Anderson localization for electric quantum walks and skew-shift CMV matrices

Abstract

We consider the spectral and dynamical properties of one-dimensional quantum walks placed into homogenous electric fields according to a discrete version of the minimal coupling principle. We show that for all irrational fields the absolutely continuous spectrum of these systems is empty, and prove Anderson localization for almost all (irrational) fields. This result closes a gap which was left open in the original study of electric quantum walks: a spectral and dynamical characterization of these systems for typical fields. Additionally, we derive an analytic and explicit expression for the Lyapunov exponent of this model. Making use of a connection between quantum walks and CMV matrices our result implies Anderson localization for CMV matrices with a particular choice of skew-shift Verblunsky coefficients as well as for quasi-periodic unitary band matrices.

Singular continuous Cantor spectrum for magnetic quantum walks

“…A similarly drastic dependence of the spectral and dynamical properties on a system parameter has recently been observed in quantum walks subject to external electric fields [25,26]. Quantum walks are discrete-time analogues of the unitary dynamics of a single particle with internal degree of freedom on a lattice constrained to finite propagation speed per time step [1,5,7,38].…”

“…there is an infinite sequence of better and better revivals alternating with farther and farther excursions [25]. On the other hand, for badly approximable fields the system displays Anderson localization which even turns out to be the typical case [26].…”

“…Taking these questions as a starting point, we discuss in this paper in detail two-dimensional quantum walks which we place into homogeneous magnetic fields using the aforementioned discrete analogue of minimal coupling from [22]. Also in the magnetic case the spectral and dynamical properties depend crucially on the rationality of the field, but this dependence turns out to be slightly less involved than for the electric walks in [25,26]: for rational fields the system is effectively translation invariant which by general arguments implies absolutely continuous spectrum and ballistic transport. For irrational fields, the spectrum is purely singular continuous which by the RAGE theorem [29] excludes any form of localization and thereby contrasts the typicality of Anderson localization for one-dimensional electric walks; see also [35] for a proof of the RAGE theorem in the discrete-time (unitary) case.…”

Quantum walks in weak electric fields and Bloch oscillations

“…(2) can be described as the composition of two effects: The displacement in quasimomentum space, followed by the action of the free walk operator. This dynamics has been examined both in quasimomentum space and in the original position space, and shows a rich behavior, depending on whether the ratio φ/(2π) is a rational or an irrational number [1,6,8,29].…”

An algorithm to factorize quantum walks into shift and coin operations

“…A shift-coin decomposition is required for a number of theoretical tasks. Firstly, to couple walks to gauge fields: Such fields are implemented as commutation phases of the shift operators [9,12,13,17]. Secondly, one would like to consider a walk as the one-particle sector of an interacting system, known abstractly as a quantum cellular automaton (QCA) [3,14,34].…”