Equation of state of partially ionized hydrogen and deuterium plasma revisited.

We present improved first-principle fermionic path integral Monte Carlo (PIMC) simulation results for a dense partially ionized hydrogen (deuterium) plasma, for temperatures in the range 15000K≤T≤400000K and densities 7×10^{-7}g/cm^{3}≤ρ_{H}≤0.085g/cm^{3} (1.4×10^{-6}g/cm^{3}≤ρ_{D}≤0.17g/cm^{3}), corresponding to 100≥r_{s}≥2, where r_{s}=r[over ¯]/a_{B} is the ratio of the mean interparticle distance to the Bohr radius. These simulations are based on the fermionic propagator PIMC (FP-PIMC) approach in the grand canonical ensemble [Filinov et al., Contrib. Plasma Phys. 61, e202100112 (2021)0863-104210.1002/ctpp.202100112] and fully account for correlation and quantum degeneracy and spin effects. For the application to hydrogen and deuterium, we develop a combination of the fourth-order factorization and the pair product ansatz for the density matrix. Moreover, we avoid the fixed node approximation that may lead to uncontrolled errors in restricted PIMC (RPIMC). Our results allow us to critically reevaluate the accuracy of the RPIMC simulations for hydrogen by Hu et al. [Phys. Rev. B 84, 224109 (2011)1098-012110.1103/PhysRevB.84.224109] and of various chemical models. The deviations are generally found to be small, but for the lowest temperature, T=15640 K they reach several percent. We present detailed tables with our first principles results for the pressure and energy isotherms. We expect our updated results will serve as a valuable benchmark for comparison with other methods.

Finite-temperature density-functional-theory investigation on the nonequilibrium transient warm-dense-matter state created by laser excitation.

We present a finite-temperature density-functional-theory investigation of the nonequilibrium transient electronic structure of warm dense Li, Al, Cu, and Au created by laser excitation. Photons excite electrons either from the inner shell orbitals or from the valence bands according to the photon energy, and give rise to isochoric heating of the sample. Localized states related to the 3d orbital are observed for Cu when the hole lies in the inner shell 3s orbital. The electrical conductivity for these materials at nonequilibrium states is calculated using the Kubo-Greenwood formula. The change of the electrical conductivity, compared to the equilibrium state, is different for the case of holes in inner shell orbitals or the valence band. This is attributed to the competition of two factors: the shift of the orbital energies due to reduced screening of core electrons, and the increase of chemical potential due to the excitation of electrons. The finite-temperature effect of both the electrons and the ions on the electrical conductivity is discussed in detail. This work is helpful to better understand the physics of laser excitation experiments of warm dense matter.

Restricted configuration path integral Monte Carlo.

Quantum Monte Carlo (QMC) belongs to the most accurate simulation techniques for quantum many-particle systems. However, for fermions, these simulations are hampered by the sign problem that prohibits simulations in the regime of strong degeneracy. The situation changed with the development of configuration path integral Monte Carlo (CPIMC) by Schoof et al. [Contrib. Plasma Phys. 51, 687 (2011)] that allowed for the first ab initio simulations for dense quantum plasmas [Schoof et al., Phys. Rev. Lett. 115, 130402 (2015)]. CPIMC also has a sign problem that occurs when the density is lowered, i.e., in a parameter range that is complementary to traditional QMC formulated in coordinate space. Thus, CPIMC simulations for the warm dense electron gas are limited to small values of the Brueckner parameter-the ratio of the interparticle distance to the Bohr radius-rs=r¯/aB≲1. In order to reach the regime of stronger coupling (lower density) with CPIMC, here we investigate additional restrictions on the Monte Carlo procedure. In particular, we introduce two different versions of "restricted CPIMC"-called RCPIMC and RCPIMC+-where certain sign changing Monte Carlo updates are being omitted. Interestingly, one of the methods (RCPIMC) has no sign problem at all, but it introduces a systematic error and is less accurate than RCPIMC+, which neglects only a smaller class of the Monte Carlo steps. Here, we report extensive simulations for the ferromagnetic uniform electron gas with which we investigate the properties and accuracy of RCPIMC and RCPIMC+. Furthermore, we establish the parameter range in the density-temperature plane where these simulations are both feasible and accurate. The conclusion is that RCPIMC and RCPIMC+ work best at temperatures in the range of Θ = kBT/EF ∼ 0.1…0.5, where EF is the Fermi energy, allowing to reach density parameters up to rs ∼ 3…5, thereby partially filling a gap left open by existing ab initio QMC methods.

