Ab Initio Path Integral Monte Research Articles

Ab initio density response and local field factor of warm dense hydrogen

We present quasi-exact ab initio path integral Monte Carlo (PIMC) results for the partial static density responses and local field factors of hydrogen in the warm dense matter regime, from solid density conditions to the strongly compressed case. The full dynamic treatment of electrons and protons on the same footing allows us to rigorously quantify both electronic and ionic exchange–correlation effects in the system, and to compare the results with those of earlier incomplete models such as the archetypal uniform electron gas or electrons in a fixed ion snapshot potential that do not take into account the interplay between the two constituents. The full electronic density response is highly sensitive to electronic localization around the ions, and our results constitute unambiguous predictions for upcoming X-ray Thomson scattering experiments with hydrogen jets and fusion plasmas. All PIMC results are made freely available and can be used directly for a gamut of applications, including inertial confinement fusion calculations and the modeling of dense astrophysical objects. Moreover, they constitute invaluable benchmark data for approximate but computationally less demanding approaches such as density functional theory or PIMC within the fixed-node approximation.

Ab initio path integral Monte Carlo simulations of warm dense two-component systems without fixed nodes: Structural properties.

We present extensive new abinitio path integral Monte Carlo (PIMC) results for a variety of structural properties of warm dense hydrogen and beryllium. To deal with the fermion signproblem-an exponential computational bottleneck due to the antisymmetry of the electronic thermal density matrix-we employ the recently proposed [Y. Xiong and H. Xiong, J. Chem. Phys. 157, 094112 (2022); T. Dornheim et al., J. Chem. Phys. 159, 164113 (2023)] ξ-extrapolation method and find excellent agreement with the exact direct PIMC reference data where available. This opens up the intriguing possibility of studying a gamut of properties of light elements and potentially material mixtures over a substantial part of the warm dense matter regime, with direct relevance for astrophysics, material science, and inertial confinement fusion research.

Ab Initio Path Integral Monte Carlo Simulations of the Uniform Electron Gas on Large Length Scales.

The accurate description of non-ideal quantum many-body systems is of prime importance for a host of applications within physics, quantum chemistry, materials science, and related disciplines. At finite temperatures, the gold standard is given by ab initio path integral Monte Carlo (PIMC) simulations, which do not require any empirical input but exhibit an exponential increase in the required computation time for Fermionic systems with an increase in system size N. Very recently, computing Fermionic properties without this bottleneck based on PIMC simulations of fictitious identical particles has been suggested. In our work, we use this technique to perform very large (N ≤ 1000) PIMC simulations of the warm dense electron gas and demonstrate that it is capable of providing a highly accurate description of the investigated properties, i.e., the static structure factor, the static density response function, and the local field correction, over the entire range of length scales.

Bound-State Breaking and the Importance of Thermal Exchange-Correlation Effects in Warm Dense Hydrogen.

Hydrogen at extreme temperatures and pressures is of key relevance for cutting-edge technological applications, with inertial confinement fusion research being a prime example. In addition, it is ubiquitous throughout our universe and naturally occurs in a variety of astrophysical objects. In the present work, we present exact ab initio path integral Monte Carlo (PIMC) results for the electronic density of warm dense hydrogen along a line of constant degeneracy across a broad range of densities. Using the well-known concept of reduced density gradients, we develop a new framework to identify the breaking of bound states due to pressure ionization in bulk hydrogen. Moreover, we use our PIMC results as a reference to rigorously assess the accuracy of a variety of exchange-correlation (XC) functionals in density functional theory calculations for different density regions. Here, a key finding is the importance of thermal XC effects for the accurate description of density gradients in high-energy-density systems. Our exact PIMC test set is freely available online and can be used to guide the development of new methodologies for the simulation of warm dense matter and beyond.

Fermionic physics from abinitio path integral Monte Carlo simulations of fictitious identical particles.

The abinitio path integral Monte Carlo (PIMC) method is one of the most successful methods in statistical physics, quantum chemistry and related fields, but its application to quantum degenerate Fermi systems is severely hampered by an exponential computational bottleneck: the notorious fermion signproblem. Very recently, Xiong and Xiong [J. Chem. Phys. 157, 094112 (2022)] have suggested to partially circumvent the signproblem by carrying out simulations of fictitious systems guided by an interpolating continuous variable ξ ∈ [-1, 1], with the physical Fermi- and Bose-statistics corresponding to ξ = -1 and ξ = 1. It has been proposed that information about the fermionic limit might be obtained by calculations within the bosonic sector ξ > 0 combined with an extrapolation throughout the fermionic sector ξ < 0, essentially bypassing the signproblem. Here, we show how the inclusion of the artificial parameter ξ can be interpreted as an effective penalty on the formation of permutation cycles in the PIMC simulation. We demonstrate that the proposed extrapolation method breaks down for moderate to high quantum degeneracy. Instead, the method constitutes a valuable tool for the description of large Fermi-systems of weak quantum degeneracy. This is demonstrated for electrons in a 2D harmonic trap and for the uniform electron gas (UEG), where we find excellent agreement (∼0.5%) with exact configuration PIMC results in the high-density regime while attaining a speed-up exceeding 11 orders of magnitude. Finally, we extend the idea beyond the energy and analyze the radial density distribution (2D trap), as well as the static structure factor and imaginary-time density-density correlation function (UEG).

