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Philos Trans A Math Phys Eng Sci
Philos Trans A Math Phys Eng Sci
RSTA
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Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
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1471-2962
The Royal Society

10.1098/rsta.2022.0217
rsta20220217
1009159Articles
Research Articles
Analysing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations
Analyzing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations
http://orcid.org/0000-0001-7293-6615
Dornheim Tobias Conceptualization Investigation Methodology Project administration Writing – original draft Writing – review & editing t.dornheim@hzdr.de
1 2
Vorberger Jan Resources Validation Writing – review & editing 2
http://orcid.org/0000-0002-9725-9208
Moldabekov Zhandos A. Validation Writing – review & editing 1 2
Böhme Maximilian Validation Writing – review & editing 1 2 3
1 Center for Advanced Systems Understanding (CASUS), D-02826 Görlitz, Germany
2 Helmholtz-Zentrum Dresden-Rossendorf (HZDR), D-01328 Dresden, Germany
3 Technische Universität Dresden, D-01062 Dresden, Germany
One contribution of 11 to a theme issue ‘Dynamic and transient processes in warm dense matter’.

21 8 2023 August 21, 2023
3 7 2023 July 3, 2023
3 7 2023 July 3, 2023
381 2253 Theme issue ‘Dynamic and transient processes in warm dense matter’ compiled and edited by David Riley, Thomas G White and Jiayu Dai 202202174 11 2022 November 4, 2022
10 1 2023 January 10, 2023
© 2023 The Authors.
2023
https://creativecommons.org/licenses/by/4.0/ Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

Rigorous diagnostics of experiments with warm dense matter are notoriously difficult. A key method is X-ray Thomson scattering (XRTS), but the interpretation of XRTS measurements is usually based on theoretical models that entail various approximations. Recently, Dornheim et al. [Nat. Commun. 13, 7911 (2022)] introduced a new framework for temperature diagnostics of XRTS experiments that is based on imaginary-time correlation functions. On the one hand, switching from the frequency to the imaginary-time domain gives one direct access to a number of physical properties, which facilitates the extraction of the temperature of arbitrarily complex materials without relying on any models or approximations. On the other hand, the bulk of theoretical work in dynamic quantum many-body theory is devoted to the frequency domain, and, to the best of our knowledge, the manifestation of physics properties within the imaginary-time density–density correlation function (ITCF) remains poorly understood. In the present work, we aim to fill this gap by introducing a simple, semi-analytical model for the imaginary-time dependence of two-body correlations within the framework of imaginary-time path integrals. As a practical example, we compare our new model to extensive ab initio path integral Monte Carlo results for the ITCF of a uniform electron gas, and find excellent agreement over a broad range of wavenumbers, densities and temperatures.

This article is part of the theme issue ‘Dynamic and transient processes in warm dense matter’.

warm dense matter
, X-ray Thomson scattering
, imaginary-time correlation functions
Bundesministerium für Bildung und Forschung http://dx.doi.org/10.13039/501100002347 CASUS Sächsisches Staatsministerium für Wissenschaft und Kunst http://dx.doi.org/10.13039/501100006114 CASUS cover-dateJuly 24, 2023
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pmc1. Introduction

The study of matter at extreme pressures (P∼1–104 MBar) and temperatures (T=104–108 K) [1,2] constitutes a highly active frontier at the interface of a number of disciplines such as plasma physics, material science and quantum chemistry. These conditions naturally occur in a host of astrophysical objects such as giant planet interiors [3,4] and brown dwarfs [5,6]. In addition, they are important for technological applications such as the discovery of novel materials [7–9] and hot-electron chemistry [10–12]. A particularly important and topical example is given by inertial confinement fusion [13,14] as it is realized, for example, at the National Ignition Facility [15]; here, the fuel capsule is predicted to traverse the aforementioned regime on its path towards ignition [16].

From a theoretical perspective, this warm dense matter (WDM) can be characterized in terms of a few dimensionless parameters that are of the order of unity [17,18]: (1) the Wigner–Seitz radius, rs=d/aB, is the ratio of the average interparticle distance to the first Bohr radius, and this density parameter also serves as the quantum coupling parameter in the WDM regime [19]; (2) the degeneracy temperature, Θ=kBT/EF, measures the thermal energy in units of the electronic Fermi energy EF [20], with Θ≪1 and Θ≫1 corresponding to the fully degenerate and semi-classical regimes, respectively.

Consequently, the rigorous theoretical description of WDM is notoriously difficult as there are no small parameters that can serve as the basis for a suitable expansion [21,22]. While the current great interest in the properties of WDM has sparked a surge of new developments regarding different methods [22–52], we are at present still far from having a complete understanding of the full WDM regime.

This unsatisfactory situation also represents a serious obstacle for the interpretation of experiments with WDM [53], as, because of the extreme conditions, often even basic parameters such as the temperature or the density cannot be directly measured and have to be inferred from other observations. A very important method for the diagnostics of WDM is X-ray Thomson scattering (XRTS) [54,55]; here, an X-ray beam is produced either from backlighter sources [56] or using free-electron X-ray lasers (XFELs), which have become available at large research facilities such as LCLS in the USA [57], SACLA in Japan [58] and the European XFEL in Germany [59]. An XRTS measurement gives one access to the scattering intensity signal, which is given by the convolution of the dynamic structure factor (DSF) S(q,ω) with the combined source and instrument function R(ω) [54], 1.1 I(q,ω)=S(q,ω)⊛R(ω).

Unfortunately, the numerical deconvolution of equation (1.1) is generally prevented by noise in the experimental measurement. Therefore, XRTS does not give one direct access to S(q,ω), which contains the sought-after physical information about the system of interest. The most widely used approach for the interpretation of XRTS experiments is to construct an approximate model for S(q,ω), which is then convolved with R(ω) and subsequently compared to the experimental signal I(q,ω). On the one hand, one can determine a priori unknown free parameters such as the temperature T by finding the best fit between theory and experiment in this way. On the other hand, the inferred parameters can depend arbitrarily strongly on the model employed for S(q,ω), which is usually based on approximations such as the widely used Chihara decomposition [55,60,61].

Recently, Dornheim et al. [62,63] suggested circumventing this obstacle by switching from the usual ω-representation to the imaginary-time domain. The required transformation is given by a two-sided Laplace transform 1.2 F(q,τ)=L[S(q,ω)]=∫−∞∞dω e−ωτS(q,ω),

which connects the DSF to the imaginary-time density–density correlation function (ITCF) F(q,τ). The latter naturally emerges in Feynman’s imaginary-time path integral picture of statistical mechanics [64,65] and corresponds to the usual intermediate scattering function evaluated at an imaginary time t=−iℏτ, where τ∈[0,β] with the inverse temperature β=1/(kBT). Note that F(q,τ) is periodic in τ with β; see §2a.

