Unifying semiclassics and quantum perturbation theory at nonlinear order

Nonlinear electrical response permits a unique window into effects of band structure geometry. It can be calculated either starting from Boltzmann approach for small frequencies, or using Kubo's formula resonances at finite frequency. However, precise connection between both approaches has not been established. Focusing on the second order nonlinear response, here we show how semiclassical limit recovered perturbation theory in velocity gauge, provided that quasiparticle lifetimes are taken account. We find matrix elements related to geometry combine this produce conductivity. demonstrate power new formalism by deriving quantum contribution conductivity which is $\tau^{-1}$ relaxation time $\tau$, principally inaccessible within approach. outline steps generalized higher orders applied perturbation, and comment about potential experimental signatures our results.