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Theory of nonlinear optical response

We present a general formalism for investigating the second-order optical response of solids to an electric field in weakly disordered crystals with arbitrarily complicated band structures based on density-matrix equations of motion, on a Born approximation treatment of disorder, and on an expansion in scattering rate to leading non-trivial order. One of the principal aims of our work is to enable extensive transport theory applications that accounts fully for the interplay between electric-field-induced interband and intraband coherence, and Bloch-state scattering. The quasiparticle bands are treated in a completely general manner that allows for arbitrary forms of the intrinsic spin-orbit coupling (SOC) and could be extended to the extrinsic SOC. According to the previous results, in the presence of the disorder potential, the interband response in conductors in addition to an intrinsic contribution due to the entire Fermi sea that captures, among other effects, the Berry curvature contribution to wave-packet dynamics includes an anomalous contribution caused by scattering that is sensitive to the presence of the Fermi surface. To demonstrate the rich physics captured by our theory, the relaxation time matrix for different strength order is considered and at the same time we explicitly solve for some electric-field response properties of simple disordered Rashba model that are known to be dominated by interband coherence contributions. The expressions we present are amenable for numerical calculations, and we demonstrate this by performing a full band-structure calculation of the interband contribution, even in metals.