Quantum version of the integral equation theory-based dielectric scheme for strongly coupled electron liquids

The three-dimensional uniform electron fluid (UEF) constitutes the simplest, yet realistic, bulk electronic system capable of exhibiting strong correlation effects.^1–3 It is often referred to as homogeneous electron gas, jellium, or quantum one-component plasma being the quantum analog of the classical one-component plasma (OCP).^4,5 Early studies focused on the ground state at metallic densities due to its relevance for valence electrons in simple metals.^2,6 These investigations have led to remarkable insights such as the Landau Fermi-liquid theory⁷ and the Bohm and Pines quasi-particle picture of collective excitations.⁸ In addition, the accurate parametrization of different UEF properties based on systematic quantum Monte Carlo (QMC) simulations^9–11 has been of vital importance for the unrivaled success of density functional theory in the description of real materials.^12,13

Increasing interest in warm dense matter (WDM),^14–16 which is an exotic yet ubiquitous state of high-temperature highly compressed matter encountered in dense astrophysical objects,^17–19 fuel compression processes during inertial confinement fusion,²⁰ and novel material fabrication,²¹ has provided the impetus for a worldwide intense research activity targeted at high-density finite-temperature UEF.^22–26 Numerous breakthroughs concerning the development of QMC methods or QMC extensions that alleviate the fermion sign problem at different WDM regions,^27–33 as well as progress regarding the correction of finite-size errors^34–36 and the numerical implementation or even full circumvention of analytic continuation,^37–39 have led to a very accurate description of the thermodynamic, static, dynamic, and non-linear behavior of the UEF in WDM conditions.^40–47 

On the other hand, the physical realization of dilute ground-state homogeneous electronic systems is known to be extremely challenging, while low-density finite-temperature inhomogeneous electron systems are inaccessible even to state-of-the-art experiments. Therefore, much less attention has been paid to the strongly coupled electron liquid regime (hence, the use of the label UEF instead of the label jellium, which typically concerns the gaseous phase), especially at finite temperatures, despite preliminary confirmations and legitimate speculations of exotic collective behavior that could be of technological importance. These include phenomena that begin to manifest themselves at the margin of the strongly coupled UEF regime. A prominent example concerns the onset of effective attraction between two electrons due to short-range order,⁴⁸ its manifestation onto the negative sign of the spin-off diagonal component of the static density-response function,⁴⁹ and its association with the possibility of Cooper pairing,⁵⁰ and thus, the potential emergence of super-conductivity without phonons.^51,52 Another closely related example concerns the existence of a minimum in the dispersion of density–density fluctuations and an associated roton feature in the dynamic structure factor,^37,38 as well as their microscopic finite-temperature interpretation in terms of an electronic pair alignment model⁵³ and their alternative ground-state interpretation in terms of an excitonic mode.^54–57 These also include phenomena that have been speculated to manifest themselves deep within the strongly coupled UEF regime such as the possibility of a charge-density wave instability,^58–61 the emergence of a spin-density wave,^61–64 and the possibility of a continuous paramagnetic to ferromagnetic transition (Stoner’s instability).^65–69 

The only exceptions to this rather discouraging state of affairs concern the thermodynamic and static properties of the strongly coupled paramagnetic UEF that were recently characterized by extensive path integral Monte Carlo (PIMC) simulations at finite temperatures.^70,71 The highly accurate results were compared to two novel schemes of the self-consistent dielectric formalism that are equipped to handle strong correlations. The hypernetted chain (HNC)-based scheme handles quantum mechanical effects at the random phase approximation level and incorporates a frequency-independent local field correction, treating strong Coulomb correlations within the classical HNC approximation.^70,72 The integral equation theory (IET)-based scheme supplements the HNC-based scheme with a near-exact classical Coulomb bridge function.^71,73 Both schemes were demonstrated to yield excellent predictions for the thermodynamic properties (benefitting from a favorable error cancellation) and accurate predictions for the position of the static structure factor peak, but quite inaccurate predictions for the magnitude of the static structure factor peak.^70,71 It is worth noting that interaction energy predictions of the consistently more accurate IET-based scheme were always within 0.68% of the exact value.⁷¹ 

In the present study, we substantially refine the treatment of quantum mechanical effects within the IET-based dielectric scheme guided by available PIMC simulations. The upgraded quantum version of the IET-based scheme features a complicated dynamic local field correction functional that leads to a substantially more involved set of equations. The associated numerical complexity proved to be rewarding, since our scheme yields excellent predictions for the static structure factor at all states except from the Wigner crystallization vicinity.

The UEF is a homogeneous model system consisting of electrons immersed in a rigid ionic neutralizing background. We focus on the paramagnetic (unpolarized) case of equal spin-up and -down electrons, whose state points are fully specified by two dimensionless quantities: (1) the quantum coupling parameter r_s = d/a_B with d = (4πn/3)^−1/3 the Wigner Seitz radius and a_B = ℏ²/m_ee² the Bohr radius and (2) the degeneracy parameter Θ = T/E_F with T the temperature in energy units, $EF=ℏ2kF2/(2me)$ the Fermi energy and $kF=(3π2n)1/3$ the Fermi wavevector. The WDM regime is roughly demarcated by 0.1 ≲ r_s, Θ ≲ 10, while the strongly coupled regime corresponds to r_s ≳ 20 and Θ ≲ 1. The theoretical treatment of the UEF in the WDM and especially the strongly coupled regime is formidable due to the coexistence of quantum effects (exchange and diffraction), Coulomb correlations, and thermal excitations, as reflected in the lack of small parameters.^24–26 

