\emph{Ab initio} Quantum Monte Carlo simulation of the warm dense electron gas
Tobias Dornheim, Simon Groth, Fionn Malone, Tim Schoof, Travis Sjostrom, W.M.C. Foulkes, Michael Bonitz
AAb initio Quantum Monte Carlo simulation of the warm dense electron gas
Tobias Dornheim, a) Simon Groth, b) Fionn D. Malone, Tim Schoof, Travis Sjostrom, W.M.C. Foulkes, and Michael Bonitz Institut f¨ur Theoretische Physik und Astrophysik, Christian-Albrechts-Universit¨at zu Kiel, D-24098 Kiel,Germany Department of Physics, Imperial College London, Exhibition Road, London SW7 2AZ,United Kingdom Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545,USA (Dated: 7 November 2018)
Warm dense matter is one of the most active frontiers in plasma physics due to its relevance for denseastrophysical objects as well as for novel laboratory experiments in which matter is being strongly compressede.g. by high-power lasers. Its description is theoretically very challenging as it contains correlated quantumelectrons at finite temperature—a system that cannot be accurately modeled by standard analytical or groundstate approaches. Recently several breakthroughs have been achieved in the field of fermionic quantum MonteCarlo simulations. First, it was shown that exact simulations of a finite model system (30 . . .
100 electrons)is possible that avoid any simplifying approximations such as fixed nodes [Schoof et al. , Phys. Rev. Lett. , 130402 (2015)]. Second, a novel way to accurately extrapolate these results to the thermodynamic limitwas reported by Dornheim et al. [Phys. Rev. Lett. , 156403 (2016)]. As a result, now thermodynamicresults for the warm dense electron gas are available that have an unprecedented accuracy on the order of0 . I. INTRODUCTION
The uniform electron gas (UEG) (i.e., electrons in auniform positive background) is inarguably one of themost fundamental systems in condensed matter physicsand quantum chemistry . Most notably, the availabilityof accurate quantum Monte Carlo (QMC) data for itsground state properties has been pivotal for the successof Kohn-Sham density functional theory (DFT) .Over the past few years, interest in the study of mat-ter under extreme conditions has grown rapidly. Exper-iments with inertial confinement fusion targets andlaser-excited solids , but also astrophysical applicationssuch as planet cores and white dwarf atmospheres ,require a fundamental understanding of the warm densematter (WDM) regime, a problem now at the forefrontof plasma physics and materials science. In this pe-culiar state of matter, both the dimensionless Wigner-Seitz radius r s = r/a (with the mean interparticle dis-tance r and Bohr radius a ) and the reduced temperature θ = k B T /E F ( E F being the Fermi energy) are of orderunity, implying a complicated interplay of quantum de-generacy, coupling effects, and thermal excitations. Be-cause of the importance of thermal excitation, the usualground-state version of DFT does not provide an ap-propriate description of WDM. An explicitly thermody-namic generalization of DFT has long been known in a) Electronic mail: [email protected] b) TD and SG contributed equally to this work. principle, but requires an accurate parametrization of theexchange-correlation free energy ( f xc ) of the UEG overthe entire warm dense regime as an input .This seemingly manageable task turns out to be a ma-jor obstacle. The absence of a small parameter preventsa low-temperature or perturbation expansion and, con-sequently, Green function techniques in the Montroll-Ward and e approximations break down. Fur-ther, linear response theory within the random phaseapproximation (RPA) and also with the additionalinclusion of static local field corrections as suggested by,e.g., Singwi, Tosi, Land, and Sj¨olander (STLS) andVashista and Singwi (VS), are not reliable. Quantumclassical mappings are exact in some known limitingcases, but constitute an uncontrolled approximation inthe WDM regime.The difficulty of constructing a quantitatively accu-rate theory of WDM leaves thermodynamic QMC simu-lations as the only available tool to accomplish the taskat hand. However, the widely used path integral MonteCarlo (PIMC) approach is severely hampered by thenotorious fermion sign problem (FSP), which limitssimulations to high temperatures and low densities andprecludes applications to the warm dense regime. In theirpioneering work, Brown et al. circumvented the FSP byusing the fixed-node approximation (an approach here-after referred to as restricted PIMC, RPIMC), which al-lowed them to present the first comprehensive results forthe UEG over a wide temperature range for r s ≥ f