Takagi topological insulator with odd $\mathcal P\mathcal T$ pairs of corner states

TL;DR

This paper introduces Takagi topological insulators (TTIs), a new class protected by sublattice and $ ext{PT}$ symmetry, featuring odd pairs of corner states and characterized by $ ext{Z}_2$ invariants in 2D and 3D.

Contribution

The paper defines TTIs protected by sublattice and $ ext{PT}$ symmetry, introduces their topological invariants, and describes their boundary corner state properties.

Findings

3D TTIs have a $ ext{Z}_2$ invariant related to Takagi's factorization.

2D TTIs are characterized by the second Stiefel-Whitney number.

3D TTIs exhibit odd pairs of corner zero-modes with a parity condition.

Abstract

We present a novel class of topological insulators, termed the Takagi topological insulators (TTIs), which is protected by the sublattice symmetry and spacetime inversion ($\mathcal P\mathcal T$) symmetry. The required symmetries for the TTIs can be realized on any bipartite lattice where the inversion exchanges sublattices. The protecting symmetries lead to the classifying space of Hamiltonians being unitary symmetric matrices, and therefore Takagi's factorization can be performed. Particularly, the global Takagi's factorization can (cannot) be done on a $3$D ($2$D) sphere. In 3D, there is a $\mathbb{Z}_2$ topological invariant corresponding to the parity of the winding number of Takagi's unitary-matrix factor over the entire Brillouin zone, where the $\mathbb Z_2$ nature comes from the $O(N)$ gauge degrees of freedom in Takagi's factorization. In 2D, the obstruction for a global Takagi's factorization is characterized by another $\mathbb{Z}_2$ topological invariant, equivalent to the second Stiefel-Whitney number. For the third-order topological phases, the $3$D TTIs feature a parity condition for corner zero-modes, i.e., there always exist odd $\mathcal P\mathcal T$ pairs of corners with zero-modes. Moreover, for any $\mathcal P\mathcal T$ invariant sample geometry, all configurations of corner zero-modes satisfying the parity condition can exist with the same nontrivial bulk topological invariant. Actually, without closing the bulk gap, the boundary phase diagram have a cellular structure, where each topological boundary phase associated with a particular (cross-order) boundary-mode pattern corresponds to a contractible cell with certain dimension in the parameter space.