbitrary impurity potential W B , assumed to be weak in the sense that ||W B || ∞ < δB, for some δ small enough. We show rigorously a phenomenon pointed out by Halperin in his work on the quantum Hall effect, namely the existence of current carrying and extended edge states in such a situation. More precisely, we show that there exist states propagating with a speed of size B 1/2 in the y-direction, no matter how fast W B fluctuates. As a result of this, we obtain that the spectrum of the Hamiltonian is purely absolutely continuous in a spectral interval of size γB (for some γ < 1) between the Landau levels of the unperturbed system (i.e. the system without edge or potential), so that the corresponding eigenstates are extended.

Moment analysis for localization in random Schrödinger operators

“…However, we do not show here that the fractional-moments of the infinite volume resolvent tend to zero for large λ (although this may still be true). [23,28]. Our use of the domain-adapted metric, dist Ω -in which exponential decay is compatible with the above picture-allows the analysis of localization to proceed even in such cases.…”

Untitled

“…Usually, Λ ′ arises from the image of Λ under the map introduced in (38) by adding further fractionally charged representations (keeping the set of representations of C fixed). The only restriction comes from the requirement that the multi-electron fields are relatively local with respect to the fields corresponding to the new representations.…”

“…An alternative approach, identifying the Hall conductivity with an index, was proposed in [33,34,35]. An approach towards understanding the quantization of the Hall conductivity in terms of edge states has been described in [36]; see also [3,37,38,39]. In all these approaches, the 2D electron gas is treated as noninteracting, which severely limits their scope.…”

Spectral asymptotics for two-dimensional Schrödinger operators with strong magnetic fields

“…By applying the Mourre theory, they proved that a part of the absolutely continuous spectrum of P (b, ω) persists. On the other hand, we can consider the model (1.1) as the quantum hall system Hamiltonian with the unbounded edge potential W (x) = ω 2 x 2 (see [4,6,20]). In this work, we give a complete asymptotic expansion in powers of b −1 of the trace of the operators f (P (b, ω)) and f (P (b, ω))F and the stationary techniques developed by M. Dimassi [7] (see also M. Dimassi-J.…”