xc required for thermo-dynamic DFT (see Refs. 24, 33, and 34), their quality has a r X i v : . [ phy s i c s . p l a s m - ph ] N ov been called into question. Firstly, RPIMC constitutes anuncontrolled approximation , which means that theaccuracy of the results for the finite model system studiedby Brown et al. was unclear. This unsatisfactory situ-ation has sparked remarkable recent progress in the fieldof fermionic QMC . In particular, a combination oftwo complementary QMC approaches has recentlybeen used to simulate the warm dense UEG withoutnodal restrictions , revealing that the nodal contraintsin RPIMC cause errors exceeding 10%. Secondly, Brown et al. extrapolated their QMC results for N = 33 spin-polarized ( N = 66 unpolarized) electrons to the macro-scopic limit by applying a finite- T generalization of thesimple first-order finite-size correction (FSC) introducedby Chiesa et al. for the ground state. As we have re-cently shown , this is only appropriate for low tempera-ture and strong coupling and, thus, introduces a secondsource of systematic error.In this paper, we give a concise overview of the currentstate of the art of quantum Monte Carlo simulations ofthe warm dense electron gas and present new results re-garding the extrapolation to the thermodynamic limit.Further, we discuss the remaining open questions andchallenges in the field.After a brief introduction to the UEG model (II) we in-troduce various QMC techniques, starting with the stan-dard path integral Monte Carlo approach (III A) and adiscussion of the origin of the FSP (III B). The sign prob-lem can be alleviated using either the permutation block-ing PIMC (PB-PIMC, III C) method, or the configura-tion PIMC algorithm (CPIMC, III D), or the density ma-trix QMC (DMQMC, III E) approach. In combination,these tools make it possible to obtain accurate resultsfor a finite model system over almost the entire warmdense regime (IV). The second key issue is the extrapo-lation from the finite to the infinite system, i.e., the de-velopment of an appropriate finite-size correction ,which is discussed in detail in Sec. V. Finally, we com-pare our QMC results for the infinite UEG to other data(V B 2) and finish with some concluding remarks and asummary of open questions. II. THE UNIFORM ELECTRON GASA. Coordinate representation of the Hamiltonian
Following Refs. 44 and 54, we express the Hamiltonian(using Hartree atomic units) for N = N ↑ + N ↓ unpolar-ized electrons in coordinate space asˆ H = − N (cid:88) i =1 ∇ i + 12 N (cid:88) i =1 N (cid:88) j (cid:54) = i Ψ( r i , r j ) + N ξ M , (1) with the well-known Madelung constant ξ M and the pe-riodic Ewald pair interactionΨ( r , s ) = 1Ω (cid:88) G (cid:54) =0 e − π G /κ e πi G ( r − s ) π G − πκ Ω + (cid:88) R erfc( κ | r − s + R | ) | r − s + R | . (2)Here R = n L and G = n /L denote the real and recip-rocal space lattice vectors, respectively, with n and n three-component vectors of integers, L the box length,Ω = L the box volume, and κ the usual Ewald parame-ter. B. Hamiltonian in second quantization
In second quantized notation using a basis of spin-orbitals of plane waves, (cid:104) r σ | k i σ i (cid:105) = L / e i k i · r δ σ,σ i , with k i = πL m i , m i ∈ Z and σ i ∈ {↑ , ↓} , the Hamiltonian,Eq. (1), becomesˆ H = 12 (cid:88) i k i ˆ a † i ˆ a i + (cid:88) i Let us consider N spinless distinguishable particles inthe canonical ensemble, with the volume Ω, the inversetemperature β = 1 /k B T , and the density N/ Ω beingfixed. The partition function in coordinate representa-tion is given by Z = (cid:90) d R (cid:104) R | e − β ˆ H | R (cid:105) , (5)where R = { r , . . . , r N } contains all 3 N particle coordi-nates, and the Hamiltonian ˆ H = ˆ K + ˆ V is given by thesum of a kinetic and a potential part, respectively. Sincethe low-temperature matrix elements of the density op-erator, ˆ ρ = e − β ˆ H , are not readily known, we exploit thegroup property e − β ˆ H = (cid:16) e − (cid:15) ˆ H (cid:17) P , with (cid:15) = β/P andpositive integers P . Inserting P − (cid:82) d R α | R (cid:105) α (cid:104) R | α into Eq. (5) leads to Z = (cid:90) d X (cid:16) (cid:104) R | e − (cid:15) ˆ H | R (cid:105) (cid:104) R | . . . | R P − (cid:105)(cid:104) R P − | e − (cid:15) ˆ H | R (cid:105) (cid:17) = (cid:90) d X W ( X ) . (6)We stress that Eq. (6) is still exact and constitutesan integral over P sets of particle coordinates (d X =d R . . . d R P − ), the integrand being a product of P density matrices, each at P times the original temper-ature T . Despite the significantly increased dimensional-ity of the integral, this recasting is advantageous as thehigh