Unifying semiclassics and quantum perturbation theory at nonlinear order

TL;DR

This paper establishes a precise connection between semiclassical and quantum perturbation theories for nonlinear electrical response, revealing new quantum contributions and providing a unified framework for analyzing band structure effects.

Contribution

It introduces a formalism that links semiclassical and quantum approaches for second order nonlinear response, including quantum effects beyond Boltzmann theory.

Findings

Derived a quantum contribution to nonlinear conductivity of order τ^{-1}

Showed how band geometry matrix elements produce semiclassical nonlinear conductivity

Outlined potential experimental signatures of the quantum effects

Abstract

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