Dynamic properties of the warm dense electron gas: an ab initio path integral Monte Carlo approach
Paul Hamann, Tobias Dornheim, Jan Vorberger, Zhandos A. Moldabekov, Michael Bonitz
DDynamic properties of the warm dense electron gas: an ab initio path integral Monte Carlo approach
Paul Hamann, Tobias Dornheim, ∗ Jan Vorberger, Zhandos A. Moldabekov,
4, 5 and Michael Bonitz Institut f¨ur Theoretische Physik und Astrophysik,Christian-Albrechts-Universit¨at zu Kiel, Leibnizstraße 15, D-24098 Kiel, Germany Center for Advanced Systems Understanding (CASUS), G¨orlitz, Germany Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, D-01328 Dresden, Germany Institute for Experimental and Theoretical Physics,Al-Farabi Kazakh National University, 71 Al-Farabi str., 050040 Almaty, Kazakhstan Institute of Applied Sciences and IT, 40-48 Shashkin Str., 050038 Almaty,Kazakhstan
There is growing interest in warm dense matter (WDM) – an exotic state on the border betweencondensed matter and plasmas. Due to the simultaneous importance of quantum and correlationeffects WDM is complicated to treat theoretically. A key role has been played by ab initio pathintegral Monte Carlo (PIMC) simulations, and recently extensive results for thermodynamic quanti-ties have been obtained. The first extension of PIMC simulations to the dynamic structure factor ofthe uniform electron gas were reported by Dornheim et al. [Phys. Rev. Lett. , 255001 (2018)].This was based on an accurate reconstruction of the dynamic local field correction. Here we extendthis concept to other dynamical quantities of the warm dense electron gas including the dynamicsusceptibility, the dielectric function and the conductivity.
I. INTRODUCTION
The uniform electron gas (UEG) is one of the mostimportant model systems in quantum physics and the-oretical chemistry [1–3]. Despite its apparent simplic-ity, it offers a wealth of interesting effects like collec-tive excitations (plasmons) [4] and Wigner crystalliza-tion at low density [6, 7]. At zero temperature, moststatic properties of the UEG have been known for decadesfrom ground-state quantum Monte Carlo (QMC) simu-lations [8–12], and the accurate parametrization of theseresults [13–17] has been pivotal for the spectacular suc-cess of density functional theory (DFT) regarding thedescription of real materials [18, 19].The recent interest in warm dense matter (WDM)—an extreme state that occurs, e.g., in astrophysical ob-jects [20–22] and on the pathway towards inertial con-finement fusion [23]—has made it necessary to extendthese considerations to finite temperatures. More specif-ically, WDM is defined by two characteristic parameters,which are both of the order of unity: a) the density pa-rameter (Wigner-Seitz radius) r s = r/a B (with r and a B being the average particle distance and first Bohr ra-dius) and b) the degeneracy temperature θ = k B T /E F (with E F being the usual Fermi energy [24]). Moreover,WDM is nowadays routinely realized in the laboratory,see Ref. [25] for a review on experimental techniques, andmany important results have been achieved over the lastyears [26–30].On the other hand, the theoretical description of WDMis most challenging [31, 32] due to the complicated inter-play of 1) thermal excitations, 2) quantum degeneracy ef-fects, and 3) Coulomb scattering. For example, the non- ∗ [email protected] negligible coupling strength rules out perturbation ex-pansions [33, 34]. Semi-classical approaches like molecu-lar dynamics using quantum potentials [35, 36] fail due tostrong quantum degeneracy effects and exchange effects.While ab initio QMC methods are, in principle, capableto take into account all of these effects exactly simultane-ously, they are afflicted with the notorious fermion signproblem (FSP, see Ref. [37] for an accessible topical intro-duction). In particular, the FSP leads to an exponentialincrease in computation time both upon increasing thesystem size and decreasing the temperature [37–39], andhas been shown to be
N P -hard for a particular class ofHamiltonians [39].For this reason, it took more than three decades af-ter the celebrated ground-state results for the UEG byCeperley and Alder [8] to obtain accurate data in theWDM regime [40–43, 45, 53]. This was achieved by devel-oping and combining new QMC methods that are avail-able in complementary parameter regions [46–54]. Theseefforts have culminated in the first accurate parametriza-tions of the exchange–correlation (XC) free energy f xc ofthe UEG [43, 55, 56], which provide a complete descrip-tion of the UEG over the entire WDM regime. Moreover,these results allow for thermal DFT simulations [57, 58]in the local density approximation, and recent studies[59, 60] have revealed that thermal XC effects are in-deed crucial to correctly describe aspects of WDM suchas microscopic density fluctuations and the behaviour ofhydrogen bonds at finite temperature.While being an important milestone, it is clear that amore rigorous theory of WDM requires to go beyond thelocal density approximation. In this context, the key in-formation is given by the response of the UEG to a time-dependent external perturbation, which is fully charac- a r X i v : . [ phy s i c s . c o m p - ph ] J u l terized by the dynamic density response function [61] χ ( q, ω ) = χ ( q, ω )1 − ˜ v ( q ) (1 − G ( q, ω )) χ ( q, ω ) . (1)Here ˜ v ( q ) = 4 π/q is the Fourier transform of theCoulomb potential, χ ( q, ω ) denotes the density responsefunction of an ideal (i.e., non interacting) Fermi gas, and G ( q, ω ) is commonly known as the dynamic local fieldcorrection (LFC). More specifically, setting G ( q, ω ) = 0in Eq. (1) leads to a mean-field description of the densityresponse (known as random phase approximation, RPA),and, consequently, G ( q, ω ) contains the full frequency-and wave-number-resolved description of XC effects.Obviously, such information is vital for many applica-tions. This includes the construction of advanced, non-local XC-functions for DFT simulations [62–65], and theexchange–correlation kernel for the time-dependent DFT(TDDFT) formalism [66]. Moreover, we mention theincorporation of XC-effects into quantum hydrodynam-ics [32, 67–69], the construction of effective ion-ion po-tentials [70–72], and the interpretation of WDM experi-ments [73, 74]. Finally, the dynamic density response ofthe UEG can be directly used to compute many materialproperties such as the electronic stopping power [75, 76],electrical and thermal conductivities [77, 78], and energytransfer rates [79].Yet, obtaining accurate data for χ ( q, ω ) and relatedquantities has turned out to be very difficult. In theground state, Moroni et al. [9, 80] obtained QMC datafor the density response function and LFC in the staticlimit (i.e., for ω = 0) by simulating a harmonically per-turbed system and subsequently measuring the actualresponse. Remarkably, this computationally expensivestrategy is not necessary at finite temperatures, as thefull wave-number dependence of the static limit of thedensity response χ ( q ) = χ ( q,
0) can be obtained from asingle simulation of the unperturbed system [81, 82], seeEq. (12) below. In this way, Dornheim et al. [82] wererecently able to provide extensive path integral MonteCarlo (PIMC) data both for χ ( q ) and G ( q ) for the warmdense UEG, which, in combination with the ground-statedata [9, 83], has allowed to construct a highly accuratemachine-learning based representation of G ( q ; r s , θ ) cov-ering the entire relevant WDM regime. Moreover, PIMCresults for the static density response have been presentedfor the strongly coupled electron liquid regime ( r s ≥ r s ≤ .
