Physical insights from imaginary-time density--density correlation functions

The accurate theoretical description of the dynamic properties of correlated quantum many-body systems such as the dynamic structure factor $S(\mathbf{q},\omega)$ constitutes an important task in many fields. Unfortunately, highly accurate quantum Monte Carlo methods are usually restricted to the imaginary time domain, and the analytic continuation of the imaginary time density--density correlation function $F(\mathbf{q},\tau)$ to real frequencies is a notoriously hard problem. In this work, we argue that no such analytic continuation is required as $F(\mathbf{q},\tau)$ contains, by definition, the same physical information as $S(\mathbf{q},\omega)$, only in an unfamiliar representation. Specifically, we show how we can directly extract key information such as the temperature or quasi-particle excitation energies from the $\tau$-domain, which is highly relevant for equation-of-state measurements of matter under extreme conditions. As a practical example, we consider \emph{ab initio} path integral Monte Carlo results for the uniform electron gas (UEG), and demonstrate that even nontrivial processes such as the \emph{roton feature} of the UEG at low density straightforwardly manifest in $F(\mathbf{q},\tau)$. In fact, directly working in the $\tau$-domain is advantageous for many reasons and holds the enticing promise for unprecedented agreement between theory and experiment.