Elsevier

Physics Reports

Volume 744, 30 May 2018, Pages 1-86
Physics Reports

The uniform electron gas at warm dense matter conditions

https://doi.org/10.1016/j.physrep.2018.04.001Get rights and content

Abstract

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–Sjölander (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.

Introduction

The uniform electron gas (UEG), often referred to as “jellium”, is one of the most important model systems in physics and quantum chemistry, and consists of Coulomb interacting electrons in a positive neutralizing background [1]. Therefore, it constitutes the quantum mechanical analogue of the classical one-component plasma (OCP) [2] and qualitatively reproduces many physical phenomena [3] such as Wigner crystallization, spin-polarization transitions, and screening. Often, it is used as a simple model system for conduction electrons in alkali metals [[1], [4]]. The investigation of the UEG at zero temperature has lead to several key insights, like the BCS theory of superconductivity [5], Fermi liquid theory [[1], [6]], and the quasi-particle picture of collective excitations [[7], [8]]. Further, as a continuous correlated electronic quantum system, it has served as a workbench for the development of countless computational many-body methods, most prominently dielectric approximations, e.g., Refs. [[8], [9], [10], [11], [12], [13], [14]] and quantum Monte Carlo (QMC) methods [[15], [16], [17], [18], [19], [20], [21], [22]]. Even though the UEG itself does not represent a real physical system, its accurate description has been of paramount importance for the unrivaled success of density functional theory (DFT) [[23], [24]], the workhorse of modern many-body simulations of realistic materials in solid state physics, quantum chemistry, and beyond [[25], [26], [27]]. Within the DFT framework, the complicated interacting many-electron system is mapped onto an effective one-particle (non-interacting) system via the introduction of an effective potential containing all exchange and correlation effects. While exact knowledge of the latter would require a complete solution of the many-body problem so that nothing was gained, it can often be accurately approximated by the exchange–correlation energy of the UEG, using a parametrization in dependence of density [[28], [29], [30]].

The first accurate data of the ferromagnetic and paramagnetic UEG were obtained in 1980 by Ceperley and Alder [16], who carried out ground state QMC simulations (see Ref. [17] for a review) covering a wide range of densities. Subsequently, these data were used as input for parametrizations, most notably by Vosko et al. [28] and Perdew and Zunger [29]. Since then, these seminal works have been used thousands of times [31] for DFT calculations in the local (spin-)density approximation (L(S)DA) and as the basis for more sophisticated gradient approximations, e.g., Refs. [[32], [33]]. Note that, in the mean time, there have been carried out more sophisticated QMC simulations [[34], [35], [36], [37], [38], [39]], with Spink et al. [39] providing the most accurate energies available.

In addition to the exchange–correlation energy, there exist many parametrizations of other quantities on the basis of QMC simulations such as pair distribution functions and static structure factors [[40], [41], [42], [43]] and the momentum distribution [[35], [36], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55]]. Finally, we mention the QMC investigation of the inhomogeneous electron gas [[56], [57], [58], [59], [60]], which gives important insights into the density response formalism, see Section 9 for more details.

Over the last decades, there has emerged a growing interest in the properties of matter under extreme conditions, i.e., at high temperature and densities exceeding those in solids by several orders of magnitude. This exotic state is usually referred to as warm dense matter (WDM) and is characterized by two parameters being of the order of unity: (i) the density parameter (Wigner–Seitz radius) rs, and (ii) the reduced temperature θ rsaB=34πn13,θ=kBTEF,with EF being the Fermi energy defined in Eq. (5). Here rs plays the role of a quantum coupling parameter: at high density (rs0), the electrons behave as an ideal Fermi gas and towards low density, the Coulomb repulsion predominates, eventually leading to a Wigner crystal [[38], [63], [64], [65]]. Further, θ can be understood as the quantum degeneracy parameter, where θ1 indicates a classical system (typically characterized by the classical coupling parameter Γ=1(rsaBkBT), cf. the red line in Fig. 1); for an overview on Coulomb correlation effects in classical systems, see Ref. [66]. On the other hand, the case θ1 characterizes a strongly degenerate quantum system. Thus, in the WDM regime, Coulomb coupling correlations, thermal excitations, and fermionic exchange effects are equally important at the same time. Naturally, this makes an accurate theoretical description of such systems most challenging [67].

