Quantum spin Hall effect in a square-lattice model under a uniform magnetic field

We study a toy square-lattice model under a uniform magnetic field. Using the Landauer--B黷tiker formula, we calculate the transport properties of the system on a two-terminal, a four-terminal, and a six-terminal device. We find that the quantum spin Hall (QSH) effect appears in energy ranges where the spin-up and spin-down subsystems have different filling factors. We also study the robustness of the resulting QSH effect and find that it is robust when the Fermi levels of both spin subsystems are far away from the energy plateaus but is fragile when the Fermi level of any spin subsystem is near the energy plateaus. These results provide an example of QSH effect with the physical origin other than time-reversal (TR) preserving spin-orbit coupling (SOC).