Extraction of the frequency moments of spectral densities from imaginary-time correlation function data
Tobias Dornheim, Damar Wicaksono, Juan E. Suarez-Cardona, P. Tolias, Maximilian Böhme, Zhandos A. Moldabekov, Michael Hecht, Jan Vorberger
Abstract
We introduce an exact framework to compute the positive frequency moments ${M}^{(\ensuremath{\alpha})}(\mathbf{q})=\ensuremath{\langle}{\ensuremath{\omega}}^{\ensuremath{\alpha}}\ensuremath{\rangle}$ 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 $S(\mathbf{q},\ensuremath{\omega})$ 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 $\ensuremath{\alpha}=1,3$, and present results for $\ensuremath{\alpha}=2,4,5$. Our idea can be straightforwardly generalized to other dynamic properties such as the single-particle spectral function $A(\mathbf{q},\ensuremath{\omega})$, and will be useful for a number of applications, including the study of ultracold atoms, exotic warm dense matter, and condensed matter systems.