Momentum distribution function and short-range correlations of the warm dense electron gas -- ab initio quantum Monte Carlo results
Kai Hunger, Tim Schoof, Tobias Dornheim, Michael Bonitz, Alexey Filinov
MMomentum distribution function and short-range correlations of the warm denseelectron gas – ab initio quantum Monte Carlo results
Kai Hunger, Tim Schoof,
1, 2
Tobias Dornheim, Michael Bonitz, and Alexey Filinov
1, 4 Institut f¨ur Theoretische Physik und Astrophysik,Christian-Albrechts-Universit¨at zu Kiel, Leibnizstraße 15, 24098 Kiel, Germany Deutsches Elektronen Synchotron (DESY), Hamburg, Germany Center for Advanced Systems Understanding (CASUS), D-02826 G¨orlitz, Germany Joint Institute for High Temperatures, Russian Academy of Sciences, Izhorskaya 13, Moscow 125412, Russia
In a classical plasma the momentum distribution, n ( k ), decays exponentially, for large k , andthe same is observed for an ideal Fermi gas. However, when quantum and correlation effects arerelevant simultaneously, an algebraic decay, n ∞ ( k ) ∼ k − has been predicted. This is of relevancefor cross sections and threshold processes in dense plasmas that depend on the number of energeticparticles. Here we present the first ab initio results for the momentum distribution of the nonidealuniform electron gas at warm dense matter conditions. Our results are based on first principlefermionic path integral Monte Carlo (CPIMC) simulations and clearly confirm the k − asymptotic.This asymptotic behavior is directly linked to short-range correlations which are analyzed via theon-top pair distribution function (on-top PDF), i.e. the PDF of electrons with opposite spin. Wepresent extensive results for the density and temperature dependence of the on-top PDF and forthe momentum distribution in the entire momentum range. PACS numbers: xxx
I. INTRODUCTION
Dense quantum plasmas and warm dense matter(WDM) are attracting growing interest in recent years.Typical for WDM are densities around solid densitiesand elevated temperatures around the Fermi tempera-ture, e.g. [1–4]. Such situations are common in astro-physical systems [5–8], including the interiors of giantplanets and white dwarf stars, or the atmosphere of neu-tron stars. In the laboratory, WDM situations are real-ized upon laser or ion beam compression of matter [9]and also in experiments on inertial confinement fusion(ICF) [10, 11]. Under WDM conditions the electrons aretypically quantum degenerate and moderately correlatedwhereas ions are classical and, possibly strongly corre-lated. These properties clearly manifest themselves inthe thermodynamic [12–14], transport and optical prop-erties [15–18] of WDM. To gain deeper understanding ofthis unusual state of matter, accurate results for struc-tural quantities are essential, including the pair distri-bution function [19, 20] and the static [21] and dynamicstructure factor [17, 22–24].Here we consider another many-particle property – themomentum distribution function n ( k ) and how it is in-fluenced by finite temperature and Coulomb interactioneffects. It is well known that, for classical systems in ther-modynamic equilibrium, n ( k ) is always of Maxwellianform regardless of the strength of the interaction. In con-trast, in a quantum system the momentum and coordi-nate dependencies do not decouple which leads to funda-mentally different behaviors of n ( k ) in ideal and nonidealquantum systems, and only for an ideal system the famil-iar Fermi distribution, n id ( k ) is being recovered (here weconsider only Fermi systems). In contrast, in a non-idealFermi system, the momentum distribution decays much slower with k , exhibiting a power law asymptotic. Theimportance of a power law asymptotic has been pointedout by Starostin and co-workers [25–27] and many oth-ers, e.g. [28], because an increased number of particlesin high-momentum states could have a significant effecton scattering and reaction cross sections, in particular onfusion reaction rates in dense plasmas. The main goal ofthe present paper is, therefore, to present accurate theo-retical results for the tail of the momentum distributionfunction. Before outlining our goals in more detail, webriefly recall the main available theoretical results on thelarge- k asymptotic of the momentum distribution func-tion.It was first demonstrated by Wigner [29] how to in-corporate quantum uncertainty between coordinate andmomentum into n ( k ). Following the development ofperturbation theory for the electron gas in the 1950’s(e.g. [30], [31]), Daniel and Vosko [32] calculated themomentum distribution for an interacting electron gas.They used the approximation due to Gell-Mann andBrueckner for the correlation energy [33] which corre-sponds to the Random Phase Approximation (RPA)).For the ground state, T = 0, they derived an analyticalexpression for the large- k asymptotic of the momentumdistribution, lim k →∞ n RPA ( k ) ∼ k , (1)i.e. they found an algebraic decay, in striking contrast tothe exponential asymptotic of an ideal classical or quan-tum system.Galitskii and Yakimets [34] used Matsubara Greenfunctions and the Kadanoff-Baym relation [35] betweenthe energy distribution in equilibrium, f EQ ( ω ) (which isalways a Fermi or Bose distribution), and the spectral a r X i v : . [ phy s i c s . p l a s m - ph ] J a n function A ( k, ω ), n ( k ) = (cid:90) dω π A ( k, ω ) f EQ ( ω ) , (2)Correlation effects enter only via the spectral function A , which is given by A id ( k, ω ) = 2 π δ [ (cid:126) ω − E ( k )], foran ideal gas. Ref. [34] computed the leading correctionto the ideal spectral function and confirmed the asymp-totic, Eq. (1). For a systematic improvement of this re-sult higher order selfenergies have been computed, e.g.by Kraeft et al . Ref. [36], and we also refer to the textbooks Refs. [35, 37, 38].The exact limiting behavior in the asymptotic (1)was found independently by Kimball [39] via a short-range ansatz to the two electron wave function, andby Yasuhara and Kawazoe [40] who analyzed the large-momentum behavior of the ladder terms in Goldstoneperturbation theory. An important result of Yasuhara etal. is the proof [40] that, at T = 0, the asymptotic can beexpressed via the on-top pair distribution function (on-top PDF), i.e. the PDF of a particle pair with differentspin projections at zero distance, g ↑↓ ( r = 0),lim k →∞ n ( k ) = 49 (cid:18) π (cid:19) / (cid:16) r s π (cid:17) k F k g ↑↓ (0) , (3)where k F denotes the Fermi momentum, and the cou-pling (Brueckner) parameter r s = ¯ r/a B is the ratio ofthe mean interparticle distance to the Bohr radius [41].A more general derivation has been presented by Hof-mann et al. [42] who have shown that Eq. (3) holds alsofor finite temperature.An extension of the results of Yasuhara et al. andKimball to arbitrary spin polarizations of the electrongas was performed by Rajagopal et al. in Ref. [43]. Theyderived the next order power in the asymptotic whichbecomes dominant in the case of a ferromagnetic electrongas because the on-top PDF vanishes: n ferro ( k ) −−−−→ k →∞
43 89 π g ↑↑ (cid:48)(cid:48) (0)2 ( αr s ) k . (4)Aside from dense plasmas, the tail of the momentumdistribution is also relevant for the electron gas in met-als, e.g. [44] as well as cold fermionic atoms [45, 46].In the latter case, however, the short-range character ofthe pair interaction leads to a modified large-momentumasymptotic, n ( k ) ∼ k − , instead of (1).A second approach to the high-momentum tail is basedon quantum Monte Carlo simulations. Here one can ei-ther directly compute the asymptotic of n ( k ) or deter-mine it from the Fourier transform of the density ma-trix. While the former requires to extend the simulationsto very large momenta and to resolve the occupationsover many orders of magnitude, the latter way is poten-tially more efficient. Here one calculates the on-top PDF(which is called “contact” in the cold atomic gas commu-nity). In addition to its use in Eq. (3), we mention that an accurate description of g ↑↓ (0) is interesting in its ownright, and is important for many other applications, likethe description of the static local field correction [47–51].Accurate QMC results for n ( k ) of the UEG in theground state were obtained by Holzmann et al. [44],whereas the on-top PDF was studied in multiple QMC-based works [44, 52, 53], most recently by Spink and co-workers [53]. At finite temperatures, the momentum dis-tribution n ( k ) has been investigated by Militzer et al. [54, 55] who carried out restricted path integral MonteCarlo (RPIMC) simulations and recently by Filinov etal. [56] based on a version of fermionic PIMC that isformulated in phase space. Furthermore, the only com-prehensive data set for g (0) in this regime was presentedby Brown et al. [57], again on the basis of the restrictedPIMC approach.While RPIMC has been shown to exhibit significantsystematic errors [58], fermionic PIMC in coordinatespace is limited to moderate degeneracy [12, 59] due tothe notorious fermion sign problem, see Ref. [60] for anaccessible topical discussion. In addition, RPIMC is sub-stantially hampered by an additional sampling problem( reference point freezing [61]) at high densities, r s (cid:46) abinitio thermodynamic results for the warm dense UEG[58]. In combination with the likewise novel permutationblocking PIMC [63–65] scheme, it was possible to avoidthe fermion sign problem and to obtain ab initio ther-modynamic results for the UEG at warm dense matterconditions [2, 66]. In addition, also ab initio results forthe static density response [67] have been obtained withCPIMC.The goal of this paper is to utilize CPIMC to obtain abinitio data for the momentum distribution of the uniformelectron gas at finite temperature and high density cor-responding to r s (cid:46) .
