  We consider the generic problem of a two-level quantum emitter near a
two-dimensional Chern insulator in the dipole approximation, and study how the
frequency-dependent response and electronic density of states of the insulator
modifies the transition rate of the emitter between the ground and excited
levels. To this end, we obtain the full real-frequency behavior of the
conductivity tensor by performing a tight-binding calculation based on the
Qi-Wu-Zhang model and using a Kubo formula, and derive the full electromagnetic
Green tensor of the system, which breaks Onsager reciprocity. This enables us
to find that for frequencies smaller than the maximum band gap, the system is
sensitive to time reversal symmetry-breaking, whereas for much larger
frequencies the system becomes insensitive, with implications for the
discrimination of the state of a circularly polarised dipole emitter. We also
study the impact of a van Hove singularity on the surface-induced correction to
the transition rate, finding that it can enhance its amplitude by a few orders
of magnitude compared to the case where the conductivity is set to its static
value. By considering configurations in which the dipole is circularly
polarised or parallel with the surface of the Chern insulator, we find that the
surface correction to the transition rate can exhibit a novel decay with sine
integral-like oscillations.
