Nonlinear Electronic Density Response in Warm Dense Matter
NNonlinear Electronic Density Response in Warm Dense Matter
Tobias Dornheim, ∗ Jan Vorberger, and Michael Bonitz Center for Advanced Systems Understanding (CASUS), G¨orlitz, Germany Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, D-01328 Dresden, Germany Institut f¨ur Theoretische Physik und Astrophysik,Christian-Albrechts-Universit¨at zu Kiel, Leibnizstraße 15, D-24098 Kiel, Germany
Warm dense matter (WDM)—an extreme state with high temperatures and densities that occurse.g. in astrophysical objects—constitutes one of the most active fields in plasma physics and mate-rials science. These conditions can be realized in the lab by shock compression or laser excitation,and the most accurate experimental diagnostics is achieved with lasers and free electron lasers whichis theoretically modeled using linear response theory. Here, we present first ab initio path integralMonte Carlo results for the nonlinear density response of correlated electrons in WDM and showthat for many situations of experimental relevance nonlinear effects cannot be neglected.
Warm dense matter (WDM) is an exotic state withextreme densities ( r s = r/a B ∼ r and a B beingthe average interparticle distance and first Bohr radius)and high temperatures ( θ = k B T /E F ∼ T and E F being the temperature and Fermi energy) that oc-curs, e.g., in astrophysical objects [1–4] and laser-excitedsolids [5, 6], and on the pathway towards inertial confine-ment fusion [7]. Consequently, WDM has emerged as oneof the most active frontiers in plasma physics and mate-rial science [8–10], and WDM conditions are routinelyrealized in experiments in large research facilities aroundthe globe (e.g., NIF, SLAC and the European XFEL),see Refs. [11–13] for review articles.On the other hand, the theoretical description of WDMconstitutes a formidable challenge [14, 15] due to thecomplicated interplay of 1) Coulomb coupling, 2) ther-mal excitations, and 3) electronic quantum degeneracyeffects. Moreover, the bulk of WDM theory assumes aweak response of the electrons to an external perturba-tion, i.e., they rely on linear response theory (LRT). Thisassumption enters, for example, in the interpretation ofXRTS experiments [9, 16], the characterization of thestopping power in WDM [17], the construction of effec-tive potentials [18–20], density functional theory (DFT)calculations [21, 22], and the computation of energy re-laxation rates [6, 23, 24]. Consequently, numerous workshave been devoted to the description of the density re-sponse of electrons both in the ground state [25–34] andat finite temperature [35–43]. These efforts have culmi-nated in the recent machine-learning representation [44]of the static electronic density response that is basedon ab initio path integral Monte Carlo (PIMC) simula-tions [45–47] and covers the entire WDM regime. More-over, even the dynamic density response can be computedfrom PIMC simulations [48, 49], and the reported nega-tive dispersion relation of a uniform electron gas (UEG)constitutes an active topic of investigation.On the other hand, very little is known about the den-sity response of correlated electrons beyond the linearregime. In particular, it is unclear up to which per-turbation strength LRT remains accurate. This ques- tion becomes increasingly urgent, as free electron lasersbecome more powerful and peak intensities of up to I ∼ W/cm [50] have been reported. Furthermore,intense VUV lasers are used to probe WDM [51]. A par-ticular promising tool are THz lasers [52] as they allowfor probing the low-frequency end of the density response,short pulse characterization, and streaking [53–55]. Yet,THz field applications might require in many cases a the-oretical description beyond LRT, as we indicate below.In this work, we go beyond linear response theory bycarrying out extensive PIMC simulations of a harmoni-cally perturbed electron gas [41, 42] (cf. Eq. (1) below)at WDM conditions. This allows us to measure the ac-tual density response of the electrons