Chiral-symmetric higher-order topological phases protected by multipole winding number invariants

We introduce novel higher-order topological phases in chiral-symmetric systems (class AIII of the ten-fold classification), most of which would be misidentified as trivial by current theories. These phases are protected by multipole winding numbers, bulk integer topological invariants that in 2D and 3D are built from sublattice multipole moment operators, as defined herein. The integer value of a multipole winding number indicates the number of degenerate zero-energy states localized at each corner of a crystal. These phases are generally boundary-obstructed and robust in the presence of disorder.