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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.

The response of the uniform electron gas (UEG) to an external perturbation is of paramount importance for many applications. Recently, highly accurate results for the static density response function and the corresponding local field correction have been provided both for warm dense matter [\textit{J.~Chem.~Phys.}~\textbf{151}, 194104 (2019)] and strongly coupled electron liquid [\textit{Phys.~Rev.~B}~\textbf{101}, 045129 (2020)] conditions based on exact \textit{ab initio} path integral Monte Carlo (PIMC) simulations. In the present work, we further complete our current description of the UEG by exploring the high energy density regime, which is relevant for, e.g., astrophysical applications and inertial confinement fusion experiments. To this end, we present extensive new PIMC results for the static density response in the range of $0.05 \leq r_s \leq 0.5$ and $0.85\leq\theta\leq8$. These data are subsequently used to benchmark the accuracy of the widely used random phase approximation and the dielectric theory by Singwi, Tosi, Land, and Sj\"olander (STLS). Moreover, we compare our results to configuration PIMC data where they are available and find perfect agreement with a relative accuracy of $0.001-0.01\%$. All PIMC data are available online.

The uniform electron gas at finite temperature is of high current interest for warm dense matter research. The complicated interplay of quantum degeneracy and Coulomb coupling effects is fully contained in the pair distribution function or, equivalently, the static structure factor. By combining exact quantum Monte Carlo results for large wave vectors with the long-range behaviour from the Singwi-Tosi-Land-Sjolander approximation, we are able to obtain highly accurate data for the static structure factor over the entire k-range. This allows us to gauge the accuracy of previous approximations and discuss their respective shortcomings. Further, our new data will serve as valuable input for the computation of other quantities.

The aim was to characterize the clearance pathways for L-glutamate from the brain interstitial fluid across the blood–brain barrier using a primary in vitro bovine endothelial/rat astrocyte co-culture. Transporter profiling was performed using uptake studies of radiolabeled L-glutamate with co-application of transporter inhibitors and competing amino acids. Endothelial abluminal L-glutamate uptake was almost abolished by co-application of an EAAT-1 specific inhibitor, whereas luminal uptake was inhibited by L-glutamate and L-aspartate (1 mM). L-glutamate uptake followed Michaelis–Menten-like kinetics with high and low affinity at the abluminal and luminal membrane, respectively. This indicated that L-glutamate is taken up via EAAT-1 at the abluminal membrane and exits at the luminal membrane via a low affinity glutamate/aspartate transporter. Metabolism of L-glutamate and transport of metabolites was examined using [U-13C] L-glutamate. Intact L-glutamate and metabolites derived from oxidative metabolism were transported through the endothelial cells. High amounts of L-glutamate-derived lactate in the luminal medium indicated cataplerosis via malic enzyme. Thus, L-glutamate can be transported intact from brain to blood via the concerted action of abluminal and luminal transport proteins, but the total brain clearance is highly dependent on metabolism in astrocytes and endothelial cells followed by transport of metabolites.

In a recent Letter [T. Dornheim et al., Phys. Rev. Lett. 121, 255001 (2018)] we have presented the first ab initio results for the dynamic structure factor $S(\mathbf{q},\omega)$ of the uniform electron gas for conditions ranging from the warm dense matter regime to the strongly correlated electron liquid. This was achieved on the basis of exact path integral Monte Carlo data by stochastically sampling the dynamic local field correction $G(\mathbf{q},\omega)$. In this paper, we introduce in detail this new reconstruction method and provide several practical demonstrations. Moreover, we thoroughly investigate the associated imaginary-time density--density correlation function $F(\mathbf{q},\tau)$. The latter also gives us access to the static density-response function $\chi(\mathbf{q})$ and static local field correction $G(\mathbf{q})$, which are compared to standard dielectric theories like the widespread random phase approximation. In addition, we study the high-frequency limit of $G(\mathbf{q},\omega)$ and provide extensive new results for the dynamic structure factor for different densities and temperatures. Finally, we discuss the implications of our findings for warm dense matter research and the interpretation of experiments.