5) [85].Finally, we mention that even the nonlinear regime hasbeen studied by the same group [86].The last unexplored dimension is then the frequency-dependence of χ ( q, ω ), which constitutes a formidablechallenge that had remained unsolved even at zero tem-perature. Since time-dependent QMC simulations sufferfrom an additional dynamical sign problem [87, 88], pre-vious results for the dynamic properties of the UEG werebased on perturbation theories like the nonequilibriumGreen function formalism at finite temperature [33, 89]or many-body theory in the ground state [90, 91]. Fortunately, PIMC simulations give direct accessto the intermediate scattering function F [defined inEq. (6)], but evaluated at imaginary times τ ∈ [0 , β ],which is related to the dynamic structure factor S ( q, ω )by a Laplace transform, F ( q, τ ) = (cid:90) ∞−∞ d ω S ( q, ω ) e − τω . (2)The numerical solution of Eq. (2) for S ( q, ω ) is a well-known, but notoriously difficult problem [92]. While dif-ferent approaches based on, e.g., Bayes theorem [93] orgenetic optimization [94, 95] exist, it was found necessaryto include additional information into this reconstructionprocedure to sufficiently constrain the results for S ( q, ω ).In order to do so, we have introduced a stochastic sam-pling procedure [81, 96] for the dynamic LFC, which al-lows to automatically fulfill a number of additional exactproperties. Thus, we were able to present the first unbi-ased results for the dynamic structure factor of the warmdense UEG without any approximation regarding XC ef-fects.In the present work, we further extend these consider-ations and adapt our reconstruction procedure to obtainother dynamic properties of the UEG such as the di-electric function (cid:15) ( q, ω ), the conductivity σ ( q, ω ) and thedensity response function χ ( q, ω ) itself. Further, we an-alyze the respective accuracy of different quantities andfind that the comparably large uncertainty in the dy-namic LFC G ( q, ω ) has only small impact on physicalproperties like χ ( q, ω ), (cid:15) ( q, ω ), and S ( q, ω ), which arewell constrained by the PIMC results. Thus, this workconstitutes a proof-of-concept investigation and opens upnew avenues for WDM theory, electron liquid theory andbeyond.The paper is organized as follows: In Sec. II we sum-marize the main formulas of linear response theory andintroduce our PIMC approach to the dynamic local fieldcorrection. In Sec. III we present our ab initio simulationresults for the local field correction, the dynamic struc-ture factor, the density response function, the dielectricfunction, and the dynamic conductivity. We concludewith a summary and outlook in Sec. IV where we give aconcise list of future extensions of our work. II. THEORY AND SIMULATION IDEAA. Path integral Monte Carlo
The basic idea of the path integral Monte Carlomethod [97, 98] is to evaluate thermodynamic expecta-tion values by stochastically sampling the density matrix, ρ ( R a , R b , β ) = (cid:104) R a | e − β ˆ H | R b (cid:105) , (3)in coordinate space, with β = 1 /k B T being the inversetemperature and R = ( r , . . . , r N ) T containing the coor-dinates of all N particles. Unfortunately, a direct eval-uation of Eq. (3) is not possible since the kinetic and τ / ε x FIG. 1. Schematic illustration of Path Integral MonteCarlo—Shown is a configuration of N = 3 electrons with P = 6 imaginary–time propagators in the x - τ plane. Dueto the single pair-exchange, the corresponding configura-tion weight W ( X ) [cf. Eq. (5)] is negative. Reprinted fromT. Dornheim et al. , J. Chem. Phys. , 014108 (2019) [100]with the permission of AIP Publishing. potential contributions ˆ K and ˆ V to the Hamiltonian donot commute, e − β ˆ H = e − β ˆ K e − β ˆ V + O (cid:0) β (cid:1) . (4)To overcome this issue, one typically employs a Trotterdecomposition [99] and finally ends up with an expressionfor the (canonical) partition function Z of the form Z = (cid:90) d X W ( X ) , (5)with the meta-variable X = ( R , . . . , R P − ) T being a so-called configuration , which is taken into account accord-ing to the corresponding configuration weight W ( X ).This is illustrated in Fig. 1, where we show an exem-plary configuration of N = 3 electrons. First and fore-most, we note that each particle is now represented byan entire path of P = 6 coordinates located on different imaginary-time slices, which are separated by a time-stepof (cid:15) = β/P . In particular, P constitutes a convergenceparameter within the PIMC method and has to be chosensufficiently high to ensure unbiased results, cf. Eq. (4).For completeness, we mention that the convergence withrespect to P can, in principle, be accelerated by usinghigher-order factorizations of the density matrix, e.g.,Refs. [101, 102]. This, however, is not advisable for thepresent study, since it would also limit the number of τ -points, on which the density–density correlation function F ( q, τ ) can be evaluated, cf. Eq. (6) below. In addition,the formulation of the PIMC method in imaginary-timeallows for a straightforward evaluation of imaginary-timecorrelation functions, such as F ( q, τ ) = 1 N (cid:104) ˆ ρ ( q, τ )ˆ ρ ( − q, (cid:105) , (6) where the two density operators are simply evaluated attwo different time slices.For the PIMC method, one uses the Metropolis algo-rithm [103] to stochastically generate a Markov chain ofconfigurations X , which can then be used to compute thethermodynamic expectation value of an arbitrary observ-able ˆ A , (cid:104) ˆ A (cid:105) MC = 1 N MC N MC (cid:88) k =1 A ( X i ) . (7)Here (cid:104) ˆ A (cid:105) MC denotes the Monte Carlo estimate, whichconverges towards the exact expectation value in thelimit of a large number of random configurations (cid:104) ˆ A (cid:105) = lim N MC →∞ (cid:104) ˆ A (cid:105) MC , (8)and the statistical uncertainty (error bar) decreases as∆ A MC ∼ √ N MC . (9)Thus, both the factorization error with respect to P andthe MC error [Eq. (9)] can be made arbitrarily