The study of matter at extreme densities and temperatures as they occur in astrophysical objects and state-of-the-art experiments with high-intensity lasers is of high current interest for many applications. While no overarching theory for this regime exists, accurate data for the density response of correlated electrons to an external perturbation are of paramount importance. In this context, the key quantity is given by the local field correction (LFC), which provides a wave-vector resolved description of exchange-correlation effects. In this work, we present extensive new path integral Monte Carlo (PIMC) results for the static LFC of the uniform electron gas, which are subsequently used to train a fully connected deep neural network. This allows us to present a representation of the LFC with respect to continuous wave-vectors, densities, and temperatures covering the entire warm dense matter regime. Both the PIMC data and neural-net results are available online. Moreover, we expect the presented combination of ab initio calculations with machine-learning methods to be a promising strategy for many applications.
We assume Hartree atomic units throughout this work.
We note that while the RPA provides a description of the density response of the system on a mean field level, it gives a nonvanishing correlation energy when inserted into the fluctuation dissipation theorem and the adiabatic connection formula (see, e.g., Ref. 98) and approaches the exact limit for rs = 0.
This is a link to both the ML representation of the static LFC and the PIMC raw data. https://github.com/ToDor90/LFC
This is, among other things, due to the softening of the Fermi surface at finite temperature and will be discussed in a separate publication.
For completeness, we note that the inclusion of the exact large-q limit at finite temperature, which is not expected to improve the quality of our representation, is the specified range of validity (0 < q < 5qF), but would potentially allow to extend it to q → ∞.