temperature matrix elements can easily be approx-imated, most simply with the primitive approximation, e − (cid:15) ˆ H ≈ e − (cid:15) ˆ K e − (cid:15) ˆ V , which becomes exact for P → ∞ .In a nutshell, the basic idea of the path integral MonteCarlo method is to map the quantum system ontoa classical ensemble of ring polymers . The resultinghigh dimensional integral is evaluated using the Metropo-lis algorithm , which allows one to sample the 3 P N -dimensional configurations X of the ring polymer accord-ing to the corresponding configuration weight W ( X ). B. The fermion sign problem To simulate N spin-polarized fermions, the partitionfunction from the previous section has to be extended toinclude a sum over all N ! permutations of particles: Z = 1 N ! (cid:88) s ∈ S N sgn( s ) (cid:90) d R (cid:104) R | e − β ˆ H | ˆ π s R (cid:105) , (7)where ˆ π s denotes the exchange operator corresponding tothe element s from the permutation group S N . Evidently,Eq. (7) constitutes a sum over both positive and negativeterms, so tht the configuration weight function W ( X ) canno longer be interpreted as a probability distribution. Toallow fermionic expectation values to be computed usingthe Metropolis Monte Carlo method, we introduce themodified partition function Z (cid:48) = (cid:90) d X | W ( X ) | , (8)and compute fermionic observables as (cid:104) O (cid:105) = (cid:104) OS (cid:105) (cid:48) (cid:104) S (cid:105) (cid:48) , (9)with averages taken over the modified probability distri-bution W (cid:48) ( X ) = | W ( X ) | and S = W ( X ) / | W ( X ) | denot-ing the sign. The average sign, i.e., the denominator inEq. (9), is a measure for the cancellation of positive andnegative contributions and exponentially decreases withinverse temperature and system size, (cid:104) S (cid:105) (cid:48) ∝ e − βN ( f − f (cid:48) ) , with f and f (cid:48) being the free energy per particle of theoriginal and the modified system, respectively. The sta-tistical error of the Monte Carlo average value ∆ O isinversely proportional to (cid:104) S (cid:105) (cid:48) ,∆ OO ∝ (cid:104) S (cid:105) (cid:48) √ N MC ∝ e βN ( f − f (cid:48) ) √ N MC . (10)The exponential increase of the statistical error with β and N evident in Eq. (10) can only be compensated byincreasing the number of Monte Carlo samples, but theslow 1 / √ N MC convergence soon makes this approach un-feasible. This is the notorious fermion sign problem ,which renders standard PIMC unfeasible even for thesimulation of small systems at moderate temperature. C. Permutation blocking path integral Monte Carlo To alleviate the difficulties associated with the FSP,Dornheim et al. recently introduced a novel simu-lation scheme that significantly extends fermionic PIMCsimulations towards lower temperature and higher den-sity. This so-called permutation blocking PIMC (PB-PIMC) approach combines: (i) the use of antisym-metrized density matrix elements, i.e., determinants ;(ii) a fourth-order factorization scheme to obtain accu-rate approximate density matrices for relatively low tem-peratures (large imaginary-time steps) ; and (iii) anefficient Metropolis Monte Carlo sampling scheme basedon the temporary construction of artificial trajectories .In particular, we use the factorization introduced inRefs. 64 and 65 e − (cid:15) ˆ H ≈ e − v (cid:15) ˆ W a e − t (cid:15) ˆ K e − v (cid:15) ˆ W − a (11) e − t (cid:15) ˆ K e − v (cid:15) ˆ W a e − t (cid:15) ˆ K , where the ˆ W operators denote a modified potential term,which combines the usual potential energy ˆ V with doublecommutator terms of the form[[ ˆ V , ˆ K ] , ˆ V ] = (cid:126) m N (cid:88) i =1 | F i | , (12)and, thus, requires the evaluation of all forces in thesystem. Furthermore, for each high-temperature factor,there appear three imaginary time steps. The final resultfor the partition function is given by Z = 1( N !) P (cid:90) d X P − (cid:89) α =0 (cid:18) e − (cid:15) ˜ V α e − (cid:15) u (cid:126) m ˜ F α (13)det( ρ α )det( ρ αA )det( ρ αB ) (cid:19) , where the determinants incorporate the three diffusionmatrices for each of the P factors , ρ α ( i, j ) = λ − Dt (cid:15) (cid:88) n exp (cid:18) − π ( r α,j − r αA,i + n L ) λ t (cid:15) (cid:19) . (14) FIG. 1. Density dependence of the average sign of a PB-PIMC simulation of the uniform electron gas. Also shown arestandard PIMC data taken from Ref. 31. The figure has beentaken from Dornheim et al. . The key problem of fermionic PIMC simulations isthe sum over permutations, where each configurationcan have a positive or a negative sign. By introduc-ing determinants, we analytically combine both positiveand negative contributions