small, andthe PIMC approach is quasi-exact .An additional obstacle is given by the fermion signproblem, which follows from the sign changes in the con-figuration weight W ( X ) due to different permutations ofparticle paths. In particular, configurations with an oddnumber of pair permutations (such an example is shownin Fig. 1) result in negative weights, which is a directconsequence of the antisymmetry of the fermionic densitymatrix under particle exchange. This leads to an expo-nential increase in computation time with increasing thesystem size N or decreasing the temperature. However, amore extensive discussion of the sign problem is beyondthe scope of the present work, and has been presentedelsewhere [37, 100].For completeness, we mention that all PIMC data pre-sented in this work have been obtained using a canoni-cal adaption [104] of the worm algorithm introduced byBoninsegni et al. [105, 106]. B. Stochastic sampling of the dynamic LFC
In this section, we describe the numerical solution ofEq. (2) based on the stochastic sampling of the dynamiclocal field correction G ( q, ω ) introduced in Refs. [81, 96].In principle, the task at hand is to find a trial solution S trial ( q, ω ), which, when being inserted into Eq. (2), re-produces the PIMC data for F ( q, τ ) within the givenMonte Carlo error bars. This, however, is a notoriouslydifficult and, in fact, ill-posed problem [92], as differ-ent trial solutions with distinct features might reproduce F PIMC ( q, τ ) within the given confidence interval.To further constrain the space of possible trial solu-tions, one might consider the frequency moments of thedynamic structure factor, which are defined as (cid:104) ω k (cid:105) = (cid:90) ∞−∞ d ω ω k S ( q, ω ) . (10)For the UEG, four frequency moments are known fromdifferent sum-rules, namely k = − , , ,
3. The corre-sponding equations are summarized in Ref. [81], and neednot be repeated here.For some applications [107, 108], the frequency mo-ments have been shown to significantly improve the qual-ity of the reconstruction procedure. For the UEG, on theother hand, the combined information from F ( q, τ ) and (cid:104) ω k (cid:105) is not sufficient to fully determine the shape andposition of the plasmon peaks.To overcome this issue, Dornheim and co-workers [81,96] proposed to further constrain the space of possible so-lutions by automatically fulfilling a number of exact prop-erties of the dynamic LFC G ( q, ω ). The central equationfor this strategy is the well-known fluctuation–dissipationtheorem [1], which gives a relation between S ( q, ω ) andthe dynamic density response function χ ( q, ω ), S ( q , ω ) = − Im χ ( q , ω ) πn (1 − e − βω ) . (11)The latter is then expressed in terms of the density re-sponse function of the noninteracting system and the dy-namic LFC, see Eq. (1) above. Therefore, using Eqs. (11)and (1), we have recast the reconstruction problem posedby Eq. (2) into the search for a suitable trial solution forthe dynamic LFC, G trial ( q, ω ).The important point is that many additional exactproperties of G ( q, ω ) are known from theory. Since,again, all formulas are listed in Ref. [81], here we givea brief summary:1. The Kramers-Kronig relations give a direct con-nection between the real and imaginary parts of G ( q, ω ) in the form of a frequency-integral.2. Re G ( q, ω ) and Im G ( q, ω ) are even and odd func-tions with respect to ω , respectively.3. The imaginary part vanishes for ω = 0 and ω → ∞ .4. The static ( ω = 0) limit of Re G ( q, ω ) can be di-rectly obtained from χ ( q ,
0) = − n (cid:90) β d τ F ( q , τ ) , (12)and Eq. (1). Further, an accurate neural-net repre-sentation of G ( q, ω = 0) was presented in Ref. [82].5. The high-frequency ( ω → ∞ ) limit of Re G ( q, ω )can be computed from the static structure fac-tor S ( q ) and the exchange–correlation contribu-tion to the kinetic energy. The latter is obtainedfrom the accurate parametrization of the exchange–correlation free energy f xc by Groth et al. [43]. To generate trial solutions for G ( q, ω ) that automat-ically fulfill these constraints, we follow an idea byDabrowski [109] and introduce an extended Pad´e formulaof the formIm G ( q , ω ) = a ω + a ω + a ω ( b + b ω ) c , (13)where a − , b , , and c are chosen randomly. The cor-responding real part of this trial solution G trial ( q, ω ) issubsequently computed from the Kramers-Kronig rela-tion, see Ref. [81] for a more detailed discussion.During the reconstruction procedure, we 1) randomlygenerate a large set of parameters T = { a i , b i , c } , 2) usethese to obtain both Im G trial ( q, ω ) and Re G trial ( q, ω ),3) Insert these into Eq. (1) to compute the correspond-ing χ trial ( q, ω ), 4) insert the latter into the fluctuation–dissipation theorem to get a dynamic structure factor S trial ( q, ω ), and 5) compare S trial ( q, ω ) to our PIMC datafor F ( q, τ ) (for all τ ∈ [0 , β ]) and the frequency moments (cid:104) ω k (cid:105) . The small subset of M trial parameters T thatreproduce both F ( q, τ ) and (cid:104) ω k (cid:105) are kept to obtain thefinal result for physical quantities of interest, like S ( q, ω )itself, but also χ ( q, ω ) and (cid:15) ( q, ω ).For example, the final solution for the dynamic struc-ture factor is given by S final ( q , ω ) = 1 M M (cid:88) i =1 S trial ,i ( q , ω ) . (14)Moreover, this approach allows for a straightforward esti-mation of the associated uncertainty as the correspondingvariance ∆ S ( q , ω ) = (cid:32) M M (cid:88) i =1 [ S trial ,i ( q , ω ) (15) − S final ( q , ω )] (cid:17) / . C. Dielectric function and inverse dielectricfunction
