Imaginary Time Formalism for Causal Nonlinear Response Functions

TL;DR

This paper rigorously establishes the connection between imaginary time-ordered Matsubara functions and causal nonlinear response functions, extending the linear response framework to nonlinear regimes through an all-orders proof and explicit representations.

Contribution

It provides a comprehensive proof linking Matsubara and causal nonlinear response functions at all perturbation orders, including explicit Lehmann representations and spectral density formulas.

Findings

Proved the all-orders connection between Matsubara and causal nonlinear responses.

Derived explicit Lehmann and spectral density representations for nonlinear response functions.

Established generalized sum rules and asymptotic expressions for harmonic generation.

Abstract

It is well established that causal linear response functions can be found by computing the much simpler imaginary time-ordered Matsubara functions and performing an analytic continuation. This principle is the basis for much of our understanding of linear response for interacting and disordered systems, via diagrammatic perturbation theory. Similar imaginary-time approaches have recently been introduced for computing nonlinear response functions as well, although the rigorous connection between Matsubara and causal nonlinear response functions has not been clearly elucidated. In this work, we provide a proof of this connection to all orders in perturbation theory. Using an equations of motion approach, we show by induction that casual nonlinear response functions at every order can be obtained from an analytic continuation of an appropriate time-ordered Matsubara function. We demonstrate this connection explicitly for second order response functions in the Lehmann representation. As a byproduct of our approach, we derive an explicit expression for the Lehmann representation of $n$-th order response functions by solving the equations of motion. We also use our result to find an analytic spectral density representation for both causal response functions and Matsubara functions. Finally, we show how our results lead to a family of generalized sum rules, focusing explicitly on the asymptotic expression for $n$-th harmonic generation rate.