Phase rotation symmetry and the topology of oriented scattering networks

TL;DR

This paper explores the topological properties of oriented scattering networks using phase rotation symmetry, revealing the origin of anomalous phases and defining bulk invariants that unify different descriptions of these systems.

Contribution

It introduces phase rotation symmetry as a new tool to analyze topological phases in unitary scattering networks and establishes the equivalence of different topological invariants.

Findings

Anomalous topological phases can be explained by phase rotation symmetry.

Bulk invariants for scattering networks are defined and shown to coincide.

The approach unifies Floquet and network model topologies.

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.