Discrete step walks reveal unconventional anomalous topology in synthetic photonic lattices

Anomalous topological phases, where edge states coexist with topologically trivial Chern bands, can only appear in periodically driven lattices. When the driving is smooth and continuous, the bulk-edge correspondence is guaranteed by the existence of a bulk invariant known as the winding number. However, in lattices subject to periodic time-step walks the existence of edge states does not only depend on bulk invariants but also on the geometry of the boundary. This is a consequence of the absence of an intrinsic time-dependence or micromotion in discrete-step walks. We report the observation of edge states and a simultaneous measurement of the bulk invariants in anomalous topological phases in a two-dimensional discrete-step walk in a synthetic photonic lattice made of two coupled fibre rings. The presence of edge states is inherent to the periodic driving and depends on the geometry of the boundary in the implemented two-band model with zero Chern number. We provide a suitable expression for the topological invariants whose calculation does not rely on micromotion dynamics.