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

We introduce an exact framework to compute the positive frequency moments M (α)(q) = 〈ωα〉
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(q, ω) of the uniform
electron gas at the electronic Fermi temperature based on ab initio path integral Monte Carlo
simulations. We find excellent agreement with known sum rules for α = 1, 3, and, to our knowledge,
present the first results for α = 2, 4, 5. Our idea can be straightforwardly generalized to other
dynamic properties such as the single-particle spectral function A(q, ω), and will be useful for a
number of applications, including the study of ultracold atoms, exotic warm dense matter, and
condensed matter systems.