Quotient symmetry protected topological phenomena

Topological phenomena are commonly studied in phases of matter which are separated from a trivial phase by an unavoidable quantum phase transition. This can be overly restrictive, leaving out scenarios of practical relevance -- similar to the distinction between liquid water and vapor. Indeed, we show that topological phenomena can be stable over a large part of parameter space even when the bulk is strictly speaking in a trivial phase of matter. In particular, we focus on symmetry-protected topological phases which can be trivialized by extending the symmetry group. The topological Haldane phase in spin chains serves as a paradigmatic example where the $SO(3)$ symmetry is extended to $SU(2)$ by tuning away from the Mott limit. Although the Haldane phase is then adiabatically connected to a product state, we show that characteristic phenomena -- edge modes, entanglement degeneracies and bulk phase transitions -- remain parametrically stable. This stability is due to a separation of energy scales, characterized by quantized invariants which are well-defined when a subgroup of the symmetry only acts on high-energy degrees of freedom. The low-energy symmetry group is a quotient group whose emergent anomalies stabilize edge modes and unnecessary criticality, which can occur in any dimension.