Extended and rigorously finite-size-corrected (Refs. 70 and 71 for details), PIMC data are available for the interaction energy, static density response χ(k) ≡ χ(k, ω = 0), static LFC G(k) ≡ G(k, ω = 0), and SSF S(k).^70,71 In particular, 34 UEF states have been simulated in the phase diagram region defined by 20 ≤ r_s ≤ 200, 0.5 ≤ Θ ≤ 4. The boundary between the WDM and liquid regimes is of little interest for the present study; thus, we focus on the 20 UEF states that belong to the 50 ≤ r_s ≤ 200, 0.5 ≤ Θ ≤ 4 region.

Systematic comparison of the predictions of the STLS, qSTLS, HNC-based, and IET-based schemes with the exact PIMC results led to the following conclusions,^70,71 (Fig. 1): (1) the IET- and HNC-based schemes yield very similar accurate predictions for the positions of the SSF and χ(k) extrema. The STLS and qSTLS schemes yield very similar predictions for the positions of the SSF and χ(k) extrema, but they significantly underestimate both. (2) The IET-based scheme result for the SSF and χ(k) peak magnitudes greatly improves the respective HNC-based result, but it still underestimates the exact PIMC result, mainly for the lowest Θ and highest r_s considered. The qSTLS scheme significantly improves the STLS result for the SSF and especially the χ(k) peak magnitudes, but the PIMC result underestimation persists. (3) The IET-based scheme consistently improves the HNC-based outcome for the G(k) peak magnitude, though there is an underestimation compared to the exact PIMC outcome. On the other hand, the STLS and the qSTLS schemes lead to static LFCs that do not feature a well-developed G(k) maximum, but rather, an extended plateau. (4) The IET- and HNC-based interaction energies are remarkably accurate owing to a favorable error cancellation (accuracies within 0.68% and 1.37%, respectively).

There are two main drawbacks of the qIET-based scheme. First, similar to the bare IET-based scheme, the classical Coulomb bridge function is introduced as a closed form b(r, Γ) parametrization and not as a b[g] functional.^88,89 This implies that the bridge function does not properly react to pure quantum mechanical effects and this also necessitates the introduction of an algebraic mapping between the classical OCP states (Γ) and quantum OCP states (r_s, Θ). The primitive mapping Γ = 2λ²(r_s/Θ) employed is clearly insufficient since it lacks a ground-state limit, does not consider the Fermi energy contribution, and does not depend on the spin polarization. Different strategies on the optimization of this mapping based on enforcing self-consistency (thermodynamic sum rules or frequency moment sum rules) will be explored in future studies. The above-mentioned shortcoming manifests itself in the deviations from the exact results that arise at very low degeneracy parameters. Second, the high-temperature limit of the dynamic LFC of the qIET-based scheme exhibits a weak frequency dependence, i.e., it is essentially static. Investigations of the classical OCP have demonstrated that a static LFC is accurate for moderate coupling but not in the vicinity of crystallization.^98,99 The solution might lie in successfully interpolating between the static LFC limit and exact high-frequency LFC limit; the latter being tightly connected with the third frequency moment sum rule.^100–102 The above-mentioned shortcoming manifests itself in the deviations from the exact results that arise at very high quantum coupling parameters.

Our novel dielectric formalism scheme is tailor made for the strongly coupled regime of the finite-temperature uniform electron fluid. Essentially, the scheme combines the interplay of quantum mechanical effects and thermal excitations as approximated in the qSTLS scheme with the interplay of strong correlations and thermal excitations as exactly described in the integral equation theory of liquids. The mathematical correspondence between the STLS and qSTLS schemes is adopted as a general recipe to appropriately quantize semi-classical dielectric schemes. The physics approximations behind our scheme are not only uncontrollable but also rather obscured behind the mathematics. It is worth emphasizing that the recipe has been guided by available exact PIMC results. The truly remarkable agreement with the PIMC static structure factors in nearly the entire strongly coupled regime (with the exception of the Wigner crystallization vicinity) is a testament to the potential of the recipe. Future work will focus on exploring the dynamic structure factor predictions of the qIET scheme in search of exotic collective behavior and improving the accuracy of the qIET scheme, especially near the liquid-crystal phase boundary.

This work was partly funded by the Swedish National Space Agency under Grant No. 143/16. This work was also partially supported by the Center for Advanced Systems Understanding (CASUS), which is financed by Germany’s Federal Ministry of Education and Research (BMBF) and the Saxon state government out of the state budget that is approved by the Saxon State Parliament. The PIMC simulations were partly carried out at the Norddeutscher Verbund für Hoch- und Höchstleistungsrechnen (HLRN) under Grant shp00026 and on a Bull Cluster at the Center for Information Services and High Performance Computing (ZIH) at Technische Universität Dresden. The dielectric schemes were numerically solved on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the NSC (Linköping University) that is partially funded by the Swedish Research Council under Grant Agreement No. 2018-05973.

The authors have no conflicts to disclose.

Panagiotis Tolias: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Writing – original draft (equal). Federico Lucco Castello: Investigation (equal); Methodology (equal); Writing – review & editing (equal). Tobias Dornheim: Methodology (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.