Jia-Xiao Dai, Kai Wang, Shengyuan A. Yang, Y. X. Zhao

We present a novel class of topological insulators, termed the Takagi topological insulators (TTIs), which is protected by the sublattice symmetry and space-time 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 $3\mathrm{D}$ ($2\mathrm{D}$) 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 $O\left(N\right)$ 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 $3\mathrm{D}$ 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 has a cellular structure, where each topological boundary phase associated with a particular (cross-order) boundary-mode pattern corresponds to a contractible cell with a certain dimension in the parameter space.