# Is Hall conductivity time-reversal-odd at finite frequency in a topological system?

In some topological materials, e.g., the quantum (anomalous) Hall state and some related variants, the Hall conductivity $$\sigma_{xy}$$ is quantized and directly related to the Chern number, which corresponds to the number of chiral edge modes. In this sense, $$\sigma_{xy}$$ must be time-reversal-odd.

But this is the $$\omega\rightarrow0$$ limit. Does the time-reversal-oddness of $$\sigma_{xy}$$ still hold at finite frequency $$\omega\neq0$$?