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We perform ab initio quantum Monte Carlo (QMC) simulations of the warm dense uniform electron gas in the thermodynamic limit. By combining QMC data with linear response theory we are able to remove finite-size errors from the potential energy over the entire warm dense regime, overcoming the deficiencies of the existing finite-size corrections by Brown et al. [PRL 110, 146405 (2013)]. Extensive new QMC results for up to N = 1000 electrons enable us to compute the potential energy V and the exchange-correlation free energy Fxc of the macroscopic electron gas with an unprecedented accuracy of |∆V |/|V |, |∆Fxc|/|F |xc ∼ 10 −3 . A comparison of our new data to the recent parametrization of Fxc by Karasiev et al. [PRL 112, 076403 (2014)] reveals significant deviations to the latter.The uniform electron gas (UEG), consisting of electrons on a uniform neutralizing background, is one of the most important model systems in physics [1]. Besides being a simple model for metals, the UEG has been central to the development of linear response theory and more sophisticated perturbative treatments of solids, the formulation of the concepts of quasiparticles and elementary excitations, and the remarkable successes of density functional theory.The practical application of ground-state density functional theory in condensed matter physics, chemistry and materials science rests on a reliable parametrization of the exchange-correlation energy of the UEG [2], which in turn is based on accurate quantum Monte Carlo (QMC) simulation data [3]. However, the charged quantum matter in astrophysical systems such as planet cores and white dwarf atmospheres [4,5] is at temperatures way above the ground state, as are inertial confinement fusion targets [6][7][8], laser-excited solids [9], and pressure induced modifications of solids, such as insulator-metal transitions [10,11]. This unusual regime, in which strong ionic correlations coexist with electronic quantum effects and partial ionization, has been termed "warm dense matter" and is one of the most active frontiers in plasma physics and materials science.The warm dense regime is characterized by the existence of two comparable length scales and two comparable energy scales. The length scales are the mean interparticle distance,r, and the Bohr radius, a 0 ; the energy scales are the thermal energy, k B T , and the electronic Fermi energy, E F . The dimensionless parameters [12] r s =r/a 0 and Θ = k B T /E F are of order unity. Because Θ ∼ 1, the use of ground-state density functional theory is inappropriate and extensions to finite T are indispensible; these require accurate exchange-correlation functionals for finite temperatures [13][14][15][16][17]. Because neither r s nor Θ is small, there are no small parameters, and weakcoupling expansions beyond Hartree-Fock such as the Montroll-Ward (MW) and e 4 (e4) approximations [18,19], as well as linear response theory within the random- phase approximation (RPA) break down [20,21]. Finite-T Singwi-Tosi-Land-Sjölander (STLS) [22,23] local-fiel...

In a recent Letter [T. Dornheim et al., Phys. Rev. Lett. 117, 156403 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.156403], we presented the first quantum Monte Carlo (QMC) results for the warm dense electron gas in the thermodynamic limit. However, a complete parametrization of the exchange-correlation free energy with respect to density, temperature, and spin polarization remained out of reach due to the absence of (i) accurate QMC results below θ=k_{B}T/E_{F}=0.5 and (ii) QMC results for spin polarizations different from the paramagnetic case. Here we overcome both remaining limitations. By closing the gap to the ground state and by performing extensive QMC simulations for different spin polarizations, we are able to obtain the first completely ab initio exchange-correlation free energy functional; the accuracy achieved is an unprecedented ∼0.3%. This also allows us to quantify the accuracy and systematic errors of various previous approximate functionals.

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 continuous representation of the LFC with respect to wave-vector, density, and temperature 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 machinelearning methods to be a promising strategy for many applications.

The uniform electron gas (UEG) at finite temperature is of key relevance for many applications in dense plasmas, warm dense matter, laser excited solids and much more. Accurate thermodynamic data for the UEG are an essential ingredient for many-body theories, in particular, density functional theory. Recently, first-principle restricted path integral Monte Carlo results became available which, however, due to the fermion sign problem, had to be restricted to moderate degeneracy, i.e. low to moderate densities with rs =r/aB 1. Here we present novel first-principle configuration PIMC results for electrons for rs ≤ 1. We also present quantum statistical data within the e 4 -approximation that are in good agreement with the simulations at small to moderate rs. [11,12]. Besides, the electron component is of crucial importance for understanding the properties of atoms, molecules and existing and novel materials. The most successful approach has been density functional theory (DFT)-combined with an approximation for the exchange-correlation potential. Its success is based on the availability of accurate zero temperature data for the UEG which is obtained from analytically known limiting cases combined with first-principle quantum Monte Carlo data [13].In recent years more and more applications have emerged where the electrons are highly excited, e.g. by compression of the material or by electromagnetic radiation (see above), which require to go beyond zero temperature DFT. This has led to an urgent need for accurate thermodynamic data of the UEG at finite temperature. One known limiting case is the highly degenerate ideal Fermi gas (IFG), and perturbation theory results around the IFG, starting with the Hartree-Fock and first order correlation corrections (Montroll-Ward) [14,15] [27]. It is well known that fermionic PIMC simulation in continuous space suffer from the fermion sign problem (FSP) which is known to be NP hard [28]. This means, with increasing quantum degeneracy, i.e. increasing parameter χ = nλ 3 DB , which is the product of density and thermal DeBroglie wave length, λ, the simulations suffer an exponential loss of accuracy. RPIMC formally avoids the FSP by an additional assumption on the nodes of the density matrix, however, it also cannot access high densities [29], r s < 1 [r s =r/a B , wherer is the mean interparticle distance, n −1 = 4πr 3 /3 and a B the Bohr radius]. Also, the quality of the simulations around r s = 1, at low temperatures Θ = k B T /E F ≤ 0.125 [E F is the Fermi energy] is unknown. However, this leaves out the high-density range that is of high importance, e.g. for deuterium-tritium implosions at NIF where mass densities of 400 gcm −3 (up to 1596 gcm −3 ) have recently been reported [9] (are expected along the implosion path [8]), corresponding to r s ≈ 0.24 (r s = 0.15), see Fig. 1.The authors of Ref.[27] also performed DPIMC simulations which confirmed that, for Θ < 0.5 and r s 4, these simulations are practically not possible, see Fig. 1. We also mention independent recent DPIMC ...