Time-reversal-symmetry-broken quantum spin Hall effect
  Quantum spin Hall (QSH) state of matter is usually considered to be protected by time-reversal (TR) symmetry. We investigate the fate of the QSH effect in the presence of the Rashba spin-orbit coupling and an exchange field, which break both inversion and TR symmetries. It is found that the QSH state characterized by nonzero spin Chern numbers $C_{\pm}=\pm 1$ persists when the TR symmetry is broken. A topological phase transition from the TR symmetry-broken QSH phase to a quantum anomalous Hall phase occurs at a critical exchange field, where the bulk band gap just closes. It is also shown that the transition from the TR symmetry-broken QSH phase to an ordinary insulator state can not happen without closing the band gap. 
Yunyou Yang, Zhong Xu, L. Sheng, Baigeng Wang, D. Y. Xing, and D. N.   Sheng
cond-mat.mes-hall cond-mat.other