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The Callan-Harvey mechanism in 2+1 D Jackiw-Rebbi model is revisited. We analyzed Callan-Harvey anomaly inflow in the massive Chern insulator (quantum anomalous Hall system) subject to external electric field. In addition to the conventional current flowing from the bulk to edge due to parity anomaly, we considered the dissipation of the edge charge due to interaction with external bosonic bath in 2+1 D and due to external bath of photons in 3+1 D. In the case of 2+1 D bosonic bath, we found the new stationary state, which is defined by the balance between Callan-Harvey current and the outgoing flow caused by the dissipation processes. In the case of 3+1 D photon bath, we found a critical electric field, below which this balance state can be achieved, but above which there is no such a balance. Furthermore, we estimated the photon-mediated transition rate between 2+1 D bulk and 1+1 D topological edge state of the order of one ns$^{-1}$ (nanosecond) at the room temperature.

The low-energy physics of two-dimensional Quantum Anomalous Hall insulators like (Hg,Mn)Te quantum wells or magnetically doped (Bi,Sb)Te thin films can be effectively described by two Chern insulators, including a Dirac, as well as a momentum-dependent mass term. Each of those Chern insulators is directly related to the parity anomaly of planar quantum electrodynamics. In this work, we analyze the finite temperature Hall conductivity of a single Chern insulator in 2+1 space-time dimensions under the influence of a chemical potential and an out-of-plane magnetic field. At zero magnetic field, this non-dissipative transport coefficient originates from the parity anomaly of planar quantum electrodynamics. We show that the parity anomaly itself is not renormalized by finite temperature effects. However, it induces two terms of different physical origin in the effective action of a Chern insulator, which is proportional to the Hall conductivity. The first term is temperature and chemical potential independent, and solely encodes the intrinsic topological response. The second term specifies the non-topological thermal response of conduction and valence band states. In particular, we show that the relativistic mass of a Chern insulator counteracts finite temperature effects, whereas its non-relativistic mass enhances these corrections. Moreover, we extend our analysis to finite magnetic fields and relate the thermal response of a Chern insulator therein to the spectral asymmetry, which is a measure of the parity anomaly in orbital fields.