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We investigate the topological properties of dynamical states evolving onperiodic oriented graphs. This evolution, that encodes the scattering processesoccurring at the nodes of the graph, is described by a single-step globaloperator, in the spirit of the Ho-Chalker model. When the successive scatteringevents follow a cyclic sequence, the corresponding scattering network can beequivalently described by a discrete time-periodic unitary evolution, in linewith Floquet systems. Such systems may present anomalous topological phaseswhere all the first Chern numbers are vanishing, but where protected edgestates appear in a finite geometry. To investigate the origin of such anomalousphases, we introduce the phase rotation symmetry, a generalization of usualsymmetries which only occurs in unitary systems (as opposed to Hamiltoniansystems). Equipped with this new tool, we explore a possible explanation of thepervasiveness of anomalous phases in scattering network models, and we definebulk topological invariants suited to both equivalent descriptions of thenetwork model, which fully capture the topology of the system. We finally showthat the two invariants coincide, again through a phase rotation symmetryarising from the particular structure of the network model.

Phase rotation symmetry and the topology of oriented scattering networks