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.

• Received 1 October 2018
• Revised 26 August 2019
• Accepted 27 November 2019

DOI:https://doi.org/10.1103/PhysRevB.101.085137