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±=±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.
@article{arxiv.1103.4473,
title = {Time-reversal-symmetry-broken quantum spin Hall effect},
author = {Yunyou Yang and Zhong Xu and L. Sheng and Baigeng Wang and D. Y. Xing and D. N. Sheng},
journal= {arXiv preprint arXiv:1103.4473},
year = {2011}
}