7. To access stronger coupling, wealso employ a recently developed approximate method –restricted CPIMC [68] as well as direct fermionic propa-gator PIMC simulations in coordinate space – an exten-sion of permutation blocking PIMC [63]. In particular, i: we verify that the high-momentum asymptotic doesindeed obey k − , and that it is solely determinedby the on-top PDF; ii: we present detailed CPIMC results for g ↑↓ (0) and an-alyze its temperature and density dependence; iii: investigate the momentum distribution function inthe vicinity of the Fermi momentum and for smallmomenta; iv: investigate the onset of the large-momentum asymp-totic.This paper is organized as follows: In Sec. II we presenta brief overview on earlier theoretical work pertaining tothe uniform electron gas, together with the main pre-dictions. This is followed by an introduction into ourquantum Monte Carlo simulations in Sec. II B and by apresentation of the numerical results in Sec. III. II. THEORY FRAMEWORKA. On-top pair distribution
Since the high-momentum tail of the momentum distri-bution function can be expressed in terms of the on-toppair distributions, cf. Eq. (3), we start by consideringthe pair distribution of electrons with spin projections σ and σ [69], g σ σ ( r , r ) = (cid:68) ˆΨ † σ ( r ) ˆΨ † σ ( r ) ˆΨ σ ( r ) ˆΨ σ ( r ) (cid:69)(cid:68) ˆΨ † σ ( r ) ˆΨ σ ( r ) (cid:69) (cid:68) ˆΨ † σ ( r ) ˆΨ σ ( r ) (cid:69) , (5)where Ψ σ ( r ) [Ψ † σ ( r )] is a fermionic field operator an-nihilating [creating] an electron in spin state | r σ (cid:105) . Notethat the two-particle density in the numerator is nor-malized to the single-particle spin densities, n σ ( r ) = (cid:104) ˆΨ † σ ( r ) ˆΨ σ ( r ) (cid:105) , in the denominator. Thus, in the absenceof correlations and exchange effects, g σ σ ( r , r ) ≡ g ↑↑ ( r , r ) ≡ g ↓↓ ( r , r ) and g ↑↓ ( r , r ) ≡ g ↓↑ ( r , r ).The total pair distribution function follows from thespin-resolved functions (5) according to g ( r , r ) = (cid:88) σ σ g σ σ ( r , r ) n σ ( r ) n σ ( r ) n ( r ) n ( r ) , (6) n ( r ) = (cid:88) σ n σ ( r ) , (7)where the normalization assures that, in the absence ofexchange and correlation effects, g ≡
1. In a spatiallyhomogeneous system, such as the UEG, the PDFs de-pend only on the distance of the pair, g σ σ ( r , r ) = g σ σ ( | r − r | . Of particular importance is the caseof zero separation. Then, the Pauli principle leads to g ↑↑ (0) ≡ g ↓↓ (0) ≡
0. On the other hand, the probabil-ity of finding two electrons with different spins “on topof each other” yields the on-top PDF, g ↑↓ (0), which isrelated to total PDF in the paramagnetic case by [cf.Eq. (6)] g (0) = g ↑↑ (0) + g ↑↓ (0)2 = 12 g ↑↓ (0) , (8)which is a fundamental property for the characterizationof short-range correlations. While in a non-interacting system ( r s → g ↑↓ id (0) = 1, Coulomb repulsion leads toa reduction of this value. Thus for the UEG a monotonicreduction with r s is expected which will directly influ-ence, via Eq. (3), the tail of the momentum distribution.There exist a variety of analytical parametrizations ofthe on-top PDF. The ground state on-top PDF of cor-related electrons was investigated in Ref. [70] by usingthe Overhauser screened Coulomb potential in the radialtwo-particle Schr¨odinger equation, see Appendix B. Theresults were parametrized for r s ≤
10 according to g ↑↓ (0) = (1 . Ar s + Br + Cr + Dr )e − Er s , (9)where A = 0 . B = 0 . C = − . D =0 . E = 0 . χ = n Λ (cid:28)
1, and Θ = k B T /E F (cid:29)
1, is alsoknown. Here n is the density depending on the meaninter-particle distance, ¯ r ∼ n − / , and Λ is the thermalDeBroglie wavelength, Λ = h / (2 π mk B T ). A quantum-mechanical expansion was given in Ref. [42], where theresult depends on the order the high-temperature limit, T → ∞ , and the classical limit, (cid:126) →
0, are taken. Thereason is the existence of a third length scale [37, 71], theBjerrum length, l B = βe , where β = ( k B T ) − , givingrise to a second dimensionless parameter, the classicalcoupling parameter, Γ = βe / ¯ r = l B / ¯ r .In the case Γ (cid:28) χ / (i.e. l B (cid:28) Λ), the result is [42] g (0) = 12 (cid:18) − √ π l B Λ + . . . (cid:19) , (10)where the behavior is still dominated by the ideal Fermigas properties with deviations scaling like Γ χ − / , or,( k B T ) − / n .On the other hand, in the case χ / (cid:28) Γ (i.e., Λ (cid:28) l B ), which corresponds to classical plasmas at moderatetemperatures, the on-top PDF becomes [42] g (0) = 4 π / / (cid:18) l B Λ (cid:19) / e − π / (cid:16) lB Λ (cid:17) / + . . . . (11)This value is exponentially small due to the moderateCoulomb repulsion and is not influenced by quantum ef-fects. Nevertheless, quantum effects (finite Λ) show upin the algebraic momentum tail, according to Eq. (3),but only on length scales much smaller than Λ or, corre-spondingly, at momenta strongly exceeding Λ − .The latter case is out of the range of WDM and notrelevant for the present analysis. Finally, there exists amore recent parametrization of the ground state on-top-PDF that is based on QMC simulations [53]: g (0; r s ) = 1 + a √ r s + br s cr s + dr s , T = 0 , (12)which will be used for comparison below. For an overviewabout different models of g (0) for the ground state, thereader is referred to the paper by Takada [51].With explicit results for the on-top PDF and, usingEq. (3), the large- k asymptotics of the momentum dis-tribution function can be reconstructed.For finite temperature one can relate the PDF to aneffective quantum pair potential, g ↑↓ ( r ) = e − βV Q ( r ) , anidea that was put forward by Kelbg [72] and further de-veloped, among others, by Deutsch, Ebeling, and Filinovand co-workers, cf. Refs. [73–76] and references therein.We will return to this issue in Sec. III B 2. B. Configuration PIMC (CPIMC) approach to g (0) and n ( k ) of the warm dense electron gas
1. Idea of CPIMC simulations
CPIMC was first formulated in Ref. [77] and appliedto the UEG in Refs. [58, 62, 78]. For a detailed descrip-tion of the CPIMC formalism we refer to the overviewarticles [2, 79] and to the recent developments [68]. Herewe only summarize the main idea. The thermodynamicexpectation value of an arbitrary operator ˆ A is deter-mined by the density operator ˆ ρ and its normalization– the partition function Z , where we use the canonicalensemble, ˆ ρ = e − β ˆ H , (13) Z ( β ) = Tr ˆ ρ , (14) (cid:104) ˆ A (cid:105) ( β ) = 1 Z Tr ˆ A ˆ ρ , (15)Since the Hamiltonian involves only one- and two-bodyoperators,ˆ H = (cid:88) ij h ij ˆ a † i ˆ a j + 12 (cid:88) ijkl w ijkl ˆ a † i a † j ˆ a l ˆ a k , (16)its expectation value can be described via the reducedone- and two-particle density matrices, d ij and d ijkl , seethe definitions (17) and (18). Here the sums are over arbitrary complete sets of single-particle states. Belowthis will be specified to momentum eigenstates. Quan-tum Monte Carlo estimators for these quantities areobtained through differentiation of the partition func-tion [79, eq. (5.88)] with respect to the single-particlematrix element d ij := (cid:104) ˆ a † i ˆ a j (cid:105) = − β ∂∂h ij ln Z (17)and the two-particle matrix element d ijkl := (cid:104) ˆ a † i ˆ a † j ˆ a k ˆ a l (cid:105) = − β ∂∂w ijkl ln Z, (18)respectively. The resulting expressions depend on theorder and choice of the indices ( i, j ) and ( i, j, k, l ), re-spectively.Let us now present explicit expressions for the one-particle and two-particle density matrices in CPIMC.Configuration PIMC is path integral Monte Carlo formu-lated in Fock space [77], i.e. in the space of N -particleSlater determinants, |{ n }(cid:105) = |{ n , n , . . . }(cid:105) , constructedfrom the single-particle orbitals | i (cid:105) where n i is the asso-ciated occupation number.In CPIMC the canonical partition function (13) is writ-ten as a Dyson series in imaginary time, for details seeref. [68]. A configuration C determining a MC state isgiven by a set of initially occupied orbitals { n } , alongwith a set of K changes κ i to this set, called kinks attheir respective times t i , 1 ≤ i ≤ K , C := { { n } , t , . . . , t K , κ , . . . , κ K } . (19)Due to the Slater-Condon rules for fermionic 2-particleoperators, each interaction matrix element yields eithera 2-particle term, corresponding to κ = ( i, j ), or a 4-particle term, κ = ( i, j, k, l ). Thus the kinks are givenby either two or four orbital indices, respectively. The kink matrix element q i,i − ( κ i ) represent the off-diagonalmatrix elements with respect to the possible choices of 2-or 4-tuples κ i . The final result for the partition functionis [68] Z ( β ) = ∞ (cid:88) K =0 K (cid:54) =1 (cid:88) { n } (cid:88) κ . . . (cid:88) κ K β (cid:90) d t β (cid:90) t d t . . . β (cid:90) t K − d t K ( − K (cid:32) K (cid:89) i =0 e − E i ( t i +1 − t i ) (cid:33) × (cid:32) K (cid:89) i =1 q i,i − ( κ i ) (cid:33) , (20)where paths with K = 1 violate the periodicity and haveto be excluded. Configurations can be sampled from thepartition function Z = (cid:88)(cid:90) C W ( C ) , (21) with the weight function W ( C ) = ( − K (cid:32) K (cid:89) i =0 e − E i ( t i +1 − t i ) (cid:33) (cid:32) K (cid:89) i =1 W i,i − (cid:33) , (22)which allows one to rewrite thermodynamic expectationvalues (15) as (cid:104) A (cid:105) = (cid:88)(cid:90) C W ( C ) A ( C ) . (23)An example configuration (path) is illustrated in Fig. 1.With three particles present, horizontal solid lines repre-sent diagonal matrix elements as given by the exponen-tial factor in the partition function (20) and the occupa-tion number state at a given time-interval is specified bythe set of all these lines in this interval. On the otherhand, the vertical solid lines represent interaction terms,where the occupation changes according to the specified kink κ i , weighted by the respective kink matrix element q i,i − ( κ i ). Due to the periodicity of the expectation val-ues (15), the kinks must add to yield the inital occupationvector at 0 < t < t again: (cid:32) K (cid:89) i =1 ˆ q i,i − ( κ i ) (cid:33) |{ n }(cid:105) = |{ n }(cid:105) (24) ⇒ K (cid:89) i =1 ˆ q i,i − ( κ i ) = ˆ . (25) t t t t t β o r b i t a l i q , ( κ ) |{ n (4) }(cid:105) FIG. 1. Illustration of a path C , Eq. (19), with five kinks.The three kinks 1, 3, 5, at times t , t , t , each involve fourorbitals: κ = (1 ,
4; 3 , κ = (1 ,
3; 0 , κ = (4 ,
5; 1 , t and t , involve twoorbitals, each: κ = (0; 1) and κ = (2; 4). The fourth Slaterdeterminant | n (4) (cid:105) exists between the imaginary “times” t and t and contains three occupied orbitals { , , } . This representation of the partition function can nowbe applied to the observables of interest. For the one-particle density matrix we obtain, for i (cid:54) = j , d ij ( C ) = − β K (cid:88) ν =1 ( − α { n ( ν ) } ,i,j q { n ( ν ) }{ n ( ν − } ( κ ν ) δ κ ν , ( i,j ) . (26)For the uniform electron gas, the off-diagonal matrix el-ements vanish in a momentum basis, whereas the diag-onal ones yield the momentum distribution, as will bediscussed in Sec. II B 2Let us now turn to the CPIMC estimator for the two-particle density matrix. Here we have to distinguishseveral cases of index combinations [80, eq. 3.14]. If i < j, k < l are pairwise distinct d ijkl ( C ) = − β K (cid:88) ν =1 ( − α { n ( ν ) } ,i,j + α { n ( ν − } ,k,l q { n ( ν ) }{ n ( ν − } ( κ ν ) δ κ ν , ( i,j,k,l ) . (27)The term under the sum (without the Kronecker-delta)will be abbreviated as the weight of the kink κ ν , W ( κ ν ) := ( − α { n ( ν ) } ,i,j + α { n ( ν − } ,k,l q { n ( ν ) }{ n ( ν − } ( κ ν ) . In the case of i = k , but with all other indices beingdifferent, d ijil ( C ) = − β K (cid:88) ν =1 ( − α { n ( ν ) } ,j,l q { n ( ν ) }{ n ( ν − } ( κ ν ) n ( ν ) i δ κ ν , ( j,l ) . (28)Finally, if i = k and j = l , but i (cid:54) = j , the matrix elementsare given by d ijij ( C ) = K (cid:88) ν =0 n ( ν ) i n ( ν ) j τ ν +1 − τ ν β . (29)The expectation value of this estimator is given by theweighted sum over all possible configurations C, d ijkl = (cid:68) ˆ a † i ˆ a † j ˆ a k ˆ a l (cid:69) = 1 Z (cid:88)(cid:90) C d ijkl ( C ) W ( C ) . (30)Due to the large single-particle basis sizes that haveto be used in the CPIMC simulations, the variances ofthese estimators may be very large for some transitions(i.e. combinations of indices ( i, j ) or ( i, j, k, l )). However,special cases can be used to derive the estimators neededto measure short-range properties of the system: Themomentum distribution and the on-top pair distributionfunction.
