Abstract
Recent developments in solid state physics give a prospect to observe the parity anomaly in (2+1)D massive Dirac systems. Here we show, that the quantum anomalous Hall (QAH) state in orbital magnetic fields originates from the Dirac mass term and induces an anomalous four-current related to the parity anomaly. This differentiates the QAH from the quantum Hall (QH) state for the experimentally relevant case of an effective constant density (seen by the gate). A direct signature of QAH phase in magnetic fields is a long σ_xy= e²/h (σ_xy= -e²/h) plateau in Cr_x(Bi_1-ySb_y)_2-xTe₃ (HgMnTe quantum wells). Furthermore, we predict a new transition between the quantum spin Hall (QSH) and the QAH state in magnetic fields, for constant effective carrier density, without magnetic impurities but driven by effective g-factors and particle-hole asymmetry. This transition can be related to the stability of edge states in the Dirac mass gap of 2D topological insulators (TIs), even in high magnetic fields.