Electronic density response of warm dense hydrogen on the nanoscale.

The properties of hydrogen at warm dense matter (WDM) conditions are of high importance for the understanding of astrophysical objects and technological applications such as inertial confinement fusion. In this work, we present extensive ab initio path integral Monte Carlo results for the electronic properties in the Coulomb potential of a fixed ionic configuration. This gives us unique insights into the complex interplay between the electronic localization around the protons with their density response to an external harmonic perturbation. We find qualitative agreement between our simulation data and a heuristic model based on the assumption of a local uniform electron gas model, but important trends are not captured by this simplification. In addition to being interesting in their own right, we are convinced that our results will be of high value for future projects, such as the rigorous benchmarking of approximate theories for the simulation of WDM, most notably density functional theory.

Physical insights from imaginary-time density–density correlation functions

An accurate theoretical description of the dynamic properties of correlated quantum many-body systems, such as the dynamic structure factor S(q, ω), is important in many fields. Unfortunately, highly accurate quantum Monte Carlo methods are usually restricted to the imaginary time domain, and the analytic continuation of the imaginary-time density–density correlation function F(q, τ) to real frequencies is a notoriously hard problem. Here, it is argued that often no such analytic continuation is required because by definition, F(q, τ) contains the same physical information as does S(q, ω), only represented unfamiliarly. Specifically, it is shown how one can directly extract key information such as the temperature or quasi-particle excitation energies from the τ domain, which is highly relevant for equation-of-state measurements of matter under extreme conditions [T. Dornheim et al., Nat. Commun. 13, 7911 (2022)]. As a practical example, ab initio path-integral Monte Carlo results for the uniform electron gas (UEG) are considered, and it is shown that even nontrivial processes such as the roton feature of the UEG at low density [T. Dornheim et al., Commun. Phys. 5, 304 (2022)] are manifested straightforwardly in F(q, τ). A comprehensive overview is given of various useful properties of F(q, τ) and how it relates to the usual dynamic structure factor. In fact, working directly in the τ domain is advantageous for many reasons and opens up multiple avenues for future applications.

Dynamic properties and the roton mode attenuation in liquid 3He: ab initio study within the self-consistent method of moments.

The dynamic structure factor and the eigenmodes of density fluctuations in liquid 3He are studied using a novel non-perturbative approach. This new version of the self-consistent method of moments invokes up to nine sum rules and other exact relations, the two-parameter Shannon information entropy maximization procedure, and the ab initio path integral Monte Carlo simulations which provide necessary reliable input information on the system static properties. Detailed analysis is performed of the collective excitations dispersion relations, the modes' decrements and the static structure factor of 3He at the saturated vapour pressure. The results are compared to available experimental data by Albergamo et al. (Albergamo et al. 2007 Phys. Rev. Lett. 99, 205301. (doi:10.1103/PhysRevLett.99.205301)) and Fåk et al. (Fåk et al. 1994 J. Low Temp. Phys. 97, 445-487. (doi:10.1007/BF00754303)). The theory reveals a clear signature of the roton-like feature in the particle-hole segment of the excitation spectrum with a significant reduction of the roton decrement in the wavenumber range [Formula: see text]. The observed roton mode remains a well-defined collective mode even in the particle-hole band, where it is strongly damped. The existence of the roton-like mode in the bulk liquid 3He is confirmed like in other quantum fluids. The phonon branch of the spectrum is in a reasonable agreement with the same experimental data. This article is part of the theme issue 'Dynamic and transient processes in warm dense matter'.

Analysing the dynamic structure of warm dense matter in the imaginary-time domain: theoretical models and simulations.

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'.

Unravelling the nonlinear ideal density response of many-body systems

Nonlinear density response theory is revisited focusing on the harmonically perturbed finite temperature uniform electron gas. Within the non-interacting limit, brute force quantum kinetic theory calculations for the quadratic, cubic, quartic and quintic responses reveal a deep connection with the linear response. Careful analysis of the static long wavelength limit led us to conjecture a canonical non-interacting form that expresses arbitrary order nonlinear responses as the weighted sum of the linear responses evaluated at all multiple harmonics. This harmonic expansion is successfully validated against ab initio path integral Monte Carlo simulations.

Energy response and spatial alignment of the perturbed electron gas.