2. Momentum distribution with CPIMC
The momentum distribution function is given by thediagonal part of the one-particle reduced density matrix,Eq. 17), if a plane wave basis is being used, cf. Sec. II B 3.For i = j , we obtain (cid:104) ˆ a † i ˆ a j (cid:105) = (cid:104) ˆ n i (cid:105) = − β ∂∂h ii log( Z )= 1 Z (cid:88)(cid:90) C (cid:32) K (cid:88) ν =0 n ( ν ) i τ i +1 − τ i β (cid:33) W ( C )= 1 Z (cid:88)(cid:90) C n i ( C ) W ( C ) , (31)where the sum runs over all occupied orbitals at eachtime-slice.
3. On-top pair distribution function with CPIMC
The definition (5) of the spin-resolved pair distributionfunction requires the two-particle density matrix in coor-dinate representation. A basis change is needed to relateit to the two-particle density matrix in momentum rep-resentation, Eq. (18). The single-particle basis of planewave orbitals is given by (cid:104) r σ | k s (cid:105) = 1 √ V e i kr δ s,σ =: ϕ k ( r ) δ s,σ . (32)To shorten the notation, the wave vector k will berepresented by an index i ↔ k i of the correspondingsingle-particle basis eigenvalue. The field operators in aposition-spin basis are related to the creation and anni-hilation operators in a momentum-spin basis | i (cid:105) := | k i s i (cid:105) by ˆΨ σ ( r ) = (cid:88) i φ i ( r , σ )ˆ a i , ˆΨ † σ ( r ) = (cid:88) i φ ∗ i ( r , σ )ˆ a † i . (33)The on-top PDF, Eq. (8), is the value of the PDF,Eq. (5), at equal position arguments r := r = r anddifferent spin projections, σ (cid:54) = σ , g ↑↓ (0) := g σ σ ( r , r ) =: g ↑↓ . (34)With the basis transformation (33) of the field operators,a straightforward calculation yields the CPIMC estima-tor for the on-top PDF, g ↑↓ ( C ) = 1 β K (cid:88) ν =1 (cid:88) k (cid:54) = i We start by giving an overview of the momentum dis-tribution in the entire k -range, and compare the CPIMCresults to the ideal case, cf. Figs. 2 and 4. After this wewill analyze the large momentum asymptotic of the mo-mentum distribution function, as obtained from CPIMCsimulations and compare the asymptotic behavior to theanalytical predictions following from Eq. (3) where thevalue for the on-top PDF is taken from CPIMC simula-tions as well. Furthermore, we compare to the availableground state data.We have performed extensive CPIMC simulations with N = 54 particles. Due to the fermion sign problem, these simulations are restricted to small coupling parameters, r s (cid:46) . 7. To extend the range of parameters, we also per-formed simulations with N = 14 parameters. As shownbefore, important structural properties, such as the staticstructure factor [14, 81] and the pair distribution functiononly weakly depend on the particle number. A quanti-tative analysis of the N -dependence of the results willbe performed for the tail of the momentum distributionin Sec. III B. The CPIMC results are complemented byrestricted CPIMC simulations [68]. To access larger val-ues of the coupling parameter, we also include fermionicPIMC simulation results in coordinate space for the on-top PDF. A. Momentum distribution 1. Overview Let us start by analyzing the general trends of the mo-mentum distribution when either the temperature or thecoupling strength are varied. In Fig. 2 we present CPIMCdata for N = 54 particles showing the entire momentumrange for moderate coupling, r s = 0 . 5, and three temper-atures. It shows that the occupation of high momentumstates in the tail region is coupled in a non-trivial way tooccupation of lower momentum states. The results showthat an increase of temperature not only leads to the fa-miliar broadening of n ( k ) around the Fermi edge and de-pletion below it, but may also lead to a lower populationof the tail (see below). The most striking observation isthe strong deviation, in the tail region, from the exponen-tial decay in case of an ideal Fermi gas. Our simulationsclearly confirm the correlation-induced enhanced popu-lation of high-momentum states. As we will see below,the simulations clearly confirm the asymptotic algebraicdecay, n ( k ) ∼ k − .Let us now turn to the dependence on the couplingparameter. To this end, we present, in Fig. 3, the mo-mentum distribution for a fixed temperature, Θ = 2, andtwo values of r s and also compare to the ideal Fermi gas.For large momenta, k (cid:38) k F , we observe an increase ofthe population when r s grows. However, for intermedi-ate momenta, k F (cid:46) k (cid:46) k F , the ideal distribution issignificantly above the correlated distributions. Finally,below the Fermi momentum, the correlated distributionsare again above the ideal momentum distribution.This behavior seems counter intuitive, and we analyzeit more in detail in the next section. 2. Interaction-induced enhanced population oflow-momentum states Let us now investigate in more detail the behavior ofthe momentum distribution in the range from k = 0 tomomenta on the order of several Fermi momenta. Tofocus on correlation effects we plot the difference of the − − − − − k/k F n ( k ) Θ = 0Θ = 1 . 5Θ = 2 . 0Θ = 4 . FIG. 2. Temperature dependence of the momentum distri-bution of moderately correlated electrons, r s = 0 . 5. CPIMCresults with N = 54 particles for three temperatures are com-pared to the ground state (solid black, data of Ref. [70]). Forcomparison, the ideal Fermi distribution is shown by dashedlines of the same color as the interacting result. − − − − k/k F n ( k ) n id ( k ) r s = 0 . r s = 0 . FIG. 3. Density dependence of the momentum distributionof moderately correlated electrons at temperature Θ = 2.CPIMC results with N = 54 particles are compared to theideal Fermi-Dirac distribution n id (full black line). For mo-menta below approximately 6 k F the correlated distributionsare indistiunguishable from n id . For comparison, the groundstate distributions, as given by Ref. [70], are shown by thedashed lines of the same color as the finite temperature re-sult. correlated distribution and the Fermi distribution. InFigure 4 we investigate the temperature dependence, forthe case of r s = 0 . 5. Clearly, we observe an enhancedpopulation of low momentum states, k (cid:46) . k F , com-pared to the Fermi function. The effect is biggest at the lowest temperature and decreases monotonically withΘ. On the other hand, it is clear that, upon further re-duction of Θ, this effect will decrease again and vanishin the ground state. The reason is that, at T = 0, alllow-momentum states are completely occupied, and, dueto the Paui principle, correlations can only enhance thepopulation of unoccupied states, at k > k F . The same n ( k ) − n i d ( k ) Θ = 1 . 5Θ = 2 . 0Θ = 4 . k/k F k { n ( k ) − n i d ( k ) } / FIG. 4. Deviation of the momentum distribution (CPIMCresults with N = 54 particles) from the ideal Fermi-Diracdistribution at moderate coupling r s = 0 . 5. Lower Panel:Difference of the distribution functions weighted with k / analysis is performed, for a fixed temperature but dif-ferent coupling parameters, in Fig. 5. Here we observea monotonic trend: with increasing r s , the difference ofthe populations increases with respect to the ideal case.This correlation-induced enhanced population of lowmomentum state has been reported before. A detailed in-vestigation by restricted PIMC simulations was presentedby Militzer and Pollock [54], and a thermodynamic Greenfunctions analysis was performed by Kraeft et al. [36].The origin of this effect is interaction-induced loweringof the energy eigenvalues, E ( k ) < E id ( k ) [54]. Here, theinteracting energy contains, in addition, an exchange anda correlation contribution, E ( k ) = E id ( k ) + ∆ E x ( k ) + ∆ E c ( k ) . (36)The behavior reported here is dominated by the exchangecontribution, i.e. by the Hartree-Fock selfenergy (theHartree term vanishes due to homogeneity and chargeneutrality) which is negative,∆ E x ( p ) = Σ HF ( p ) = − (cid:90) d q (2 π (cid:126) ) w ( | p − q | ) n ( q ) . (37)The negative Hartree-Fock selfenergy shift is largest atsmall momenta and decreases monotonically with k . As aconsequence, the system tends to increase the populationof low-momentum states.An interesting consequence of this population increaseis that the mean kinetic energy of the correlated electrongas may be lower than that of the ideal electron gas atthe same temperature [36, 54]. Our simulations clearlyconfirm this prediction. This effect is illustrated in thelower panels of Figs. 4 and 5 where we plot the k -resolveddifference of kinetic energy densities. For the parametersshown in theses figures, the excess kinetic energy (com-pared to the ideal UEG) concentrated in low-momentumstates (positive difference) is smaller than the kinetic en-ergy reduction (negative difference) at larger momenta.This is evident from the areas under the curves in thelower panels of Figs. 4 and 5. As a result the total kineticenergy difference of the interacting system compared tothe ideal system is negative for a broad range of parame-ters. The corresponding kinetic energies for the interact-ing and ideal systems are presented in the appendix, intables II and III, for 54 and 14 particles, respectively. n ( k ) − n i d ( k ) r s = 0 . r s = 0 . r s = 0 . k/k F k { n ( k ) − n i d ( k ) } / FIG. 5. Same as Fig. 4, but for a fixed temperature, Θ = 2,and three densities. The different ordering of the curves in thelower panel arises from the r s -dependence of the horizontalscale, k F ∝ r − s . Our argument, so far, was based on the negative signof the Hartree-Fock selfenergy. However, for a completepicture we also need to consider the energy shift dueto correlations, ∆ E c . In contrast to the Hartree-Fockshift, the correlation corrections to the energy dispersionare typically positive, but smaller, as was shown for theBorn approximation (Montroll-Ward approximation), inRef. [36]. However, this result applies only for weak cou-pling. For stronger coupling, in particular, r s (cid:38) 1, atleast T-matrix selfenergies would be required. An alter-native are QMC simulations, as presented in Ref. [54],which allow one to map out the range of density andtemperature parameters where the difference of corre-lated and ideal kinetic energies changes sign.The present CPIMC simulations are not directly appli-cable to the range r s (cid:38) 1. However, we can take advan- θ r s Militzer -0.12-0.1-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 FIG. 6. Parameter range of interaction-induced loweringof the kinetic energy. The heat-map shows the exchange–correlation contribution to the kinetic energy K xc computedfrom the parametrization by Groth et al. [66], see Eq. (38).The solid black line indicates the r s -Θ-combinations where K xc vanishes, and the dotted lines an uncertainty interval of5 × − Ha. The solid blue line depicts the sign change in K xc that was predicted by restricted PIMC in Ref [54]. Red(green) pluses: CPIMC results for kinetic energy decrease (in-crease) compared to ideal case. Red circles (green crosses):CPIMC data points where the occupation of the lowest or-bital, n (0), is higher (lower) than in the ideal case, i.e. n id (0).Extensive data for the kinetic energy are presented in the ta-bles in the Appendix. tage of the accurate parametrization of the exchange–correlation free energy f xc of Groth et al. [66] thatis based on a combination of CPIMC, PB-PIMC andground state QMC results. In particular, the exchange–correlation contribution to the kinetic energy is obtainedby evaluating [22] K xc = − f xc ( r s , θ ) − θ ∂f xc ( r s , θ ) ∂θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r s (38) − r s ∂f xc ( r s , θ ) ∂r s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) θ , and the corresponding results are depicted in Fig. 6. Thisresult nicely confirms the line where the kinetic energydifference changes sign [54], for r s (cid:38) 1. At the sametime we find significant deviations at smaller r s and lowertemperatures, see Fig. 6.Let us return once more to the correlation effects on theoccupation numbers and focus on the lowest energy state, k = 0. It is interesting to compare the parameter valueswhere the kinetic energy difference changes sign to theoccupation of the zero-momentum state, n (0), relativeto the ideal distribution, n id (0). For most temperaturesconsidered, the interacting zero-momentum state n (0)has larger population than the corresponding ideal state.Only for the lowest temperatures, Θ ∈ { / , / } , weobserve larger occupations of the ideal zero-momentumstate compared to the correlated result. 3. High-momentum asymptotics of n ( k ) In Figs. 7–9 we present data for low to moderate tem-peratures focusing on momenta beyond the Fermi edge.We directly compare the CPIMC data to the asymptopicbehavior where a k − tail is expected, with the coeffi-cient determined by the on-top PDF g (0), cf. Eq. (3),where g (0) is taken from the same CPIMC simulation.As can be seen in these figures, the CPIMC data clearlyexhibit the expected algebraic decay, for sufficiently large k . To make a quantitative comparison, we also plot, inthe lower panels, the relative difference between CPIMCdata, n ( k ), and the asymptotic, n ∞ ( k ), according to δ ∞ ( k ) = n ( k ) n ∞ ( k ) − . (39)The results for δ ∞ clearly confirm that our ab initio − − − n ( k ) CPIMC / (2325 k ) Ground State . 58 2 . 51 3 . 98 6 . 31 10 − k/k F δ ∞ ( k ) FIG. 7. Log-log plot of the large momentum behavior of themomentum distribution, for r s = 0 . . Top :blue line. CPIMC results for N = 54 particles, pink line: bestfit to the asymptotic, Eq. (3) with the contact-value takenfrom CPIMC data; green: ground state value from Ref. [82]. Bottom : relative difference of CPIMC and the ground statedata from the asymptotic (pink line in top plot), according toEq. (39). CPIMC data approach the asymptotic. Moreover, wecan estimate the momentum range where the asymptoticbehavior dominates. For low temperatures of E F / 16, theasymptotic is reached at about 6 k F , cf. Fig. 7. With in-creasing temperature, the asymptotic is approached only − − − − n ( k ) CPIMC / (2399 k ) Ground State . 98 5 . 01 6 . 31 7 . 94 10 − k/k F δ ∞ ( k ) FIG. 8. Same as Fig. 7, but for r s = 0 . at larger momenta, e.g. for Θ = 1, around 8 k F , cf. Fig. 8,and for Θ = 2, around 11 k F , cf. Fig. 9. A systematicanalysis of the onset of the asymptotic will be given inSec. III C.In these figures we also included ground state data forthe momentum distribution (green lines) which allows usto analyze finite temperature effects. In all figures weobserve that the finite temperature distribution, n ( k ; Θ),intersects the ground state function, n ( k ; 0), coming fromabove, before it reaches the asymptotic. In the rangeof the algebraic tail the finite temperature function isalways below the ground state result, for the same r s and k , in agreement with Fig. 2. This behavior is, atfirst sight, counter intuitive because one expects that fi-nite temperature effects increase the population of highmomentum states. As we will show in Sec. III B 2 thistemperature dependence is, in fact, non-monotonic andis due to a competition between Coulomb repulsion andexchange effects.Finally, we note that our simulations reveal that the k − asymptotic is observed independently of the particlenumber, in agreement with the predictions of Refs. [42,83]. We will return to the question of the particle numberdependence in Sec. III B 1. B. Ab initio results for g (0) After analyzing CPIMC data for the large momentumtail of the distribution function we now concentrate onthe coefficient in front of the asymptotic k − term. Ac-cording to Eq. (3), this coefficient is entirely determinedby the on-top PDF g (0) which is directly accessible inquantum Monte Carlo simulations. For PIMC in coor-dinate space, the straightforoward way is to analyze the0 − − − n ( k ) CPIMC / (458 k ) Ground State . 31 7 . 94 10 12 . − k/k F δ ∞ ( k ) FIG. 9. Same as Fig. 7, but for r s = 0 . . ∞ -1 -1 -1 -1 r s =1, θ =4 g ( r ) r/a B E max CPIMCPIMCRPIMC FIG. 10. Determination of the ontop-PDF, g ↑↓ (0) = 2 g (0), for N = 54 electrons at r s = 1 and θ = 4, using different methods.The red circles are CPIMC results for g ↑↓ (0) for different val-ues of the momentum cutoff E max , see the top x -axis, and thesolid red line corresponds to a linear fit. The green crosses arestandard PIMC data for the distance-dependent PDF, g ↑↓ ( r ),(bottom x -axis), and the solid green curve corresponds to alinear fit. The blue diamonds are restricted PIMC data fromRef. [57] for the same conditions, but N = 66. r -dependence of the PDF and subsequently extrapolateto r = 0. Typical results are shown in Fig. 10 for directfermionic (labeled “PIMC”) and restricted (“RPIMC”)PIMC simulations. In contrast, in CPIMC a direct esti-mator for the on-top PDF is available, cf. Eq. 35, andthe results are included in Fig. 10 with the red symbols.These results depend on the size of the single-particle basis and the corresponding cut-off energy E max (top x -axis). Overall, for a sufficiently large basis, very goodagreement of the two independent fermionic simulations– PIMC and CPIMC – is observed for the parametercombinations where both are feasible.This gives additional support for our CPIMC data, inparticular for its use at low temperatures, where CPIMCprovides the only ab initio approach. In fact, CPIMCdata for g (0) were already used for comparisons above.In this section we investigate the density and tempera-ture dependence of g (0). But first we explore how sensi-tive this value depends on the number of particles in thesimulation cell. 1. Particle Number Dependence We have performed extensive CPIMC simulations for g (0) for a broad range of particle numbers, from N = 14to N = 66. Three typical examples are shown, forΘ = 0 . . 5, in Fig. 12, and forΘ = 2, in Fig. 13. In these figures we use the case N = 14as the reference for comparison because, for this number,the widest range of parameters is feasible, although, nat-urally, simulations with larger N are more accurate. Allfigures confirm that finite size effects are very small in g (0). Thus, where possible, we will use 54 or 66 particles.But in some cases, we can extend the range of the sim-ulations by using just N = 14 particles with a relativeerror of g (0) not exceeding 2%. Regarding simulationswith the two approximate CPIMC variants that were dis-cussed above [68], the analysis reveals that RCPIMC+ isreliable for intermediate temperatures, 0 . (cid:46) Θ (cid:46) . N = 54 are close to CPIMCsimulations for N = 54 particles (and more accurate thanCPIMC for N = 14) and, therefore, can be well used forlarger r s -values, where CPIMC is not possible, due to thesign problem. At the same time, RCPIMC [68] turns outto be not sufficiently accurate for computing g (0) and isnot being used in this paper. 2. Temperature Dependence We now turn to the temperature dependence of the on-top PDF. In Figs. 14 and 15, we plot g (0) from CPIMCdata over a broad range of temperatures for r s = 0 . . ≤ r s ≤ . 7, respectively. The figures displayan interesting non-monotonic behavior: the on-top PDFincreases, both towards low and high temperatures. Thisis easy to understand: At very low temperatures, thesystem approaches an almost ideal Fermi gas for which g (0) would be exactly 0 . 5. The (weak) Coulomb repulsiongives rise to an additional depletion of zero distance pairstates. This is confirmed by the lower absolute values of g (0) when r s is increased from r s = 0 . . . r s F S E [ % ] N=54N=38 (TA)RCPIMC+, N=54RCPIMC, N=54 FIG. 11. Influence of the particle number on the on-top pairdistribution of the unpolarized UEG at Θ = 0 . N = 66, N = 54,and 38. Shown is the relative deviation of each respectivedata point from the corresponding CPIMC result for N = 14which corresponds to the horizontal line at 0. Further, simu-lation results from the approximate RCPIMC and RCPIMC+methods [68] are included. r s F S E [ % ] N=54N=38 (TA)RCPIMC+, N=54RCPIMC, N=54 FIG. 12. Same as Fig. 11, but for Θ = 0 . On the other hand, for increasing temperature, inthe range where the electron gas is dominated by clas-sical behavior (Θ > k B T /E F ∼ . 