Higher-order topological insulators have a modified bulk-boundary correspondence compared to other topological phases: instead of gapless edge or surface states, they have gapped edges and surfaces, but protected modes at corners or hinges. Here, we explore symmetry-protected topological phases in strongly interacting many-body systems with this generalized bulk-boundary correspondence. We introduce several exactly solvable bosonic lattice models as candidates for interacting higher-order symmetry-protected topological (HOSPT) phases protected by spatial symmetries, and develop a topological field theory that captures the nontrivial nature of the gapless corner and hinge modes. We show how, for rotational symmetry, this field theory leads to a natural relationship between HOSPT phases and conventional SPT phases with an enlarged internal symmetry group. We also explore the connection between bosonic and fermionic HOSPT phases in the presence of strong interactions, and comment on the implications of this connection for the classification of interacting fermionic HOSPT phases. Finally, we explore how gauging internal symmetries of these phases leads to topological orders characterized by nontrivial braiding statistics between topological vortex excitations and geometrical defects related to the spatial symmetry.
Bibliographical noteFunding Information:
We are grateful to T. Hughes, M Hermele, and X.-Y. Song for insightful comments and discussions. Y.Y. is supported by PCTS Fellowship at Princeton University. F.J.B. is grateful for the financial support of NSF-DMR 1352271 and the Sloan Foundation FG-2015-65927. This work (Y.Y., F.J.B.) was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. T.N. acknowledges support from the Swiss National Science Foundation (Grant No. 200021_169061) and from the European Union's Horizon 2020 research and innovation program (ERC-StG-Neupert-757867-PARATOP).