into a single configurationweight (hence the label ’permutation blocking’). There-fore, parts of the cancellation are carried out before-hand and the average sign of our simulations [Eq. (9)]is significantly increased. Since this effect diminisheswith increasing P , we employ the fourth-order factoriza-tion, Eq. (11), to obtain sufficient (although limited , | ∆ V | /V (cid:46) . S ) as a function of the density pa-rameter r s for a UEG simulation cell containing N = 33spin-polarized electrons subject to periodic boundaryconditions. The red, blue, and black curves correspondto PB-PIMC results for three isotherms and exhibit aqualitatively similar behavior. At high r s , fermionic ex-change is suppressed by the strong Coulomb repulsion,which means that almost all configuration weights arepositive and S is large. With increasing density, the sys-tem becomes more ideal and the electron wave functionsoverlap, an effect that manifests itself in an increasednumber of negative determinants. Nevertheless, the valueof S remains significantly larger than zero, which meansthat, for the three depicted temperatures, PB-PIMC sim-ulations are possible over the entire density range. Incontrast, the green curve shows the density-dependentaverage sign for standard PIMC simulations at θ = 1and exhibits a significantly steeper decrease with density,limiting simulations to r s ≥ D. Configuration path integral Monte Carlo For CPIMC , instead of performing the trace overthe density operator in the coordinate representation [seeEq. (5)], we trace over Slater determinants of the form |{ n }(cid:105) = | n , n , . . . (cid:105) , (15)where, in case of the uniform electron gas, n i denotes thefermionic occupation number ( n i ∈ { , } ) of the i -thplane wave spin-orbital | k i σ i (cid:105) . To obtain an expressionfor the partition function suitable for Metropolis MonteCarlo, we split the Hamiltonian into diagonal and off-diagonal parts, ˆ H = ˆ D + ˆ Y (with respect to the chosenplane wave basis, see Sec. II), and explore a perturbationexpansion of the density operator with respect to ˆ Y : e − β ˆ H = e − β ˆ D ∞ (cid:88) K =0 β (cid:90) dτ β (cid:90) τ dτ . . . β (cid:90) τ K − dτ K ( − K ˆ Y ( τ K ) ˆ Y ( τ K − ) · . . . · ˆ Y ( τ ) , (16)with ˆ Y ( τ ) = e τ ˆ D ˆ Y e − τ ˆ D . In this representation the par-tition function becomes Z = ∞ (cid:88) K =0( K (cid:54) =1) (cid:88) { n } (cid:88) s ...s K − β (cid:90) dτ β (cid:90) τ dτ . . . β (cid:90) τ K − dτ K (17)( − K e − K (cid:80) i =0 D { n ( i ) } ( τ i +1 − τ i ) K (cid:89) i =1 Y { n ( i ) } , { n ( i − } ( s i ) . The matrix elements of the diagonal and off-diagonal op-erators are given by the Slater-Condon rules D { n ( i ) } = (cid:88) l k l n ( i ) l + (cid:88) l Instead of sampling contributions to the partition func-tion, as in path integral methods, DMQMC samples the(unnormalized) thermal density matrix directly by ex-panding it in a discrete basis of outer products of Slaterdeterminantsˆ ρ = (cid:88) { n } , { n (cid:48) } ρ { n } , { n (cid:48) } |{ n }(cid:105)(cid:104){ n (cid:48) }| , (21)where ρ { n } , { n (cid:48) } = (cid:104){ n }| e − β ˆ H |{ n (cid:48) }(cid:105) . The density-matrixcoefficients ρ { n } , { n (cid:48) } appearing in Eq. (21) are found bysimulating the evolution of the Bloch equation, d ˆ ρdβ = − (cid:16) ˆ ρ ˆ H + ˆ H ˆ ρ (cid:17) , (22)which may be finite-differenced as ρ { n } , { n (cid:48) } ( β + ∆ β ) = ρ { n } , { n (cid:48) } ( β ) − (23)∆ β (cid:88) { n (cid:48)(cid:48) } (cid:2) ρ { n } , { n (cid:48)(cid:48) } ( β ) H { n (cid:48)(cid:48) } , { n (cid:48) } + H { n } , { n (cid:48)(cid:48) } ρ { n (cid:48)(cid:48) } , { n (cid:48) } ( β ) (cid:3) . The matrix elements of the Hamiltonian are as given asin Eqs. (18) and (19).Following Booth and coworkers , we note thatEq. (23) can be interpreted as a rate equation and can besolved by evolving a set of positive and negative walkerswhich stochastically undergo birth and death processesthat, on average, reproduce the full solution. The rulesgoverning the evolution of the walkers, as derived fromEq. (23), can be found elsewhere . The form of ˆ ρ isknown exactly at infinite temperature ( β = 0, ˆ ρ = ˆ1),providing an initial condition for Eq. (22). For the elec-tron gas, however, it turns out that simulating a differ-ential equation that evolves a mean-field density matrixat inverse temperature β to the exact density matrix atinverse temperature β is much more efficient than solvingEq. (22), an insight that leads to the ‘interaction picture’version of DMQMC used throughout this work.The sign problem manifests itself in DMQMC as an