A particular advantage of the τ-domain is given by the well-known convolution theorem, 1.3 L[S(q,ω)]=L[S(q,ω)⊛R(ω)]L[R(ω)],

which makes the deconvolution trivial; in practice, it is easy to compute the Laplace transform of the XRTS intensity and then divide it by the Laplace transform of the instrument function R(ω). Accurate knowledge of R(ω) is usually available from source monitoring at XFEL facilities or from the characterization of backlighter emission spectra [56]. In this way, equation (1.3) gives one direct access to physical information, which enables accurate inference of the temperature of arbitrarily complex systems in thermodynamic equilibrium without the use of any model, simulation or approximation [62].

The two-sided Laplace transform defined in equation (1.2) is a unique transformation, which means that the ITCF F(q,τ) contains exactly the same information as the usual DSF S(q,ω). Indeed, it has been demonstrated in reference [64] that F(q,τ) gives one direct access to a wealth of physical information, such as the excitation energies of quasi-particles. Moreover, even complex physical processes, such as the exchange–correlation-induced alignment of pairs of electrons [66] that leads to a roton-type minimum in the dispersion ω(q) of the DSF, can be observed and interpreted in the τ-domain. At the same time, we note that the bulk of dynamic quantum many-body theory has been developed in the frequency domain to describe S(q,ω) and related properties.

Here, we aim to partly remedy this unsatisfactory situation by presenting a simple, semi-analytical model for the imaginary-time diffusion process. It is able to accurately capture the dependence of the ITCF on τ over a broad range of parameters. As a practical application, we consider the uniform electron gas (UEG) [17,20,67], which is the archetypical model for interacting electrons and has provided important insights in a number of different contexts. The availability of many properties of the UEG in WDM conditions [17,31,68] allows us to compare our new model—in addition to other models such as the well-known random phase approximation (RPA)—to highly accurate ab initio path integral Monte Carlo (PIMC) simulation results for F(q,τ).

We believe that these new insights into the imaginary-time dependence of electron–electron correlations in the WDM regime will form an important basis for future studies of real WDM applications that include both electrons and ions. This article is organized as follows. In §2, we introduce the relevant theoretical background, starting with an introduction to the PIMC method and its natural connection to the ITCF in §2a. Section 2b is devoted to a brief introduction to linear response theory, followed by a concise overview of a few important properties of the ITCF in §2c. The theoretical background is concluded by our new imaginary-time diffusion model, which is introduced in §2d. In §3, we present an extensive analysis of different properties of the ITCF, starting with a discussion of its dependence on the imaginary time τ in §3a; the subsequent §§3b and 3c focus on the wavenumber q and temperature Θ, respectively. This article concludes with a discussion and future outlook in §4.

2. Theory

We assume Hartree atomic units throughout. A detailed introduction to the UEG model, including the UEG Hamiltonian in different representations, can be found, for example, in references [17,20]. Note that many details about the PIMC method and the estimation of the ITCF have already been introduced elsewhere [17,64,69,70], and §2a can be skipped by readers that are familiar with these concepts.

(a) Imaginary-time PIMC

Since its inception as a method for the description of ultracold bosonic He 4 some six decades ago [71,72], the ab initio PIMC method [73–75] has emerged as one of the most successful tools for the description of non-ideal quantum many-body systems in thermodynamic equilibrium. Since a detailed introduction to PIMC has been presented elsewhere [76], we will here restrict ourselves to outlining the main idea and how it relates to the estimation of imaginary-time correlation functions such as F(q,τ).

As a starting point, we express the canonical partition function of N=N↑+N↓ electrons in a cubic volume Ω=L3 and at an inverse temperature β=1/T as 2.1 Zβ,N,Ω=1N↑!N↓!∑σ↑∈SN↑∑σ↓∈SN↓sgn(σ↑,σ↓)∫Ω dR ⟨R| e−βH^|π^σ↑π^σ↓R⟩,

where the variable R=(r1,…,rN)T contains the coordinates of all N↑ majority and N↓ minority electrons. We restrict ourselves to the unpolarized (i.e. paramagnetic) case of N↑=N↓=N/2 throughout this work. The summation over all elements σ↑,↓ of the respective permutation group SN↑,↓, with π^σ↑,↓ being the corresponding permutation operator, takes into account the antisymmetry of the partition function with respect to the exchange of particle coordinates of identical fermions. The sign function sgn(σ↑,σ↓) is positive (negative) for an even (odd) number of pair exchanges in a particular combination of σ↑ and σ↓. Note that the corresponding bosonic partition function can be obtained by setting the sign function to unity.

Unfortunately, direct evaluation of the matrix elements of the density operator ρ^=e−βH^ is precluded by the non-commutativity of the kinetic (K^) and potential (V^) contributions to the total Hamiltonian H^=K^+V^. A well-known way of overcoming this obstacle is based on the utilization of the exact semi-group property of the density operator, which eventually leads to 2.2 Zβ,N,Ω =1N↑!N↓!∑σ↑∈SN↑∑σ↓∈SN↓sgn(σ↑,σ↓) ×∫Ω dR0… dRP−1⟨R0| e−ϵH^|R1⟩⋯⟨RP−1| e−ϵH^|π^σ↑π^σ↓R0⟩.

We have thus replaced the evaluation of a single-density matrix at the temperature T with the evaluation of P density matrices at P times the original temperature. Moreover, each corresponding exponential function can be viewed as a propagation in imaginary time by an amount of Δt=−iϵ (with ϵ=β/P); this is the origin of Feynman’s celebrated imaginary-time path integral formalism that is illustrated in figure 1. Shown is a configuration of N=4 electrons in the τ–x plane, with P=6 high-temperature factors, corresponding to P imaginary-time slices of length Δτ=ϵ. Each particle is represented by a closed path along the imaginary time. We have thus mapped the quantum system of interest onto a classical system of interacting ring polymers; this concept is often referred to as classical isomorphism in the literature [77]. The closed nature of the polymers is a direct consequence of the definition of the partition function as the trace of the density operator, basically leading to the same start and end points R. The situation becomes somewhat more interesting for the cases of indistinguishable quantum particles (i.e. fermions or bosons), where the application of the permutation operator π^σ↑,↓ in equation (2.2) leads to polymers containing more than a single particle. Such an example is depicted in figure 1, where the two particles in the centre form a single permutation cycle [78]. The presence of this single pair exchange results in a negative sign function sgn(σ↑,σ↓). This is the root cause of the notorious fermion sign problem [69,79–81], the main computational bottleneck in PIMC simulations of WDM, which is discussed in more detail below. Figure 1. Schematic illustration of Feynman’s imaginary-time path integral formalism. Shown is a configuration X that consists of N=4 electrons on P=6 imaginary-time slices. The yellow Gaussian represents the ideal kinetic density matrix (equation (2.5)), which effectively acts as a harmonic spring between beads on adjacent slices. The dashed green lines illustrate the definition of the ITCF F(q,τ) as the correlated evaluation between densities in reciprocal space at different imaginary times, which can naturally be estimated within the PIMC formalism. Owing to the single-particle exchange of the two electrons in the centre, the configuration weight of the depicted configuration is actually negative, W(X)<0, thereby contributing to the fermion sign problem [69]. Taken from reference [64] with permission of the authors.