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 analyse the joint probability to simultaneously find cycles of different lengths within a single configuration. Again, we find that finite-size effects predominate the observed behaviour. 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.

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 (non-ideality) 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-Sj\"olander approximation, and compared to the MD simulation data obtained considering the electronic screening in the random phase approximation and using the Yukawa potential. %This allows us to gain insight into the impact of the electronic non-ideality on the ionic DSF. 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 non-negligible 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.

The response of the uniform electron gas (UEG) to an external perturbation is of paramount importance for many applications. Recently, highly accurate results for the static density response function and the corresponding local field correction have been provided both for warm dense matter [J. Chem. Phys. 151, 194 104 (2019)] and strongly coupled electron liquid [Phys. Rev. B 101, 045 129 (2020)] conditions based on exact ab initio path integral Monte Carlo (PIMC) simulations. In the present work, we further complete our current description of the UEG by exploring the high energy density regime, which is relevant for, e.g. astrophysical applications and inertial confinement fusion experiments. To this end, we present extensive new PIMC results for the static density response in the range of 0.05 ≤ r s ≤ 0.5 and 0.85 ≤ θ ≤ 8. These data are subsequently used to benchmark the accuracy of the widely used random phase approximation and the dielectric theory by Singwi, Tosi, Land, and Sjölander (STLS). Moreover, we compare our results to configuration PIMC data where they are available and find perfect agreement with a relative accuracy of 0.001 − 0.01%. All PIMC data are available online.

[EN] Several sum rules and other exact relations are employed to determine both the static and the dynamic properties of strongly coupled, partially and completely degenerate one-component plasmas. Emphasis is placed on the electron gas, both at zero and finite temperatures. The procedure is based on the self-consistent method of moments, recently developed in Phys. Rev. Lett., 2017, 119, 045001, that provides a neat expression for the loss function valid at strong couplings. An input value of the method in its classical version is the static structure factor, whose accuracy is shown to insignificantly affect the resulting numerical data. Starting from the Cauchy-Bunyakovsky-Schwarz inequality, a criterion is proposed to verify the quality of various approaches to the evaluation of the static characteristics of one-component, strongly coupled plasmas., This research was supported by Grants PTsF‐BR05236730, AP05132333 and AP05132677 (Ministry of Education and Science, Kazakhstan), No. BO1366-10 (Deutsche Forschungsgemeinschaft, Germany), and Grant No. ESP2013-41078R (Ministerio de Economía y Competitividad, Spain).

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 $S(\mathbf{q},\ensuremath{\omega})$ 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 $S(\mathbf{q},\ensuremath{\omega})$ 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 $G(\mathbf{q},\ensuremath{\omega})$. In addition, we find that the so-called static approximation constitutes a promising opportunity to obtain high-quality data for $S(\mathbf{q},\ensuremath{\omega})$ over substantial parts of the warm dense matter regime.