Ion energy-loss characteristics and friction in a free-electron gas at warm dense matter and nonideal dense plasma conditions.

We investigate the energy-loss characteristics of an ion in warm dense matter (WDM) and dense plasmas concentrating on the influence of electronic correlations. The basis for our analysis is a recently developed ab initio quantum Monte Carlo- (QMC) based machine learning representation of the static local field correction (LFC) [Dornheim et al., J. Chem. Phys. 151, 194104 (2019)JCPSA60021-960610.1063/1.5123013], which provides an accurate description of the dynamical density response function of the electron gas at the considered parameters. We focus on the polarization-induced stopping power due to free electrons, the friction function, and the straggling rate. In addition, we compute the friction coefficient which constitutes a key quantity for the adequate Langevin dynamics simulation of ions. Considering typical experimental WDM parameters with partially degenerate electrons, we find that the friction coefficient is of the order of γ/ω_{pi}=0.01, where ω_{pi} is the ionic plasma frequency. This analysis is performed by comparing QMC-based data to results from the random-phase approximation (RPA), the Mermin dielectric function, and the Singwi-Tosi-Land-Sjölander (STLS) approximation. It is revealed that the widely used relaxation time approximation (Mermin dielectric function) has severe limitations regarding the description of the energy loss of ions in a correlated partially degenerate electrons gas. Moreover, by comparing QMC-based data with the results obtained using STLS, we find that the ion energy-loss properties are not sensitive to the inaccuracy of the static local field correction (LFC) at large wave numbers, k/k_{F}>2 (with k_{F} being the Fermi wave number), but that a correct description of the static LFC at k/k_{F}≲1.5 is important.

Ultrafast dynamics of strongly correlated fermions-nonequilibrium Green functions and selfenergy approximations.

This article presents an overview on recent progress in the theory of nonequilibrium Green functions (NEGF). We discuss applications of NEGF simulations to describe the femtosecond dynamics of various finite fermionic systems following an excitation out of equilibrium. This includes the expansion dynamics of ultracold atoms in optical lattices following a confinement quench and the excitation of strongly correlated electrons in a solid by the impact of a charged particle. NEGF, presently, are the only ab initio quantum approach that is able to study the dynamics of correlations for long times in two and three dimensions. However, until recently, NEGF simulations have mostly been performed with rather simple selfenergy approximations such as the second-order Born approximation (SOA). While they correctly capture the qualitative trends of the relaxation towards equilibrium, the reliability and accuracy of these NEGF simulations has remained open, for a long time. Here we report on recent tests of NEGF simulations for finite lattice systems against exact-diagonalization and density-matrix-renormalization-group benchmark data. The results confirm the high accuracy and predictive capability of NEGF simulations-provided selfenergies are used that go beyond the SOA and adequately include strong correlation and dynamical-screening effects. With an extended arsenal of selfenergies that can be used effectively, the NEGF approach has the potential of becoming a powerful simulation tool with broad areas of new applications including strongly correlated solids and ultracold atoms. The present review aims at making such applications possible. To this end we present a selfcontained introduction to the theory of NEGF and give an overview on recent numerical applications to compute the ultrafast relaxation dynamics of correlated fermions. In the second part we give a detailed introduction to selfenergies beyond the SOA. Important examples are the third-order approximation, the [Formula: see text] approximation, the T-matrix approximation and the fluctuating-exchange approximation. We give a comprehensive summary of the explicit selfenergy expressions for a variety of systems of practical relevance, starting from the most general expressions (general basis) and the Feynman diagrams, and including also the important cases of diagonal basis sets, the Hubbard model and the differences occuring for bosons and fermions. With these details, and information on the computational effort and scaling with the basis size and propagation duration, readers will be able to choose the proper basis set and straightforwardly implement and apply advanced selfenergy approximations to a broad class of systems.

