Classification and construction of higher-order symmetry-protected topological phases of interacting bosons

Motivated by the recent discovery of higher-order topological insulators, we study their counterparts in strongly interacting bosons: "higher-order symmetry-protected topological (HOSPT) phases." While the usual (first-order) SPT phases in d spatial dimensions support anomalous (d−1)-dimensional surface states, HOSPT phases in d dimensions are characterized by topological boundary states of dimension (d−2) or smaller, protected by certain global symmetries and robust against disorders. Based on a dimensional reduction analysis, we show that HOSPT phases can be built from lower-dimensional SPT phases in a way that preserves the associated crystalline symmetries. When the total symmetry is a direct product of global and crystalline symmetry groups, we are able to classify the HOSPT phases using the Künneth formula of group cohomology. Based on a decorated domain-wall picture of the Künneth formula, we show how to systematically construct the HOSPT phases, and demonstrate our construction with many examples in two and three dimensions.