Rotational symmetry protected edge and corner states in Abelian topological phases

Spatial symmetries can enrich the topological classification of interacting quantum matter and endow systems with nontrivial strong topological invariants (protected by internal symmetries) with additional ``weak'' topological indices. In this paper, we study the edge physics of systems with a nontrivial shift invariant, which is protected by either a continuous U (1) ₑ or discrete C₍ rotation symmetry, along with internal U (1) ₂ charge conservation. Specifically, we construct an interface between two systems that have the same Chern number but are distinguished by their Wen-Zee shift, and, through analytic arguments supported by numerics, we show that the interface hosts counterpropagating gapless edge modes that cannot be gapped by arbitrary local symmetry-preserving perturbations. Using the Chern-Simons field theory description of two-dimensional Abelian topological orders, we then prove sufficient conditions for continuous rotation symmetry protected gapless edge states using two complementary approaches. One relies on the algebraic Lagrangian subalgebra framework for gapped boundaries, while the other uses a more physical flux insertion argument. For the case of discrete rotation symmetries, we extend the field theory approach to show the presence of fractional corner charges for Abelian topological orders with gappable edges, and we compute them in the case in which the Abelian topological order is placed on the two-dimensional surface of a Platonic solid. Our work paves the way for studying the edge physics associated with spatial symmetries in strongly interacting symmetry enriched topological phases.