without any a-prioriassumptions (including the fluctuation dissipation theo-rem) and, thus, to unambiguously characterize the valid-ity range of LRT. In addition, going beyond the linearregime allows us to gauge the systematic errors of LRTas a function of perturbation strength, and to report thefirst results for the cubic response function χ ( q ) over theentire relevant wave number range for different densitiesand temperatures including all exchange–correlation ef-fects . Therefore, our results provide the basis for a gen-eralized theory of the electronic density response beyondLRT, extending earlier work for classical plasmas [56, 57]and moderately coupled quantum plasmas [58, 59].Our investigation of the nonlinear density response ofelectrons in WDM should be relevant for many otherfields, and spark similar investigations in other domainsas we note that LRT is one of the most successful con-cepts in physics [60, 61]. It is of paramount impor-tance in many fields, such as for describing phonons insolid state physics [62, 63], excitations in systems of ul-tracold atoms [64, 65], and screening or quasiparticleexcitations in plasmas [66, 67]. Moreover, it has al-lowed for profound physical insights into, e.g., superflu-idity [65, 68], collective excitations [25, 69], and quantumdynamics [21, 70].All PIMC data are available online [71] and can beused to benchmark theoretical models and approximatesimulation techniques like DFT. a r X i v : . [ c ond - m a t . s t r- e l ] A p r -0.02-0.01 0 0 0.2 0.4 0.6 0.8 1 a) ρ A PIMCLRTcubic fi t χ , fi t -0.03 0 0.03 0.06 0 0.2 0.4 0.6 0.8 1 b) Δ χ / χ A LRTcubic
FIG. 1. Density response of the UEG for N = 14, r s = 2, and θ = 1 with q ≈ . q F in dependence of the perturbation am-plitude A [cf. Eq. (1)]. Panel a) shows the PIMC data for theinduced density ρ (green crosses), the prediction from LRT[solid red, cf. Eq. (3)], and a cubic fit [dotted blue, cf. Eq. (4)]as well as the linear component thereof (dashed black). Panelb) shows the deviation of LRT (black squares) and the cubicfit (blue diamonds) from the PIMC data. The vertical greydashed line corresponds to the maximum A -value that hasbeen included in the fit. Results.
We simulate a harmonically perturbedelectron gas governed by the Hamiltonian (we assumeHartree atomic units throughout this work)ˆ H = ˆ H UEG + 2 A N (cid:88) k =1 cos (ˆ r k · ˆ q ) , (1)with ˆ H UEG being the usual (unperturbed) UEG Hamil-tonian [47, 61, 72], A being the perturbation amplitude,and the wave vector q = 2 π/L ( n x , n y , n z ) T (with n i ∈ Z ,and L being the length of the simulation box). Note thatwe use a canonical adaption [73] of the worm algorithmby Boninsegni et al. [74, 75] without any assumptions onthe nodal structure of the thermal density matrix. There-fore, our simulations are computationally involved due tothe fermion sign problem [76, 77], but are exact withinthe given statistical uncertainty.To measure the density response, we compute the in-duced density ρ ( q , A ) := (cid:104) ˆ ρ q (cid:105) A = 1 V (cid:42) N (cid:88) k =1 e − i q · ˆ r k (cid:43) A , (2)where (cid:104) . . . (cid:105) A indicates the expectation value computedfrom Eq. (1). The PIMC results for Eq. (2) are shownin Fig. 1a) as the green crosses for the electron gas witha metallic density ( r s = 2) at the Fermi temperature, θ = 1, for a wave number of q ≈ . q F . For small A , LRT is accurate and it holds ρ ( q , A ) = χ ( q ) A , and thedensity response function χ ( q ) does not depend on A . Inthis context, we mention that the linear response functioncan be computed from a simulation of the unperturbedUEG via the imaginary-time version of the fluctuation–dissipation theorem, which states that χ ( q ) = − n (cid:90) β d τ F ( q , τ ) , (3)with F ( q, τ ) being the usual intermediate scattering func-tion [9] evaluated at an imaginary time argument τ ∈ [0 , β ], see Ref. [44] for details. The LRT result for ρ as obtained from Eq. (3) is depicted by the solid redline and is in excellent agreement to the PIMC data for A (cid:46) .
