Extraction of the frequency moments of spectral densities from imaginary-time correlation function data

Abstract

We introduce an exact framework to compute the positive frequency moments $M^{(\alpha )}(\mathbf{q})=⟨{\omega }^{\alpha }⟩$ of different dynamic properties from imaginary-time quantum Monte Carlo data. As a practical example, we obtain the first five moments of the dynamic structure factor $S(\mathbf{q},\omega )$ of the uniform electron gas at the electronic Fermi temperature based on \emph{ab initio} path integral Monte Carlo simulations. We find excellent agreement with known sum rules for $\alpha =1,3$, and, to our knowledge, present the first results for $\alpha =2,4,5$. Our idea can be straightforwardly generalized to other dynamic properties such as the single-particle spectral function $A(\mathbf{q},\omega )$, and will be useful for a number of applications, including the study of ultracold atoms, exotic warm dense matter, and condensed matter systems.

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