In nature, WDM occurs in astrophysical objects such as giant planet interiors [[68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78]], brown and white dwarfs [[79], [80], [81], [82], [83]] and neutron star crusts [84], see Refs. [[61], [85]] for a recent review. Further areas of interest contain the physics of meteor impacts [82] and nuclear stewardship [86]. Another highly important aspect of warm dense matter research is the concept of inertial confinement fusion [[87], [88], [89], [90], [91], [92]], which could become a potentially nearly infinite source of clean energy in the future.

WDM conditions are now routinely realized at large research facilities such as the National Ignition Facility (NIF) at Lawrence Livermore National Lab, California [[93], [94]], the Omega laser facility at the Laboratory for Laser Energetics in Rochester [95], the Z Pulsed Power Facility (Z-machine) at the Sandia National Labs in New Mexico [[96], [97], [98], [99], [100]], the Linac Coherent Light Source (LCLS) in Stanford, California [[101], [102], [103]], FLASH and the European X-FEL (free electron laser) in Hamburg, Germany [[104], [105]] and other laser and free electron laser laboratories. Moreover, we mention shock-compression experiments, e.g. [[97], [106], [107]]. Of particular importance is X-ray Thomson scattering (XRTS), e.g. Refs. [[104], [108], [109], [110], [111], [112], [113]], which provides a widespread diagnostics for warm dense matter experiments, see Ref. [114] for a review. More specifically, it allows for the direct measurement of the dynamic structure factor, which can subsequently be used to obtain, for example, the temperature [114]. Finally, we stress that WDM experiments allow for the investigation of many other quantities, such as the dielectric function [[115], [116]], electrical and thermal conductivities [[117], [118], [119], [120]], the electron–ion temperature equilibration [121] and even the formation of transient nonequilibrium states [[119], [122]]. As a schematic overview, in Fig. 1 various important applications are depicted in the density–temperature plane around the warm dense matter regime. For a recent text book overview we refer to [123].

Despite the remarkable experimental progress, a thorough theoretical description of warm dense matter is still lacking (even in the case of thermodynamic equilibrium), and it is well-known that simple analytic models do not sufficiently reproduce experimental measurements [[124], [125]]. Naturally, an exact quantum mechanical treatment that incorporates all correlation and excitation effects is not feasible. Unfortunately, quantum Monte Carlo methods which often allow for accurate results in the ground state are not straightforwardly extended to the simulation of fermionic matter at finite temperature. More specifically, exact fermionic path integral Monte Carlo (PIMC) simulations (see Section 5.2) are severely hampered by the so-called fermion sign problem; nevertheless, there has been made some progress in direct fermionic QMC simulations by Filinov and co-workers [[126], [127], [128], [129], [130], [131], [132], [133], [134]]. To avoid the fermion sign problem, usually the fixed node approximation is utilized [[135], [136], [137]] (also “Restricted PIMC”, RPIMC, see Section 5.3) which, however, breaks down at low temperature and high density. Therefore, RPIMC is not available over substantial parts of the warm dense regime, and the accuracy is, in general, unknown.

The probably most widespread simulation technique for warm dense matter is the combination of molecular dynamics (for the heavy ions) with a thermal density functional theory description of the electrons [[138], [139], [140]], usually denoted as DFT-MD [[141], [142], [143], [144], [145]]. Naturally, the decoupling of the ionic and electronic systems according to the Born–Oppenheimer approximation might not be appropriate in all situations. In addition, similar to the ground state, the accuracy of the DFT calculation itself strongly relies on the specific choice of the exchange–correlation functional [[146], [147]]. An additional obstacle for thermal DFT calculations is the explicit dependence of the exchange–correlation functional on temperature [[148], [149]], a topic which has only recently attracted serious attention, but might be crucial to achieve real predictive capability [[31], [67]]. Even worse, at moderate to high temperature, the usual thermal Kohn–Sham (KS) treatment of DFT becomes unfeasible, due to the increasing number of orbitals necessary to reach convergence. For this reason, Militzer and co-workers have proposed to combine RPIMC at high temperature with DFT elsewhere, and successfully applied this idea to the simulations of many different materials at warm dense matter conditions [[150], [151], [152], [153], [154], [155]]. A possible extension of KS-DFT towards stronger excitations is given by the so-called orbital free (OF) DFT [[156], [157], [158], [159], [160], [161]], where the total electronic density is not represented by Kohn–Sham orbitals. While being computationally cheap and, in principle, still exact, in practice orbital free DFT relies on an approximation for the ideal part of the (free) energy [162], whereas the latter is treated exactly within KS-DFT. Since the ideal part usually constitutes the largest contribution, it is widely agreed that OF-DFT does not provide sufficient accuracy, and, therefore, cannot give a suitable description of warm dense matter [161]. A recent, more promising, strategy to extend KS-DFT towards higher temperature has been introduced by Zhang and co-workers, see Refs. [[161], [163], [164]] for details.