Having obtained ab initio results for the density re-sponse function χ ( q, ω ), it is straightforward to obtainthe dynamic retarded dielectric function [110] as well asthe inverse dielectric function, (cid:15) ( q, ω ) = 1 − ˜ v ( q )Π( q, ω ) , (16) (cid:15) − ( q, ω ) = 1 + ˜ v ( q ) χ ( q, ω ) , (17)where Π is the retarded polarization function. Its relationto the density response function isΠ( q, ω ) = (cid:15) ( q, ω ) χ ( q, ω ) = χ ( q, ω )1 + ˜ v ( q ) χ ( q, ω ) . (18)In the limiting case of an ideal Fermi gas, χ → χ , where χ is the Lindhard response function (but at finite tem-perature), and Eq. (18) yields the RPA polarization, andthe dielectric function becomes (cid:15) RPA ( q, ω ) = 1 − ˜ v ( q ) χ ( q, ω ) . (19)Correlation effects, i.e. deviations from χ , can beexpressed in terms of the dynamic local field correction G ( q, ω ), and the dielectric function can be written as (cid:15) ( q, ω ) = 1 − ˜ v ( q ) χ ( q, ω )1 + ˜ v ( q ) G ( q, ω ) χ ( q, ω ) . (20)While (cid:15) − is commonly used in linear response theory, (cid:15) emerges naturally in electrodynamics, e.g. Ref. [111], andit is of prime importance for the description of plasmaoscillations.Let us summarize a few definitions and importantproperties of the retarded dielectric function.1. Since χ ( q, ω ) describes a causal response, (cid:15) − ( q, ω )is an analytic function in the upper frequency half-plane. Real and imaginary parts are connected viathe Kramers-Kronig relation for real frequencies.2. If G ( q, ω ) is computed via an ab initio QMC proce-dure [96], also the dielectric function has ab initio quality. We will call this result (cid:15)
DLFC ( q, ω ).3. Another important approximation is obtained byreplacing, in Eq. (20), G ( q, ω ) → G ( q,
0) = G ( q ).This is still a dynamic dielectric function which willbe denoted by (cid:15) SLFC ( q, ω ). Comparison to the fulldynamic treatment revealed that this static approx-imation provides an accurate description of the dy-namic structure factor for r s (cid:46)
4, for all wave num-bers [96].4. Since the static limit of the response function is realand negative: χ ( q, ≤ , (21)which is a necessary prerequisite for the stabilityof any system, it immediately follows for the staticdielectric function that: (cid:15) ( q, − < , (22)which implies (cid:15) ( q, (cid:54)∈ (0 , K , K − = n ∂µ∂n (23)via lim q → (cid:15) ( q,
0) = 1 + ˜ v ( q ) n K , (24)where n is the density and µ the chemical potential[1]. The practical evaluation of the compressibilityis given by Eq. (32) below. D. Dynamic conductivity
Having the dielectric function at hand, it is straightfor-ward to compute further dynamic linear response quan-tities. An example is the dynamical conductivity σ ( q, ω ),that follows from the response of the current density toan electric field with the result (cid:15) ( q, ω ) = 1 + 4 πiω σ ( q, ω ) . (25)This can be transformed into an expression for the con-ductivity in terms of the RPA response function and thedynamic local field correction, using Eq. (20), σ ( q, ω ) = i ω π ˜ v ( q ) χ ( q, ω )1 + ˜ v ( q ) G ( q, ω ) χ ( q, ω ) (26)= i ω π ˜ v ( q )Π( q, ω ) , where Π is the longitudinal polarization function (18).The analytical properties of the polarization functionin RPA at finite temperature were thoroughly investi-gated in many papers, including Refs. [112–114]. Fol-lowing these works, it is straightforward to find variouslimiting cases for the conductivity in RPA which are valu-able for comparison to the correlated results presented inSec. III F.1.) At q (cid:28) q F (and for arbitrary frequency), we havefor the real part of the conductivity in RPA:Re σ ( q, ω ) ω p = 2 √ π r / s (cid:16) qq F (cid:17) − (cid:16) ωω p (cid:17) (cid:104) αr s θ ( ω/ω p ) ( q/q F ) − η (cid:105) , (27)where α = 3(4 / π ) / (cid:39) .
221 and η = βµ .2.) At ω (cid:28) qv F (i.e., ω/ω p (cid:28) ( q/q F ) × . / √ r s ) andarbitrary wavenumber, the real part of the conductivityin RPA readsRe σ ( q, ω ) ω p = 2 √ π r / s (cid:16) qq F (cid:17) − (cid:16) ωω p (cid:17) (cid:104) ( q/q F ) θ − η (cid:105) . (28)3.) For the imaginary part of the conductivity in RPA,at high frequencies, ω (cid:29) (cid:126) q / (2 m ) and ω (cid:29) qv F , we findthe following result:Im σ ( q, ω ) ω p = 14 π (cid:16) ω p ω (cid:17) (29)+ (cid:34) r s (cid:18) π (cid:19) / (cid:104) v (cid:105) v F (cid:18) qq F (cid:19) + 364 r s (cid:18) π (cid:19) / (cid:18) qq F (cid:19) (cid:35) (cid:16) ω p ω (cid:17) + 9 π r s (cid:18) π (cid:19) / (cid:104) v (cid:105) v F (cid:18) qq F (cid:19) (cid:16) ω p ω (cid:17) + ..., τ ) τ / β q/q F FIG. 2. PIMC data for the imaginary-time density–densitycorrelation function F ( q, τ ) for N = 34 unpolarized electronsat r s = 4 and θ = 1 (WDM conditions). The red crossescorrespond to the static structure factor S ( q ) = F ( q, where (cid:104) ... (cid:105) indicates an average with the finite tempera-ture Fermi function. Analytical parametrizations for themoments, (cid:104) v (cid:105) v F and (cid:104) v (cid:105) v F , are given in the Appendix.If one neglects terms of the order O (cid:0) ( ω p /ω ) (cid:1) andhigher, i.e. retains only the O (cid:0) ( ω p /ω ) (cid:1) order term, theoften used high frequency limit for the RPA conductivityis recovered (e.g., in the Drude conductivity)[115],Im σ ( q, ω ) ω p = 14 π (cid:16) ω p ω (cid:17) . (30) III. NUMERICAL RESULTSA. Density correlation function
Let us begin the investigation of our numerical resultswith a brief discussion of the imaginary-time density–density correlation function F ( q, τ ), which constitutesthe most important input for the reconstruction proce-dure. To this end, we show F ( q, τ ) in the q - τ -plane for r s = 4 and θ = 1 in Fig. 2. Since a physically meaning-ful interpretation of this quantity is rather difficult, herewe restrict ourselves to a summary of some basic prop-erties. First and foremost, we note that F approachesthe static structure factor (see the red crosses in Fig. 2)in the limit of small τ , F ( q,
0) = S ( q ). Moreover, F issymmetric in the imaginary time around τ = β/ F ( q, τ ) = F ( q, β − τ )], and it is thus fully sufficient toshow only the range of τ ∈ [0 , β/ . . . . . . τ /β . . . . . F ( q , τ ) DLFCRPASLFCPIMC 0 .
40 0 .
45 0 . . . FIG. 3. Imaginary-time density-density correlation function F ( q, τ ) for r s = 4, θ = 1 at q/q F ≈ .
