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

## TL;DR

This paper demonstrates that one-dimensional electric quantum walks exhibit Anderson localization for almost all irrational electric fields, providing a spectral and dynamical characterization and connecting these results to CMV matrices with skew-shift Verblunsky coefficients.

## Contribution

It establishes Anderson localization for electric quantum walks under irrational fields and derives an explicit Lyapunov exponent, linking quantum walks to CMV matrices with skew-shift coefficients.

## Key findings

- Absolutely continuous spectrum is empty for irrational fields.
- Anderson localization holds for almost all irrational fields.
- Explicit Lyapunov exponent expression derived.

## 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.