Higher order symmetry-protected topological states for interacting bosons and fermions

TL;DR

This paper investigates higher-order symmetry-protected topological phases in strongly interacting bosonic and fermionic systems, introducing solvable models and a topological field theory to understand their protected corner and hinge modes.

Contribution

It presents exactly solvable models for interacting HOSPT phases, develops a topological field theory, and explores the relationship between bosonic and fermionic HOSPT phases under strong interactions.

Findings

Exactly solvable bosonic lattice models for HOSPT phases

Topological field theory capturing corner and hinge modes

Connection between HOSPT and conventional SPT phases with enlarged symmetry

Abstract

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 non-trivial 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.