If there exist electromagnetic fields in solids, electrons can feel interactions like $\vec{j} \cdot \vec{A}$ or $\rho A^2$ (these are not regarded as perturbations). But these are not gauge invariant. Then we add some other perturbation fields to study their corresponding linear response. I can't see whether brute-force Kubo formula evaluation gives gauge invariant result.

This post imported from StackExchange Physics at 2014-08-22 05:09 (UCT), posted by SE-user huotuichang