Adiabatic theorems and applications to the quantum hall effect

Abstract

We study an adiabatic evolution that approximates the physical dynamics and describes a natural parallel transport in spectral subspaces. Using this we prove two folk theorems about the adiabatic limit of quantum mechanics: 1. For slow time variation of the Hamiltonian, the time evolution reduces to spectral subspaces bordered by gaps. 2. The eventual tunneling out of such spectral subspaces is smaller than any inverse power of the time scale if the Hamiltonian varies infinitly smoothly over a finite interval. Except for the existence of gaps, no assumptions are made on the nature of the spectrum. We apply these results to charge transport in quantum Hall Hamiltonians and prove that the flux averaged charge transport is an integer in the adiabatic limit.

An Adiabatic Theorem without a Gap Condition

“…We recall the notion of adiabatic evolution [33,5]. Let U A (s) be the solution of the initial value problem:…”

“…No assumption on the spectral type of H(s) restricted to RangeP ⊥ (s) need be made, Fig. 3 [5,42], to P (s) that need not be associated with an eigenvalue, and whose rank could also be infinite. In particular, the initial data could lie in a subspace corresponding to an energy band provided it is separated by a gap from the rest of the spectrum.…”

Geometric Properties of Transport in Quantum Hall Systems

“…We aim now at the adiabatic theorem of quantum mechanics, following the article J. E. Avron, R. Seiler, L. G. Ya e (1987). Even though the theorem itself is rather old | its rst formulation goes back to Born and Fock (M. Born, V. Fock (1928)) | its proper formulation was found years later by T. Kato (1950) in the context of pertubation theory of linear operators.…”

“…Thouless (1987), J. E. Avron, R. Seiler, L. G. Ya e (1987)). The con guration space is a compact domain in the two dimensional plain with two holes, with two Aharonov-Bohm uxes 1 and 2 running through the holes, and again a strong magnetic eld B perpendicular to the plane, see Fig.…”

Towards the fractional quantum Hall effect: a noncommutative geometry perspective

“…[2]) then shows that the adiabatic evolution approximates well the physical evolution, for large values of the adiabatic parameter τ → ∞, via an estimate of the form…”