Fig. 1. Advantages of working with F(q, τ) in the context of x-ray Thomson scattering (XRTS) experiments. Traditionally (left column), XRTS experiments have been interpreted by constructing an approximate model for the dynamic structure factor (DSF) S(q, ω) and then convolving it with the instrument function R(ω) for comparison with the measured intensity I(q, ω) [cf. Eq. (4)]; a deconvolution to obtain experimental data directly for S(q, ω) is generally impossible. In contrast, working in the τ domain makes the deconvolution trivial [cf. Eq. (5)], which allows the direct extraction of important system parameters such as the temperature without any models or simulations.²⁸ Furthermore, one can compare these experimental results directly with exact quantum Monte Carlo (QMC) data for F(q, τ), which opens up the possibility for unprecedented agreement between theory and experiment.

Fig. 2. Illustration of path-integral Monte Carlo (PIMC) method and corresponding estimation of imaginary-time density–density correlation function (ITCF) F(q, τ). Shown is a configuration of N = 4 particles in the x–τ plane with P = 6 imaginary-time propagators. The two horizontal dashed green lines depict the evaluation of a pair of density operators at τ = 0 and τ₁. The yellow Gaussian curve on the right corresponds to the kinetic part of the configuration weight [Eq. (13)]. Adapted from Ref. 82.

Fig. 3. Top: synthetic Gaussian [Eq. (29)] DSFs S(ω) with β = 1, ω₀ = 1, and different variance σ; the units of all properties are arbitrary. Bottom: corresponding two-sided Laplace transform F(τ), Eq. (30). Note that we have normalized both S(ω) and F(τ) to F(0).

Fig. 4. Top: synthetic Gaussian [Eq. (29)] DSFs S(ω) with β = 1, σ = 0.1, and different values of ω₀; the units of all properties are arbitrary. Bottom: corresponding two-sided Laplace transform F(τ), Eq. (30). Note that we have normalized both S(ω) and F(τ) to F(0).

Fig. 5. Top: synthetic Gaussian [Eq. (29)] DSFs S(ω) with ω₀ = 1, σ = 0.5, and slightly different values of the inverse temperature β; the units of all properties are arbitrary. Bottom: corresponding two-sided Laplace transform F(τ), Eq. (30). Note that we have normalized both S(ω) and F(τ) to F(0).

Fig. 6. Ab initio PIMC results for ITCF F(q, τ) in q–τ plane. Shown are results for the unpolarized UEG with N = 34 at r_s = 10 and different values of the degeneracy temperature Θ.

Fig. 7. Snapshots of ab initio PIMC simulation of N = 14 unpolarized electrons at r_s = 10 for Θ = 1 (left) and Θ = 4 (right). Top row: paths of spin-up (red) and spin-down (blue) electrons in x–τ plane. Bottom row: depiction of configurations as paths in 3D simulation cell.

Fig. 8. Imaginary-time dependence of F(q, τ) for two selected q values taken from the full dataset shown in Fig. 6. The solid colored curves correspond to the different values of Θ, and the dashed black lines show a linear expansion evaluated from the exact f-sum rule, Eq. (23). Note that we have not normalized the τ axis by the respective values of β.

Fig. 9. Ab initio PIMC results for F(q, τ) in the q–τ plane. Shown are results for the unpolarized UEG with N = 34 at Θ = 1 and different values of the density parameter r_s.

Fig. 10. Slices along q direction from full ITCF shown in Fig. 9. Top: static structure factor S(q) = F(q, 0). Bottom: F(q, β/2).

Fig. 11. Measure of relative τ decay [cf. Eq. (35)] for same conditions as in Fig. 9. The vertical bars in the top panel indicate the q → 0 limit given by a sharp plasmon excitation at the plasma frequency; see Eq. (33). Bottom: same information but normalized by the respective value at q = 0, with the horizontal bars indicating the single-particle limit for q → ∞.

Fig. 12. Position of maximum in DSF, ω(q), for same parameters as in Fig. 9. Data taken from Ref. 41.