63. The PIMC data iscompared to the reconstruction result.
Regarding physical parameters, the density–temperature combination depicted in Fig. 2 correspondsto a metallic density in the WDM regime. This issomewhat reflected in Fig. 2 by the amount of structurein the surface plot, in particular the maximum around q ≈ . q F . For example, for larger coupling strengththe UEG forms an electron liquid and F ( q, τ ) exhibitsa more pronounced structure with several maxima andminima [84]. For decreasing r s , on the other hand,electronic correlation effects become less important andone approaches the high-energy-density regime, where F ( q, τ ) exhibits even less structure than for the presentexample, and the only maximum is shifted to smallervalues of q/q F , see Ref. [85].Let us next briefly touch upon the utility of theimaginary-time density–density correlation function forthe reconstruction of dynamic quantities like the dynamicstructure factor S ( q, ω ). This is illustrated in Fig. 3,where we show the τ -dependence of F for a fixed wavenumber q ≈ . q F . The blue points correspond to ourPIMC data, and the three curves have been obtainedby inserting different solutions for the dynamic structurefactor into Eq. (2), see the bottom left panel of Fig. 4for the corresponding depiction of S ( q, ω ). Let us startwith the green curve, which shows the random phase ap-proximation (RPA). Evidently, the mean field descriptionexhibits severe deviations from the exact PIMC data andis too low by ∼
10% over the entire τ -range. Moreover,this shift is not constant, and the RPA curve exhibits afaster decay with τ compared to the blue points. This isconsistent with previous studies [3, 42, 50] of the staticstructure factor S ( q ), where RPA has been shown to givesystematically too low results for all wave numbers.In contrast, the dashed black curve has been obtainedon the basis of the static approximation , i.e., by setting FIG. 4. Dynamic structure factor S ( q, ω ) at θ = 1 for three wave numbers. Top: r s = 10, bottom: r s = 4. Green: RPA,dashed: SLFC, red: DLFC. G ( q, ω ) = G ( q,
0) in Eq. (1). Evidently, this leads toa substantially improved imaginary-time density–densitycorrelation function, and the black curve is within theMonte-Carlo error bars over the entire τ -range. Finally,the solid red curve has been obtained by stochasticallysampling the full frequency-dependence of G ( q, ω ) as de-scribed in Sec. II B. While this does lead to an even betteragreement to the PIMC data, one cannot decide betweenthe two solutions on the basis of F ( q, τ ) alone. This fur-ther illustrates the need for the incorporation of the ex-act constraints on the stochastic sampling of G ( q, ω ), asthe static approximation and full frequency-dependenceof the LFC lead to substantially different dynamic struc-ture factors, but similar F ( q, τ ). B. Dynamic structure factor
The next quantity of interest is the dynamic structurefactor S ( q, ω ) itself, which is shown in Fig. 4 at the Fermitemperature for two different densities and for three dif-ferent wave numbers each. In this context, we recall that S ( q, ω ) constitutes a key quantity for the full reconstruc-tion of any dynamical property, as it is used as a measureof quality of the dynamic LFC G ( q, ω ) in the stochasticsampling procedure, see Sec. II B. Moreover, it is directlyaccessible in XRTS experiments [73] and is of paramountimportance for plasma diagnostics [74], like the deter-mination of the electronic temperature. For this rea-son, S ( q, ω ) has been extensively investigated in previousstudies [81, 96, 118]. Very recently, Dornheim and Vor-berger [118] have found that the ab initio PIMC resultsfor S ( q, ω ) at the Fermi temperature are not afflicted with any significant finite-size error even for as few as N = 14unpolarized electrons. In addition, Dornheim et al. [96]have presented results going from the WDM regime to thestrongly coupled electron liquid regime, where S ( q, ω ) ex-hibits a negative dispersion relation (estimated from themaximum in the DSF), which might indicate the onsetof an incipient excitonic mode [90, 91, 119]. For this rea-son, here we restrict ourselves to a brief discussion of themost important features.The top row of Fig. 4 shows results for r s = 10 and theleft, center, and right panels corresponds to q ≈ . q F , q ≈ . q F , and q ≈ . q F , respectively. For the small-est wave number, we find a relatively sharp peak slightlyabove the plasma frequency, which can be identified asthe plasmon of the UEG. Yet, while it is well knownthat this collective excitation is correctly described onthe mean-field level (i.e., within RPA, green curve) for q →
0, we find substantial deviations between the threedepicted data sets for q ≈ . q F . More specifically, thestatic approximation (dashed black) leads to a red-shift compared to RPA, and a correlation-induced broadening.In addition, this trend becomes even more pronouncedfor the full reconstructed solution (red curve), which re-sults in an almost equal position of the peak as the staticsolution, but is much broader.The center panel corresponds to an intermediate wavenumber, and we find an even more pronounced red-shift.Remarkably, the static approximation performs very welland can hardly be distinguished from the full dynamicsolution within the given confidence interval (red shadedarea). Finally, the right panel shows the DSF for approx-imately thrice the Fermi wave number, where we alsoobserve some interesting behaviour. First and foremost,all three spectra exhibit a similar width, which is muchbroader compared to the previous cases, as it is expected.Furthermore, including a local field correction in Eq. (1)leads to a red-shift, which is most pronounced for thered curve. Yet, the full frequency dependence of G ( q, ω )leads to a nontrivial shape in S ( q, ω ), which is not cap-tured by the black curve and leads to a substantially morepronounced negative dispersion relation compared to thestatic approximation [96].The bottom row of Fig. 4 corresponds to r s = 4, whichis a metallic density in the WDM regime. The moststriking difference to the electron liquid example aboveis that the spectra are comparably much broader for thesame value of q/q F , which is a direct consequence of theincreased density. Moreover, all three curves are in rela-tively good agreement, we observe only a small red-shiftcompared to RPA, and the static approximation fully de-scribes the DSF. The same holds for the two larger wavenumbers shown in the center and right panels, althoughhere the red-shift to the mean-field description is some-what larger.In accordance with Ref. [96], we thus conclude thatthe static approximation provides a nearly exact descrip-tion of S ( q, ω ) for weak to moderate coupling ( r s (cid:46) χ ( q, ω ) [Sec. III D] and dielectric function (cid:15) ( q, ω )[Sec. III E] is one of the central goals of this work. C. Local field correction
Before we move on to the investigation of χ ( q, ω ) and (cid:15) ( q, ω ), it is well worth to briefly examine the recon-structed dynamic local field correction. In particular,the dynamic LFC constitutes the basis for the compu-tation of all other dynamic properties from the ab ini-tio PIMC data and, thus, is of central importance forour reconstruction scheme. Since the stochastic samplingand subsequent elimination/verification of trial solutionsfor G ( q, ω ) has been extensively discussed by Groth etal. [81], here we restrict ourselves to the discussion of atypical example shown in Fig. 5 for θ = 1, r s = 6, and q ≈ . q F . These parameters are located at the mar-gins of the WDM regime with a comparably large im-pact of electronic correlation effects and can be realizedexperimentally in hydrogen jets [120] and evaporation ex-periments, e.g. at the Sandia Z-machine [60, 121–123].Furthermore, the selected wave number is located in themost interesting regime, where the position of the max-imum of S ( q, ω ) exhibits a non-monotonous behaviourand the impact of G ( q, ω ) is expected to be most pro-nounced.In the top panel, we show the frequency-dependence ofthe real part of G , which exhibits a fairly nontrivial pro-gression: starting from the exact static limit G ( q,
0) di- . . . . . . R e G ω/ω P . . . . . I m G FIG. 5. Average solution obtained for the real and imaginaryparts of the dynamic local field correction, G ( q, ω ), at θ = 1, r s = 6, and q ≈ . q F . The shaded area corresponds to therange of valid reconstructions. rectly known from our PIMC data for χ ( q ), the dynamicLFC exhibits a maximum around ω ≈ . ω p followed byshallow minimum around ω ∼ ω p and then monoton-ically converges to the also exactly known ω → ∞ limitfrom below. Moreover, the associated uncertainty inter-val (light grey shaded area) is relatively small, and themaximum appears to be significant, whereas the mini-mum is probably not. At this point, we mention that thedynamic LFC is afflicted with the largest relative uncer-tainty of all reconstructed quantities considered in thiswork, which can be understood as follows: by design,the LFC contains the full information about exchange–correlation effects beyond RPA. In the limit of small andlarge wave numbers, the RPA is already exact and, con-sequently, the dynamic LFC has no impact on the re-constructed solutions for S ( q, ω ) that are compared tothe PIMC data using Eq. (2). Naturally, the same alsoholds for increasing temperature and density where theimportance of G ( q, ω ) also vanishes [85].In the bottom panel of Fig. 5, we show the correspond-ing imaginary part of G for the same conditions. Inter-estingly, Im G exhibits a seemingly less complicated be-haviour featuring a single maximum around ω ≈ ω p ,and vanishing both, in the high and low frequency limitsfrom above.We thus conclude that our reconstruction scheme al-lows us to obtain accurate results for the dynamic LFC − q/q F ≈ . R e χ · ω P / n a) − . − . . . q/q F ≈ . − . − . . q/q F ≈ . ω/ω p − I m χ · ω P / n ω/ω p . . . ω/ω p . . . . . − . . . q/q F ≈ . R e χ · ω P / n d) − . . q/q F ≈ . − . − . . . q/q F ≈ . ω/ω p . . . − I m χ · ω P / n ω/ω p . . . .