The static local field correction of the warm dense electron gas: An ab initio path integral Monte Carlo study and machine learning representation.

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.

Path integral Monte Carlo simulation of degenerate electrons: Permutation-cycle properties.

Being motivated by the surge of fermionic quantum Monte Carlo simulations at finite temperature, we present a detailed analysis of the permutation-cycle properties of path integral Monte Carlo (PIMC) simulations of degenerate electrons. Particular emphasis is put onto the uniform electron gas in the warm dense matter regime. We carry out PIMC simulations of up to N = 100 electrons and investigate exchange-cycle frequencies, which are found not to follow any simple exponential law even in the case of ideal fermions due to the finite size of the simulation box. Moreover, we introduce a permutation-cycle correlation function, which allows us to analyze the joint probability to simultaneously find cycles of different lengths within a single configuration. Again, we find that finite-size effects predominate the observed behavior. Finally, we briefly consider an inhomogeneous system, namely, electrons in a 2D harmonic trap. We expect our results to be of interest for the further development of fermionic PIMC methods, in particular, to alleviate the notorious fermion sign problem.

Dynamical structure factor of strongly coupled ions in a dense quantum plasma.

The dynamical structure factor (DSF) of strongly coupled ions in dense plasmas with partially and strongly degenerate electrons is investigated. The main focus is on the impact of electronic correlations (nonideality) on the ionic DSF. The latter is computed by carrying out molecular dynamics (MD) simulations with a screened ion-ion interaction potential. The electronic screening is taken into account by invoking the Singwi-Tosi-Land-Sjölander approximation, and it is compared to the MD simulation data obtained considering the electronic screening in the random phase approximation and using the Yukawa potential. We find that electronic correlations lead to lower values of the ion-acoustic mode frequencies and to an extension of the applicability limit with respect to the wave-number of a hydrodynamic description. Moreover, we show that even in the limit of weak electronic coupling, electronic correlations have a nonnegligible impact on the ionic longitudinal sound speed. Additionally, the applicability of the Yukawa potential with an adjustable screening parameter is discussed, which will be of interest, e.g., for the interpretation of experimental results for the ionic DSF of dense plasmas.

Self-diffusion in two-dimensional quasimagnetized rotating dusty plasmas.

The self-diffusion phenomenon in a two-dimensional dusty plasma at extremely strong (effective) magnetic fields is studied experimentally and by means of molecular dynamics simulations. In the experiment the high magnetic field is introduced by rotating the particle cloud and observing the particle trajectories in a corotating frame, which allows reaching effective magnetic fields up to 3000 T. The experimental results confirm the predictions of the simulations: (i) superdiffusive behavior is found at intermediate timescales and (ii) the dependence of the self-diffusion coefficient on the magnetic field is well reproduced.

Structural characteristics of strongly coupled ions in a dense quantum plasma.

The structural properties of strongly coupled ions in dense plasmas with moderately to strongly degenerate electrons are investigated in the framework of the one-component plasma model of ions interacting through a screened pair interaction potential. Special focus is put on the description of the electronic screening in the Singwi-Tosi-Land-Sjölander (STLS) approximation. Different cross-checks and analyses using ion potentials obtained from ground-state quantum Monte Carlo data, the random phase approximation (RPA), and existing analytical models are presented for the computation of the structural properties, such as the pair distribution and the static structure factor, of strongly coupled ions. The results are highly sensitive to the features of the screened pair interaction potential. This effect is particularly visible in the static structure factor. The applicability range of the screened potential computed from STLS is identified in terms of density and temperature of the electrons. It is demonstrated that at r_{s}>1, where r_{s} is the ratio of the mean interelectronic distance to the Bohr radius, electronic correlations beyond RPA have a nonnegligible effect on the structural properties. Additionally, the applicability of the hypernetted chain approximation for the calculation of the structural properties using the screened pair interaction potential is analyzed employing the effective coupling parameter approach.

