Localization Without Disorder: Quantum Walks on Structured Graphs

TL;DR

This paper analytically investigates localization in continuous-time quantum walks on symmetric graphs, revealing how spectral degeneracy and modular structure influence confinement and transport properties.

Contribution

It provides a complete analytical characterization of localization phenomena on specific structured graphs, linking spectral properties to quantum transport behavior.

Findings

Localization is driven by degenerate subspaces and their hybridization.

Dynamical IPR can surpass eigenstate IPRs due to superposition effects.

Connectivity alone can predict quantum walk localization patterns.

Abstract

Continuous-time quantum walks (CTQWs) exhibit localization phenomena that differ fundamentally from their classical counterparts, yet the precise relationship between network structure, spectral degeneracy, and confined dynamics remains incompletely understood. In this work, we present a complete analytical characterization of localization in CTQWs on two highly symmetric graph families: barbell graphs and star-of-cliques graphs. These networks combine pronounced spectral degeneracy with modular structure, enabling exact diagonalization and explicit computation of both eigenstate and dynamical inverse participation ratios (IPRs). Our analysis reveals that localization is governed by the interplay between degenerate subspaces, which generate families of confined modes, and hybridization between invariant subspaces, which redistributes spectral weight. Notably, the dynamical IPR can exceed expectations based solely on eigenstate IPRs, demonstrating that coherent superposition within degenerate eigenspaces enhances confinement. By connecting IPR values to the effective number of vertices visited, we provide a structural diagnostic for predicting quantum transport outcomes in modular networks, establishing that connectivity alone can determine where and how strongly a quantum walk localizes.