75 0 10 20 ω/ω p . . . FIG. 6. Dynamic density response function at θ = 1, for three wave numbers. a)–c): r s = 10; d)–f): r s = 4. Top (bottom) ofeach figure: real (minus imaginary part). Comparison of RPA (green), and PIMC simulation results using the static (SLFC,dashed) and dynamic (DLFC, red) local field correction. when it has impact on physical observables like S ( q, ω ),i.e., precisely when it is needed in the first place. Thisallows for the intriguing possibility to construct a both q - and ω -dependent representation of G for some param-eters, which could then be used for many applicationslike a real time-dependent DFT simulation without theadiabatic approximation. D. Dynamic density response function χ The present procedure to reconstruct the dynamicstructure factor via a reconstruction of G ( q, ω ) can bestraightforwardly extended to other dynamic quantities.The first example is the density response function de-fined by Eq. (1). Here we extend our preliminary results[32] and present more extensive data for two densities of interest.The top half of Fig. 6 corresponds to r s = 10 (strongcoupling), and panels a)-c) to three interesting wave num-bers. Similarly to the dynamic LFC discussed in the pre-vious section, we find that the real part of χ exhibitsa more complicated behaviour compared to the imagi-nary part. More specifically, the latter vanishes bothin the high- and low-frequency limits, whereas the for-mer attains a finite value in the static case. In addition,Re χ ( q, ω ) has a remarkable structure in between, with apole-like structure owing to the Kramers-Kronig relationbetween imaginary and real part. This is the excitationof density fluctuations visible as a peak in the imaginarypart, as translated to the real part and this structure.For the smallest depicted wave number, the excitationrange is narrow and the position of the zero crossing ofRe χ ( q, ω ) almost exactly coincides with the position of0 q/q F ≈ . − Im (cid:15) − Re (cid:15) . . . . . . ω/ω p q/q F ≈ . − Im (cid:15) − Re (cid:15) q/q F ≈ . (cid:15) . . . . . . ω/ω p q/q F ≈ . (cid:15) FIG. 7.
Left : Real part of the dielectric function (cid:15) ( q , ω ) and imaginary part of the inverse dielectric function. Right : Imaginarypart of the dielectric function, for r s = 6, θ = 1. The peak of -Im (cid:15) − [and of S ( q , ω )] is in the vicinity of the second root ofRe (cid:15) (if roots exist, as in the upper figure). Green lines: RPA, red (dashed) lines: dynamic (static) PIMC results, for detailssee text. the maximum in both Im χ ( q, ω ) and S ( q, ω ). In contrast,a broader excitation at larger values of q leads to a shiftedfeature in the real part for q ≈ . q F and q ≈ . q F .Furthermore, we note that the imaginary part closely re-sembles the dynamic structure factor S ( q, ω ) [which isa direct consequence of the fluctuation–dissipation the-orem, Eq. (11)], and the associated physics, thus, neednot be further discussed at this point.Let us next examine the difference between the threedifferent depicted solutions for the dynamic density re-sponse function. Due to the strong electronic correla-tions, the RPA only provides a qualitative description,as it is expected. Including a local field correction doesnot only lead to a red-shift, but also to a significantlychanged shape, and a substantially different static limitfor Re χ . For example, the mean field description of thedynamic density response predicts a shallow minimum inthe real part around ω = 1 . ω p for q ≈ . q F (panel b),which is not present for both the static and the dynamicLFC. Finally, we note that the static LFC leads to a clearimprovement over RPA in particular in the description ofthe peak position, but–as in the case of S ( q, ω )–cannotcapture the nontrivial behaviour of χ ( q, ω ) at q ≈ . q F .The bottom half of Fig. 6 corresponds to r s = 4 and θ = 1, which is located in the WDM regime. As for S ( q, ω ), both the real and imaginary part of χ ( q, ω ) arenot as sharply peaked as for r s = 10, as it is expected.Overall, the RPA seems to provide a somewhat better de- scription of Re χ ( q, ω ) than of Im χ ( q, ω ), although thereare substantial deviations for ω →
0. Moreover, the staticapproximation is highly accurate and cannot be distin-guished from the full solution for all three wave numbers.