Ab initio Path Integral Monte Carlo Results for the Dynamic Structure Factor of Correlated Electrons: From the Electron Liquid to Warm Dense Matter.

The accurate description of electrons at extreme density and temperature is of paramount importance for, e.g., the understanding of astrophysical objects and inertial confinement fusion. In this context, the dynamic structure factor S(q,ω) constitutes a key quantity as it is directly measured in x-ray Thomson scattering experiments and governs transport properties like the dynamic conductivity. In this work, we present the first ab initio results for S(q,ω) by carrying out extensive path integral Monte Carlo simulations and developing a new method for the required analytic continuation, which is based on the stochastic sampling of the dynamic local field correction G(q,ω). In addition, we find that the so-called static approximation constitutes a promising opportunity to obtain high-quality data for S(q,ω) over substantial parts of the warm dense matter regime.

Spontaneous generation of temperature anisotropy in a strongly coupled magnetized plasma.

A magnetic field was recently shown to enhance field-parallel heat conduction in a strongly correlated plasma whereas cross-field conduction is reduced. Here we show that in such plasmas, the magnetic field has the additional effect of inhibiting the isotropization process between field-parallel and cross-field temperature components, thus leading to the emergence of strong and long-lived temperature anisotropies when the plasma is locally perturbed. An extended heat equation is shown to describe this process accurately.

Cage correlation and diffusion in strongly coupled three-dimensional Yukawa systems in magnetic fields.

The influence of an external homogeneous magnetic field on the quasilocalization of the particles-characterized quantitatively by cage correlation functions-in strongly coupled three-dimensional Yukawa systems is investigated via molecular dynamics computer simulations over a wide domain of the system parameters (coupling and screening strengths, and magnetic field). The caging time is found to be enhanced by the magnetic field B. The anisotropic migration of the particles in the presence of magnetic field is quantified via computing directional correlation functions, which indicate a more significant increase of localization in the direction perpendicular to B, while a moderate increase is also found along the B field lines. Associating the particles' escapes from the cages with jumps of a characteristic length, a connection is found with the diffusion process: the diffusion coefficients derived from the decay time of the directional correlation functions in both the directions perpendicular to and parallel with B are in very good agreement with respective diffusion coefficients values obtained from their usual computation based on the mean-squared displacement of the particles.

Ab Initio Thermodynamic Results for the Degenerate Electron Gas at Finite Temperature.

The uniform electron gas 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 uniform electron gas are an essential ingredient for many-body theories, in particular, density-functional theory. Recently, first-principles restricted path integral Monte Carlo results became available, which, however, had to be restricted to moderate degeneracy, i.e., low to moderate densities with r_{s}=r[over ¯]/a_{B}≳1. Here we present novel first-principles configuration path integral Monte Carlo results for electrons for r_{s}≤4. We also present quantum statistical data within the e^{4} approximation that are in good agreement with the simulations at small to moderate r_{s}.

Resolving structural transitions in spherical dust clusters.

Finite systems in confining potentials are known to undergo structural transitions similar to phase transitions. However, these systems are inhomogeneous, and their "melting" point may depend on the position in the trap and vary with the particle number. Focusing on three-dimensional Coulomb systems in a harmonic trap a rich physics is revealed: in addition to radial melting we demonstrate the existence of intrashell disordering and intershell angular melting. Our analysis takes advantage of a novel melting criterion that is based on the spatial two- and three-particle distribution functions and the associated reduced entropy which can be directly measured in complex plasma experiments.

