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. Novelly, the higher-order topological boundary modes can not only be tuned by the parameters but also the geometric shape of the sample.