Antiferromagnetic Quantum Spin Hall and Quantum Anomalous Hall Effect

The Quantum Spin Hall Effect (QSHE) and Quantum Anomalous Hall Effect (QAHE) are typically experimentally realized in topological insulators and magnetic topological insulators with ferromagnetic order. The ferromagnetic order breaks the time-reversal symmetry (TRS) and opens a nontrivial gap in the magnetic topological insulator for QAHE, characterized by the quantized Chern number. On the other hand, the TRS in topological insulators is preserved for the QSHE, characterized by the Z2 number. Here, we extend the Kane-Mele model to include an antiferromagnetic order that breaks the TRS. The system is found to exhibit the QAH phase with a robust edge state and, remarkably, a QSHE phase with a vanishing total Chern number. Our model realizes both the TRS-broken QSHE and QAHE within an antiferromagnetic honeycomb lattice, providing the insights for exploring the potential of antiferromagnetic topological materials.