Fermionic path-integral Monte Carlo results for the uniform electron gas at finite temperature.

The uniform electron gas (UEG) at finite temperature has recently attracted substantial interest due to the experimental progress in the field of warm dense matter. To explain the experimental data, accurate theoretical models for high-density plasmas are needed that depend crucially on the quality of the thermodynamic properties of the quantum degenerate nonideal electrons and of the treatment of their interaction with the positive background. Recent fixed-node path-integral Monte Carlo (RPIMC) data are believed to be the most accurate for the UEG at finite temperature, but they become questionable at high degeneracy when the Brueckner parameter rs=a/aB--the ratio of the mean interparticle distance to the Bohr radius--approaches 1. The validity range of these simulations and their predictive capabilities for the UEG are presently unknown. This is due to the unknown quality of the used fixed nodes and of the finite-size scaling from N=33 simulated particles (per spin projection) to the macroscopic limit. To analyze these questions, we present alternative direct fermionic path integral Monte Carlo (DPIMC) simulations that are independent from RPIMC. Our simulations take into account quantum effects not only in the electron system but also in their interaction with the uniform positive background. Also, we use substantially larger particle numbers (up to three times more) and perform an extrapolation to the macroscopic limit. We observe very good agreement with RPIMC, for the polarized electron gas, up to moderate densities around rs=4, and larger deviations for the unpolarized case, for low temperatures. For higher densities (high electron degeneracy), rs≲1.5, both RPIMC and DPIMC are problematic due to the increased fermion sign problem.

Effect of correlations on heat transport in a magnetized strongly coupled plasma.

In a classical ideal plasma, a magnetic field is known to reduce the heat conductivity perpendicular to the field, whereas it does not alter the one along the field. Here we show that, in strongly correlated plasmas that are observed at high pressure and/or low temperature, a magnetic field reduces the perpendicular heat transport much less and even enhances the parallel transport. These surprising observations are explained by the competition of kinetic, potential, and collisional contributions to the heat conductivity. Our results are based on first-principle molecular dynamics simulations of a one-component plasma.

Dynamics of two-dimensional one-component and binary Yukawa systems in a magnetic field.

We consider two-dimensional Yukawa systems in a perpendicular magnetic field. Computer simulations of both one-component and binary systems are used to explore the equilibrium particle dynamics in the fluid state. The mobility is found to scale with the inverse of the magnetic field strength (Bohm diffusion), for strong fields (ωc/ωp≳1). For bidisperse mixtures, the magnetic field dependence of the long-time mobility depends on the particle species, providing an external control of their mobility ratio. At large magnetic fields, the highly charged particles are almost immobilized by the magnetic field and form a porous matrix of obstacles for the mobile low-charge particles.

Magnetic field blocks two-dimensional crystallization in strongly coupled plasmas.

Crystallization in a two-dimensional strongly coupled plasma from a rapidly cooled fluid is found to be efficiently blocked by an external magnetic field. Beyond a threshold of the magnetic field strength B, the relaxation time to the equilibrium crystal increases exponentially with B, which is attributed to an impeded conversion of potential to kinetic energy. Our finding is opposed to the standard picture of two-dimensional freezing of one-component systems which does not exhibit a nucleation barrier and opens the way to keep two-dimensional fluids metastable over long times.

Wave spectra of a strongly coupled magnetized one-component plasma: quasilocalized charge approximation versus harmonic lattice theory and molecular dynamics.

Two different approaches to the calculation of the wave spectra of magnetized strongly coupled liquid one-component plasmas are analzyed: the semianalytical quasilocalized charge approximation (QLCA) and the angle-averaged harmonic lattice (AAHL) theory. Both theories are benchmarked against the numerical evidence obtained from molecular dynamics simulations. It is found that not too far from the melting transition (Γ≳100), the AAHL theory is superior to the QLCA, while further away from the transition, the QLCA performs comparably to or better than the AAHL theory.