ex-ponential growth in the number of walkers required forthe sampled density matrix to emerge from the statisticalnoise . Working in a discrete Hilbert space helps toreduce the noise by ensuring a more efficient cancellationof positive and negative contributions, enabling largersystems and lower temperatures to be treated than wouldotherwise be possible. Nevertheless, at some point thewalker numbers required become overwhelming and ap-proximations are needed. Recently, we have applied the . r s − − − − − E x c · r s θ = 2 θ = 0 . i − DMQMC − . FIG. 2. Exchange-correlation energy of N = 33 spin-polarized electrons as a function of the density parameter r s for two isotherms. Shown are results from CPIMC andPB-PIMC taken from Ref. 51, restricted PIMC from Ref. 31,and DMQMC from Ref. 39. For θ = 0 . 5, all data has beenshifted by 0 . 05 Hartree. In the case of DMQMC, the initiatorapproximation is used. initiator approximation to DMQMC ( i − DMQMC).In principle, at least, this allows a systematic approachto the exact result with increasing walker number. Moredetails on the use of the initiator approximation inDMQMC and its limitations can be found in Ref. 39. IV. SIMULATION RESULTS FOR THE FINITE SYSTEM The first step towards obtaining QMC results for thewarm dense electron gas in the thermodynamic limit isto carry out accurate simulations of a finite model sys-tem. In Fig. 2, we compare results for the density depen-dence of the exchange correlation energy E xc of the UEGfor N = 33 spin-polarized electrons and two differenttemperatures. The first results, shown as blue squares,were obtained with RPIMC for r s ≥ 1. Subsequently,Groth, Dornheim and co-workers showed that thecombination of PB-PIMC (red crosses) and CPIMC (redcircles) allows for an accurate description of this systemfor θ ≥ . 5. In addition, it was revealed that RPIMCis afflicted with a systematic nodal error for densitiesgreater than the relatively low value at which r s = 6.Nevertheless, the FSP precludes the use of PB-PIMC atlower temperatures and, even at θ = 0 . r s = 2, thestatistical uncertainty becomes large. The range of ap-plicability of DMQMC is similar to that of CPIMC andthe DMQMC results (green diamonds) fully confirm theCPIMC results . Further, the introduction of the ini-tiator approximation (i-DMQMC) has made it possibleto obtain results up to r s = 2 for θ = 0 . 5. Although i-DMQMC is, in principle, systematically improvable andcontrolled, the results suggest that the initiator approxi-mation may introduce a small systematic shift at higherdensities.In summary, the recent progress in fermionic QMCmethods has resulted in a consensus regarding the finite- N UEG for temperatures θ ≥ . 5. However, there re-mains a gap at r s ≈ − θ < . V. FINITE SIZE CORRECTIONS In this section, we describe in detail the finite-size cor-rection scheme introduced in Ref. 47 and subsequentlypresent detailed results for two elucidating examples. A. Theory As introduced above (see Eq. (1) in Sec. II A), the po-tential energy of the finite simulation cell is defined as theinteraction energy of the N electrons with each other, theinfinite periodic array of images, and the uniform posi-tive background. To estimate the finite-size effects, itis more convenient to express the potential energy in k -space. For the finite simulation cell of N electrons, theexpression obtained is a sum over the discrete reciprocallattice vectors G : V N N = 12Ω (cid:88) G (cid:54) = [ S N ( G ) − 1] 4 πG + ξ M , (24)where S ( k ) is the static structure factor. In the limit asthe system size tends to infinity and ξ M → 0, this yieldsthe integral v = 12 (cid:90) k< ∞ d k (2 π ) [ S ( k ) − 1] 4 πk . (25)Combining Eqs. (24) and (25) yields the finite-size errorfor a given QMC simulation:∆ V N N [ S ( k ) , S N ( k )] = v − V N N (26)= 12 (cid:90) k< ∞ d k (2 π ) [ S ( k ) − 1] 4 πk (cid:124) (cid:123)(cid:122) (cid:125) v − L (cid:88) G (cid:54) = [ S N ( G ) − 1] 4 πG + ξ M (cid:124) (cid:123)(cid:122) (cid:125) V N /N . (27)The task at hand is to find an accurate estimate of thefinite-size error from Eq. (26), which, when added to theQMC result for V N /N , gives the potential energy in thethermodynamic limit. As a first step, we note that theMadelung constant may be approximated by ξ M ≈ L (cid:88) G (cid:54) = πG e − (cid:15)G − π ) (cid:90) k< ∞ d k πk e − (cid:15)k , (28) an expression that becomes exact in the limit as (cid:15) → S ( k ) by its finite-size equivalent S N ( k ); and (ii) the approximation of thecontinuous integral by a discrete sum, resulting in a dis-cretization error. As we will show in Sec. V B, S N ( k )exhibits