Let us postpone the discussion of the other elements in figure 1 and return to the partition function equation (2.2). For sufficiently large P, the matrix elements of e−ϵH^ can be evaluated using a suitable high-temperature approximation. For the parameters that are of interest in the present work, it is most convenient to make use of the primitive factorization e−ϵH^≈e−ϵK^ e−ϵV^, which becomes exact in the limit of large P as O(P−2) [82]. The incorporation of higher-order factorizations into PIMC simulations has been discussed extensively in the literature [83–86] and becomes advisable for lower temperatures.

In the end, one can express the partition function as 2.3 Z=∑∫ΩdX W(X),

where the configuration variable X=(R0,…,RP−1)T contains the coordinates of all N particles on each of the P imaginary-time slices. Note that the symbolic notation ∑∫ΩdX includes both the integration over all coordinates and the summation over all possible permutation configurations.

From a practical perspective, the partition function has been recast as a sum over all possible paths (including all permutation topologies) X, where each path contributes according to its configuration weight W(X), which is a function that we can straightforwardly evaluate. Specifically, we can factorize the weight contribution of each imaginary-time slice α into a kinetic and a potential contribution, WαK(X) and WαV(X). The latter is simply determined from the total potential energy V(Rα) on slice α, and we find that 2.4 WαV(X)=e−ϵV(Rα).

The kinetic part follows from the ideal single-particle density matrix for the reduced inverse temperature ϵ, 2.5 ρ0(rα,rα+1;ϵ)=λϵ−3 e−(π/λϵ2)(rα−rα+1)2,

and is given by 2.6 WαK(X)=∏l=1Nρ0(rl,α,rl,α+1;ϵ).

Returning once more to the illustration of the imaginary-time path integral picture in figure 1, we can say that the beads of the polymers interact with each other via the usual pair potential on a given time slice; the pair potential does not act between sets of coordinates for different τ. Beads of the same polymer on adjacent time slices are effectively linked via a harmonic spring potential by the ideal single-particle density matrix, equation (2.5). This is indicated by the yellow Gaussian curve on the right-hand side of figure 1. The width of the Gaussian is proportional to λϵ=2πϵ, with ϵ=β/P. Therefore, the Gaussian becomes infinitely narrow in the classical limit of ϵ→0, where the paths are given as straight lines, corresponding to point particles in coordinate space. Conversely, the Gaussian becomes increasingly broad with decreasing temperature, which leads to more extended paths. As we shall see, this imaginary-time diffusion process decisively shapes the τ-dependence of the ITCF F(q,τ) and can thus be used to explain and interpret observations from XRTS experiments.

The basic idea of the PIMC method [76] is to randomly generate every path X with a probability that is proportional to its respective configuration weight W(X) via the Metropolis algorithm [87,88]. This is relatively straightforward for bosons and also hypothetical distinguishable particles that are sometimes referred to as Boltzmannons in the literature [89]. Indeed, modern sampling techniques such as the worm algorithm of Boninsegni et al. [90,91] allow efficient exploration of different permutation topologies, and exact PIMC simulations of up to N∼104 quantum particles are feasible. The situation dramatically changes for the case of fermions, such as the electrons in the warm dense UEG in which we are interested in the present work. In particular, the aforementioned antisymmetry of the fermionic thermal density matrix with respect to the exchange of particle coordinates implies that the configuration weight can be both positive and negative; one has W(X)<0 for the configuration depicted in figure 1. Therefore, W(X) cannot correspond to a proper probability distribution, which must be strictly non-negative. While this problem can be formally circumvented [69] by sampling the configurations proportional to the modulus weight |W(X)| and keeping track of the respective sign of W(X), it also means that the fermionic partition function Z is given as the sum over a large number of positive and negative contributions, which could cancel to a large degree. This cancellation is the origin of the fermion sign problem [69,79,80] and leads to an exponential increase in the required computation time with increasing β (i.e. decreasing the temperature T) and with increasing system size N. The sign problem is the main computational bottleneck in our simulations and limits the regime of feasible simulation parameters such as N and Θ. A detailed review of the fermion sign problem in direct PIMC simulations has been presented elsewhere [69,81] and is beyond the scope of this article.

The definition of the ITCF as the intermediate scattering function [54] evaluated at imaginary times t=−iτ is 2.7 F(q,τ)=⟨n^(q,0)n^(−q,τ)⟩,

with τ∈[0,β]. It is easy to see that we can evaluate equation (2.7) in the path integral formalism for integer multiples of the imaginary time step ϵ by inserting the respective density operators at a distance of η∈{0,…,P−1} into equation (2.2): 2.8 F(q,τj) =1Zβ,N,Ω1N↑!N↓!∑σ↑∈SN↑∑σ↓∈SN↓sgn(σ↑,σ↓) ×∫ dR0…dRP−1 ⟨R0|n^(q) e−ϵH^|R1⟩⟨R1| e−ϵH^|R2⟩⋯ ⋯⟨Rj|n^(−q) e−ϵH^|Rj+1⟩⋯⟨RP−1| e−ϵH^|π^σ↑π^σ↓R0⟩.

This is illustrated in figure 1 by the horizontal green dashed lines. In practice, F(q,τ) is obtained by computing the correlation between density operators at different imaginary times, which, in thermodynamic equilibrium, depends only on the difference of imaginary-time arguments τ. Furthermore, it directly follows from the closed nature of the paths (see figure 1) that F(q,τ) is periodic in τ with respect to β.

We conclude this subsection with a note regarding the PIMC estimation of F(q,τ) for fermions. Specifically, it is well known that the utilization of antisymmetric imaginary-time propagators alleviates the sign problem for a small number of time slices P [92–95]. Therefore, it has been shown that the combination of this idea with a higher-order factorization of the thermal density matrix allows one to obtain accurate results for parameters that are beyond the scope of direct PIMC due to the fermion sign problem [17,31]. Yet, the small number of time slices means that F(q,τ) will only be available on a sparse τ-grid. Such data can still be useful for the evaluation of τ-decay measures (equation (2.21)), which constitute the natural analogue of the dispersion relation ω(q) of the DSF in the imaginary-time domain. Yet, they will likely not suffice for the full estimation of other physical properties such as the static linear density response function χ(q) (equation (2.12)) or for the numerical inversion of equation (1.2) to reconstruct S(q,ω) via an analytic continuation [96–98]. The in-depth investigation of the application of advanced fermionic PIMC methods to the investigation of imaginary-time properties is therefore an important topic for future research.

(b) Linear response theory

Let us next consider a modified Hamiltonian that includes an external harmonic perturbation of wavevector q and frequency ω, 2.9 H^=H^UEG+Aϕext(q,ω),

with H^UEG being the usual UEG Hamiltonian; see references [17,99] for more details on the latter. In the limit of small perturbation amplitudes A, the response of the UEG is fully described by the linear density response function, which can be expressed as follows [20,100]: 2.10 χ(q,ω)=χ0(q,ω)1−(4π/q2)[1−G(q,ω)]χ0(q,ω).

