Gauge invariance and Ward identities in nonlinear response theory

TL;DR

This paper develops a formal framework for analyzing nonlinear response functions using correlation functions, extending linear response theory, and derives gauge-invariant relations between second-order responses in different electromagnetic gauges.

Contribution

It generalizes linear response theory to nonlinear regimes and establishes exact gauge-invariant relations between second-order response functions.

Findings

Causal nonlinear response functions can be obtained from analytic continuation of time-ordered functions.

Derived gauge invariance relations between density and current response functions.

Established non-perturbative identities connecting nonlinear optics calculations in different gauges.

Abstract

We present a formal analysis of nonlinear response functions in terms of correlation functions in real- and imaginary-time domains. In particular, we show that causal nonlinear response functions, expressed in terms of nested commutators in real time, can be obtained from the analytic continuation of time-ordered response functions, which are more easily amenable to diagrammatic calculation. This generalizes the well-known result of linear response theory. We then use gauge invariance arguments to derive exact relations between second-order response functions in density and current channels. These identities, which are non-perturbative in the strength of inter-particle interactions, allow us to establish exact connections between nonlinear optics calculations done in different electromagnetic gauges.