a remarkably fast convergence with system size,which leaves explanation (ii). In particular, about adecade ago, Chiesa et al. suggested that the main con-tribution to Eq. (26) stems from the G = 0 term thatis completely missing from the discrete sum. To remedythis shortcoming, they made use of the random phaseapproximation (RPA) for the structure factor, which be-comes exact in the limit k → 0. The leading term in theexpansion of S RPA ( k ) around k = 0 is S RPA0 ( k ) = k ω p coth (cid:18) βω p (cid:19) , (29)with ω p = (cid:112) /r s being the plasma frequency. The finite- T generalization of the Chiesa et al. FSC, hereafter calledthe BCDC-FSC, is :∆ V BCDC ( N ) = lim k → S RPA0 ( k )4 π L k = ω p N coth (cid:18) βω p (cid:19) . (30)Eq. (30) would be sufficient if (i) S RPA0 ( k ) were accuratefor k (cid:46) π/L , and (ii) all contributions to Eq. (26) be-yond the G = term were negligible. As is shown inSec. V B, both conditions are strongly violated in partsof the warm dense regime.To overcome the deficiencies of Eq. (30), we need a con-tinuous model function S model ( k ) to accurately estimatethe discretization error from Eq. (27):∆ V N [ S model ( k )] = ∆ V N N [ S model ( k ) , S model ( k )] . (31)A natural choice would be to combine the QMC resultsfor k ≥ k min , which include all short-ranged correlationsand exchange effects, with the STLS structure factor forsmaller k , which is exact as k → S STLS ( k ) [or the full RPA structure factor S RPA ( k )] forall k is sufficient to accurately estimate the discretizationerror. B. Results1. Particle number dependence To illustrate the application of the different FSCs,Fig. 3 shows results for the unpolarized UEG at θ = 2 Δ V /| V | a) Δ N [S STLS ] Δ V BCDC -0.46-0.45-0.44-0.43-0.42-0.41-0.4-0.39 V / N [ H a ] b) QMCQMC+ Δ V BCDC QMC+ Δ N [S comb ] QMC+ Δ N [S STLS ] -0.397-0.395-0.393-0.391-0.389 0 0.005 0.01 0.015 0.02 0.025 0.03 V / N [ H a ] c) S k [a ] d) N=66N=40N=34 θ =2, r s =1 RPA(k=0)STLSRPAQMC d) N=66N=40N=34 θ =2, r s =1 FIG. 3. Finite-size correction for the UEG at θ = 2 and r s = 1: a) N dependence of the FSCs; b) potential energy perparticle, V /N ; the dotted grey line corresponds to the TDL value where the ∆ N [ S STLS ] had been substracted; c) extrapolationof the residual finite-size error; and d) corresponding static structure factors S ( k ) from QMC (for N = 34 , , k = 0, Eq. (29). and r s = 1. The green crosses in panel b) correspondto the raw, uncorrected QMC results that, clearly, arenot converged with system size N . The raw data pointsappear to fall onto a straight line when plotted as a func-tion of 1 /N . This agrees with the BCDC-FSC formula,Eq. (30), which also predicts a 1 /N behavior, and sug-gests the use of a linear extrapolation (the green line).However, while the linear fit does indeed exhibit goodagreement with the QMC results, the computed slopedoes not match Eq. (30). Further, the points that havebeen obtained by adding ∆ V BCDC to the QMC results,i.e., the yellow asterisks, do not fall onto a horizontalline and do not agree with the prediction of the linearextrapolation (see the horizontal green line). To resolvethis peculiar situation, we compute the improved finite-size correction [Eq. (31)] using both the static structurefactor from STLS ( S STLS ) and the combination of STLSwith the QMC data ( S comb ) as input. The resulting cor-rected potential energies are shown as black squares andred diamonds, respectively, and appear to exhibit almostno remaining dependence on system size. In panel c)we show a segment of the corrected data, magnified inthe vertical direction. Any residual finite-size errors [dueto the QMC data for S ( k ) not being converged with re-spect to N , see panel d)] can hardly be resolved withinthe statistical uncertainty and are removed by an addi-tional extrapolation. In particular, to compute the finalresult for V /N in the thermodynamic limit, we obtain alower bound via a linear extrapolation of the correcteddata (using S STLS ) and an upper bound by performinga horizontal fit to the last few points, all of which areconverged to within the error bars. The dotted grey line in panel b), which connects to the extrapolated result,shows clearly that the results of this procedure deviatefrom the results of a naive linear extrapolation.Finally, in panel d) of Fig. 3, we show results for thestatic structure factor S ( k ) for the same system. As ex-plained in Sec. V A, momentum quantization limits theQMC results to discrete k values above a minimum value k min = 2 π/L . Nevertheless, the N dependence of the k grid is the only apparent change of the