On the other hand, even at relatively low temperature, when the electrons are in the ground state, a DFT description for the electronic component is often not sufficient [[146], [147]]. For this reason, Ceperley, Pierleoni and co-workers proposed to combine a classical Monte Carlo (instead of MD) for the heavy ions, with highly accurate ground-state QMC calculations for the electrons. This so-called coupled electron–ion QMC (CEIMC) method [[165], [166], [167], [168]] has subsequently been applied, e.g., to the (controversially discussed, see also the recent experiments in Ref. [169]) liquid–liquid phase transition in hydrogen [[170], [171]]. Note that, within the CEIMC approach, quantum effects of the ions can easily be included, e.g., Ref. [171]. In a similar spirit, Sorella and co-workers [[172], [173], [174], [175], [176], [177]] introduced a combination of electronic ground state QMC calculations with classical MD for the ions, although, to our knowledge, no consensus with CEIMC (and, for that matter, with DFT-MD) simulations has been reached so far regarding liquid hydrogen.

In addition, there has been remarkable recent progress in the development of real time-dependent DFT calculations [[178], [179], [180], [181]], which would also give direct access to the dynamic properties of the electrons, although this topic remains in its infancy mainly due to the difficulty constructing XC potentials beyond the adiabatic LDA.

Finally, we mention the possibility of so-called quantum–classical mappings employed by Dharma-wardana et al. [[182], [183], [184], [185]], where the complicated quantum mechanical system of interest is mapped onto a classical model system with an effective “quantum temperature”, see Section 4.2 for more details.

Of particular interest for the theoretical description of WDM are the properties of the warm dense uniform electron gas. As mentioned above, an accurate parametrization of the exchange–correlation free energy with respect to temperature θ, density rs, and spin-polarization ξ is of paramount importance for thermal DFT simulation both in the local (spin) density approximation or as a basis for more sophisticated gradient approximations [[33], [186]]. Further, direct applications of such a functional include astrophysical models [[187], [188], [189], [190], [191], [192]], quantum hydrodynamics [[193], [194], [195]], and the benchmark for approximations, such as finite-temperature Green function methods [[196], [197]], for a recent study see Ref. [198].

However, even the description of this simple model system, without an explicit treatment of the ionic component, has turned out to be surprisingly difficult. Throughout the eighties of the last century, Ebeling and co-workers [[199], [200], [201], [202], [203]] proposed various interpolations between different known limits (i.e., high temperature, weak coupling, and the ground state). A more sophisticated approach is given by the dielectric formalism, which, at finite temperature, has been extensively developed and applied to the UEG by Ichimaru, Tanaka, and co-workers, see Refs. [[204], [205], [206], [207], [208], [209]]. For a more comprehensive discussion of recent improvements in this field, see Section 3. In addition, we mention the classical-mapping based scheme by Perrot and Dharma-wardana [[210], [211]], the application of which is discussed in Section 4.2.1. Unfortunately, all aforementioned results contain uncontrolled approximations and systematic errors of varying degrees, so that their respective accuracy has remained unclear.

While, in principle, thermodynamic QMC methods allow for a potentially exact description, their application to the warm dense UEG has long been prevented by the so-called fermion sign problem, see Section 5. For this reason, the first QMC results for this system were obtained by Brown et al. [212] in 2013 by employing the fixed node approximation (i.e., RPIMC). While this strategy allows for QMC simulations without a sign problem, this comes at the cost of the exact ab-initio character and it has been shown that results for different thermodynamic quantities are not consistent [213]. Nevertheless, these data have subsequently been used as the basis for various parametrization [[213], [214], [215]].