63, with a depth of 0 . 42, for r s = 0 . . 63, with a depth of 0 . r s = 0 . 4, andaround Θ = 0 . 63, with a depth of 0 . r s = 0 . r s F S E [ % ] CPIMC, N=54CPIMC, N=38CPIMC, N=66 FIG. 13. Same as Fig. 11, but for the temperature Θ = 2. This minimum can be understood as due to the balanceof two opposite trends: depletion of g (0), due to Coulombrepulsion and increase of g (0), due to quantum delocal-ization effects. At high temperatures and low densities,the PDF can be expressed in binary collision (ladder)approximation as [37] g ↑↓ ( r ) = e − βV ( r ) , (40)where V is the Coulomb potential, which reproduces thebehavior right of the minimum. At small interparticledistances, r (cid:46) Λ, however, quantum effects have to betaken into account in the pair interaction. Averagingover the finite spatial extension of electrons leads to thereplacement of the Coulomb potential by the Kelbg po-tential (quantum pair potential) [72, 84, 85], V K ( r ) = V ( r ) (cid:26) − e − r + √ π r ˜Λ (cid:20) − erf (cid:18) r ˜Λ (cid:19)(cid:21)(cid:27) (41)where ˜Λ = Λ. Note that V K has the asymptotic V K (0; β ) = e Λ( β ) ∼ T / which removes the Coulomb sin-gularity at zero separation. While this potential has thecorrect derivative, dV K (0) /dr = − e Λ , its value at r = 0is accurate only at weak coupling. At the same time,this potential can be extended to arbitrary coupling byretaining the same analytical form, but correcting thestandard thermal DeBroglie wavelength Λ (referring toan ideal gas) to the wave length of interacting particles,which gives rise to the so-called improved Kelbg potential[74, 75], Λ → ˜Λ = Λ · γ , (42) V K (0; β ) → V IK (0; β ) = e Λ( β ) γ ( β ) . (43)At low temperature the effective wavelength of the elec-trons increases, γ ( β ) ∼ T − / , which ensures that2 g ↑↓ IK (0) = e − βV IK (0) is finite. Accurate values for thefunction γ in a two-component plasma and for differ-ent spin projections were presented in Refs. [74, 75] froma fit to PIMC data. In similar manner, the present abinitio QMC results for the on-top PDF can be used tocompute an effective DeBroglie wavelength of the warmdense uniform electron gas, and the concept of an effec-tive quantum pair potential allows for a simple physicalinterpretation of some of its thermodynamic properties. Θ g ( ) CPIMC, N = 54 RCPIMC+, N = 54 FIG. 14. Temperature dependence of the on-top pair distribu-tion for r s = 0 . N = 54 par-ticles. Very good agreement of RCPIMC+ [68] with CPIMCis confirmed. − − − . . . . . . Θ g ( ) r s = 0 . r s = 0 . ( +0 . ) r s = 0 . ( +0 . ) FIG. 15. Temperature dependence of the on-top PDF for r s = 0 . 5, from CPIMC simulations with 14 particles. Twistangle averaging has been applied. Shaded area indicates thestatistical error. The minimum temperature is set by thefermion sign problem. For better visibility, the curves for r s = 0 . . As we already saw for the example of three densities,the location of the minimum changes with the couplingstrength r s . This effect is analyzed systematically inFig. 16. We observe an increase of the minimum posi-tion, Θ min , with r s (full squares, left axis). The reason isthat, with increasing coupling, the interaction strengthincreases, as is seen by the increasing depth of the min-imum (open symbols, right axis). Therefore, the mono-tonic increase of g ( r ) with temperature sets in already ata higher temperature, when r s is increased. In additionto CPIMC data which are restricted to r s (cid:46) r s = 8. More infor-mation on this approximation is given in the discussionof Fig. 18. r s r s g ( ) CPIMCESA g (0, ) 0.2 0.4 0.6 0.80.40.60.81.0 0.20.30.40.5 FIG. 16. Analysis of the minimum of the on-top PDF. Filledsymbols correspond to the location of the minimum in theΘ − r s –plane (left axis Θ). Open symbols correspond to theminimum value of the OT-PDF (right axis). Orange circles:CPIMC results for N = 14 particles. The green line representsthe values of g (0) at a fixed temperature Θ = 0 . 3. Density Dependence Let us now discuss the density dependence of the on-top PDF. As we have seen above, with increasing cou-pling strength, r s , the value of g ↑↓ (0) decreases, due tothe increased interparticle repulsion. This connection canbe qualitatively understood from Eq. (40) if it is usedwith an effective potential that includes many-body ef-fects beyond the pair interaction. This monotonic de-crease with r s is confirmed by our simulations for alltemperatures. As an illustration, we show in Figs. 17and 18 the behavior for Θ = 0 . g (0) is very weak, cf. Fig. 17, in agree-ment with Fig. 15. At θ = 1, finite temperature effectsincrease the particle repulsion due to stronger localiza-3tion of electrons, and g (0) falls slightly below the groundstate value, cf. Fig. 18. This confirms the non-monotonictemperature dependence of g (0) discussed above, sincethis temperature is in the vicinity of the minimum of g (0). r s g ( ) RCPIMC+, N = 54 CPIMC, N = 54 Overhauser-ModelCalmels et al. FIG. 17. Density dependence of the on-top PDF for Θ = 1 / N = 54 particles. Lines: results of ground statemodels, i.e. Eq. (9) (Overhauser model) and of Calmels et al. Ref. [87]. Let us now discuss the consequences of this density andtemperature dependence of g (0) for the high-momentumasymptotics of n ( k ). According to Eq. (3), the numberof electrons occupying large- k states is proportional to n ( k ; r s , Θ) ∝ r s · g (0; r s , Θ) k F ( r s ) k , where we made thedependence on the coupling parameter explicit. Takinginto account that k F ∝ n / ∼ r − s , the absolute valueof the asymptotic occupation number, at a given k andfixed Θ, scales as n ( k ) ∝ r − s g (0; r s , Θ) · k − . On the otherhand, considering the occupation number as a functionof the momentum normalized to the Fermi momentum, κ = k/k F , the density dependence becomes n ( κ ) → s ( r s , Θ) · κ − , (44) s ( r s , Θ) = 92 α r s · g (0; r s , Θ) , α := (cid:18) π (cid:19) . (45)Given the monotonic decrease of g (0) with r s , the func-tion n ( κ ) may exhibit non-monotonic behavior as a func-tion of r s , including a maximum at an intermediate r s -value. This is clearly seen in Fig. 19 for the temperaturesΘ = 2 , s ( r s )increases monotonically, for small r s , starting from zero.The decrease, governed by the monotonic decrease of g (0)sets in only at large r s where CPIMC simulations arenot possible any more. On the other hand, an extensiveset of restricted PIMC data [57] for g ( r ) is available, for − . . . . r s g ↑↓ ( ) CPIMCDMCFP-PIMCCalmels et al.ESA . . . . . . . . . . FIG. 18. Density dependence of the on-top PDF for Θ = 1.Red squares: CPIMC data [ r s ≤ . N = 54 particles; r s ≥ . N = 14]. Blue circles: FP-PIMC data [0 . < r s ≤ N = 66; 0 . ≤ r s ≤ . N = 34]. Black full (dashed) lines:ground state DMC simulations [53] and Eq. (9), respectively.Inset: zoom into the high-density range (linear scale). . . . . . . . . . . r s g ↑↓ ( ) Calmels et al.ESACPIMCPIMC . . . . r s s ( r s ) FIG. 19. Top: On-top PDF (top), bottom: the function (45)for two temperatures: Θ = 4 (orange line and symbols) andΘ = 2 (green line and symbols). Triangles: CPIMC data for N = 54, squares: FP-PIMC with N = 66 particles. Dot-ted lines: parametrization of Dornheim et al . [86], black line:ground state parametrization of Calmels et al . Note the ex-tended r s -range in the lower figure. For more data on themaximum of s ( r s ), see Tab. I. Θ r maxs s max Θ r maxs s max Θ r maxs s max s ( r s ), Eq. (45), as a function of temperature. Results arebased on the parametrization of the on-top-PDF by Dornheim et al . [86], see also Fig. 19. ≤ r s ≤ 40, which has recently been used by Dornheim et al. [86] to construct an analytical parametrization of g (0; r s , θ ). The results are denoted as ESA because theyconstitute an important ingredient to the effective staticapproximation for the static local field correction thatwas presented in Ref. [86].An example is shown in the lower part of Fig. 19 fortwo temperatures, Θ = 2 and Θ = 4. The maximum of s is observed around r s = 4, for Θ = 2 and r s ≈ 5, forΘ = 4. We have performed a systematic parameter scanon the basis of the analytical fit (ESA) over a broad rangeof temperatures. The results are collected table I. Theseresults show that the maximum of s ( r s ) is generally lo-cated in the range 3 . (cid:46) r s (cid:46) . 0. Interestingly r max s –the r s -value where the maximum is located – exhibits anon-monotonic temperature dependence. The reason isthe non-monotonic temperature dependence of g (0) thatwas discussed in detail in Sec. III B 2. Finally, the com-parison with the ab initio results contained in Fig. 19suggests that the ESA fit can be further improved usingour CPIMC and FP-PIMC data. C. Onset of the large- k asymptotic of n ( k ) Let us now find an approximate value of the momen-tum k ∞ where the k − -asymptotic starts to dominatethe behavior of the distribution function. In particular,we are interested to understand how this value dependson density and temperature. First, we observe that thesignificant broadening of the low-momentum part of thedistribution that is observed when the temperature isincreased pushes the value k ∞ to larger momenta. Fig-ure 2 suggests that this onset is near the intersection ofthe asymptotic, Eq. (3), n ∞ ( k ) with the ideal MDF givenby the Fermi-Dirac distribution function n id ( k ): n id ( k ∞ ) ! = n ∞ ( k ∞ ) . (46)This approach is demonstrated in Fig. 20, and the resultsare presented for a broad range of densities, in the rangeof r s = 0 . . . . . 