Phase rotation symmetry and the topology of oriented scattering networks

Abstract

We investigate the topological properties of dynamical states evolving on periodic oriented graphs. This evolution, that encodes the scattering processes occurring at the nodes of the graph, is described by a single-step global operator, in the spirit of the Ho-Chalker model. When the successive scattering events follow a cyclic sequence, the corresponding scattering network can be equivalently described by a discrete time-periodic unitary evolution, in line with Floquet systems. Such systems may present anomalous topological phases where all the first Chern numbers are vanishing, but where protected edge states appear in a finite geometry. To investigate the origin of such anomalous phases, we introduce the phase rotation symmetry, a generalization of usual symmetries which only occurs in unitary systems (as opposed to Hamiltonian systems). Equipped with this new tool, we explore a possible explanation of the pervasiveness of anomalous phases in scattering network models, and we define bulk topological invariants suited to both equivalent descriptions of the network model, which fully capture the topology of the system. We finally show that the two invariants coincide, again through a phase rotation symmetry arising from the particular structure of the network model.

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• Topological characterization of periodically-driven quantum systems
• Exploring Topological Phases With Quantum Walks
• Observation of photonic anomalous Floquet Topological Insulators
• Experimental observation of anomalous topological edge modes in a slowly-driven photonic lattice
• Measurement of a topological edge invariant in a microwave network
• Robustness of topologically protected edge states in quantum walk experiments with neutral atoms
• Edge-state enhanced transport in a 2-dimensional quantum walk
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• Symmetry Analysis of Anomalous Floquet Topological Phases
• Strongly Disordered Floquet Topological Systems
• Topological boundary invariants for Floquet systems and quantum walks
• Effective vacua for Floquet topological phases: A numerical perspective on switch-function formalism
• Topological chiral modes in random scattering networks
• Quantum Hall network models as Floquet topological insulators
• Network model for periodically strained graphene
• Effective Floquet model for minimally twisted bilayer graphene
• Network model for higher-order topological phases
• Topological chiral interface states beyond insulators
• Edge-dependent anomalous topology in synthetic photonic lattices subject to discrete step walks
• Topological delocalization in the completely disordered two-dimensional quantum walk
• Engineering stable quantum currents at bulk boundaries
• Chirality induced Interface Currents in the Chalker Coddington Model
• Network model for magnetic higher-order topological phases
• Anomalous nonreciprocal topological networks: stronger than Chern insulators
• Supermetal-insulator transition in a non-Hermitian network model
• Effects of spin-orbit coupling in a valley chiral kagomé network
• Chiral electronic network within skyrmionic lattice on topological insulator surfaces
• Examples for stable quantum currents