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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 $\omega\neq0$?

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