Spectral Transition for Random Quantum Walks on Trees

TL;DR

This paper investigates spectral transitions in random quantum walks on homogeneous trees, showing how the spectrum type depends on the internal unitary matrix, with implications for understanding quantum dynamics in disordered systems.

Contribution

It demonstrates the existence of spectral transitions driven by the internal unitary matrix in quantum walks on trees, providing a detailed spectral analysis for degrees 3 and 4.

Findings

Existence of pure point spectrum for certain matrices in U(q).

Existence of absolutely continuous spectrum for other matrices in U(q).

Spectral transition driven by the internal unitary matrix C.

Abstract

We define and analyze random quantum walks on homogeneous trees of degree $q\geq 3$. Such walks describe the discrete time evolution of a quantum particle with internal degree of freedom in $\C^q$ hopping on the neighboring sites of the tree in presence of static disorder. The one time step random unitary evolution operator of the particle depends on a unitary matrix $C\in U(q)$ which monitors the strength of the disorder. We prove for any $q$ that there exist open sets of matrices in $U(q)$ for which the random evolution has either pure point spectrum almost surely or purely absolutely continuous spectrum, thereby showing the existence of a spectral transition driven by $C\in U(q)$. For $q\in\{3,4\}$, we establish properties of the spectral diagram which provide a description of the spectral transition.