15. This can be seen particularly well in Fig. 1b),where the black squares correspond to the relative devi-ation between the PIMC data and LRT. We note thatLRT systematically overestimates the density response,and the deviation to LRT appears to be parabolic in thedepicted A -range.Indeed, it is well known [31, 32] that the first termbeyond χ ( q ) is cubic in A and can be obtained by fittingthe PIMC data to ρ ( q , A ) = χ ( q ) A + χ ( q ) A , (4)where χ ( q ) and χ ( q ) are the free parameters. The re-sults for Eq. (4) are included in Fig. 1a) as the dashedblue curve, and exhibit a significantly improved agree-ment with the PIMC data as compared to LRT. Thevertical dashed grey line corresponds to the maximum A -value that has been included into the fit, but Eq. (4)remains accurate for significantly larger perturbationstrengths, see also the blue diamonds in panel b). Forcompleteness, we mention that it is, in principle, redun-dant to obtain χ ( q ) from the PIMC data, as it is alreadyknown from Eq. (3). On the other hand, comparing thetwo allows to check the consistency of our approach, andthe two independent estimations of the LRT function arein perfect agreement with an uncertainty interval of 0 . q . This is shown in Fig. 2a) where the top and bottomhalf correspond to the cubic and linear response, respec-tively. The red symbols correspond to the usual LRTfunction computed from Eq. (3) for N = 14 (diamonds)and N = 20 (stars), and the dashed red line to χ ( q )computed in the thermodynamic limit ( N → ∞ ) fromthe neural-net representation given in Ref. [44]. We notethat they are in good agreement, as finite-size effects aresmall in this regime [44]. The black and blue symbolshave been obtained from our new PIMC simulations ofthe perturbed system as χ ( q , A ) = ρ ( q , A ) A (5) -0.04-0.02 0 0.02 0.04 0 1 2 3 4 r s =2 θ =1 χ a) q/q F LRTcubic A=0.5A=0.2 -0.01-0.00500.250.50.75 0 1 2 3 4 r s =6 θ =1 b) q/q F A=0.03 A=0.1 -0.04-0.02 0 0.02 0.04 0 1 2 3 4 r s =2 c) q/q F θ =4 θ =2 θ =1 FIG. 2. Density response of an electron gas to an external harmonic perturbation at different conditions. Panels a) and b)show results for θ = 1 and r s = 2 and r s = 6, respectively. The top halfs correspond to the cubic response function χ ( q )[computed via fits, cf. Eq. (4)], and the bottom half to the (pseudo-) linear response function χ ( q ) from LRT [red, cf. Eq. (3)],and for different perturbation amplitudes [black and blue, cf. Eq. (5)]. Panel c) corresponds to r s = 2 for θ = 4 (blue), θ = 2(green), and θ = 1 (red) and shows χ and χ (from LRT) in the top and bottom half. The diamonds, stars, and crossescorrespond to N = 14 , , and 34. The dashed curves depict the LRT prediction computed from a recent machine-learningrepresentation [44] of the static local field correction. such that this pseudo response function converges to LRTin the limit of small perturbations, lim A → χ ( q , A ) = χ ( q ). For A = 0 . q ,but systematically deviates around q ∼ q F . For A = 0 . q -range, and the discrepancy is again mostpronounced for intermediate wave numbers, with a max-imum deviation of ∼ A -scanssuch as depicted in Fig. 1 for different q -values over theentire relevant wave number range [71]. This has allowedus to obtain the first results for the cubic response func-tion χ ( q ), which are shown in the top half of Fig. 2a) asthe green data points. As a side note, we mention thata single χ ( q ) point requires 10 −
15 independent PIMCsimulations of Eq. (1) with different A values for eachwave number, which results in a total computation costof O (cid:0) (cid:1) CPU hours.Overall, χ ( q ) qualitatively somewhat mirrors χ ( q ), al-though with some pronounced differences. First and fore-most, we find that no finite-size effects can be resolvedwithin the given error bars, and the results for N = 14and N = 20 exhibit a smooth progression. The maindifference is that they are available at different q -points,which is a direct consequence of the momentum quanti-zation in the finite simulation cell, see, e.g., Refs. [78, 79].Moreover, χ ( q ) always has the opposite sign of χ ( q ), asthe system cannot react arbitrarily strong to the per-turbation, and the response eventually saturates. Whileboth the linear and the cubic response function vanish in the large- and small- q limits, this happens significantlysooner for the latter function. Heuristically, this can beunderstood as follows: for large q -values, only single-particle effects contribute to the response, the systemas a whole remains hardly affected, and LRT is suffi-cient; similarly, the response is suppressed by the perfectscreening [80] in the small- q limit. Lastly, we find thatthe maximum in χ ( q ) appears to be slightly shifted tolarger q -values compared to χ ( q ), see also panel b) forthe same trend at r s = 6.In summary, our results predict that nonlinear effectsin the electronic density response manifest in an effec-tively damped response function [cf. the blue symbols inFig. 