  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.