This overall unsatisfactory situation has sparked remarkable recent progress in the field of fermionic QMC simulations of the UEG at finite temperature. The first new development in this direction has been the configuration path integral Monte Carlo method (CPIMC, see Section 5.5), which, in contrast to standard PIMC, is formulated in second quantization with respect to plane waves, and has been developed by Schoof, Groth and co-workers [[216], [217], [218]]. In principle, CPIMC can be viewed as performing a Monte Carlo simulation on the exact, infinite perturbation expansion around the ideal system. Therefore, it excels at high density and strong degeneracy, but breaks down around rs1 and, thus, exhibits a complementary nature with respect to standard PIMC in coordinate space. Surprisingly, the comparison of exact CPIMC data [219] for N=33 spin-polarized electrons with the RPIMC data by Brown et al. [212] revealed systematic deviations exceeding 10% towards low temperature and high density, thereby highlighting the need for further improved simulations. Therefore, Dornheim and co-workers [[220], [221]] introduced the so-called permutation blocking PIMC (PB-PIMC, see Section 5.4) paradigm, which significantly extends standard PIMC both towards lower temperature and higher density. In combination, CPIMC and PB-PIMC allow for an accurate description of the UEG over the entire density range down to half the Fermi temperature [[218], [222]]. Soon thereafter, these results were fully confirmed by a third independent method. This density matrix QMC (DMQMC, see Section 5.6) [[223], [224], [225]] is akin to CPIMC by being formulated in Fock space. Hence, there has emerged a consensus regarding the description of the electron gas with a finite number of particles [226]. The next logical step is the extrapolation to the thermodynamic limit, i.e., to the infinite system at a constant density, see Section 6. As it turned out, the extrapolation scheme utilized by Brown et al. [212] is not appropriate over substantial parts of the warm dense regime. Therefore, Dornheim, Groth and co-workers [[222], [227]] have developed an improved formalism that allows to approach the thermodynamic limit without the loss of accuracy over the entire density–temperature plane.

Finally, these first ab initio results have very recently been used by the same authors to construct a highly accurate parametrization of the exchange–correlation free energy of the UEG covering the entire WDM regime [228], see Section 8. Thereby, a complete thermodynamic description of the uniform electron gas at warm dense matter conditions has been achieved.

  • In Section 2, we start by providing some important definitions and physical quantities that are of high relevance for the warm dense UEG. Further, we discuss the jellium Hamiltonian for a finite number of electrons in a box with periodic boundary conditions, and the corresponding Ewald summation.

  • In Section 3, we give an exhaustive introduction to the dielectric formalism within the density–density linear response theory and its application to the uniform electron gas, both in the ground state and at finite temperature. Particular emphasis is put on the STLS approach, which is extensively used throughout this work. Most importantly, it is a crucial ingredient for the accurate extrapolation of QMC data to the thermodynamic limit, see Section 6. In addition, we summarize all relevant equations that are required for the implementation and numerical evaluation of various dielectric approximations.

  • In Section 4, we briefly discuss other approximate methods that have been applied to the warm dense UEG. This includes the finite-temperature Green function approach, as well as two different classical mapping formalisms.

  • In Section 5, we provide an all-encompassing discussion of the application of quantum Monte Carlo methods to the uniform electron gas at warm dense matter conditions. We start with a brief problem statement regarding the calculation of thermodynamic expectation values in statistical physics. The solution is given by the famous Metropolis algorithm, which constitutes the backbone of most finite-temperature quantum Monte Carlo methods (Section 5.1). Undoubtedly, the most successful among these is the path integral Monte Carlo method (Section 5.2), which, unfortunately, breaks down for electrons in the warm dense matter regime due to the notorious fermion sign problem (Section 5.2.3). Two possible workarounds are given by our novel permutation blocking PIMC (Section 5.4) and configuration PIMC (Section 5.5) methods, which we both introduce in detail. Further mentioned are the approximate restricted PIMC method (Section 5.3) and the recent independent density matrix QMC approach (Section 5.6). The section is concluded with a thorough comparison between results for different quantities by all of these methods for a finite number of electrons (Section 5.7).

  • In Section 6, we discuss the extrapolation of QMC data that has been obtained for a finite number of electrons to the thermodynamic limit. A brief introduction and problem statement (Section 6.1) is followed by an exhaustive discussion of the theory of finite-size effects (Section 6.2). Due to the demonstrated failure of pre-existing extrapolation schemes, in Section 6.3 we present our improved finite-size correction and subsequently illustrate its utility over the entire warm dense matter regime (Section 6.4).