2a)], with a maximum that is shifted to smaller wavenumbers.Let us next investigate the dependence of the cubicresponse on the density parameter r s . To this end, werepeat our previous study for r s = 6, and the results areshown in Fig. 2b) for θ = 1. While such low densitiesare not typical for WDM applications, they can be real-ized experimentally in hydrogen jets [51] and evaporationexperiments, e.g. at the Sandia Z-machine [81–84]. Onthe other hand, these conditions are highly interestingfrom a theoretical point of view, as electronic exchange–correlation effects are even more important due to theincreased coupling strength [47, 85, 86].First and foremost, we find that the nonlinear be-haviour of the density response appears for significantlysmaller perturbation amplitudes as compared to r s = 2,which is due to the different energy scales in the sys-tem [87]. For example, for A = 0 . -0.02-0.01 0 0 0.2 0.4 0.6 0.8 1 ρ A θ =4 θ =2 θ =1LRT FIG. 3. Density response of the UEG for N = 14 and r s = 2with q ≈ . q F in dependence of the perturbation amplitude A [cf. Eq. (1)]. Shown are PIMC data for the induced density ρ for θ = 1 (red circles), θ = 2 (green crosses), and θ = 4 (bluediamonds), as well as the corresponding predictions from LRT[dotted black, cf. Eq. (3)]. hardly any effect would be noticed at the higher densityin this case. Overall, both χ ( q ) and χ ( q ) exhibit a simi-lar structure as for r s = 2, but are somewhat more sym-metric around the maximum at q ≈ q F . Moreover, χ ( q )nearly vanishes for the smallest depicted q -value (the left-most green cross, corresponding to N = 34) and we finda value more than two orders of magnitude smaller thanfor q = 2 q F . Again, no system-size dependence of χ ( q )can be resolved within the given confidence interval evenfor N = 34 electrons (crosses).Another interesting question is how nonlinear effectsare influenced by the temperature. To this end, we re-turn to r s = 2 for θ = 1 (red), θ = 2 (green), and θ = 4(blue) in Fig. 2c). With increasing temperature, the lin-ear response function monotonically decreases in magni-tude as it is expected, see the bottom half. The same alsoholds for the cubic response function, where this trend isdrastically more pronounced compared to χ ( q ). Whilethe maximum in χ ( q ) is reduced by a factor of 3 upongoing from θ = 1 to θ = 4, the cubic response is reducedby a factor of 20.This behaviour is further illustrated in Fig. 3, wherewe show the A -dependence of the induced density forthe three temperatures at q ≈ . q F , i.e., around themaximum of the density response. The different symbolscorrespond to our PIMC data, and the dotted lines to theprediction from LRT, i.e., Eq. (3). There are two domi-nant trends: 1) the actual density response is smaller forlarge θ and 2) LRT remains accurate for larger A .Let us conclude this investigation by briefly touch-ing upon the impact of our findings on state-of-the-artWDM experiments. A typical free electron laser witha frequency corresponding to a photon energy of 8 keVand an intensity of I ∼ − W/cm corresponds toan approximate perturbation amplitude on the order of A ∼ − − − (see the Supplemental Material [71]for details), which falls safely into the LRT regime even for low densities. On the other hand, intensities of upto I = 10 W/cm have been reported recently by em-ploying the novel seeding technique [50], which results in A ∼ r s = 2 and r s = 6. Even current VUV lasers like Flashare capable to reach the nonlinear regime [51, 71]. An-other application of our findings concerns the experimen-tal probing of the low-frequency response of WDM usingTHz lasers [52]. For example, the recently reported setupwith an intensity of 600kV/cm at around 1 THz leadsto a perturbation amplitude of A = 0 .
