On stable quantum currents

Abstract

We study transport properties of discrete quantum dynamical systems on the lattice, in particular Coined Quantum Walks and the Chalker-Coddington model. We prove existence of a non trivial charge transport implying that the absolutely continuous spectrum covers the whole unit circle under mild assumptions. We discuss anomalous quantum charge transport. For Quantum Walks we exhibit explicit constructions of coins which imply existence of stable directed quantum currents along classical curves. The results are of topological nature and independent of the details of the model.

One-dimensional Discrete Dirac Operators in a Decaying Random Potential I: Spectrum and Dynamics

“…We aim to obtain a full unitary analogue of the radial transfer matrix set formalism in [49] for finite hopping unitary operators, including higher dimensional quantum walks in Z d and Chalker-Coddington models. Note that other techniques have been used to get delocalization for random operators on tree graphs in the Hermitian case [1,3,20,32,34,36,44,46] and the unitary case [24], and for unitary network models [6,7,8,9].…”

Absolutely Continuous Edge Spectrum of Hall Insulators on the Lattice

“…Proposition 4.1 (Theorem 2.1. of [2]). Let U be a unitary operator on a Hilbert space and P an orthogonal projection such that [U, P ] is trace class, then the index Ind(U, P ) is a well-defined finite integer.…”

“…The topological index characterizing the edge system in this bulk-edge correspondence is a property of a unitary operator which is defined in terms of the edge states. Using a result from [2] one immediately gets that this unitary has absolutely continuous spectrum if the topological index is nonzero. We then show that the Hamiltonian of the half-plane system inherits this absolutely contiuous spectrum.…”

Absolutely continuous edge spectrum of topological insulators with an odd time-reversal symmetry

“…The proof of Theorem 2.1 builds on the following time-reversal symmetric version of Theorem 2.1 of [3] which is of independent interest: Theorem 2.3. Let U be a unitary and P a projection such that A = U P U * − P is compact.…”

“…We appeal to the bulk-edge correspondence for time-reversal invariant topological insulators [15,13,2,6] which links the bulk index to an edge index associated to a time-reversal symmetric unitary acting on the edge modes. Inspired by [3], we prove a symmetric Wold decomposition for such unitaries which implies in particular that the absolutely continuous spectrum of this unitary covers the whole unit circle if the edge index is non-trivial. The Hamiltonian describing the system with edge is then shown to inherit this absolutely continuous spectrum.…”