E. Dynamic dielectric function
Proceeding in a similar way as for the dynamic struc-ture factor and the density response function, we nowturn to a reconstruction of the dynamic dielectric func-tion (cid:15) ( q, ω ). This function is particularly interesting, asit gives direct access to the spectrum of collective ex-citations of the plasma. Using Eq. (20), the dynamicdielectric function is directly expressed by the local fieldcorrection to which we have access in our ab initio sim-ulations. Thus, it is straightforward to directly comparethe RPA dielectric function to correlated results that useeither the static or dynamic LFC.A first typical result for the dielectric function of thecorrelated electron gas is shown in Fig. 7, for the caseof r s = 6 and θ = 1. In the left (right) panel we showthe real (imaginary) part of the dielectric function fortwo wave numbers. At large frequencies, ω (cid:38) ω p thecorrelated results are in close agreement with the RPA.However, strong deviations occur below ω p . The peakof the imaginary part narrows and shifts to much lowerfrequencies. Due to the Kramers-Kronig relations, the10 . . . . . − − q/q F ≈ . R e (cid:15) q/q F ≈ . R e (cid:15) ω/ω p . . . q/q F ≈ . R e (cid:15) FIG. 8. Real part of the dielectric function at different valuesof q for r s = 10, θ = 1. Green lines: RPA, red (dashed) lines:dynamic (static) PIMC results. For small wave vectors, inthe ab initio results, the static limit becomes negative with alarge absolute value, for details see text. same trend is observed for the real part. The statisti-cal uncertainty of the reconstruction of G ( q, ω ) leads toan uncertainty in the region of the peak of Im (cid:15) that isindicated by the red band. Interestingly, the static ap-proximation is very close to the full dynamic results atthe present parameters. In the left part of the figure,we also show the imaginary part of the inverse dielectricfunction, -Im (cid:15) − which is proportional to the dynamicstructure factor, cf. Eq. (11). At the lower wave number(top left figure), its peak is close to the zero of the realpart of (cid:15) .Let us now proceed to stronger coupling, to the mar-gins of the electron liquid regime ( r s = 10 and θ = 1),and discuss the behavior of the dynamic dielectric func-tion. Of particular interest is the possibility of instabili-ties [1]. In Fig. 8, we show the frequency dependence ofRe (cid:15) ( q, ω ), for q ≈ . q F (top), q ≈ . q F (center), and q ≈ . q F (bottom). For the two larger wave numbers,we find similar trends as for r s = 6, although the differ- 0 . . . . . q/q F ≈ . I m (cid:15) . . . q/q F ≈ . I m (cid:15) ω/ω p . . . . q/q F ≈ . I m (cid:15) FIG. 9. Imaginary part of the dielectric function for r s = 10, θ = 1. Green lines: RPA, red (dashed) lines: dynamic (static)PIMC results. ences between the RPA and the LFC based curves is sub-stantially larger, in particular around q = 2 q F . The toppanel, on the other hand, exhibits a peculiar behavior,which deserves special attention: while the RPA predictsa positive static limit, as it is expected, the red and blackcurves attain a comparably large (though finite) negativevalue, for ω = 0.To understand the implications of this nontrivial find-ing, we show the wave number dependence of the staticlimit of Re χ ( q, ω ) in Fig. 10. The RPA curve (green line)converges to 1 from above, for large q and diverges topositive infinity for q →
0, as it is known from theory [1].The static approximation (dashed black), on the otherhand, leads to an altogether different behaviour. Whileit eventually attains the same limit for large q -values,it exhibits a highly nontrivial structure with two polesaround q ≈ q F and q ≈ . q F , and remains finite for q = 0 (with a small local maximum around q ≈ . q F ).Let us first briefly touch upon the implications of this be-haviour for the stability of the system. As we have notedin Sec. II C, the static dielectric function needs to remain2 -400-200 0 200 400 600 800 0 1 2 ε q/q F SLFCRPACSRPIMC
FIG. 10. Static limit of the dielectric function at r s = 10 and θ = 1. The blue diamonds depict PIMC data for N = 34electrons and the solid green and dashed black curves theRPA results and the neural-net representation of the staticLFC from Ref. [82]. Moreover, the solid red curve has beenobtained by using for G ( q ) the compressibility sum-rule givenby Eq. (32). outside the interval between zero and one, (cid:15) ( q, (cid:54)∈ (0 , q → q → (cid:15) ( q,
0) = lim q → − πq χ ( q, πq G ( q, χ ( q,
0) (31)= lim q → − πχ ( q, q [1 + 4 πCχ ( q, , where the second equality follows from the well-knownlimit of the static local field correctionlim q → G ( q,
0) = q C , (32)see, e.g., Refs. [81, 82] for details. Naturally, this limitwas incorporated into the training procedure for theneural-network representation of G ( q,
0) in Ref. [82].However, directly utilizing Eq. (32) to compute a dielec-tric function (which becomes exact for q →
0) leads tothe red curve in Fig. 10, which does indeed diverge to-wards positive infinity as predicted by Eq. (24). Theexplanation for the finite value at q = 0 (and also the un-smooth behaviour for small q ) of the dashed black curveis given by the construction of the neural-network rep-resentation itself. As any deviations both from Eq. (32)for small q , and the ab initio PIMC input data elsewhere, -0.2 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 K / K r s GDSMFBSTLSVSRPIMC
FIG. 11. Density dependence of the ratio of the ideal to theinteracting compressibility for θ = 1. The solid green line hasbeen obtained from the parametrization of f xc by Groth etal. [43], and the dashed black, dotted blue, and dash-dottedred lines have been taken from Ref. [124] and correspond toSTLS, VS, and restricted PIMC, respectively. were equally “punished” by the loss function, the result-ing neural network exhibits an overall absolute accuracyof ∆ G ∼ .
01 and, thus, does not exactly go to zero inthe long wave-number limit. We, thus, conclude that us-ing the neural-network representation from Ref. [46] leadsto the exact q → χ ( q ) and S ( q ), but becomes inaccurate for effective,dielectric properties like (cid:15) ( q ) and Π( q ) in this regime.Let us conclude this section with a brief discussion ofthe eponymous quantity of Eq. (24), i.e., the compress-ibility K . In Fig. 11, we show the r s -dependence of theratio of the noninteracting to the interacting compress-ibility, for θ = 1, and the solid green line has been ob-tained from the accurate PIMC-based parametrization of f xc by Groth and co-workers [43]. In addition, we alsoshow results from (static) dielectric theories investigatedby Sjostrom and Dufty [124], namely STLS [125, 126](dashed black) and VS [127, 128] (dotted blue). Fi-nally, the dash-dotted red curve has been obtained bythe same authors on the basis of the restricted PIMCdata by Brown et al. [40].First and foremost, we note that all four curves exhibita qualitatively similar trend and approach the correctlimit for r s →
0, where the UEG becomes ideal. Withincreasing coupling strength, the compressibility is re-duced as compared to K . Moreover, K does eventuallybecome negative around r s ≈ .
5. This has some inter-esting implications and indicates, that the static dielec-tric function (cid:15) ( q,
0) converges towards negative infinity inthe long wavelength limit. Lastly, we compare the threeapproximate curves from Ref. [124] against the accurateGDSMFB-benchmark and find that RPIMC and STLSexhibit relatively small systematic deviations, whereasthe VS-curve deviates the most. This is certainly remark-3able as the closure relation of the VS-formalism stronglydepends on a consistency relation of K towards both G ( q,
0) and f xc , see Ref. [124] for a detailed discussionof this point. F. Ab initio results for the dynamic conductivity
Let us start our investigation of the dynamic conduc-tivity with an analysis of the four asymptotics introducedin Sec. II D. This is shown in Fig. 12 corresponding to r s = 2 ,
4, and r s = 10 and three wavenumbers. Theagreement to the full RPA solution (solid green curve) isvery good. First, we note that it is evident from Eq. (27)that the real part of the RPA conductivity tends to zero,if the limit q → q/q F ≈ .
63. Second, Re σ RPA tends to zero as ω if the limit ω → q (grey lines), cf. Eqs. (27) and (28). Third,the asymptotic behaviour from Eq. (29) (cf. dotted bluelines) correctly reproduces the behavior of Im σ RPA forfrequencies larger than the first positive maximum, forall q , and with further frequency increase merges withEq. (30) (solid blue line) which correctly captures thehigh frequency limit.Having ab initio results for the local field correctionand the longitudinal polarization available allows us toproduce accurate results for the conductivity includingexchange and correlation effects as well. Simulation re-sults for the conductivity within the static and dynamicLFC approximation are presented in Fig. 12 by the blackdashes lines and red bands, respectively. First, we ob-serve that the static and dynamic results for the conduc-tivity are in very good agreement with each other for allcases. Small deviations are visible mainly for r s = 10.Second, the agreement of the RPA conductivity with thecorrelated approximations is reasonable for r s = 2. Atstronger coupling, r s = 4, good agreement is observedonly at the largest wavenumber, q/q F ≈ .
