Localisation, delocalisation, and topological transitions in disordered 2D quantum walks

Abstract

We investigate time-independent disorder on several two-dimensional discrete-time quantum walks. We find numerically that, contrary to claims in the literature, random onsite phase disorder, spin-dependent or otherwise, cannot localise the Hadamard quantum walk; rather, it induces diffusive spreading of the walker. In contrast, split-step quantum walks are generically localised by phase disorder. We explain this difference by showing that the Hadamard walk is a special case of the split-step quantum walk, with parameters tuned to a critical point at a topological phase transition. We show that the topological phase transition can also be reached by introducing strong disorder in the rotation angles. We determine the critical exponent for the divergence of the localisation length at the topological phase transition, and find $\nu =2.6$, in both cases. This places the two-dimensional split-step quantum walk in the universality class of the quantum Hall effect.

9 pages, 7 figures

References in corpus (26)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Mechanics helps in searching for a needle in a haystack
• Anderson Transitions
• Quantum random walks - an introductory overview
• Topological characterization of periodically-driven quantum systems
• A Quantum Random Walk Search Algorithm
• Exploring Topological Phases With Quantum Walks
• Decoherence and disorder in quantum walks: From ballistic spread to localization
• Symmetries, Topological Phases and Bound States in the One-Dimensional Quantum Walk
• Scattering theory of topological insulators and superconductors
• Localization of Two-Dimensional Quantum Walks
• Critical exponent for the quantum Hall transition
• Network Models of Photonic Floquet Topological Insulators
• Quantum Walks on a Random Environment
• Observation of the Anderson Metal-Insulator Transition with Atomic Matter Waves: Theory and Experiment
• Optimized quantum random-walk search algorithms
• Topological phases and delocalization of quantum walks in random environments
• Quantum Hall effect in a one-dimensional dynamical system
• Disordered Quantum Walks in one lattice dimension
• Almost Commuting Matrices, Localized Wannier Functions, and the Quantum Hall Effect
• Edge-state enhanced transport in a 2-dimensional quantum walk
• Simulating Anderson localization via a quantum walk on a one-dimensional lattice of superconducting qubits
• Finite Size Scaling of the Chalker-Coddington Model
• Metallic phase of the quantum Hall effect in four-dimensional space
• Implementation of a spatial two-dimensional quantum random walk with tunable decoherence

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• Localization, delocalization, and topological phase transitions in the one-dimensional split-step quantum walk
• Creating cat states in one-dimensional quantum walks using delocalized initial states
• Attractor-repeller pair of topological zero-modes in a nonlinear quantum walk
• Persistence of Topological Phases in Non-Hermitian Quantum Walks
• Bulk-edge correspondence in nonunitary Floquet systems with chiral symmetry
• Bloch-Landau-Zener dynamics induced by a synthetic field in a photonic quantum walk
• Quantum Hall criticality in Floquet topological insulators
• Edge states in a two-dimensional quantum walk with disorder
• Inhibition of spreading in quantum random walks due to quenched Poisson-distributed disorder
• Complete classification of trapping coins for quantum walks on the 2D square lattice
• Anomalous levitation and annihilation in Floquet topological insulators
• Dynamics of discrete-time quantum walk with time-correlated unitary noise
• Parrondo's paradox in quantum walks with inhomogeneous coins
• Second-order topological insulator in a coinless discrete-time quantum walk
• Relative homotopy approach to topological phases in quantum walks
• Topological delocalization in the completely disordered two-dimensional quantum walk
• Interference-induced localization in quantum random walk on clean cyclic graph
• Quantum walk on a square lattice with identical particles
• Noisy Cyclic Quantum Random Walk