Session TO06: HED: Warm Dense Matter Theory

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Here we use a fully nonlinear hydrodynamic simulation framework, SPARC, to model ablation of metal surfaces by \textasciitilde ps length, \textasciitilde mJ Class lasers. SPARC is a set of modules within a larger, TurboWAVE code, with expanded Equation-of-State (EOS) capabilities that include solid, liquid, and gaseous states. In problems such as these the laser deposits energy into the conduction band electrons at a much faster rate than the electron and ion temperatures equilibrate, and so a two-temperature model (TTM) is necessary to resolve the physics. The rapid heating of the electrons result in thermionic emission, followed by an electrostatic pull which initiates the ablation process before the ion temperature reaches the melting point. Our choice of EOS for both species will be discussed and elaborated. Special consideration is given to the fact that these electrons are neither a degenerate gas or an ideal plasma, but rather exist in the Warm Dense Matter (WDM) regime. The results of the simulations are benchmarked against experimental results, from which we can extrapolate some of the behaviors of EOS quantities in this intermediate regime while preserving the necessary limiting behaviors.   

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Shock and quasi-isentropic compression of solid-state matter via laser-ablation affords the creation of high energy density states of matter, with pressures and temperatures of relevance to core conditions within planets in our own solar system and beyond. Crystallographic phase and density can be discerned via ultra-fast x-ray diffraction, whilst pressure is deduced from VISAR measurements. Temperature is more difficult to determine, but techniques based on inelastic scattering from phonons are being considered [1]. It is in this context that we present here multi-million atom molecular dynamics simulations of the phonons present in fcc crystals shocked beyond their elastic limit. Despite high dislocation densities behind the shock front, distinct phonon modes can still easily be discerned, though such defects do contribute to the quasi-elastic peak that will compete with any inelastic scattering signal in a real experiment. Changes in the dispersion curves due to compression and the high number of stacking faults can also be observed. [1] E.E. McBride et al., Rev. Sci. Instrum. 89, 10F104 (2018)   

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The computational demands of modelling large numbers of coupled electrons and ions have long been considered insurmountable, despite advances and refinements in density functional theory (DFT) calculations. However, a different approach to modelling quantum interactions, via application of the Bohmian trajectories formalism, can overcome this hurdle. We present further results from a new Bohm - molecular dynamics approach (Bohm MD). The static results of our simulations are validated by DFT results -- our static ion-ion structure factor of aluminium at 5.2 g cm$^{\mathrm{-3}}$ and 3.5 eV shows excellent agreement with both orbital free and Kohn Sham DFT. We then use Bohm MD to extract dynamic results, not only the ion-ion dynamic structure factor which provides a direct link to experimental observables, but also, unprecedentedly, the ion-electron and electron-electron dynamic structure factors. Thus Bohm MD provides a self-consistent approach to non-adiabatic investigation of dynamic modes in systems of thousands of particles.   

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The dielectric function calculated within the Random Phase Approximation (RPA) has been applied in many studies to determine the response properties of plasmas, like dynamic structure factors and stopping powers. However, the RPA does not take into account short range electron-electron and electron-ion interactions which can be important in the warm dense matter regime. To go beyond the RPA, we modify the dielectric function by incorporating electron-ion collisions and electronic correlations. The collisions are calculated self-consistently using an average-atom model, while the correlations come from the static local field correction of the uniform electron gas, which was computed by using a path integral Monte Carlo based neural network [Dornheim et al., J. Chem. Phys. 151, 194104 (2019)]. We show how the modified dielectric function changes both dynamic structure factors and stopping powers for charged particles traveling through a plasma and compare with results from experimental data and other theoretical models.   

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Quantum Monte Carlo is the most accurate method for simulating the homogeneous electron gas under warm dense matter conditions [T. Dornheim et al., Phys. Reports 744, 1-86 (2018)] However, due to the notorious fermion sign problem, previous ab initio approaches were restricted to temperatures above half the Fermi temperature, i.e. $\Theta=k_BT/E_F \ge 0.5$. Also, the method of choice for the degenerate Fermi gas -- configuration path integral Monte Carlo (CPIMC) [T. Schoof et al., Contrib. Plasma Phys. 51, 687 (2011)] -- is restricted to high density, i.e. $r_s=\bar r/a_B \le 1$. Here, we construct two new approximations -- RCPIMC and RCPIMC+ -- that neglect some classes of Monte Carlo updates. While RCPIMC+ reduces the sign problem, RPCIMC completely eliminates it, at the price of a systematic error. We investigate the magnitude of the errors by comparing new finite size corrected [T. Dornheim et al., Phys. Rev. Lett. 117, 156403 (2016)] simulations to the parametrization [S. Groth et al., Phys. Rev. Lett. 115, 135001(2017)]. As a result we conclude that RCPIMC+ allows for accurate simulations of thermodynamic properties (deviations of less than $1\%$) at least up to $r_s = 3$ and $0.05 \le \Theta \le 0.3$, significantly extending the range of CPIMC.   

