Quantum spin Hall phase transition in the α−T3 lattice

Abstract

We report a theoretic study of the topological properties of the $\ensuremath{\alpha}\text{\ensuremath{-}}{T}_{3}$ lattice by taking the intrinsic spin orbit interaction into account. It is shown that when $0<\ensuremath{\alpha}<1$, the electronic band structure is spin-valley split and the original flat band is distorted to display a nonzero dispersion that imparts the system to be metallic. The gapped valence or conduction band shows the quantum spin Hall phase and experiences a topological phase transition from the spin Chern number ${C}_{s}=1$ to ${C}_{s}=2$ at the critical point $\ensuremath{\alpha}=\frac{1}{2}$. When a staggered magnetization order is considered in the system, a bunch of topological phases appear such as the quantum spin quantum anomalous Hall phase with variation of $\ensuremath{\alpha}$ or the magnetization strength.