Fate of Quantum Anomalous Hall Effect in the Presence of External Magnetic Fields and Particle-Hole Asymmetry

The quantum anomalous Hall (QAH) effect, a condensed matter analog of the parity anomaly, is characterized by a quantized Hall conductivity in the absence of an external magnetic field. However, it has been recently shown that, even in the presence of Landau levels, the QAH effect can be distinguished from the conventional quantum Hall (QH) effect due to the parity anomaly. As a signature of this effect, we predicted coexistent counterpropagating QAH and QH edge states. In the present work, we generalize these findings to QAH insulators with broken particle-hole symmetry. In particular, we derive the connection to the spectral asymmetry, which is a topological quantity arising in the context of Dirac-like systems. Moreover, it is shown that, depending on the magnetic field direction, particle-hole asymmetry strengthens or weakens the hybridization of the counterpropagating QH and QAH edge states. Implications for ferro- or paramagnetic topological insulators are derived which paves the way towards identifying signatures of the QAH effect even in external magnetic fields.