6, and temperatures Θ ≤ 4, in Fig. 21.For this procedure, to obtain the asymptotic n ∞ we usedthe value of g (0) that was computed in CPIMC simula-tions.This figure shows that, with an increase of correlations(increase of r s ) the onset of the asymptotic is shifted − − − − k/k F n ( k ) Θ = 1 . 5Θ = 2 . 0Θ = 4 . FIG. 20. Illustration of the prescription (46) to determine theonset of the large-momentum asymptotic from the intersec-tion of the ideal Fermi function, f id ( k ), (dashes), with the k − asymptotic, n ∞ ( k ), (full lines of the same color). The asymp-totic is determined from CPIMC simulations of the on-topPDF for N = 54 particles. Θ k ∞ / k F r s = 0 . r s = 0 . r s = 1 . r s = 1 . FIG. 21. Onset k ∞ of the large-momentum asymptotic, as cal-culated from Eq. (46). The procedure is illustrated in Fig. 20.CPIMC simulations with N = 14 particles. The lower limit ofΘ, for the different curves, is set by the fermion sign problem. to lower momenta, even though the dependence is weak.The figure also shows that an algebraic tail of the momen-tum distribution exists also in a weakly quantum degen-erate plasma with Θ > 1. With increasing temperature,the onset of this asymptotic is pushed to larger momentawith k ∞ /k F increasing slightly faster than Θ . .5 IV. SUMMARY AND OUTLOOKA. Summary In this paper we have performed an analysis of themomentum distribution function of the correlated warmdense electron gas using recently developed ab initio quantum Monte Carlo methods. We have presented ex-tensive data obtained with CPIMC, for small r s . Thiswas complemented with new fermionic propagator PIMCdata, for r s (cid:38) 1, so the entire density range hase beencovered. Our CPIMC results for the momentum distribu-tion of the warm uniform electron gas achieve an unprece-dented accuracy – the asymptotic is resolved up to theeleventh digit for momenta up to approximately 15 k F ,cf. Figs. 2 and 3. For all parameters the existence ofthe 1 /k asymptotic is confirmed. Moreover, based onaccurate data for the on-top PDF the absolute value of n ( k ) in the asymptotic is obtained.While the value of the on-top PDF decreases monoton-ically with r s , it exhibits an interesting non-monotonictemperature dependence with a minimum around Θ =0 . k F (cid:46) k (cid:46) k F . Thisnon-trivial re-distribution of electrons may give rise to acounter-intuitive interaction-induced decrease of the ki-netic energy of the finite temperature electron gas, Thisconfirms earlier results [36, 54] and, at the same time, wepresent extensive data that allow to quantify this effectfor a broad range of parameters. B. Outlook Part of our results for the on-top PDF were obtainedwith help of the recent extended static approximation(ESA) [86]. Its advantage is that it allows for relativelyeasy parameter scans in a broad range of densities andtemperatures. Therefore, an important task is to fur-ther improve this approximation with the present high-quality data for g (0). The present simulations concen-trated on the range of r s (cid:46) 10 which is of relevance forwarm dense matter. At the same time the jellium modelis also of interest for the strongly correlated electron liq-uid, e.g. [88, 89]. It will, therefore, be interesting to ex- tend this analysis to larger r s -values, which should bestraightforward based on an analysis of the on-top PDF.Finally, the momentum distribution function is ofcrucial importance for realistic two-component plasmasfor which extensive restricted PIMC simulations, e.g.[11, 20] and fermionic PIMC simulations, e.g. [12, 90]have been performed. Therefore, an extension of thepresent analysis of the on-top PDF two two-componentQMC simulations if of high interest.This will also be the basis for the application of thepresent results to estimate the effect of power law tailsin n ( k ) in fusion rates and other inelastic processes, thatinvolve the impact of energetic particles. An example forthe latter are electron impact excitation and ionizationrates of atoms in a dense plasma. Such effects were pre-dicted for various chemical reactions in Ref. [27] basedon a approximate treatment of collision rates and phe-nomenological Lorentzian-type broadening of the elec-tron spectral function in Eq. (2). However, such approx-imations were shown to violate energy conservation, e.g.[91]. The present approach to n ( k ) makes such approxi-mations obsolete and, moreover, eliminates the multipleintegrations over the energy variables substantially sim-plifying the expressions for the rates.Finally, the relevance of algebraic tails of n ( k ) for nu-clear fusion rates in dense plasmas was discussed by manyauthors, e.g. [25, 27, 28, 92], but the agreement with ex-perimental data remains open. The results of the presentwork are, applicable to many fusion reactions of fermionicparticles, such as the proton-proton or He– He fusion re-actions in the sun or supernova stars that were consideredin Ref. [92]. For quantitative comparisons the presentsimulations should to be extended to multi-componentelectron-ion plasmas and include screening effects of theion-ion interactions which should not pose a principalproblem. ACKNOWLEDGMENTS This work has been supported by the DeutscheForschungsgemeinschaft via project BO1366-15/1. TDacknowledges financial support by the Center for Ad-vanced Systems Understanding (CASUS) which is fi-nanced by the German Federal Ministry of Educationand Research (BMBF) and by the Saxon Ministry forScience, Art, and Tourism (SMWK) with tax funds onthe basis of the budget approved by the Saxon State Par-liament.We gratefully acknowledge CPU-time atthe Norddeutscher Verbund f¨ur Hoch- undH¨ochstleistungsrechnen (HLRN) via grant shp00026and on a Bull Cluster at the Center for InformationServices and High Performance Computing (ZIH) atTechnische Universit¨at Dresden. REFERENCES [1] F. Graziani, M. P. Desjarlais, R. Redmer, and S. B.Trickey. Frontiers and Challenges in Warm Dense Mat-ter . Springer, 2014.[2] Tobias Dornheim, Simon Groth, and Michael Bonitz. Theuniform electron gas at warm dense matter conditions. Phys. Rep. , 744:1 – 86, 2018.[3] V. E. Fortov. Extreme States of Matter (High EnergyDensity Physics, Second Edition) . Springer, Heidelberg,2016.[4] M. Bonitz, T. Dornheim, Zh. A. Moldabekov, S. Zhang,P. Hamann, H. K¨ahlert, A. Filinov, K. Ramakrishna, andJ. Vorberger. Ab initio simulation of warm dense matter. Physics of Plasmas , 27(4):042710, 2020.[5] Gilles Chabrier. Quantum effects in dense Coulumbicmatter - Application to the cooling of white dwarfs. As-trophys. J. , 414:695, September 1993.[6] M. Schlanges, M. Bonitz, and A. Tschttschjan. Plasmaphase transition in fluid hydrogen–helium mixtures. Con-trib. Plasma Phys. , :109, 1995.[7] V. Bezkrovniy, V. S. Filinov, D. Kremp, M. Bonitz,M. Schlanges, W. D. Kraeft, P. R. Levashov, and V. E.Fortov. Monte Carlo results for the hydrogen Hugoniot. Phys. Rev. E , 70:057401, Nov 2004.[8] Nadine Nettelmann, Robert P¨ustow, and Ronald Red-mer. Saturn layered structure and homogeneous evolu-tion models with different EOSs. Icarus , 225(1):548–557,July 2013.[9] Ralph Ernstorfer, Maher Harb, Christoph T.Hebeisen, Germ´an Sciaini, Thibault Dartigalongue,and R. J. Dwayne Miller. The formation of warm densematter: Experimental evidence for electronic bondhardening in gold. Science , 323(5917):1033–1037, 2009.[10] O. A. Hurricane, D. A. Callahan, D. T. Casey, E. L.Dewald, T. R. Dittrich, T. D¨oppner, S. Haan, D. E.Hinkel, L. F. Berzak Hopkins, O. Jones, A. L. Kritcher,S. Le Pape, T. Ma, A. G. MacPhee, J. L. Milovich,J. Moody, A. Pak, H.-S. Park, P. K. Patel, J. E. Ralph,H. F. Robey, J. S. Ross, J. D. Salmonson, B. K. Spears,P. T. Springer, R. Tommasini, F. Albert, L. R. Benedetti,R. Bionta, E. Bond, D. K. Bradley, J. Caggiano, P. M.Celliers, C. Cerjan, J. A. Church, R. Dylla-Spears,D. Edgell, M. J. Edwards, D. Fittinghoff, M. A. Bar-rios Garcia, A. Hamza, R. Hatarik, H. Herrmann,M. Hohenberger, D. Hoover, J. L. Kline, G. Kyrala,B. Kozioziemski, G. Grim, J. E. Field, J. Frenje,N. Izumi, M. Gatu Johnson, S. F. Khan, J. Knauer,T. Kohut, O. Landen, F. Merrill, P. Michel, A. Moore,S. R. Nagel, A. Nikroo, T. Parham, R. R. Rygg, D. Sayre,M. Schneider, D. Shaughnessy, D. Strozzi, R. P. J. Town,D. Turnbull, P. Volegov, A. Wan, K. Widmann, C. Wilde,and C. Yeamans. Inertially confined fusion plasmas dom-inated by alpha-particle self-heating. Nat. Phys. , 12:800–806, April 2016.[11] S. X. Hu, B. Militzer, V. N. Goncharov, and S. Skupsky.Strong coupling and degeneracy effects in inertial con-finement fusion implosions. Phys. Rev. Lett. , 104:235003,Jun 2010.[12] V S Filinov, M Bonitz, W Ebeling, and V E Fortov.Thermodynamics of hot dense H-plasmas: path integralMonte Carlo simulations and analytical approximations. Plasma Phys. Control. Fusion , (6):743, 2001. [13] B. Militzer, W. B. Hubbard, J. Vorberger, I. Tamblyn,and S. A. Bonev. A Massive Core in Jupiter Predictedfrom First-Principles Simulations. Astrophys. J. Lett. ,688(1):L45, November 2008.[14] Tobias Dornheim, Simon Groth, Travis Sjostrom,Fionn D. Malone, W. M. C. Foulkes, and Michael Bonitz.Ab Initio Quantum Monte Carlo Simulation of the WarmDense Electron Gas in the Thermodynamic Limit. Phys.Rev. Lett. , 117:156403, 2016.[15] B. B. L. Witte, L. B. Fletcher, E. Galtier, E. Gamboa,H. J. Lee, U. Zastrau, R. Redmer, S. H. Glenzer, andP. Sperling. Warm dense matter demonstrating non-drude conductivity from observations of nonlinear plas-mon damping. Phys. Rev. Lett. , 118:225001, May 2017.[16] Paul Hamann, Tobias Dornheim, Jan Vorberger, Zhan-dos Moldabekov, and Michael Bonitz. Dynamic Proper-ties of the warm dense Electron gas: an ab initio path in-tegral Monte Carlo approach. Phys. Rev. B , 102:125150,2020.