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Takagi Topological Insulator on the Honeycomb Lattice

Recently, real topological phases protected by $PT$ symmetry have been actively investigated. In two dimensions, the corresponding topological invariant is the Stiefel-Whitney number. A recent theoretical advance is that in the presence of the sublattice symmetry, the Stiefel-Whitney number can be equivalently formulated in terms of Takagi's factorization. The topological invariant gives rise to a novel second-order topological insulator with odd $PT$-related pairs of corner zero modes. In this article, we review the elements of this novel second-order topological insulator, and demonstrate the essential physics by a simple model on the honeycomb lattice.