Motivated by the current high interest in the field of warm dense matter research, in this article we review the uniform electron gas (UEG) at finite temperature and over a broad density range relevant for warm dense matter applications. We provide an exhaustive overview of different simulation techniques, focusing on recent developments in the dielectric formalism (linear response theory) and quantum Monte Carlo (QMC) methods. Our primary focus is on two novel QMC methods that have recently allowed us to achieve breakthroughs in the thermodynamics of the warm dense electron gas: Permutation blocking path integral MC (PB-PIMC) and configuration path integral MC (CPIMC). In fact, a combination of PB-PIMC and CPIMC has allowed for a highly accurate description of the warm dense UEG over a broad density–temperature range. We are able to effectively avoid the notorious fermion sign problem, without invoking uncontrolled approximations such as the fixed node approximation. Furthermore, a new finite-size correction scheme is presented that makes it possible to treat the UEG in the thermodynamic limit without loss of accuracy. In addition, we in detail discuss the construction of a parametrization of the exchange–correlation free energy, on the basis of these data — the central thermodynamic quantity that provides a complete description of the UEG and is of crucial importance as input for the simulation of real warm dense matter applications, e.g., via thermal density functional theory. A second major aspect of this review is the use of our ab initio simulation results to test previous theories, including restricted PIMC, finite-temperature Green functions, the classical mapping by Perrot and Dharma-wardana, and various dielectric methods such as the random phase approximation, or the Singwi–Tosi–Land–Sjolander (both in the static and quantum versions), Vashishta–Singwi and the recent Tanaka scheme for the local field correction. Thus, for the first time, thorough benchmarks of the accuracy of important approximation schemes regarding various quantities such as different energies, in particular the exchange–correlation free energy, and the static structure factor, are possible. In the final part of this paper, we outline a way how to rigorously extend our QMC studies to the inhomogeneous electron gas. We present first ab initio data for the static density response and for the static local field correction.

Recently a number of theoretical studies of the uniform electron gas (UEG) at finite temperature have appeared that are of relevance for dense plasmas, warm dense matter and laser excited solids and thermodynamic density functional theory simulations. In particular, restricted path integral Monte Carlo (RPIMC) results became available which, however, due to the Fermion sign problem, are confined to moderate quantum degeneracy, i.e. low to moderate densities. We have recently developed an alternative approach—configuration PIMC [T. Schoof et al., Contrib. Plasma Phys. 51, 687 (2011)] that allows one to study the so far not accessible high degeneracy regime. Here we present the first step towards UEG simulations using CPIMC by studying implementation and performance of the method for the model case of N = 4 particles. We also provide benchmark data for the total energy. Copyright line will be provided by the publisher

Precise knowledge of the static density response function (SDRF) of the uniform electron gas (UEG) serves as key input for numerous applications, most importantly for density functional theory beyond generalized gradient approximations. Here we extend the configuration path integral Monte Carlo (CPIMC) formalism that was previously applied to the spatially uniform electron gas to the case of an inhomogeneous electron gas by adding a spatially periodic external potential. This procedure has recently been successfully used in permutation blocking path integral Monte Carlo simulations (PB-PIMC) of the warm dense electron gas [Dornheim \textit{et al.}, Phys. Rev. E in press, arXiv:1706.00315], but this method is restricted to low and moderate densities. Implementing this procedure into CPIMC allows us to obtain exact finite temperature results for the SDRF of the electron gas at \textit{high to moderate densities} closing the gap left open by the PB-PIMC data. In this paper we demonstrate how the CPIMC formalism can be efficiently extended to the spatially inhomogeneous electron gas and present the first data points. Finally, we discuss finite size errors involved in the quantum Monte Carlo results for the SDRF in detail and present a solution how to remove them that is based on a generalization of ground state techniques.

In a recent Letter [T.~Dornheim \textit{et al.}, Phys. Rev. Lett. \textbf{117}, 156403 (2016)], we presented the first \textit{ab initio} quantum Monte-Carlo (QMC) results of 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 $\theta=k_\text{B}T/E_\text{F}=0.5$ and (ii) of 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 complete \textit{ab initio} exchange-correlation free energy functional; the accuracy achieved is an unprecedented $\sim 0.3\%$. This also allows us to quantify the accuracy and systematic errors of various previous approximate functionals.

The static density response of the uniform electron gas is of fundamental importance for numerous applications. Here, we employ the recently developed \textit{ab initio} permutation blocking path integral Monte Carlo (PB-PIMC) technique [T.~Dornheim \textit{et al.}, \textit{New J.~Phys.}~\textbf{17}, 073017 (2015)] to carry out extensive simulations of the harmonically perturbed electron gas at warm dense matter conditions. In particular, we investigate in detail the validity of linear response theory and demonstrate that PB-PIMC allows to obtain highly accurate results for the static density response function and, thus, the static local field correction. A comparison with dielectric approximations to our new \textit{ab initio} data reveals the need for an exact treatment of correlations. Finally, we consider a superposition of multiple perturbations and discuss the implications for the calculation of the static response function.