[17] T. Dornheim, S. Groth, J. Vorberger, and M. Bonitz.Ab initio Path Integral Monte Carlo Results for the Dy-namic Structure Factor of Correlated Electrons: Fromthe Electron Liquid to Warm Dense Matter. Phys. Rev.Lett. , 121:255001, Dec 2018.[18] Paul Hamann, Tobas Dornheim, Jan Vorberger, ZhandosMoldabekov, and Michael Bonitz. Ab initio results for theplasmon dispersion and damping of the warm dense elec-tron gas. Contrib. Plasma Phys. , 60:e202000147, 2020.[19] V.S. Filinov, V.E. Fortov, M. Bonitz, and D. Kremp.Pair distribution functions of dense partially ionized hy-drogen. Phys. Lett. A , 274(5):228 – 235, 2000.[20] B. Militzer and D. M. Ceperley. Path IntegralMonte Carlo Calculation of the Deuterium Hugoniot. Phys. Rev. Lett. , 85(9):1890–1893, August 2000.[21] Tobias Dornheim, Simon Groth, and Michael Bonitz. Abinitio results for the static structure factor of the warmdense electron gas. Contrib. Plasma Phys. , 57(10):468–478, 2017.[22] S. Groth, T. Dornheim, and J. Vorberger. Ab initio pathintegral monte carlo approach to the static and dynamicdensity response of the uniform electron gas. Phys. Rev.B , 99:235122, Jun 2019.[23] D Kraus, B Bachmann, B Barbrel, R W Falcone, L BFletcher, S Frydrych, E J Gamboa, M Gauthier, D OGericke, S H Glenzer, S G¨ode, E Granados, N J Hart-ley, J Helfrich, H J Lee, B Nagler, A Ravasio, W Schu-maker, J Vorberger, and T D¨oppner. Characterizing theionization potential depression in dense carbon plasmaswith high-precision spectrally resolved x-ray scattering. Plasma Physics and Controlled Fusion , 61(1):014015, nov2018.[24] Siegfried H. Glenzer and Ronald Redmer. X-ray thomsonscattering in high energy density plasmas. Rev. Mod.Phys. , 81:1625–1663, Dec 2009.[25] A. N. Starostin, A. B. Mironov, N. L. Aleksandrov, N. J.Fisch, and R. M. Kulsrud. Quantum corrections tothe distribution function of particles over momentum indense media. Physica A: Statistical Mechanics and itsApplications , 305(1):287–296, March 2002.[26] A. N. Starostin, A. G. Leonov, Yu. V. Petrushevich,and Vl. K. Rerikh. Quantum corrections to the particle distribution function and reaction rates in dense media. Plasma Physics Reports , 31(2):123–132, 2005.[27] A. N. Starostin, V.K. Gryaznov, and Yu. V. Petrushe-vich. Quantum corrections to the distribution functionof particles over momentum in dense media. JETP ,125(5):940–947, 2017.[28] V. I. Savchenko. Quantum, multibody effects and nuclearreaction rates in plasmas. Physics of Plasmas , 8(1):82–91, 2001.[29] Eugene Wigner. On the quantum correction for thermo-dynamic equilibrium. Physical review , 40(5):749, 1932.[30] David Bohm and David Pines. A Collective Descriptionof Electron Interactions: III. Coulomb Interactions in aDegenerate Electron Gas. 92(3):609–625, 1953.[31] P. Nozi`eres and D. Pines. Correlation energy of a freeelectron gas. 111(2):442–454.[32] E. Daniel and S. H. Vosko. Momentum Distribution of anInteracting Electron Gas. Physical Review , 120(6):2041–2044, December 1960.[33] Murray Gell-Mann and Keith A Brueckner. Correlationenergy of an electron gas at high density. Physical Review ,106(2):364, 1957.[34] V. M. Galitskii and V. V. Yakimets. Particle relaxationin a maxwell gas. SOVIET PHYSICS JETP , 24(3), 1967.[35] L.P. Kadanoff and G. Baym. Quantum Statistical Me-chanics . Addison-Wesley Publ. Co. Inc., 2nd edition,1989.[36] W. D. Kraeft, M. Schlanges, J. Vorberger, and H. E.DeWitt. Kinetic and correlation energies and distributionfunctions of dense plasmas. Phys. Rev. E , 66:046405, Oct2002.[37] M. Bonitz. Quantum Kinetic Theory . Teubner-Texte zurPhysik. Springer, Cham, 2 edition, 2016.[38] K. Balzer and M. Bonitz. Nonequilibrium Green’s Func-tions Approach to Inhomogeneous Systems . Springer,Berlin Heidelberg, 2013.[39] J. C. Kimball. Short-range correlations and the struc-ture factor and momentum distribution of electrons.8(9):1513.[40] H. Yasuhara and Y. Kawazoe. A note on the momen-tum distribution function for an electron gas. PhysicaA: Statistical Mechanics and its Applications , 85(2):416–424, January 1976.[41] Torben Ott, Hauke Thomsen, Jan Willem Abraham, To-bias Dornheim, and Michael Bonitz. Recent progress inthe theory and simulation of strongly correlated plas-mas: phase transitions, transport, quantum, and mag-netic field effects. The European Physical Journal D ,72(5):84, May 2018.[42] Johannes Hofmann, Marcus Barth, and Wilhelm Zw-erger. Short-distance properties of Coulomb systems.87(23):235125.[43] A. K. Rajagopal, J. C. Kimball, and M. Banerjee. Short-ranged correlations and the ferromagnetic electron gas.18(5):2339–2345.[44] Markus Holzmann, Bernard Bernu, Carlo Pierleoni,Jeremy McMinis, David M. Ceperley, Valerio Olevano,and Luigi Delle Site. Momentum distribution of the ho-mogeneous electron gas. Phys. Rev. Lett. , 107:110402,Sep 2011.[45] S. Jensen, C. N. Gilbreth, and Y. Alhassid. Contactin the Unitary Fermi Gas across the Superfluid PhaseTransition. Phys. Rev. Lett. , 125:043402, Jul 2020.[46] Elmer V. H. Doggen and Jami J. Kinnunen. Momentum- resolved spectroscopy of a Fermi liquid. Scientific Re-ports , 5.[47] Tobias Dornheim, Attila Cangi, Kushal Ramakrishna,Maximilian B¨ohme, Shigenori Tanaka, and Jan Vor-berger. Effective static approximation: A fast and reli-able tool for warm-dense matter theory. Phys. Rev. Lett. ,125:235001, Dec 2020.[48] A. Holas. Exact asymptotic expression for the static di-electric function of a uniform electron liquid at large wavevector. In F.J. Rogers and H.E. DeWitt, editors, StronglyCoupled Plasma Physics . Plenum, New York, 1987.[49] Travis Sjostrom and J´erˆome Daligault. Gradient correc-tions to the exchange-correlation free energy. Phys. Rev.B , 90:155109, Oct 2014.[50] T. Dornheim, J. Vorberger, S. Groth, N. Hoffmann,Zh.A. Moldabekov, and M. Bonitz. The static local fieldcorrection of the warm dense electron gas: An ab initiopath integral Monte Carlo study and machine learningrepresentation. J. Chem. Phys , 151:194104, 2019.[51] Yasutami Takada. Emergence of an excitonic collectivemode in the dilute electron gas. Phys. Rev. B , 94:245106,Dec 2016.[52] G. Ortiz and P. Ballone. Correlation energy, struc-ture factor, radial distribution function, and momentumdistribution of the spin-polarized uniform electron gas. Phys. Rev. B , 50:1391–1405, Jul 1994.[53] G. G. Spink, R. J. Needs, and N. D. Drummond. Quan-tum Monte Carlo study of the three-dimensional spin-polarized homogeneous electron gas. Phys. Rev. B ,88:085121, Aug 2013.[54] B. Militzer and E. L. Pollock. Lowering of the kineticenergy in interacting quantum systems. Phys. Rev. Lett. ,89:280401, Dec 2002.[55] B. Militzer, E.L. Pollock, and D.M. Ceperley. Path inte-gral Monte Carlo calculation of the momentum distribu-tion of the homogeneous electron gas at finite tempera-ture. High Energy Density Physics , 30:13 – 20, 2019.[56] A.S. Larkin and V.S. Filinov. Quantum tails in themomentum distribution functions of non-ideal fermi sys-tems. Contributions to Plasma Physics , 58(2-3):107–113,2018.[57] Ethan W. Brown, Bryan K. Clark, Jonathan L. DuBois,and David M. Ceperley. Path-Integral Monte Carlo Sim-ulation of the Warm Dense Homogeneous Electron Gas. Phys. Rev. Lett. , 110:146405, Apr 2013.[58] T. Schoof, S. Groth, J. Vorberger, and M. Bonitz. AbInitio thermodynamic results for the degenerate electrongas at finite temperature. Phys. Rev. Lett. , 115:130402,2015.[59] V. S. Filinov, V. E. Fortov, M. Bonitz, and Zh. Mold-abekov. Fermionic path-integral Monte Carlo results forthe uniform electron gas at finite temperature. Phys.Rev. E , 91:033108, 2015.[60] T. Dornheim. Fermion sign problem in path integralmonte carlo simulations: Quantum dots, ultracold atoms,and warm dense matter. Phys. Rev. E , 100:023307, Aug2019.[61] Ethan Brown, Miguel A. Morales, Carlo Pierleoni, andDavid Ceperley. Quantum monte carlo techniques andapplications for warm dense matter. In Frank Graziani,Michael P. Desjarlais, Ronald Redmer, and Samuel B.Trickey, editors, Frontiers and Challenges in WarmDense Matter , pages 123–149, Cham, 2014. Springer In-ternational Publishing. [62] T. Schoof, S. Groth, and M. Bonitz. Towards ab initiothermodynamics of the electron gas at strong degeneracy. Contrib. Plasma Phys. , 55:136–143, 2015.[63] Tobias Dornheim, Simon Groth, Alexey Filinov, andMichael Bonitz. Permutation blocking path integralMonte Carlo: a highly efficient approach to the simu-lation of strongly degenerate non-ideal fermions. New J.Phys. , 17(7):073017, 2015.[64] Tobias Dornheim, Tim Schoof, Simon Groth, Alexey Fil-inov, and Michael Bonitz. Permutation blocking path in-tegral Monte Carlo approach to the uniform electron gasat finite temperature. J. Chem. Phys. , 143(20):204101,2015.[65] Tobias Dornheim, Simon Groth, and Michael Bonitz.Permutation blocking path integral monte carlo simula-tions of degenerate electrons at finite temperature. Con-tributions to Plasma Physics , 59(4-5):e201800157, 2019.[66] Simon Groth, Tobias Dornheim, Travis Sjostrom,Fionn D. Malone, W. M. C. Foulkes, and Michael Bonitz.Ab initio Exchange-Correlation Free Energy of the Uni-form Electron Gas at Warm Dense Matter Conditions. Phys. Rev. Lett. , 119:135001, 2017.[67] Simon Groth, Tobias Dornheim, and Michael Bonitz.Configuration path integral Monte Carlo approach to thestatic density response of the warm dense electron gas. J. Chem. Phys. , 147(16):164108, 2017.[68] Arif Yilmaz, Kai Hunger, Tobias Dornheim, SimonGroth, and Michael Bonitz. Restricted configurationpath integral Monte Carlo. accepted for publication inJ. Chem. Phys. , 2020.[69] G. Giuliani, G. Vignale, and Cambridge University Press. Quantum Theory of the Electron Liquid . Masters Seriesin Physics and Astronomy. Cambridge University Press,Leiden, 2005.[70] Paola Gori-Giorgi and John P. Perdew. Short-range cor-relation in the uniform electron gas: Extended Over-hauser model. Phys. Rev. B , 64(15):155102, 2001.[71] W.-D. Kraeft, D. Kremp, W. Ebeling, and G. R¨opke. Quantum Statistics of Charged Particle Systems .Akademie-Verlag, Berlin, 1986.