Here χ0(q,ω) describes the density response of an ideal Fermi gas, which is sometimes known as the (temperature-dependent) Lindhard function in the literature; it can readily be computed by numerically evaluating a simple one-dimensional integral [20]. The dynamic local field correction G(q,ω) contains the full wavevector- and frequency-resolved information about electronic exchange–correlation effects; it is formally equivalent to the dynamic exchange–correlation kernel Kxc(q,ω) that is used in the context of linear response time-dependent density functional theory simulations [101]. Setting G(q,ω)≡0 in equation (2.10) corresponds to the RPA, which describes the dynamic density response on the mean-field level. Consequently, G(q,ω) constitutes the key input for a gamut of applications such as the construction of effective potentials [102–104], quantum fluid models [48,105,106] or the construction of advanced, non-local exchange–correlation functionals for density functional theory [47,107]. The development of approximate local field corrections is an active field of research [108–120]. For the UEG, highly accurate results for G(q,ω) have been presented by Hamann et al. [121] based on the framework for analytic continuation of imaginary-time PIMC data developed in references [97,98,122]. First results for the static local field factor of real materials are currently being developed on the basis of either PIMC simulations [123,124] or density functional theory [125].

In the context of the present work, the main utility of χ(q,ω) is given by the fluctuation–dissipation theorem, which provides a straightforward connection between the density response and the DSF [20]: 2.11 S(q,ω)=−Imχ(q,ω)πn(1−e−βω).

In combination with equation (1.2), any dielectric theory for the local field correction can thus be used to compute the ITCF F(q,τ).

A final interesting relation from linear response theory is the imaginary-time version of the fluctuation–dissipation theorem [64,126], which states that 2.12 χ(q,0)=−n∫0βdτ F(q,τ).

Equation (2.12) can thus be used to compute the full wavenumber dependence of the static linear density response function from a single simulation of the unperturbed system from PIMC data for F(q,τ) by solving a simple one-dimensional integral for each value of q; this approach provides the basis for a number of studies of the UEG across different regimes [19,98,127–129]. Equation (2.12) also offers a feasible way to obtain χ(q,0) from XRTS experiments. In particular, the direct evaluation of the inverse frequency sum rule 2.13 ⟨ω−1⟩=∫−∞∞dω S(q,ω) ω−1=−12nχ(q,0)

is usually prevented by the convolution of the DSF with the instrument function in the measured XRTS intensity. The convolution theorem for the two-sided Laplace transform (equation (1.3)), on the other hand, allows us to exactly remove the influence of R(ω) and thus opens up the way to obtain the static density response of a given system [63].

(c) Properties of the ITCF

In thermodynamic equilibrium, the DSF obeys the detailed balance relation between positive and negative frequencies [20], 2.14 S(q,−ω)=S(q,ω) e−βω.

Inserting equation (2.14) into the definition of the two-sided Laplace transform (equation (1.2)) gives the important symmetry relation [62,63] 2.15 F(q,τ)=∫0∞dω S(q,ω){e−ωτ+e−ω(β−τ)}=F(q,β−τ).

In particular, Dornheim et al. [62] have recently shown that equation (2.15) allows for an exact, simulation-free diagnostic of the temperature of any given system from an XRTS measurement by locating the minimum of F(q,τ) at τ=β/2=1/(2T); further development of this idea is a topic of active research [63,64].

A further important set of properties related to the description of the dynamics of quantum many-body systems is given by the frequency moments of the DSF, which we define as follows: 2.16 Mα=⟨ωα⟩=∫−∞∞dω S(q,ω) ωα.

It is straightforward to show that all positive frequency moments can be obtained from τ-derivatives of F(q,τ) around the origin [64], 2.17 Mα=(−1)α∂α∂ταF(q,τ)|τ=0.

We note that the odd frequency moments can alternatively be expressed in terms of nested commutator expressions [130], which are known as sum rules in the literature [131]. In the case of α=1, we have the well-known f-sum rule, 2.18 M1=⟨ω1⟩=q22.

An important insight into the physical origin of the observed τ-dependence of F(q,τ) can be obtained from the spectral representation of the DSF, which is given by [20] 2.19 S(q,ω)=∑m,lPm‖nml(q)‖2δ(ω−ωlm).

Equation (2.19) states that the DSF is the sum over all possible transitions between the eigenstates l and m (with ωlm being the energy difference) of the full N-body Hamiltonian, where Pm is the occupation probability of the initial state and ‖nml(q)‖2 denotes the corresponding transition matrix element. Inserting equation (2.19) into equation (1.2) then gives [64] 2.20 F(q,τ)=∑m,lPm‖nml(q)‖2 e−τωlm.

In other words, the τ-decay is shaped by the energy difference between the eigenstates for transitions that are important at a particular wavevector q. More specifically, large energy differences as they occur in the non-collective single-particle regime where the dispersion relation of the DSF is ω(q)∼q2 lead to a pronounced τ-decay, with F(q,τ) almost vanishing around the minimum at τ=β/2. Conversely, energetically low-lying transitions such as the roton feature of the UEG [66,132] manifest as a reduced τ-decay; see references [63,64] for an extensive discussion of this effect. In other words, the existence of a low-energy excitation is equivalent to the stability of density correlations throughout the imaginary time. Dornheim et al. [64] recently suggested quantifying this mechanism via a relative decay measure of the form 2.21 ΔFτ(q)=F(q,0)−F(q,τ)F(q,0),

which we will also employ in the present work.

(d) Single-particle delocalization model

To gain further insight into the physical meaning of F(q,τ), we will present a simple model for its τ-dependence. It has recently been reported [133] based on extensive, spin-resolved PIMC results for the ITCF that the τ-dependence is almost exclusively due to single-particle imaginary-time diffusion effects; density–density correlations between electrons of different spin-orientation are nearly unaffected by the thermal delocalization of the paths. On the basis of this empirical observation, we decompose the total ITCF into 2.22 F(q,τ)=S(q)+ΔF(q,τ),

where the static structure factor (SSF) corresponds to the τ→0 limit of the ITCF, S(q)=F(q,0). The full τ-dependence is thus, by definition, contained in the function ΔF(q,τ)=F(q,τ)−F(q,0).

As the next step, we consider the imaginary-time diffusion of a single particle at the inverse temperature β in a volume Ω=L3, with L≫λβ. In this case, the appropriate imaginary-time propagator is simply given by the diagonal elements of the ideal thermal density matrix ρ0(r,r;β); see equation (2.5). To estimate the corresponding single-particle ITCF FSP(q,τ′), we insert a single intermediate time slice at τ′. This can be expressed as 2.23 ρ0(r,r;β)=∫Ωdr′ ρ0(r,r′;τ′)ρ0(r′,r;β−τ′).

Inserting the corresponding density operators n^(q) and n^(−q) at the appropriate imaginary times and performing a few straightforward transformations then gives a simple expression for the single-particle ITCF: 2.24 FSP(q,τ′)=∫ΩdΔr P(Δr,τ′) cos(q⋅Δr),

with Δr being the displacement between the position of the particle at τ=0 (which is the same as at τ=β owing to the diagonal nature of the trace, leading to closed trajectories in the path integral picture) and τ=τ′. The probability of a particular displacement is given by the three-dimensional Gaussian distribution 2.25 P(Δr,τ)=(λβλτλβ−τ)3exp(−πλβ2λτ2λβ−τ2Δr2),

which takes into account the distance in imaginary-time to the time slice of interest starting both from τ=0 and from τ=β.