Warm dense matter is one of the most active frontiers in plasma physics due to its relevance for dense astrophysical objects as well as for novel laboratory experiments in which matter is being strongly compressed e.g. by high-power lasers. Its description is theoretically very challenging as it contains correlated quantum electrons at finite temperature---a system that cannot be accurately modeled by standard analytical or ground state approaches. Recently several breakthroughs have been achieved in the field of fermionic quantum Monte Carlo simulations. First, it was shown that exact simulations of a finite model system ($30 \dots 100$ electrons) is possible that avoid any simplifying approximations such as fixed nodes [Schoof {\em et al.}, Phys. Rev. Lett. {\bf 115}, 130402 (2015)]. Second, a novel way to accurately extrapolate these results to the thermodynamic limit was reported by Dornheim {\em et al.} [Phys. Rev. Lett. {\bf 117}, 156403 (2016)]. As a result, now thermodynamic results for the warm dense electron gas are available that have an unprecedented accuracy on the order of $0.1\%$. Here we present an overview on these results and discuss limitations and future directions.

Die dynastisch begründete Periodisierung des Mittelalters in Karolinger und Ottonen (und Salier und Staufer) als grundlegendes Ordnungsprinzip ist in der (deutschen) Mediävistik üblich. Doch nur selten wurde nach den Grundlagen und Folgen dieser Binnendifferenzierung gefragt. Vor diesem Hintergrund untersucht der Band auf den beiden abgrenzbaren Feldern Herrschaftsfolge und Herrschaftsraum die Konsequenzen dieser Einteilung und bezieht sich dabei gleichgewichtig auf das empirische Material der zeitgenössischen Quellen wie auf die mediävistische Forschung seit dem 19. Jahrhundert. Die Verbindung zwischen dem Wechsel der Person des Königs und dem Prozess der Raumbildung bündelt dabei zwei in Wechselwirkung stehende unmittelbare Bereiche der politischen Ordnung erstmals zu einem gemeinsamen Untersuchungsgegenstand.

Riassunto Il complesso e intrecciato rapporto tra impero, papato e Regnum Italicum, che aveva segnato la storia altomedievale a partire dalla notte di natale dell’800, subì un notevole cambiamento con l’incoronazione a imperatore di Ottone I. Mentre sotto i suoi predecessori il regno d’Italia e l’impero avevano continuato a essere due entità autonome, ora si creò un collegamento diretto tra essi. Ottone I pose il Regnum Italicum sotto il suo dominio imperiale (senza chiamarsi Re d’Italia, come aveva invece fatto ancora Carlo Magno), e durante la sua terza campagna d’Italia cercò di spingersi verso l’Italia meridionale. L’Italia, diventata in tal modo centro del suo dominio, forse era assorto per l’imperatore a simbolo della tanto ambita parità con Bisanzio che il matrimonio di Ottone II con una principessa bizantina avrebbe reso palese. Un titolo onorifico che in fondo non aveva comportato nessuna reale crescita di potere, né un legame con un territorio, si era trasformato in un motivo per chiedere il dominio sul territorio italiano, inteso di conseguenza non più come un regnum tra gli altri, ma come nucleo centrale di un impero. Rispetto ai suoi diretti predecessori sul trono imperiale, anche i suoi rapporti con la Sede Apostolica erano molto più condizionati dalla rivendicazione di supremazia e ricordano le tendenze presenti ai tempi di Carlo Magno.