[72] G. Kelbg. Ann. Phys. (Leipzig) , 12:219, 1963.[73] C. Deutsch. Nodal expansion in a real matter plasma. Physics Letters A , 60(4):317 – 318, 1977.[74] A. Filinov, M. Bonitz, and W. Ebeling. Improved Kelbgpotential for correlated Coulomb systems. J. Phys. A:Math. Gen. , :5957–5962, 2003.[75] A. V. Filinov, V. O. Golubnychiy, M. Bonitz, W. Ebeling,and J. W. Dufty. Temperature-dependent quantum pairpotentials and their application to dense partially ionizedhydrogen plasmas. Phys. Rev. E , 70:046411, Oct 2004.[76] W. Ebeling, A. Filinov, M. Bonitz, V Filinov, andT. Pohl. The method of effective potentials in thequantum-statistical theory of plasmas. J. Phys. A: Math.Gen. , 39(17):4309, 2006.[77] T. Schoof, M. Bonitz, A. Filinov, D. Hochstuhl, and J.W.Dufty. Configuration path integral Monte Carlo. Contrib.Plasma Phys. , 84:687–697, 2011.[78] S. Groth, T. Schoof, T. Dornheim, and M. Bonitz. Ab ini-tio quantum Monte Carlo simulations of the uniform elec-tron gas without fixed nodes. Phys. Rev. B , 93:085102, 2016.[79] Tim Schoof, Simon Groth, and Michael Bonitz. Intro-duction to Configuration Path Integral Monte Carlo. InMichael Bonitz, Jose Lopez, Kurt Becker, and HaukeThomsen, editors, Complex Plasmas , volume 82 of Springer Ser. At., Opt., Plasma Phys. , pages 153–194.Springer International Publishing, 2014.[80] Tim Schoof. Configuration path integral Monte Carlo:Ab inition simulations of fermions in the warm densematter regime, 12 2016.[81] Simone Chiesa, David M. Ceperley, Richard M. Mar-tin, and Markus Holzmann. Finite-size error in many-body simulations with long-range interactions. Phys.Rev. Lett. , 97:076404, Aug 2006.[82] Paola Gori-Giorgi and Paul Ziesche. Momentum distribu-tion of the uniform electron gas: Improved parametriza-tion and exact limits of the cumulant expansion. PhysicalReview B , 66(23), December 2002.[83] Marcus Barth. Few-Body Correlations in Many-BodyPhysics . Dissertation, Technische Universit¨at M¨unchen,2015.[84] G. Kelbg. Ann. Phys. (Leipzig) , 13:354, 1963.[85] G. Kelbg. Ann. Phys. (Leipzig) , 14:394, 1964.[86] Tobias Dornheim, Attila Cangi, Kushal Ramakrishna,Maximilian B¨ohme, Shigenori Tanaka, and Jan Vor-berger. Effective Static Approximation: A fast and re-liable tool for warm dense matter theory. submitted forpublication (2020).[87] L. Calmels and A. Gold. Pair-correlation function of theelectron gas with long-range Coulomb interaction: Lad-der theory. Physical Review B , 57(3):1436–1443.[88] Tobias Dornheim, Jan Vorberger, and Michael Bonitz.Nonlinear Electronic Density Response in Warm DenseMatter. Phys. Rev. Lett. , 125:085001, 2020.[89] Tobias Dornheim, Travis Sjostrom, Shigenori Tanaka,and Jan Vorberger. The Strongly Coupled Electron Liq-uid: ab initio Path Integral Monte Carlo Simulations andDielectric Theories. Phys. Rev. B , 101:045129, 2020.[90] M. Bonitz, V. S. Filinov, V. E. Fortov, P. R. Lev-ashov, and H. Fehske. Crystallization in Two-ComponentCoulomb Systems. Phys. Rev. Lett. , 95:235006, Dec 2005.[91] M. Bonitz, D. Semkat, and H. Haug. Non-Lorentzianspectral functions for Coulomb quantum kinetics. Europ.Phys. J. B , 9:309, 1999.[92] N.J. Fisch, M.G. Gladush, Yu.V Petrushevich,P Quarati, and A.N Starostin. Enhancement offusion rates due to quantum effects in the particlesmomentum distribution in nonideal plasma media. Eur.Phys. Journal D , 66:154, 2012.[93] A. W. Overhauser. Pair-correlation function of an elec-tron gas. Canadian Journal of Physics , 73(11-12):683–686, 1995. Appendix A: Derivation of the CPIMC-Estimatorfor the on-top PDF, Eq. (35) We start by expressing the field operators in termsof the creation and annihilation operators in momentumrepresentation, cf. Eqs. (33),9ˆΨ † σ ( r ) ˆΨ † σ ( r ) ˆΨ σ ( r ) ˆΨ σ ( r ) = (cid:32)(cid:88) i φ ∗ i ( r , σ )ˆ a † i (cid:33) (cid:88) j φ ∗ j ( r , σ )ˆ a † j (cid:32)(cid:88) k φ k ( r , σ )ˆ a k (cid:33) (cid:32)(cid:88) l φ l ( r , σ )ˆ a l (cid:33) = (cid:88) ijkl φ ∗ i ( r , σ ) φ ∗ j ( r , σ ) φ k ( r , σ ) φ l ( r , σ )ˆ a † i ˆ a † j ˆ a k ˆ a l = (cid:88) ijkl ϕ ∗ i ( r ) ϕ ∗ j ( r ) ϕ k ( r ) ϕ l ( r ) δ s i ,σ δ s j ,σ δ s k ,σ δ s l ,σ ˆ a † i ˆ a † j ˆ a k ˆ a l . (A1)The Kronecker deltas in the last line imply that the spins of the orbitals i, l and j, k must be pairwise equal.Since the equation is symmetric with respect to the two possible choices of the spin projections ( σ = ↑ , σ = ↓ ) and( σ = ↓ , σ = ↑ ), we extend the sum over the two possibilities. Since we are interested in the case of antiparallel spins, σ (cid:54) = σ , we consider the following relations of the summation indices, { i, j, k, l ∈ Z | s i = σ = s l , s j = σ = s k } ∪ { i, j, k, l ∈ Z | s i = σ = s l , s j = σ = s k } = { i, j, k, l ∈ Z | s i = s l (cid:54) = s j = s k } = { i, j, k, l ∈ Z | s i = s l , s j = s k } \ { i, j, k, l ∈ Z | s i = s l = s j = s k } . (A2)Thus the last line of Eq. (A1) can be replaced by the sum over the sets in the last line of the above Eq. (A2). Sinceboth possible choices of the spins are allowed in the latter relation, the sum is twice the value of one definite choice,ˆΨ † σ ( r ) ˆΨ † σ ( r ) ˆΨ σ ( r ) ˆΨ σ ( r ) = (cid:88) ijkl ϕ ∗ i ( r ) ϕ ∗ j ( r ) ϕ k ( r ) ϕ l ( r ) δ s i ,σ δ s j ,σ δ s k ,σ δ s l ,σ ˆ a † i ˆ a † j ˆ a k ˆ a l = 12 (cid:88) ijkl ϕ ∗ i ( r ) ϕ ∗ j ( r ) ϕ k ( r ) ϕ l ( r ) δ s i ,s l δ s j ,s k (1 − δ s i ,s j )ˆ a † i ˆ a † j ˆ a k ˆ a l (A3)The statistical expectation value of this four-operator product can be expressed via the momentum representationof the two-particle density matrix, d ijkl , (cid:68) ˆΨ † σ ( r ) ˆΨ † σ ( r ) ˆΨ σ ( r ) ˆΨ σ ( r ) (cid:69) = 12 (cid:88) ijkl ϕ ∗ i ( r ) ϕ ∗ j ( r ) ϕ k ( r ) ϕ l ( r ) δ s i ,s l δ s j ,s k (1 − δ s i ,s j ) (cid:104) ˆ a † i ˆ a † j ˆ a k ˆ a l (cid:105) = d ijkl . (A4)We further need two-operator products that give riseto the spin densities appearing in the denominatorof Eq. (5). Applying again the basis transformation,Eq. (33), we obtainˆΨ † σ ( r ) ˆΨ σ ( r ) = (cid:88) ij φ i ( r , σ ) φ j ( r , σ )ˆ a † i ˆ a j = (cid:88) ij ϕ i ( r ) ϕ j ( r ) δ s i ,σ δ s j ,σ ˆ a † i ˆ a j . (A5)In the uniform electron gas, momentum conservationleads to (cid:104) ˆ a † i ˆ a j (cid:105) = (cid:104) ˆ n i (cid:105) δ i,j and, consequently, (cid:68) ˆΨ † σ ( r ) ˆΨ σ ( r ) (cid:69) = (cid:88) ij ϕ i ( r ) ϕ j ( r ) δ s i ,σ δ s j ,σ (cid:68) ˆ a † i ˆ a j (cid:69) = (cid:88) ij ϕ i ( r ) ϕ j ( r ) δ s i ,σ δ s j ,σ (cid:104) ˆ n i (cid:105) δ i,j = (cid:88) i | ϕ i ( r ) | δ s i ,σ (cid:104) ˆ n i (cid:105) (A6) The expectation value, Eq. (A4), and the spin density,Eq. (A6), contain products of plane wave single-particleorbitals (32). The product of two orbitals directly yieldsthe normalization factor. | ϕ i ( r ) | = ϕ ∗ i ( r ) ϕ i ( r ) = 1 V (A7)Further, due to momentum conservation, ϕ ∗ i ( r ) ϕ ∗ j ( r ) ϕ k ( r ) ϕ l ( r ) =1 V e i =0 ( k k + k l − k i − k j ) r = 1 V . (A8)With the definition (5) of the spin-resolved pair distribu-tion function and the results from Eqs. (A4) and (A6),the on-top PDF may be expressed via quantities that aredirectly accessible in CPIMC simulations,0 g ↑↓ = g σ σ ( r , r ) = V (cid:80) ijkl δ s i ,s l δ s j ,s k (1 − δ s i ,s j ) d ijkl (cid:18) V (cid:80) i δ s i ,σ (cid:104) ˆ n i (cid:105) (cid:19) (cid:18) V (cid:80) i δ s i ,σ (cid:104) ˆ n i (cid:105) (cid:19) = 12 (cid:80) ijkl δ s i ,s l δ s j ,s k (1 − δ s i ,s j ) d ijkl (cid:88) i δ s i ,σ (cid:104) ˆ n i (cid:105) N σ (cid:88) i δ s i ,σ (cid:104) ˆ n i (cid:105) N σ = 1 Z (cid:88)(cid:90) C N σ ( C ) N σ ( C ) (cid:88) ijkl δ s i ,s l δ s j ,s k (1 − δ s i ,s j ) d ijkl ( C ) W ( C ) (A9)In the last line, the expression for the CPIMC expecta-tion value of the two-particle density matrix, Eq. (30),has been inserted. Also the spin-resolved particle num-bers, N σ , were introduced and their dependence on the configuration C has been written explicitly in the lastline. The estimator can be read off the expression in thebraces, g ↑↓ ( C ) = 12 N σ ( C ) N σ ( C ) (cid:88) ijkl δ s i ,s l δ s j ,s k (1 − δ s i ,s j ) d ijkl ( C ) := g ijkl ( C ) , (A10)where the sum can be rearranged as12 (cid:88) ijkl g ijkl = (cid:88) k (cid:54) = i Gori-Giorgi and Perdew presented a parametrizationof the ground state on-top pair distribution function [70]that was given in the main text, cf. Eq. (9). Here webriefly recall the idea of this model. The authors solvedthe radial two-particle Schr¨odinger equation for the uni-form electron gas by replacing the bare Coulomb poten- tial by a screened potential that was introduced by Over-hauser in Ref. [93] v ( r ) = r s (cid:18) r s r + r r − (cid:19) , r ≤ r s , , r > r s . (B1)This potential is the sum of the Coulomb potential ofan electron and the statically screened potential aroundit that is due to the uniform positive background. Forthe latter a mean-field result (solution of Poisson’s equa-tion) is being used. The main approximations are relatedto setting this potential to zero outside a sphere of ra-dius r s and neglecting correlations. The solution of theSchr¨odinger equation allows to compute the on-top PDFleading to the parametrization (9).1 N Θ r s (cid:104) T (cid:105) id (cid:104) T (cid:105) 54 0.0625 0.1 108.8100 108.8246 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± − . Appendix C: Modification of the kinetic energy byinteraction effects The influence of Coulomb interaction on the kinetic en-ergy of the warm dense uniform electron gas was studiedin the main text in Sec. III A 2, see in particular Figs. 4and 5. In this Appendix we provide extensive data for thekinetic energy of the UEG compared to the kinetic energyof the ideal system, based on ab initio CPIMC simula-tions, for temperatures 0 . ≤ Θ ≤ r s ≤ N Θ r s (cid:104) T (cid:105) id (cid:104) T (cid:105) 14 0.0625 0.1 12.0017 12.0066 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±±