29 Ha, such thata thorough theoretical interpretation of a correspondingscattering signal would most likely require to take intoaccount nonlinear effects.
Summary.
We have carried out extensive ab initio
PIMC simulations of the harmonically perturbed electrongas. This has allowed us to 1) unambiguously character-ize the validity range of LRT and 2) to obtain the firstresults for the cubic response function χ ( q ) of the warmdense electron gas, including all exchange–correlation ef-fects. Firstly, we have found that including χ ( q ) sig-nificantly improves the accuracy of the density responsefunction for larger perturbation amplitudes. Moreover,nonlinear effects are particularly important for interme-diate wave numbers q ∼ q F , whereas χ ( q ) vanishes bothin the small- and large- q regimes. Regarding physicalparameters, we have found that nonlinear effects becomemore important at lower densities due to the intrinsicenergy scale of the system. This makes materials of rela-tively low density a highly interesting laboratory to studythe interplay of nonlinearity with electronic exchange–correlation effects, and a challenging benchmark for the-ory.In addition, we have found that nonlinear effects areseverely affected by the electronic temperature and van-ish upon increasing θ . While our current simulations arelimited to temperatures down to the Fermi temperature( θ = 1), this is a strong indication that nonlinear effectsmight be even more important for lower temperatures θ = 0 . ... .
5, where many WDM experiments are located.Our findings are particularly relevant for state-of-the-art WDM experiments with intense free electron lasers inthe x-ray or VUV regime, and for low-frequency probingin the THz regime, where the diagnostics methods rely ontheory input for the response functions [50–52]. Finally,our results will also be important for nonlinear opticaldiagnostics such as Raman or four-wave mixing spec-troscopy, e.g. [88–90], or THz streaking [91] that couldprovide additional information on correlation effects inwarm dense matter.All PIMC data are available online [71] and can be usedto benchmark approximate theories like DFT. Moreover,our new data are exact within the given confidence inter-val and thus provide the basis for a more general theoryof the electronic density response beyond LRT thus fur-ther completing our understanding of the electron gas asa fundamental model system [47, 92].
ACKNOWLEDGMENTS
We acknowledge stimulating discussions with RichardPausch and Dominik Kraus, and helpful comments byMichael Bussmann. This work was partly funded by theCenter of Advanced Systems Understanding (CASUS)which is financed by Germany’s Federal Ministry of Edu-cation and Research (BMBF) and by the Saxon Ministryfor Science and Art (SMWK) with tax funds on the basisof the budget approved by the Saxon State Parliament,and by the Deutsche Forschungsgemeinschaft (DFG) viaproject BO1366/13. The PIMC calculations were car-ried out at the Norddeutscher Verbund f¨ur Hoch- undH¨ochstleistungsrechnen (HLRN) under grant shp00015,on a Bull Cluster at the Center for Information Servicesand High Performace Computing (ZIH) at TechnischeUniversit¨at Dresden, on the clusters hypnos and hemera at Helmholtz-Zentrum Dresden-Rossendorf (HZDR), andat the computing center (Rechenzentrum) of Kiel univer-sity. ∗ [email protected][1] B. Militzer, W. B. Hubbard, J. Vorberger, I. Tam-blyn, and S. A. 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