QMC resultsfor S ( k ) with system size and no difference between theresults for the three particle numbers studied can beresolved within the statistical uncertainty (see also themagnified segment in the inset). The STLS curve (red)is known to be exact in the limit k → k thereappears an almost constant shift. The full RPA curve(grey) exhibits a similar behavior, albeit deviating moresignificantly at intermediate k . Finally, the RPA expan-sion around k = 0 [Eq. (29), light blue] only agrees withthe STLS and full RPA curves at very small k and doesnot connect to the QMC data even for the largest systemsize simulated.To further stress the importance of our improved finite-size correction scheme, Fig. 4 shows results again for θ = 2 but at higher density, r s = 0 . 1. In this regime,the CPIMC approach (and also DMQMC) is clearly su-perior to PB-PIMC and simulations of N = 700 unpolar-ized electrons in N b = 189234 basis functions are feasible.Due to the high density, the finite-size errors are drasti-cally increased compared to the previous case and exceed50% for N = 38 particles [see panels a) and b)]. Further,we note that the BCDC-FSC is completely inappropriate Δ V /| V | a) Δ N [S STLS ] Δ V BCDC -3.2-2.7-2.2-1.7-1.2 V / N [ H a ] b) QMCQMC+ Δ V BCDC QMC+ Δ N [S comb ] QMC+ Δ N [S STLS ] -2.1-2.095-2.09-2.085-2.08 0 0.005 0.01 0.015 0.02 0.025 0.03 V / N [ H a ] c) S k [a ] d) N=700N=300N=66 θ =2, r s =0.1 RPA(k=0)STLSRPAQMC d) N=700N=300N=66 θ =2, r s =0.1 FIG. 4. Finite-size correction for the UEG at θ = 2 and r s = 0 . 1: a) N dependence of the FSCs; b) potential energy perparticle, V /N ; c) extrapolation of the residual finite-size error; and d) corresponding static structure factors S ( k ) from QMC(for N = 66 , , k = 0, Eq. (29). for the N values considered, as the yellow asterisks areclearly not converged and differ even more strongly fromthe correct TDL than the raw uncorrected QMC data.Our improved FSC, on the other hand, reduces thefinite-size errors by two orders of magnitude (both with S STLS and S comb ) and approaches Eq. (30) only in thelimit of very large systems [ N (cid:38) ; see panel a)].The small residual error is again extrapolated, as shownin panel c).Finally, we show the corresponding static static struc-ture factors in panel d). The RPA expansion is againinsufficient to model the QMC data, while the full RPAand STLS curves smoothly connect to the latter. 2. Comparison to other methods To conclude this section, we use our finite-size cor-rected QMC data for the unpolarized UEG to analyzethe accuracy of various other methods that are commonlyused. In Fig. 5 a), the potential energy per particle, V /N ,is shown as a function of r s for the isotherm with θ = 2.Although all four depicted curves exhibit qualitativelysimilar behavior, there are significant deviations betweenthem [see panel b), where we show the relative deviationsfrom a fit to the QMC data in the TDL]. Let us startwith the QMC results: the black squares correspond tothe uncorrected raw QMC data for N = 66 particles (seeRef. 52) and the red diamonds to the finite-size correcteddata from Ref. 47. As expected, the finite-size effectsdrastically increase with density from | ∆ V | /V ≈ r s = 10, to | ∆ V | /V ≥ r s = 0 . 1. This -0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2 ( r s V ) / N a) N=66TDLRPASTLSKSDT -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.1 1 10 Δ V / V r s b) FIG. 5. Potential energy per particle of the uniform electrongas at θ = 2–simulations versus analytical models. Squares:QMC results for N = 66 particles , (red) rhombs: finite-sizecorrected QMC data (TDL) , green (yellow) curves: RPA(STLS) data , blue: results of the parametrization of Ref. (KSDT). Lower Fig.: relative deviations of all curves from thefit to the thermodynamic QMC results. again illustrates the paramount importance of accuratefinite-size corrections for QMC simulations in the warmdense matter regime. The RPA calculation (green curve)is accurate at high density and weak coupling. How-ever, with increasing r s the accuracy quickly deterioratesand, already at moderate coupling, r s = 1, the system-atic error is of the order of 10%. The yellow asterisksshow the SLTS result which agrees well with the simu-lations (the systematic error does not exceed 3%) overthe entire r s -range considered, i.e., up to r s = 10. Fi-nally, the blue curve has been obtained from the recentparametrization of f xc by Karasiev et al. (KSDT), forwhich RPIMC data have been used as an input. Whilethere is a reasonable agreement with our new data for r s (cid:38) | ∆ V | /V ∼ r s , which only vanish for