Nonlinear terahertz electro-optical responses in centrosymmetric electronic systems

TL;DR

This paper develops a rigorous field theoretical framework to compute finite frequency nonlinear electro-optical responses in centrosymmetric electronic systems, with applications to terahertz spectroscopy and correlated materials.

Contribution

It introduces a formalism linking response functions to correlation functions, deriving sum rules and constraints, and applies it to noninteracting electrons with disorder, extending to symmetry broken phases.

Findings

Derived exact mapping between response and correlation functions.

Proved a generalized $f$-sum rule for nonlinear conductivity.

Computed gauge invariant nonlinear conductivity for disordered electrons.

Abstract

Motivated by the recent developments in terahertz spectroscopy using pump-probe setups to study correlated electronic materials, we review the field theoretical formalism to compute finite frequency nonlinear electro-optical responses in centrosymmetric systems starting from basic time dependent perturbation theory. We express the nonlinear current kernel as a sum of several causal response functions. These causal functions cannot be evaluated using perturbative field theory methods, since they are not contour ordered. Consequently, we associate each response function with a corresponding imaginary time ordered current correlation function, since the latter can be factorized using Wick's theorem. The mapping between the response functions and the correlation functions, suitably analytically continued to real frequencies, is proven exactly. We derive constraints satisfied by the nonlinear current kernel and we prove a generalized $f$-sum rule for the nonlinear conductivity, all of which are consequences of particle number conservation. The constraints guarantee that the nonlinear static responses are free from spurious divergences. We apply the theory to compute the gauge invariant nonlinear conductivity of a system of noninteracting electrons in the presence of weak disorder. As special cases of this generalized nonlinear response, we discuss its third harmonic and its instantaneous terahertz Kerr signals. The formalism can be used to compute the nonlinear conductivity in symmetry broken phases of electronic systems such as superconductors, density waves and nematic states.