The basic idea behind equation (2.24) is schematically illustrated in figure 2a. The blue beads correspond to the start and end points at τ=0 and τ=β, which are always equal. The green curves depict the Gaussian probability distributions from equation (2.25) for different values of τ; they are symmetric with respect to τ=β/2. In other words, the symmetry relation (2.15) of the ITCF not only can be seen as a consequence of the detailed balance of the DSF but also naturally emerges as a consequence of the β-periodicity within the imaginary-time path integral formalism. The yellow star at τ=4ϵ indicates a particular realization of the displacement Δr(τ), which is weighted with the probability P(Δr,τ). The final expectation value for FSP(q,τ) then follows from an integral over all possible values of Δr, which we evaluate numerically. Figure 2. Schematic illustration of the single-particle diffusion through the imaginary time and its estimation via equation (2.24). The green curves show the Gaussian probability distribution equation (2.25) of a diffused particle coordinate at a difference τ from its origin (blue beads), and the yellow star represents a particular realization.

In figure 3, we analyse the integrand of equation (2.24), which is depicted along the x-component of Δr at y=z=0 for N=34 and Θ=1. We note that such single-particle expressions do not, by definition, depend on the density parameter rs for a given value of Θ. The different panels of figure 3 correspond to different wavevectors q=(q,0,0)T, and the sinusoidal black lines show the cosine from equation (2.24). Figure 3a has been obtained for the minimum possible wavenumber q=2π/L, figure 3b for q=2qmin, figure 3c for q=5qmin and figure 3d for q=10qmin. In addition, the solid red, dotted blue and dashed green lines show the products of the cosine and the thermal probability function P(Δr,τ) for different values of τ. Note that we employ standard periodic boundary conditions such that x→x+L for x<0. Evidently, the Gaussian becomes increasingly narrow with decreasing τ, since the corresponding path has only a shorter imaginary-time interval to diffuse away from its position at τ=0 in that case. In the limit of τ→0, P(Δr,τ) becomes a delta distribution. Consequently, the single-particle ITCF attains unity for τ=0 and τ=β for all q. Figure 3. Integrand of the estimator for the single-particle ITCF FSP(q,τ), equation (2.24), for N=34 and Θ=1 for (a) q=0.63qF, (b) q=1.25qF, (c) q=3.13qF and (d) q=6.27qF. The solid black sinusoidal lines show the respective cosine functions, and the solid red, dotted blue and dashed green curves represent the full integrands for τ=β/2, τ=β/4 and τ=β/10, respectively.

For q=qmin, the cosine is positive over the entire x-range where the thermal distribution has significant values. Consequently, the oscillating nature of the cosine function has a small impact, as it is almost constant in the relevant x-interval. Still, the small decay of the cosine function for x≤L/2 becomes more important with increasing τ≤β/2, since the thermal distribution is spread out to larger x in this case. This is the origin of the comparably small τ-decay of the corresponding single-particle ITCF, which is depicted as the dotted blue line in figure 4. Figure 4. Single-particle ITCF (equation (2.24)) for different wavevectors q for N=34 and Θ=1.

In figure 3b, we show the integrand of equation (2.24) for a moderate wavevector, q=1.25qF. In this case, the cosine function oscillates twice as fast as for q=0.63qF, which has a noticeable impact on the integrand. Indeed, the latter even becomes slightly negative for x≳L/8; the resulting cancellation becomes more pronounced with increasing τ, which explains the larger τ-decay of the corresponding single-particle ITCF (dashed red curve) in figure 4. Further increasing the wavenumbers to q=3.13qF and q=6.27qF leads to the integrands shown in figure 3c,d, which exhibit an even more substantial cancellation of positive and negative contributions. Consequently, the τ-decay of FSP(q,τ), shown in figure 4 as the dash–dotted black and solid green curves, is even sharper. The sharp decay in the single-particle limit of q≫qF is thus the result of competition between the delta distribution limit of the thermal distribution equation (2.25) for τ→0 and the cancellation of the integrand as a consequence of the rapidly oscillating cosine function for large q at any finite value of τ.

3. Results

All PIMC results were obtained using the extended ensemble approach introduced in reference [70], which is a canonical adaptation of the worm algorithm of Boninsegni et al. [90,91]. We typically employ P=200 imaginary-time slices, which is sufficient with respect to the convergence of the factorization error [98] and sufficient to resolve F(q,τ) on a useful τ-grid.

(a) Dependence on the imaginary time

We begin our in-depth analysis of the τ-dependence of the ITCF of the warm dense UEG at the electronic Fermi temperature (i.e. Θ=1) for a metallic density of rs=4, which is shown in figure 5. It is sufficient to show F(q,τ) in the half-range of 0≤τβ/2, since all curves are symmetric around τ=β/2; see equation (2.15) above. The solid green line shows our exact PIMC results; we note that the statistical uncertainty of these data is smaller than the width of the curve and can be neglected here. The dash–dotted black curve depicts the static approximation [97] and was obtained by setting Gstatic(q,ω)≡G(q,0) in equation (2.10). For completeness, we note that we have used the neural-net parametrization of G(q,ω=0;rs,θ) from reference [127], but the analytic parametrization of the local field factor [134] within the recently introduced effective static approximation [45] would have worked equally well for this purpose. The static approximation provides a very accurate description of F(q,τ) for all depicted wavevectors and over the entire τ-range. This is consistent with previous investigations of the DSF S(q,ω), which have reported the same trend [97]. The only deviations that are visible on the depicted scale appear for the intermediate wavenumbers of q=1.25qF and q=1.88qF that are shown in figure 5b,c, but the systematic errors are very small. Figure 5. Imaginary-time dependence of the ITCF F(q,τ) for different wavenumbers q for the unpolarized UEG with N=34, rs=4 and Θ=1. Solid green: exact PIMC results; solid yellow: single-particle ITCF FSP(q,τ), equation (2.24); long-dashed purple: combination of SSF with single-particle ITCF, equation (2.22); dashed blue: quasi-particle (QP) expansion, equation (3.2); dotted red: RPA; dash–dotted black: static approximation, i.e. Gstatic(q,ω)≡G(q,0); dash–double-dotted grey: ideal Fermi gas.