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Quantum Monte Carlo belongs to the most accurate simulation techniques. For Fermi systems, however, its applicability at strong degeneracy is limited by the fermion sign problem. Recently, a significant progress has been achieved for uniform electron gas with configuration path integral Monte Carlo (CPIMC) and permutation blocking (PB-PIMC)~[1]. Both methods are free from uncontrolled errors introduced by the fixed nodes approximations. Here we develop a generalization of the PB-PIMC~[1] suitable for the grandcanonical ensemble and, in combination with the improved Kelbg potential~[2], perform simulations for hydrogen and helium plasmas down to temperatures 3$\cdot 10^{4}$~K. The obtained isotherms of pressure and internal energy are compared with the restricted PIMC~[3] allowing us to conclude on the accuracy of the fixed-node approximation at weak and strong degeneracy, and how the bound state formation affects the results. Some thermodynamic properties are compared with finite temperature DFT simulations~[4]. [1] T.~Dornheim \textit{et al.}, Phys.Rep. {\bf 744}, 1-86(2018); A.~Filinov \textit{et al.}, Phys.Rev.E {\bf 70}, 046411 (2004); [3] B.~Milizer \textit{et al.}, Phys.Rev.B {\bf 84}, 224109 (2011); [4] D.~Knyazev and, P.~Levashov, Phys. Plasmas {\bf 23}, 102708 (2016).   

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The Nonequilibrium Green Functions (NEGF) method is a powerful tool to compute time-dependent expectation values of single-particle observables in correlated quantum many-body systems. Its unfavorable $N_t^3$-scaling with propagation time $N_t$ could be reduced to $N_t^2$ by introduction of the Generalized Kadanoff--Baym Ansatz (GKBA)[1]. Recently, an exact time-local ($N_t^1$) reformulation of the GKBA, the G1--G2 scheme [2,3], has been found for various self energies, which makes this method viable for long time simulations.\\ In a general basis the G1--G2 scheme has a computationally expensive scaling with basis size ($N_b^5$-$N_b^6$). For the uniform electron gas (UEG) however, we found an advantageous $N_b^3 N_t^1$ scaling for both second-order and GW selfenergies, which makes this scheme particularly interesting for this system. Here, we present first relaxation results in 1 and 2 dimensions.\\ \quad [1] P. Lipavsk\'y, V. \v{S}pi\v{c}ka, Velick\'y, B, \textit{Phys. Rev. B}34, 6933 (1986)\\ \quad [2] N. Schl\"unzen, Jan-Philip Joost, Michael Bonitz, \textit{Phys. Rev. Lett.} 124, 076601 (2020)\\ \quad [3] M. Bonitz, \textit{Quantum Kinetic Theory} (Springer, 2016)   

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In warm dense matter, many of the properties are strongly influenced by the charge states of the ions. This is often taken into account by way of the mean ionization or the average charge state populations. However, even in steady state, the number of electrons remaining bound to a particular ion is not constant. Stochastic collisional ionisation and recombination will cause the charge states of ions to fluctuate around their mean. In this work, we use a random walk model to investigate the properties of charge state fluctuations in warm dense matter. A strong dependence on the atomic shell structure is predicted. We also examine the possibility that charge state fluctuations contribute to electron-ion temperature equilibration by interacting with ion density fluctuations. Despite its importance, temperature equilibration remains poorly understood, with many experimental results yielding disagreements with theory. Preliminary calculations suggest that charge state fluctuations could be a significant factor in understanding temperature relaxation.   

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Dense plasmas can be subject to electron degeneracy, strong Coulomb coupling, and varying degrees of partial ionization, which makes them difficult to model. We present a method for predicting dense plasma transport using the quantum Boltzmann equation of Uehling and Uhlenbeck, in which the scattering potential is the potential of mean force which is determined by the equilibrium state of the plasma. The dynamics are therefore reduced to those of a binary collision, whereas the potential of mean force can be calculated by any suitable equilibrium method. The method is thus faster than fully dynamical simulations while still containing much of the relevant physics. We apply the method to the calculation of electrical conductivity in dense plasmas; specifically compressed hydrogen and solid density aluminum, each over a range of temperatures spanning above and below the Fermi temperature. Results are compared to alternative models in addition to quantum molecular dynamics simulations in order to validate the model. This work was supported by the U.S. Department of Energy, Office of Fusion Energy Sciences under Award Number DE-SC0016159.   

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First-principles methods based on orbital-dependent and orbital-free density functional theory take into account the electron quantum effects and provide a compromise between reliability and computational efficiency for simulations of matter across extreme conditions of pressure and temperature in the warm-dense matter (WDM) regime. With that, the standard molecular dynamics approach treats ions classically within the Born--Oppenheimer approximation, omitting nuclear quantum effects (NQE’s). The NQE’s at high pressure are not negligible in a wide range of temperatures and must be taken into account for accurate predictions. In this talk we will discuss recent progress in the development of meta-generalized gradient approximation (meta-GGA) exchange- correlation functional enhanced by the GGA-level thermal corrections providing improved accuracy across the temperature regimes. Together with quantum treatment of ions via path integral molecular dynamics, our approach provides a systematically improved accuracy of WDM simulations, as we show in an example of dense hydrogen and deuterium plasmas.   

