Quantum Spin Hall State on Square-like Lattice

TL;DR

This paper demonstrates that quantum spin Hall states can exist on square-like lattices, expanding the types of materials and structures where topological insulators can be realized, with detailed analysis of band inversion and topological invariants.

Contribution

It introduces a new class of QSH insulators on square-like lattices derived from TMD haeckelites, with a comprehensive theoretical framework and topological characterization.

Findings

QSH states can be realized on square-like lattices.

Band inversion is controlled by hopping parameters.

SOC opens a band gap at Dirac cones, leading to topological insulating behavior.

Abstract

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.