Let us next consider the dotted red curve, which corresponds to the RPA. It is well known that the RPA becomes exact in the limit as q→0, where the UEG is dominated by the collective plasmon feature; see equation (3.1). For q=0.63qF, which is the smallest wavenumber that is accessible for simulations with N=34 particles, one is already within the pair continuum [66], and the DSF substantially differs from the plasmon shape with respect to both its peak position and the peak shape. In this regime, the RPA becomes increasingly inaccurate, and the dotted red curve noticeably deviates from the solid green PIMC results. With increasing q, the systematic error of the RPA becomes more pronounced and attains a maximum around intermediate wavenumbers, q∼1.5qF. In the single-particle limit of q≫qF, the RPA becomes exact again, and no systematic deviations are visible for q=4.39qF and q=6.27qF (figure 5e,f). Interestingly, the main source of error in the RPA at intermediate wavenumbers seems to be a constant shift in F(q,τ) that does not depend on the imaginary time. In fact, it is easy to see that the first derivative of F(q,τ) around the origin is exactly reproduced by the RPA, since it fulfils the f-sum rule; cf. equations (2.17) and (2.18). From equation (2.22), it is then clear that the bulk of the RPA error should already be present in the SSF S(q), which is shown in figure 6. The green squares show our exact PIMC results for rs=4, and the green dashed line represents the corresponding RPA. Evidently, the RPA underestimates the true magnitude of S(q) for intermediate q, which explains the observed trends in figure 5. Figure 6. PIMC results for the SSF of the UEG with N=34 at Θ=1 for rs=4 (green squares) and rs=20 (red circles). The dashed curves show corresponding results within the RPA, and the dotted line depicts the SSF of the ideal Fermi gas, which is independent of rs.

The dashed blue curves in figure 5 were computed from a simple quasi-particle ansatz. In this model, it is assumed that the DSF consists of a single delta-like peak around the quasi-particle excitation energy ωQP, 3.1 SSP(q,ω)=A{δ(ω−ωQP)+e−βωQPδ(ω+ωQP)},

and the second term on the right-hand side corresponds to the respective contribution at negative frequencies and obeys the detailed balance relation equation (2.14). Inserting equation (3.1) into the two-sided Laplace transform equation (1.2) then gives the corresponding ITCF, 3.2 FQP(q,τ)=A{e−τωQP+e−(β−τ)ωQP}.

For the case of a plasmon excitation in the UEG, we have ωQP=ωp (with ωp=3/rs3 being the usual plasma frequency [20]) for q→0, and the normalization A of equations (3.1) and (3.2) is determined by the SSF S(q)=F(q,0), 3.3 A=S(q)1+e−βωQP.

In the limit of small wavevectors, the latter can be computed analytically from the parabolic expansion [135] 3.4 S(q)=q22ωpcoth(βωp2).

As a final ingredient in computing the quasi-particle ITCF FQP(q,τ), we require some information about the wavenumber dependence of the excitation energy, which can be expanded as [136] 3.5 ω2(q)ωp2=1+B2(rs,Θ)(qqF)2

within the RPA in the limit of small q. We note that the coefficient B2(rs,Θ) has been parametrized by Hamann et al. [136]. For q=0.63qF, equation (3.2) gives the correct qualitative description of F(q,τ), but it substantially overestimates the static limit of τ→0; this is a direct consequence of the overestimation of the true SSF by the q→0 expansion given in equation (3.4). For q=1.25qF, the validity of this simple plasmon approximation breaks down, and neither the τ-decay nor the static limit is described accurately. Therefore, we omit the corresponding results from the other panels for even larger values of q.

Let us next consider the results for the semi-analytical single-particle model FSP(q,τ) that we introduced in §2d, which is depicted by the solid yellow curves in figure 5. We find that FSP(q,τ) exhibits the correct qualitative τ-decay for all q. Yet it always exhibits a static limit of FSQ(q,0)≡1, which leads to large systematic errors for small q. From a physical perspective, this deficiency can be attributed to the lack of screening effects in the underlying single-particle imaginary-time diffusion process. Combining these results for the thermal delocalization with the exact description of static correlation effects (i.e. by using our PIMC results for S(q)=F(q,τ)) via equation (2.22) leads to the long-dashed purple curves. Clearly, equation (2.22) leads to a spectacular improvement over the bare FSP(q,τ) for q≲2qF, i.e. in the regime where the SSF S(q) has not yet reached the single-particle limit of S(q→∞)=1. This finding constitutes one of the central observations of the present work, as it implies that the complicated dynamic behaviour of the quantum UEG can be accurately approximated by a phenomenologically simple combination of static correlations, which can easily be estimated from equilibrium simulations, with the simple Gaussian imaginary-time diffusion of a single particle.

In fact, we can show that the τ-dependence of the single-particle ITCF, ΔFSP(q,τ), does become exact in the limit as τ→0 by numerically evaluating the derivative of FSP(q,τ) with respect to τ around the origin. The results are shown as the black squares in figure 7 and compared to the exact f-sum rule (equation (2.18)), which is depicted by the solid red line via equation (2.17). We find perfect agreement between the numerical results obtained from the single-particle ITCF and the exact sum rule for all q. Returning once more to figure 5, we find that FSP(q,τ) even becomes exact for large wavenumbers where S(q)→1, and we find perfect agreement with the PIMC reference data for q=6.27qF. Figure 7. First frequency moment of the UEG at rs=4 and Θ=1. Solid red: exact f-sum rule, equation (2.18); black diamonds: numerical derivative of the single-particle ITCF FSP(q,τ), as given by equation (2.17).

Finally, the grey dash–double-dotted curves in figure 5 show results for the ITCF of the ideal Fermi gas [137], F0(q,τ), and we observe similar, though not the same, trends as in FSP(q,τ). In particular, fermionic exchange effects lead to density correlations between electrons of the same spin-orientation, and the static limit F0(q,0) that is depicted as the dotted black curve in figure 6 deviates from unity for q≲2qF. Therefore, F0(q,τ) is somewhat closer to the PIMC results for q≲2qF than is FSP(q,τ), even though they exhibit nearly identical dependence on the imaginary time.

We next repeat this analysis at a lower density, rs=20, the results of which are shown in figure 8. These conditions are characterized by the onset of strong electronic correlations and constitute the boundary to the strongly coupled electron liquid regime [128]. As a consequence, the dispersion relation of the DSF exhibits a pronounced roton feature [97,138], which has recently been explained by Dornheim et al. [66] in terms of a new pair alignment model. Since we find overall similar trends of F(q,τ) to those for rs=4, we here restrict ourselves to a brief discussion of the main differences between the two regimes. The most pronounced difference comes from the static limit F(q,0)=S(q), which is shown in figure 6 as the red circles and red dashed line for PIMC and RPA. We find that S(q) attains smaller values than in the rs=4 case for q≲2qF, which is directly reflected by our results for F(q,τ) in figure 8. In addition, the RPA is less accurate as electronic exchange–correlation effects are substantially more important in the electron liquid regime. Surprisingly, we find that the RPA is even less accurate than the bare single-particle ITCF FSP(q,τ) or the corresponding function of the ideal Fermi gas F0(q,τ) for intermediate wavenumbers; this is a direct consequence of the overestimation of the Coulomb repulsion in the RPA, which is discussed in more detail in reference [66], for example. Figure 8. Imaginary-time dependence of the ITCF F(q,τ) for different wavenumbers q for the unpolarized UEG with N=34, rs=20 and Θ=1. Solid green: exact PIMC results; solid yellow: single-particle ITCF FSP(q,τ), equation (2.24); long-dashed purple: combination of SSF with single-particle ITCF, equation (2.22); dashed blue: quasi-particle (QP) expansion, equation (3.2); dotted red: RPA; dash-dotted black: static approximation, i.e. Gstatic(q,ω)≡G(q,0); dash–double-dotted grey: ideal Fermi gas.

