Wannier charge center, spin resolved bulk polarization, and corner modes in a strained quantum spin Hall insulator

Topological invariants are a significant ingredient in the study of topological phases of matter that intertwines the supposedly contradicting concepts of bulk and boundary. The nature of the invariants differs depending on the dimensionality of the boundary at which the topologically nontrivial states manifest themselves. The primary motivation of this work is to study two distinct scenarios of topological phases, differing in the dimensionality of their boundary states, and study the associated bulk topological invariants that characterize them. In this regard, we study the band-engineered Kane-Mele model which originally is a prototypical example of a system that hosts quantum spin Hall effect on a honeycomb lattice. Under a smooth band deformation caused by varying one of the nearest-neighbor hopping amplitudes (say t₁) as compared to the other two (say t), we observe that the system transits from its first-order topological insulating state (or quantum spin Hall state) to a second-order topological insulating (SOTI) state via a gap-closing transition. This transition occurs when the system crosses a particular threshold of the deformation parameter t₁/t (namely, t₁/t=2), known as the semi-Dirac limit in literature. We show the presence of edge and corner modes as a signature of first- and second-order topology, respectively. Further, we observe the evolution of the Wannier charge center (WCC), a bulk property as a function of the deformation parameter t₁/t. It is seen that the behavior of the WCC is entirely different in the quantum spin Hall (QSH) phase as compared to the second-order topological state. We also find that, while the Z₂ invariant successfully characterizes the QSH state, it cannot characterize higher-order topology (second order here). The model being mirror invariant, we also calculate mirror winding number to show that it is rendered trivial in the SOTI phase as well, while being nontrivial in the QSH phase. Finally, we observe that the spin resolved bulk polarization correctly establishes the appearance of second-order topological corner modes and thus categorizes this phase as an obstructed atomic insulator.