Anderson localization for electric quantum walks and skew-shift CMV matrices
arXiv:1906.11931 · doi:10.1007/s00220-021-04204-w
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.
18 pages, 2 figures
References in corpus (10)
- Universal computation by quantum walk
- Exponential algorithmic speedup by quantum walk
- Quantum Walk in Position Space with Single Optically Trapped Atoms
- A 2D Quantum Walk Simulation of Two-Particle Dynamics
- Asymptotic evolution of quantum walks with random coin
- The topological classification of one-dimensional symmetric quantum walks
- Quantum walks in external gauge fields
- Complete homotopy invariants for translation invariant symmetric quantum walks on a chain
- Eigenvalue Measurement of Topologically Protected Edge states in Split-Step Quantum Walks
- Quantum walks: Schur functions meet symmetry protected topological phases