Electrical conductivity $\sigma(q,\omega)$ can be frequency and momentum dependent in general for electric fields with a spatiotemporal variation. Experimentally, is it possible to measure such a quantity with both finite $q$ and $\omega$ on the surface of some material?
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It depends on the specific situation: material, geometry, how the current is driven, etc.
E.g., ac response (i.e., $\sigma(\omega)$) is routinely measured by applying time-dependent bias of frequency $\omega$ and measuring the response: $$\mathbf{j}(\omega)=\sigma(\omega)\mathbf{E}(\omega)\leftrightarrow I(\omega)=G(\omega)V(\omega) $$
Measuring spatial dependence might be trickier, since it requires excitation and sensors for controlling and detecting spatial variation. In surface layers of materials this could be achieved, e.g., via excitation by electromagnetic waves.
Note also that calculating $\sigma(\omega,\mathbf{k})$ becomes meaningless at length scales shorter than the coherence length, where current reponse is a non-local and requires quantum treatment see this post and the references in it.)
Roger V.
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