Localization and Fractality in Inhomogeneous Quantum Walks with Self-Duality

TL;DR

This paper introduces inhomogeneous, self-dual quantum walks on a 1D lattice, revealing localization phenomena and fractal spectra akin to the Aubry-Andre9 model, with implications for quantum transport and spectral analysis.

Contribution

It presents a novel class of inhomogeneous quantum walks that are self-dual under Fourier transform, connecting quantum walk dynamics with fractal spectral properties.

Findings

Quantum walks exhibit localization at the origin for incommensurate coin periods.

The eigenvalue spectrum displays a Hofstadter butterfly fractal pattern.

The models demonstrate a self-duality property analogous to the Aubry-Andre9 model.

Abstract

We introduce and study a class of discrete-time quantum walks on a one-dimensional lattice. In contrast to the standard homogeneous quantum walks, coin operators are inhomogeneous and depend on their positions in this class of models. The models are shown to be self-dual with respect to the Fourier transform, which is analogous to the Aubry-Andr\'e model describing the one-dimensional tight-binding model with a quasi-periodic potential. When the period of coin operators is incommensurate to the lattice spacing, we rigorously show that the limit distribution of the quantum walk is localized at the origin. We also numerically study the eigenvalues of the one-step time evolution operator and find the Hofstadter butterfly spectrum which indicates the fractal nature of this class of quantum walks.