  We present extensive new ab initio path integral Monte Carlo (PIMC) results
for an electron gas at warm dense matter conditions that is subject to multiple
harmonic perturbations. In addition to the previously investigated nonlinear
effects at the original wave number [Dornheim \emph{et al.}, PRL \textbf{125},
085001 (2020)] and the excitation of higher harmonics [Dornheim \emph{et al.},
PRR \textbf{3}, 033231 (2021)], the presence of multiple external potentials
leads to mode-coupling effects, which constitute the dominant nonlinear effect
and lead to a substantially more complicated density response compared to
linear response theory. One possibility to estimate mode-coupling effects from
a PIMC simulation of the unperturbed system is given in terms of generalized
imaginary-time correlation functions that have been recently introduced by
Dornheim \emph{et al.}~[JCP \textbf{155}, 054110 (2021)]. In addition, we
extend our previous analytical theory of the nonlinear density response of the
electron gas in terms of the static local field correction [Dornheim \emph{et
al.}, PRL \textbf{125}, 235001 (2020)], which allows for a highly accurate
description of the PIMC results with negligible computational cost.
