Quantum Spin Hall State on Square-like Lattice

We find that quantum spin Hall (QSH) state can be obtained on a square-like or rectangular lattice, which is generalized from two-dimensional (2D) transition metal dichalcogenide (TMD) haeckelites. Band inversion is shown to be controled by hopping parameters and results in Dirac cones with opposite or same vorticity when spin-orbit coupling (SOC) is not considered. Effective k$\cdot$p model has been constructed to show the merging or annihilation of these Dirac cones, supplemented with the intuitive pseudospin texture. Similar to graphene based honeycomb lattice system, the QSH insulator is driven by SOC, which opens band gap at the Dirac cones. We employ the center evolution of hybrid Wannier function from Wilson-loop method, as well as the direct integral of Berry curvature, to identify the $Z_2$ number. We hope our detailed analysis will stimulate further efforts in searching for QSH insulators in square or rectangular lattice, in addition to the graphene based honeycomb lattice.