r s < − . VI. SUMMARY AND OPEN QUESTIONS Let us summarize the status of ab initio thermody-namic data for the uniform electron gas at finite tem-perature. The present paper has given an overview ofrecent progress in ab initio finite temperature QMC sim-ulations that avoid any additional simplifications such asfixed nodes. While these simulations do not “solve” thefermion sign problem, they provide a reasonable and effi-cient way how to avoid it , in many practically relevant sit-uations, by combining simulations that use different rep-resentations of the quantum many-body state: the coor-dinate representation (direct PIMC and PB-PIMC) andFock states (CPIMC, DMQMC). With this it is now pos-sible to obtain highly accurate results for up to N ∼ θ (cid:38) . 5. As a second step we demonstrated that thesecomparatively small simulation sizes are sufficient to pre-dict results for the macroscopic uniform electron gas notsignificantly loosing accuracy . This unexpected resultis a consequence of a new highly accurate finite-size cor-rection that was derived by invoking STLS results for thestatic structure factor.With this procedure it is now possible to obtain ther-modynamic data for the uniform electron gas with an ac-curacy on the order of 0 . et al. (KSDT) the accuracy and errors of which can nowbe unambiguously quantified. We found that among thetested models, the STLS is the most accurate one. Wewish to underline that even though exchange-correlationeffects are often small compared to the kinetic energy,their accurate treatment is important to capture theproperties of real materials, see e.g. In the following we summarize the open questions andoutline future research directions.1. Construction of an improved fit for the exchange-correlation free energy due to their key relevance as input for finite-temperature DFT. Such fits arestraightforwardly generated from the current re-sults but require a substantial extension of the sim-ulations to arbitrary spin polarization. This workis currently in progress.2. The presently available accurate data are limitedto temperatures above half the Fermi energy, as aconsequence of the fermion sign problem. A majorchallenge will be to advance to lower temperatures,Θ < . ab initio results allow for an entirelynew view on previous theoretical models. For thefirst time, a clear judgement about the accuracybecomes possible which more clearly maps out thesphere of applicability of the various approaches,e.g. . Moreover, the availability of our data will al-low for improvements of many of these approachesvia adjustment of the relevant parameters to theQMC data. This could yield, e.g., improved staticstructure factors, dielectric functions or local fieldcorrelations.4. Similarly, our data may also help to improve alter-native quantum Monte Carlo concepts. In partic-ular, this concerns the nodes for Restricted PIMCsimulations which can be tested against our data.This might help to extend the range of validity ofthose simulations to higher density and lower tem-perature. Since this latter method does not have asign problem it may allow to reach parameters thatare not accessible otherwise.5. A major challenge of Metropolois-based QMC sim-ulations that are highly efficient for thermodynamicand static properties is to extend them to dynamicquantities. This can, in principle, be done via an-alytical continuation from imaginary to real times(or frequencies). However, this is known to be anill-posed problem. Recently, there has been sig-nificant progress by invoking stochastic reconstruc-tion methods or genetic algorithms. For example,for Bose systems, accurate results for the spectralfunction and the dynamics structure factor couldbe obtained, e.g. and references therein whichare encouraging also for applications to the uniformelectron gas, in the near future.6. Finally, there is a large number of additional ap-plications of the presented ab initio simulations.0This includes the 2D warm dense UEG where ther-modynamic results of similar accuracy should bestraightforwardly accessible. Moreover, for theelectron gas, at high density, r s (cid:46) . 1, rela-tivistic corrections should be taken into account.Among the presented simulations, CPIMC is per-fectly suited to tackle this task and to provide abinitio data also for correlated matter at extremedensities. ACKNOWLEDGEMENTS This work was supported by the Deutsche Forschungs-gemeinschaft via project BO1366-10 and via SFB TR-24 project A9 as well as grant shp00015 for CPUtime at the Norddeutscher Verbund f¨ur Hoch- undH¨ochstleistungsrechnen (HLRN). 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