$(d-2)$-dimensional edge states of rotation symmetry protected topological states

TL;DR

This paper investigates $(d-2)$-dimensional edge states in rotation symmetry protected topological phases with time-reversal symmetry in 2D and 3D, providing both non-interacting and interacting system insights.

Contribution

It introduces a comprehensive framework for understanding $(d-2)$-dimensional edge states in rotation symmetry protected topological states, including a formula for topological invariants and models for interacting systems.

Findings

Nontrivial topology manifests as $(d-2)$-dimensional edge states in 2D and 3D.

Bulk invariants can be calculated using a Fu-Kane-like formula with inversion symmetry.

Explicit models demonstrate robust edge states in strongly interacting systems.

Abstract

We study fourfold rotation invariant gapped topological systems with time-reversal symmetry in two and three dimensions ($d=2,3$). We show that in both cases nontrivial topology is manifested by the presence of the $(d-2)$-dimensional edge states, existing at a point in 2D or along a line in 3D. For fermion systems without interaction, the bulk topological invariants are given in terms of the Wannier centers of filled bands, and can be readily calculated using a Fu-Kane-like formula when inversion symmetry is also present. The theory is extended to strongly interacting systems through explicit construction of microscopic models having robust $(d-2)$-dimensional edge states.