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Understanding many processes depends strongly on microscopic physics modeling of warm dense matter (WDM) and hot dense plasma. This regime challenges both experiment and analytical modeling, necessitating predictive \emph{ab initio} atomistic computation, typically based on quantum mechanical Kohn-Sham Density Functional Theory (KS-DFT). However, cubic computational scaling with temperature and system size prohibits the use of DFT through much of the WDM regime. A recently-developed stochastic approach to KS-DFT can be used at high temperatures, with the exact same accuracy as the deterministic approach, but the stochastic error can converge slowly and it remains expensive for intermediate temperatures. We have developed a universal mixed stochastic-deterministic algorithm for DFT at any temperature. This approach leverages the physics of KS-DFT to seamlessly integrate the best aspects of these different approaches. We demonstrate that this method significantly accelerated self-consistent field calculations for temperatures from 3 to 50 eV, while producing stable molecular dynamics and accurate diffusion coefficients. LA-UR-20-22738   

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Recently, it has been shown that explicit dependence on temperature $T$ in the exchange-correlation (XC) free-energy density functional is important in density functional theory studies of warm dense matter. As a first approximation, the finite-$T$, non-empirical KSDT local spin-density approximation (LSDA) functional was constructed by analytical parametrization of the XC free energy of the homogeneous electron gas. Consequently, the KDT16 generalized gradient approximation (GGA) functional, which captures non-homogeneity effects and shows better agreement with experiments, was constructed. Here, we present progress in climbing the (finite-$T$) Jacob's ladder of XC functional approximations beyond the GGA, by developing a finite-$T$ extension of the well-established PBE0 and HSE06 hybrids. Application to static calculations of electronic band gap at a wide range of $T$ for various systems of interest to high-energy-density physics show that thermal hybrids provide a significant improvement to the LSDA and GGA rung XC functionals and to the ground-state PBE0 and HSE06 hybrids.   

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While being ubiquitous in astrophysics and highly compressed laboratory plasmas, warm dense matter is very difficult to describe theoretically, due to the intricate interplay of quantum degeneracy, correlations and thermal excitations. Ab-initio thermodynamic results for the electronic component were obtained only recently [Dornheim et al., Phys. Reports 744, 1--86 (2018)], using novel path integral Monte Carlo (PIMC) methods. These, however, are limited to static quantities. The investigation of dynamic properties, such as density correlation functions, leading to the dynamic structure factor $S(q,\omega)$ -- a key quantity measured in x-ray Thomson scattering experiments -- is only feasible in terms of an imaginary time. By carrying out extensive PIMC simulations and developing a new reconstruction method, based on stochastic sampling of the dynamic local field correction, we have recently obtained the first exact quantum Monte Carlo results for the dynamic structure factor [Dornheim et al., Phys. Rev. Lett. 121, 255001 (2018)]. Here we extend this approach to other dynamical quantities of the warm dense electron gas, including the dynamic susceptibility, optical conductivity, and dielectric function.   

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The thermodynamics of warm dense matter (WDM) have gained high interest in the past decade due to its significance for technological and astrophysical applications. Theoretically, the electrons pose the most difficulties since quantum and correlation effects are not negligible and can not be approximated with decent accuracy. Recently we have developed the first ab initio Configuration Path Integral Monte Carlo (CPIMC) [1] method for the model of the uniform electron gas (UEG) at finite temperature [2]. Here we extend the CPIMC simulations to the momentum distribution function (MDF) and the static structure factor of the UEG. These structural properties, in particular, a non-exponential high-momentum asymptotics (``quantum tail'') of the MDF, are of crucial importance for scattering cross sections, transport and optical properties. Our simulations confirm the $p^{-8}$ asymptotics that was predicted for the ground state and its relation to the so-called on-top pair distribution function [3]. We also present extensive data for the MDF of electrons in the WDM regime, including its values around the Fermi edge. [1] T. Schoof et al., Phys. Rev. Lett. 115, 130402 (2015) [2] T. Dornheim et al., Phys. Reports 744, 1-86 (2018) [3] J.C. Kimball, J. Phys. A: Math. General 8, 1513 (1975)   

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Density functional theory (DFT) is widely used to investigate the properties of warm dense matter (WDM), including the equation of state, structural properties, as well as transport properties such as thermal and electrical conductivity. However, the common usage of ground state exchange correlation functionals undermines the reliability of DFT calculations since the electron temperature in the WDM regime is far beyond zero. Applying the Kubo-Greenwood formula, we present a DFT investigation of the electrical conductivity of warm dense lithium, aluminum, copper and gold with peculiar electronic structure that has electron holes for a two-temperature situation, which can be realized in the lab by isochoric heating techniques. Comparing the novel finite-temperature exchange correlation functional (FTXC) of Ref.~\footnote{S. Groth \textit{et al.}, Phys. Rev. Lett. \textbf{119}, 135001 (2017)} with the widely used ground state expressions, we reexamine the temperature effect of electrons in WDM\footnote{M. Bonitz \textit{et al.}, Phys. Plasmas \textbf{27}, 042710 (2020)}. This work is helpful to better understand the physics of related WDM experiments.