In addition, the combination of our new single-particle imaginary-time diffusion model for FSP(q,τ) with the SSF via equation (2.22) overall exhibits a comparable accuracy to that for rs=4. An exception occurs only for the smallest depicted wavenumber, q=0.63qF, where the corresponding long-dashed purple curve exhibits a too-pronounced decay towards τ=β/2. Finally, we find exactly the same functional form of all models for F(q,τ) as for rs=4 at q=6.27qF, as electron–electron correlations do not influence density correlations in the single-particle regime.

(b) Dependence on the wavenumber

To gain better qualitative insight into the dependence of imaginary-time density–density correlations on the wavenumber q, Dornheim et al. [64] recently suggested analysing the relative decay measure ΔFτ(q) defined in equation (2.21). The results are shown in figure 9 for rs=4 (top) and rs=20 (bottom) based on exact PIMC simulation data for F(q,τ). More specifically, the green squares, red circles and blue crosses show ΔFτ(q) for τ=β/10, τ=β/4 and τ=β/2, respectively. For rs=4, the three depicted datasets exhibit qualitatively similar behaviour. In the limit of small q, they approach the exact plasmon quasi-particle limit that directly follows from FQP(q→0,τ); see equation (3.2). Upon increasing q, all three curves monotonically increase and eventually attain the single-particle limit of ΔFτ(q→∞)=1 for large q; this is a direct consequence of the steeper decay of F(q,τ) with τ for 0≤τ≤β/2 that is depicted in figures 5f and 8f. Furthermore, it is easy to see that the onset of the single-particle limit happens for increasingly large q for decreasing imaginary-time distances τ in ΔFτ(q). Figure 9. Ab initio PIMC results for the relative decay measure ΔFτ(q) for τ=β/2 (blue crosses), τ=β/4 (red circles) and τ=β/10 (green squares) of the UEG at Θ=1 for rs=4 (top) and rs=20 (bottom).

For rs=20, the relative deviation measure exhibits an even more interesting dependence on q. While again all three datasets attain their respective plasmon limits for q→0, ΔFτ(q) has a minimum for intermediate wavenumbers of q∼2qF for all depicted values of τ; for larger q, we then recover the same single-particle limit as for rs=4. From a physical perspective, the minimum in the relative decay measure signals a reduced decay of correlations along the imaginary time; in other words, electronic correlations remain more stable throughout the imaginary-time diffusion process described in §2d in the strongly coupled case of rs=20. Indeed, Dornheim et al. [64] have suggested that such a reduced τ-decay of F(q,τ) can directly be translated to a shift of spectral weight in the DSF towards lower frequencies; this effect manifests as a roton-type minimum in the dispersion relation ω(q) of S(q,ω); cf. figure 10. This effect appears when the wavelength λq=2π/q is comparable to the average interparticle distance d. It is explained in reference [66] in terms of the spontaneous alignment of pairs of electrons, which was further substantiated in the subsequent work [132]. Figure 10. PIMC results for the dispersion relation ω(q) of the DSF (black diamonds, taken from reference [97]) and relative decay measure of the ITCF ΔFτ(q) for τ=β/2 (dotted blue), τ=β/4 (dashed red) and τ=β/10 (solid green) for the unpolarized UEG with N=34 at Θ=1 for different values of the density parameter rs. All curves have been renormalized to their respective q→0 limits, given by the plasma frequency ωp in the case of ω(q).

The connection between the relative measure of τ-decay ΔFτ(q) and the dispersion ω(q) of S(q,ω) is investigated in more detail in figure 10, where we overlay both quantities at Θ=1 for different values of the density parameter rs; note that each curve has been rescaled by its respective q→0 limit, which is given by the plasma frequency ωp in the case of the dispersion relation. For the metallic density of rs=2, the PIMC-based results for ω(q) taken from reference [97] exhibit a smooth, monotonic crossover between the collective regime, where the DSF corresponds to a sharp plasmon feature (equation (3.1)), and the single-particle regime with λq≪d, where ω(q)∼q2. The corresponding results for the decay measure ΔFτ(q) exhibit very similar behaviour; the main difference from ω(q) arises from the fact that the single-particle limit is given by a constant instead of a parabolic divergence.

For rs=4, we find a very similar picture to that of rs=2 for both ω(q) and the three datasets for ΔFτ(q). Upon further decreasing the density to rs=10, the dispersion starts to exhibit the incipient roton minimum around intermediate wavenumbers. This effect is qualitatively reflected in the relative τ-decay measure ΔFτ(q) as a minimum around q∼2qF, although its overall behaviour is less complicated than that of ω(q), as it does not exhibit the local maximum of the latter around q=qF.

Finally, we show results for the strongly coupled case of rs=20 in figure 10d. In this case, both the roton minimum in ω(q) and the reduced decay in ΔFτ(q) are more pronounced, as expected. Interestingly, the physical mechanism of the roton—the spontaneous alignment of pairs of electrons [66]—manifests in ΔFτ(q) with a similar magnitude for all three considered values of τ. This finding is of practical importance for the consideration of different approximate theories for F(q,τ), which are discussed next.

In figure 11, we show the wavenumber dependence of ΔFτ(q) for both rs=4 (left column) and rs=20 (right column) for τ=β/2 (top row) and τ=β/10 (bottom row). The green squares are results obtained from our ab initio PIMC simulations and provide an unassailable benchmark for the other depicted theoretical approaches. Figure 11. Relative imaginary-time decay measure ΔFτ(q) (equation (2.21)) of the unpolarized UEG at rs=4 (left) and rs=20 (right) for Θ=1. The top and bottom rows correspond to τ=β/2 and τ=β/10. Green squares: ab initio PIMC results; solid yellow: single-particle ITCF model FSP(q,τ) (equation (2.24)); long-dashed purple: single-particle τ-decay combined with PIMC results for the SSF S(q) via equation (2.22); dashed blue: quasi-particle ITCF FQP(q,τ) (equation (3.2)); dotted red: RPA; dash–dotted black: static approximation using the local field correction (LFC) from reference [127] as input.

The dashed blue curves were computed from the quasi-particle approximation for the ITCF, FQP(q,τ), defined in equations (3.2) and (3.5). For rs=4, the corresponding decay measure qualitatively follows the PIMC data; the limits of both large and small wavenumbers are correctly reproduced, although this happens only in the limit for very large q in the case of τ=β/10. In addition, ΔFτ(q) exhibits a smooth and monotonic transition between these collective and single-particle limits, which is phenomenologically correct but quantitatively strongly deviates from the green squares. In particular, we find that while FQP(q,τ) becomes, by definition, exact in the plasmon limit as q→0, the change in the position of the delta peak described by equation (3.5) does not constitute a good first-order correction to this limit. In other words, the finite width of the true DSF for q>0 appears to be more important than the plasmon shift away from the plasma frequency ωp. This trend can be seen particularly well for the more strongly coupled case of rs=20, where the true ΔFτ(q) has a negative slope for small q; in contrast, the quasi-particle approximation FQP(q,τ) exhibits the opposite behaviour for both depicted values of τ.