The screened ion potential in non-ideal dense quantum plasmas is investigated by invoking the Singwi-Tosi-Land-Sj\"olander approximation for the electronic local field correction at densities $r_s\lesssim 2$ and degeneracy parameters $\theta\lesssim 1$, where $r_s$ is the ratio of the mean inter-particle distance to the first Bohr radius, and $\theta$ is the ratio of the thermal energy to the Fermi energy of the electrons. Various cross-checks with ion potentials obtained from ground state quantum Monte-Carlo data, the random phase approximation, as well as with existing analytical models are presented. Further, the importance of the electronic correlation effects for the dynamics in strongly coupled ionic subsystems for $0.1\leq r_s\leq 2$ is discussed.

The uniform electron gas is a key model system in the description of matter, including dense plasmas and solid state systems. However, the simultaneous occurence of quantum, correlation, and thermal effects makes the theoretical description challenging. For these reasons, over the last half century many analytical approaches have been developed the accuracy of which has remained unclear. We have recently obtained the first \textit{ab initio} data for the exchange correlation free energy of the uniform electron gas [T. Dornheim \textit{et al.}, Phys.~Rev.~Lett.~\textbf{117}, 156403 (2016)] which now provides the opportunity to assess the quality of the mentioned approaches and parametrizations. Particular emphasis is put on the warm dense matter regime, where we find significant discrepancies between the different approaches.

In a recent publication [S. Groth \textit{et al.}, PRB (2016)], we have shown that the combination of two novel complementary quantum Monte Carlo approaches, namely configuration path integral Monte Carlo (CPIMC) [T. Schoof \textit{et al.}, PRL \textbf{115}, 130402 (2015)] and permutation blocking path integral Monte Carlo (PB-PIMC) [T. Dornheim \textit{et al.}, NJP \textbf{17}, 073017 (2015)], allows for the accurate computation of thermodynamic properties of the spin-polarized uniform electron gas (UEG) over a wide range of temperatures and densities without the fixed-node approximation. In the present work, we extend this concept to the unpolarized case, which requires non-trivial enhancements that we describe in detail. We compare our new simulation results with recent restricted path integral Monte Carlo data [E. Brown \textit{et al}., PRL \textbf{110}, 146405 (2013)] for different energy contributions and pair distribution functions and find, for the exchange correlation energy, overall better agreement than for the spin-polarized case, while the separate kinetic and potential contributions substantially deviate.

The uniform electron gas (UEG) at finite temperature is of key relevance for many applications in the warm dense matter regime, e.g. dense plasmas and laser excited solids. Also, the quality of density functional theory calculations crucially relies on the availability of accurate data for the exchange-correlation energy. Recently, new benchmark results for the N = 33 spin-polarized electrons at high density, r_s = r/a_B

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 $r_s={\bar r}/a_B \gtrsim 1$. Here we present novel first-principle configuration PIMC results for electrons for $r_s \leq 1$. We also present quantum statistical data within the $e^4$-approximation that are in good agreement with the simulations at small to moderate $r_s$., Comment: Revision in response to referee comments. New and more accurate data including extrapolation to macroscopic limit

In low-temperature high-density plasmas quantum effects of the electrons are becoming increasingly important. This requires the development of new theoretical and computational tools. Quantum Monte Carlo methods are among the most successful approaches to first-principle simulations of many-body quantum systems. In this chapter we present a recently developed method—the configuration path integral Monte Carlo (CPIMC) method for moderately coupled, highly degenerate fermions at finite temperatures. It is based on the second quantization representation of the \(N\)-particle density operator in a basis of (anti-)symmetrized \(N\)-particle states (configurations of occupation numbers) and allows to tread arbitrary pair interactions in a continuous space. We give a detailed description of the method and discuss the application to electrons or, more generally, Coulomb-interacting fermions. As a test case we consider a few quantum particles in a one-dimensional harmonic trap. Depending on the coupling parameter (ratio of the interaction energy to kinetic energy), the method strongly reduces the sign problem as compared to direct path integral Monte Carlo (DPIMC) simulations in the regime of strong degeneracy which is of particular importance for dense matter in laser plasmas or compact stars. In order to provide a self-contained introduction, the chapter includes a short introduction to Metropolis Monte Carlo methods and the second quantization of quantum mechanics.