  • In Section 7, we use our new data for the thermodynamic limit to gauge the accuracy of the most important existing approaches, both for the interaction energy and the static structure factor.

  • In Section 8, we give a concise introduction (Section 8.1) of the state of the art of parametrizations of the exchange–correlation energy of the warm dense uniform electron gas, and of their respective construction (Section 8.2). Particular emphasis is put on the parametrization of the spin-dependence, Section 8.3. Finally, we provide exhaustive comparisons (Section 8.4) of fxc itself, and of derived quantities, which allows us to gauge the accuracy of the most widely used functionals.

  • In Section 9, we extend our QMC simulations to the inhomogeneous electron gas. This allows us to obtain highly accurate results for the static density response function and the corresponding local field correction (Section 9.1). As a demonstration, we give two practical examples at strong coupling using PB-PIMC (Section 9.3.1) and at intermediate coupling using CPIMC (Section 9.3.2). Further, we employ our parametrization of fxc to compute the long-range asymptotic behavior of the local field correction via the compressibility sum-rule and find excellent agreement to our QMC results.

  • In Section 10, we provide a summary and give an outlook about future tasks and open questions regarding the warm dense electron gas.

Section snippets

Basic parameters of the warm dense UEG

In the following, we introduce the most important parameters and quantities regarding the warm dense electron gas. Observe, that Hartree atomic units are assumed throughout this work, unless explicitly stated otherwise. Of high importance is the above mentioned density parameter (often denoted as Wigner–Seitzradius, or Brueckner parameter) rs=34πn13,which is independent of temperature and spin-polarization and solely depends on the combined density of both spin-up and -down electrons, n=n+n.

Finite-temperature (Matsubara) Green functions

An alternative derivation of the dielectric function encountered in the previous section can be achieved within the framework of quantum kinetic theory [240]. In this formalism, correlation effects are usually incorporated by approximating the collision integrals, which take the role of the local field correction in the dielectric formulation. For instance, completely neglecting collisions gives the random phase approximation, whereas invoking the relaxation time approximation [[258], [259]]

Quantum Monte Carlo methods

In the following section, we will discuss in detail various quantum Monte Carlo methods and discuss the fermion sign problem, which emerges for the simulations of electrons. In particular, we introduce the Metropolis algorithm [270], which constitutes the backbone of all subsequent path integral Monte Carlo methods except the density matrix QMC paradigm. Not mentioned are the multilevel blocking idea by Mak, Egger and co-workers [[271], [272], [273], [274], [275]] and the expanded-ensemble

Introduction and problem statement

The big advantage of using the quantum Monte Carlo methods introduced in Section 5 is that they – in stark contrast to the dielectric approximations or quantum–classical mappings – allow to obtain an exact solution to the UEG Hamiltonian, Eq. (7). However, this is only possible for a model system with a finite number of particles N and a finite box length L. In practice, we are interested in the thermodynamic limit [328], i.e., the limit where both L and N go to infinity while the density n

Benchmarks of other methods

The improved finite-size correction introduced in the previous section has subsequently been used to obtain an exhaustive and very accurate data set for the interaction energy for different temperature–density combinations and four different spin-polarizations (ξ=0, ξ=13, ξ=0.6, and ξ=1), see Refs. [[227], [228]]. The computational effort for a single data point is entirely determined by the actual value of the average sign which depends on rs,Θ,ξ, and N, cf. the detailed discussion in Section 

Summary and discussion

The present work has been devoted to the thermodynamic description of the uniform electron gas at warm dense matter conditions — a topic of high current interest in many fields including astrophysics, laser plasmas and material science. Accurate thermodynamic data for these systems are crucial for comparison with experiments and for the development of improved theoretical methods. Of particular importance are such data as input for many-body simulations such as the ubiquitous density functional

Acknowledgments

We are grateful to Tim Schoof and W.M.C. Foulkes for many stimulating discussions. Moreover, we acknowledge Jan Vorberger for providing the Montroll–Ward and e4 data for the interaction energy shown in Fig. 28, Fig. 30, Travis Sjostrom for the Vashista–Singwi data for the interaction energy shown in Figs. 28 and 30, the static structure factor, Fig. 31, and the local field correction depicted in Fig. 43, and Shigenori Tanaka for the results from his HNC-based dielectric method for the static

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