94, whereasfor lower q deviations are growing. At r s = 10 agree-ment is observed only at large frequencies, whereas thebehavior around the main peak of Re σ as well as forfrequencies below the peak shows dramatic influences ofcorrelations, except for the largest q .We note that the present analysis is complementaryto the approaches for the conductivity executed in theoptical limit, q →
0. In this limit, the focus is usu-ally on incorporating electron-ion correlations that canbe shown to be the dominant non-ideality contributionin that limit. Examples include linear response calcula-tions as well as DFT approaches using Kubo-Greenwoodrelations [77, 78, 115, 116, 122, 123]. Electron-electronand electron-ion correlations have been taken into ac-count simultaneously, but only for the non-degenerate,weakly coupled case [77]. The current work opens up thepossibility to improve this description and extend it tothe warm dense matter regime with all its correlationsand quantum effects causing a non-Drude form of the conductivity.
IV. SUMMARY AND DISCUSSION
In this work, we have investigated in detail the cal-culation of dynamic properties based on ab initio
PIMCdata for the warm dense electron gas. More specifically,we have discussed the imaginary-time version of the in-termediate scattering function F ( q, τ ), which is used asa starting point for the reconstruction of the dynamicstructure factor S ( q, ω ). This is achieved by a recentstochastic sampling scheme of the dynamic local fieldcorrection [81, 96], which, in principle, gives the com-plete wave-number- and frequency- resolved descriptionof exchange–correlation effects in the system.In particular, we have demonstrated that such knowl-edge of G ( q, ω ) allows for the subsequent accurate calcu-lation of other dynamic quantities such as the dynamicdensity response function, χ ( q, ω ), the (inverse) dielec-tric function (cid:15) ( q, ω ) [1 /(cid:15) ( q, ω )], and the dynamic conduc-tivity σ ( q, ω ). Therefore, our new results will open upavenues for future research, as they contain key infor-mation about different properties of the system: χ ( q, ω )fully describes the response of the system to an externalperturbation, e.g. by a laser beam [1, 86]; the dielectricfunction is of paramount importance in electrodynamicsand gives access to the full spectrum of collective excita-tions [110]. This is a fundamental point, which will beexplored in detail in a future publication [129]; the dy-namic conductivity in warm dense matter is of particularimportance for magneto-hydrodynamics, e.g., planetarydynamos [117].An additional key point of this paper is the inves-tigation of the so-called static approximation [81, 96],where the the exact dynamic LFC is replaced by its ex-act static limit G ( q,
0) that has recently become availableas a neural-net representation [82]. Here we found thatthis approach allows basically for exact results for all dy-namic quantities mentioned above over the entire q - and ω -range for r s (cid:46)
4, i.e., over substantial parts of theWDM regime. This has important applications for manyaspects of WDM theory such as the on-the-fly interpre-tation of XRTS experiments [73, 74], as it comes withno additional computational cost compared to RPA. Forlarger values of r s , the static approximation does inducesignificant deviations to the exact results, but it neverthe-less reproduces the most important trends of the variousdynamical properties that are absent in an RPA-baseddescription.Our investigation of the dynamic dielectric functionhas uncovered that electronic exchange–correlation ef-fects [either by using G ( q, ω ) or G ( q, (cid:15) ( q, ω ), where the static limit can ac-tually become negative for certain wave numbers. Thisis i) a correct physical behaviour and ii) does not signalthe onset of an instability, and neither does the nega-tive compressibility depicted in Fig. 11. In contrast, our4 FIG. 12. Real and imaginary part of the dynamic conductivity at θ = 1 for three densities – Top: r s = 2, middle: r s = 4,bottom: r s = 10 – and three wavenumbers – left: q/q F ≈ .
63, middle q/q F ≈ .
25, and right: q/q F ≈ . Ab initio simulation results using the static (black dashed line) and dynamic (red bands) LFC are compared to the RPA (full green line)and analytical asymptotics: For Re σ approximations Eq. (28) and Eq. (27) are shown by the solid gray and dotted green line(for q/q F ≈ .
63 only), respectively. For Im σ the approximations Eq. (29) and Eq. (30) are shown by the dotted blue andsolid blue lines, respectively. (cid:15) ( q, G ( q,
0) does induce an artificial,unphysical behaviour in the q → S ( q, ω ), S ( q ), or χ ( q ) are not afflicted with thisissue, as here the impact of the LFC vanished for smallwave numbers. Lastly, the conductivity is afflicted thesame way as the dielectric function, but this might bealleviated if e-i scattering is included [115].Future extensions of our research include the imple-mentation of other imaginary-time correlation functionsinto our PIMC simulations. Possible examples are givenby the Matsubara Green function [105, 107] or the ve-locity autocorrelation function [130], which would giveaccess to the single-particle spectrum or a dynamical dif-fusion constant, respectively. Furthermore, the combina-tion of PIMC simulations with the reconstruction schemeexplored in this work can potentially be applied to realelectron-ion-plasmas, which would allow for the first timeto compute ab initio results of, e.g., XRTS signals thatcan be directly compared to state-of-the-art WDM ex-periments. ACKNOWLEDGMENTS
This work is supported by the German Science Founda-tion (DFG) via grant BO1366-13. T. Dornheim acknowl-edges support by the Center of Advanced Systems Under-standing (CASUS) which is financed by Germanys Fed-eral Ministry of Education and Research (BMBF) andby the Saxon Ministry for Science, Culture and Tourism (SMWK) with tax funds on the basis of the budget ap-proved by the Saxon State Parliament. Zh.A. Mold-abekov acknowledges support via the Grant AP08052503by the Ministry of Education and Science of the Republicof Kazakhstan.All PIMC calculations were carried out on the clus-ters hypnos and hemera at Helmholtz-Zentrum Dresden-Rossendorf (HZDR), the computing centre of Kiel uni-versity, at the Norddeutscher Verbund f¨ur Hoch- undH¨ochstleistungsrechnen (HLRN) under grant shp00015,and on a Bull Cluster at the Center for Information Ser-vices and High Performance Computing (ZIH) at Tech-nische Universit¨at Dresden.
APPENDIX
Here we present approximate results for the second andfourth moments of the finite temperature Fermi function.For the second moment, one finds (cid:104) v (cid:105) v F = 32 θ / I / ( η ) (cid:39) (cid:0) + θ (cid:1) / − .
14 [ e − θ − e − . θ ] , (33)where I ν is the Fermi integral of order ν . Similarly, forthe fourth moment follows (cid:104) v (cid:105) v F = 32 θ / I / ( η ) (cid:39) (cid:16) . θ . (cid:17) . . (34)The parametrizations in Eqs. (33) and (34) agree withthe exact numerical results with a precision better than3%, in the entire range of θ . REFERENCES [1] G. Giuliani and G. Vignale, Quantum Theory of theElectron Liquid, Cambridge University Press (2008)[2] P.-F. Loos and P.M.W. Gill, The uniform electron gas,
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