The full RPA is represented by the dotted red curves in figure 11. It is exact in the limits of both q→0 and q≫qF, and represents a substantial improvement over the simple quasi-particle ansatz in between. For rs=4, the RPA performs comparably well over the entire q-range, with a maximum relative error around intermediate wavenumbers q∼2qF. In addition, we note that the τ-decay measure within the RPA is more accurate for τ=β/10 than for τ=β/2. This can be explained by the fact that while the SSF S(q)=F(q,0) in the RPA exhibits systematic errors for intermediate q (see figure 6), it does attain the correct first derivative given by the well-known f-sum rule (equation (2.18)). For rs=20, the qualitative accuracy of the RPA noticeably deteriorates as electron–electron exchange–correlation effects become more important. In particular, the RPA, too, does not reproduce the negative slope of the exact PIMC data for ΔFτ(q) for small q. This is consistent with the inability of the RPA to capture the roton minimum in the dispersion relation of the DSF, which has been investigated in detail in references [66,97].

This systematic shortcoming of the mean-field-based RPA is remedied by including exact PIMC results for the static local field correction via equation (2.10), and the results for ΔFτ(q) are shown as the dash–dotted black curve in figure 11. For a metallic density of rs=4, no systematic deviations between the static approximation and the PIMC reference results can be resolved on the depicted scale. For the electron liquid at rs=20, too, we find that including the static local field correction leads to a spectacular improvement, and the dash–dotted black curve nicely captures the roton minimum of ΔFτ(q) around q=2qF. At the same time, small deviations from the green squares can be resolved in particular for τ=β/2 in this regime; yet the effect is significantly smaller than the corresponding underestimation of the true depth of the roton minimum of S(q,ω) due to the static approximation, which is discussed in more detail in reference [66].

Let us conclude this analysis by considering our new single-particle imaginary-time diffusion model FSP(q,τ) (see equation (2.24) in §2d), which is included in figure 11 as the solid yellow curve. Evidently, this simple, semi-analytical model becomes exact in the single-particle regime of q≥3qF but does not reproduce either the collective plasmon limit or the roton minimum for intermediate q at rs=20. The combination of FSP(q,τ) with the correct static structure F(q,0)=S(q) via equation (2.22) (see the long-dashed purple curves in figure 11) substantially mitigates this deficiency. More specifically, our model is very accurate at rs=4 over the entire q-range. Indeed, no systematic errors can be resolved on the depicted scale for the smaller value of the imaginary time τ=β/10, since equation (2.22) becomes exact for τ=0 both with respect to F(q,0) and with respect to the first slope, which is given by the f-sum rule that has been investigated in detail in figure 7. Even in the more complicated case of rs=20, the combination of the correct static structure with the simple single-particle model for the imaginary-time diffusion discussed in §2d is capable of reproducing the roton minimum of ΔFτ(q) with remarkable precision; it only breaks down for q≲qF, where it fails to attain the true plasmon limit in the collective regime. For τ=β/10, even this flaw is basically removed, and the long-dashed purple curve gives a very good description of the entire wavenumber range.

(c) Dependence on the temperature

We conclude this investigation of the ITCF of the warm dense UEG by briefly touching upon its dependence on the temperature. In figure 12, we show F(q,τ) at rs=4 for q=0.63qF (top), q=2.51qF (centre) and q=4.39qF (bottom). The solid red lines correspond to exact PIMC simulation data for Θ=4, Θ=2 and Θ=1, and the curves are ordered with respect to temperature as indicated in figure 12a. Overall, we find a similar trend for all depicted wavenumbers: the RPA (dotted green) underestimates the static limit of F(q,0)=S(q) but exhibits the correct qualitative decay with respect to τ. The combination of our new single-particle model for the imaginary-time diffusion and the exact static limit via equation (2.22) overall fits even better to the PIMC reference data and captures the correct trends for all depicted temperatures and wavenumbers. In fact, it becomes more accurate with increasing temperature, which can be understood intuitively in the following way. As we have previously shown in this work, equation (2.22) becomes exact in the limit of τ→0 with respect to both F(q,0) and the slope at the origin. For higher temperatures, the imaginary-time propagation described in §2d is carried out over a shorter distance in the τ-domain. Therefore, systematic errors have a shorter distance to accumulate, which explains the increased relative accuracy for larger values of Θ. Figure 12. ITCF for different Θ. Solid red: PIMC; dotted green: RPA; dashed black: combination of FSP(q,τ) with S(q) via equation (2.22).

4. Discussion

In this work, we have introduced a simple, semi-analytic model that gives new insights into the dependence of the ITCF F(q,τ) on the imaginary time τ. In practice, we find that the combination of the static structure S(q)=F(q,0) with the simple single-particle imaginary-time diffusion function FSP(q,τ) (equation (2.22)) gives a highly accurate description of the ITCF over a broad range of densities, temperatures and wavenumbers. Remarkably, our model is even capable of capturing the roton feature of the strongly coupled electron liquid [64,66,97], which emerges from the exchange–correlation-induced alignment of pairs of electrons. Even in this regime, the effect of electron–electron correlations on the imaginary-time diffusion is relatively small.

In addition, our model also very accurately captures the behaviour of the ITCF with respect to the temperature parameter Θ and generally agrees well with the PIMC results for the decay measure ΔFτ(q).

We believe that our new results represent an important first step towards a more thorough theoretical understanding of dynamic quantum many-body effects formulated in imaginary time. Such an improved understanding will be of immediate use for the interpretation of ab initio PIMC simulations of WDM [17,123,124,127], which give direct access to the ITCF. While previous direct PIMC simulations have mostly been restricted to the UEG, future efforts will produce results for the ITCF of real materials such as hydrogen [123,124]. In addition, we note that previous models for S(q,ω) such as the Chihara decomposition [60,61], but also time-dependent density functional theory calculations [36,44], can easily be translated to the imaginary-time domain and benchmarked against upcoming PIMC results. In addition, having an improved understanding of the physical properties of F(q,τ) will be helpful for the interpretation of XRTS experiments with WDM. As mentioned earlier, the latter give straightforward access to the ITCF [62] and contain the same physical information as the DSF. Performing the corresponding analysis of the observed XRTS signal in the τ-domain, therefore, has the potential to yield physical insights beyond temperature diagnostics without any models or approximations [64].

Data accessibility

This article has no additional data.

Authors' contributions

T.D.: conceptualization, investigation, methodology, project administration, writing—original draft and writing—review and editing; J.V.: resources, validation, writing—review and editing; Z.A.M.: validation, writing—review and editing; M.B.: validation and writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

This work was partially supported by the Center for Advanced Systems Understanding (CASUS), which is financed by Germany’s Federal Ministry of Education and Research (BMBF) and by the Saxon state government out of the state budget approved by the Saxon State Parliament. The PIMC calculations for the UEG were carried out at the Norddeutscher Verbund für Hoch- und Höchstleistungsrechnen (HLRN) under grant no. shp00026 and on a Bull Cluster at the Center for Information Services and High Performance Computing (ZIH) at Technische Universität Dresden.
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