We study the linear energy response of the uniform electron gas to an external harmonic perturbation with a focus on resolving different contributions to the total energy. This has been achieved by carrying out highly accurate abinitio path integral Monte Carlo (PIMC) calculations for a variety of densities and temperatures. We report a number of physical insights into effects such as screening and the relative importance of kinetic and potential energies for different wave numbers. A particularly interesting finding is obtained from the observed non-monotonic behavior of the induced change in the interaction energy, which becomes negative for intermediate wave numbers. This effect is strongly dependent on the coupling strength and constitutes further direct evidence for the spatial alignment of electrons introduced in earlier works [T. Dornheim et al., Commun. Phys. 5, 304 (2022)]. The observed quadratic dependence on the perturbation amplitude in the limit of weak perturbations and the quartic dependence of perturbation amplitude corrections are consistent with linear and nonlinear versions of the density stiffness theorem. All PIMC simulation results are freely available online and can be used to benchmark new methods or as input for other calculations.

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

A novel dielectric scheme is proposed for strongly coupled electron liquids, which handles quantum mechanical effects beyond the random phase approximation level and treats electronic correlations within the integral equation theory of classical liquids. The self-consistent scheme features a complicated dynamic local field correction functional and its formulation is guided by abinitio path integral Monte Carlo simulations. Remarkably, our scheme is capable of providing unprecedently accurate results for the static structure factor with the exception of the Wigner crystallization vicinity, despite the absence of adjustable or empirical parameters.

Correlation properties of a one-dimensional repulsive Bose gas at finite temperature

We present a comprehensive study shedding light on how thermal fluctuations affect correlations in a Bose gas with contact repulsive interactions in one spatial dimension. The pair correlation function, the static structure factor, and the one-body density matrix are calculated as a function of the interaction strength and temperature with the exact ab-initio Path Integral Monte Carlo method. We explore all possible gas regimes from weak to strong interactions and from low to high temperatures. We provide a detailed comparison with a number of theories, such as perturbative (Bogoliubov and decoherent classical), effective (Luttinger liquid) and exact (ground-state and thermal Bethe Ansatz) ones. Our Monte Carlo results exhibit an excellent agreement with the tractable limits and provide a fundamental benchmark for future observations which can be achieved in atomic gases, cavity quantum-electrodynamic and superconducting-circuit platforms.

Ab initio path integral Monte Carlo simulations of hydrogen snapshots at warm dense matter conditions.

We combine ab initio path integral Monte Carlo (PIMC) simulations with fixed ion configurations from density functional theory molecular dynamics (DFT-MD) simulations to solve the electronic problem for hydrogen under warm dense matter conditions [Böhme et al., Phys. Rev. Lett. 129, 066402 (2022)0031-900710.1103/PhysRevLett.129.066402]. The problem of path collapse due to the Coulomb attraction is avoided by utilizing the pair approximation, which is compared against the simpler Kelbg pair potential. We find very favorable convergence behavior towards the former. Since we do not impose any nodal restrictions, our PIMC simulations are afflicted with the notorious fermion sign problem, which we analyze in detail. While computationally demanding, our results constitute an exact benchmark for other methods and approximations within DFT. Our setup gives us the unique capability to study important properties of warm dense hydrogen such as the electronic static density response and exchange-correlation kernel without any model assumptions, which will be very valuable for a variety of applications such as the interpretation of experiments and the development of new XC functionals.

The uniform electron gas at high temperatures: ab initio path integral Monte Carlo simulations and analytical theory

We present extensive new Ab initio path integral Monte Carlo (PIMC) simulations of the uniform electron gas (UEG) in the high-temperature regime, 8≤θ=kBT/EF≤128. This allows us to study the convergence of different properties towards the classical limit. In particular, we investigate the classical relation between the static structure factor S(q) and the static local field correction G(q), which is only fulfilled at low densities. Moreover, we compare our new results for the interaction energy to the parametrization of the UEG by Groth et al. (2017), which interpolates between PIMC results for θ≤8 and the Debye–Hückel limit, and to higher order analytical virial expansions. Finally, we consider the momentum distribution function n(q) and find an interaction-induced increase in the occupation of the zero-momentum state even for θ≳32. All PIMC data are freely available online, and can be used as input for improved parametrizations and as a rigorous benchmark for approximate methods.

Abnormal quantum moment of inertia and structural properties of electrons in 2D and 3D quantum dots: an ab initio path-integral Monte Carlo study

We present extensive new direct path-integral Monte Carlo results for electrons in quantum dots in two and three dimensions. This allows us to investigate the nonclassical rotational inertia (NCRI) of the system, and we find an abnormal negative quantum moment of inertia (2014 Phys. Rev. Lett. 112 235301) under some conditions. In addition, we study the structural properties by computing a re-normalized, angular-resolved center-two particle correlation function. Remarkably, we find no connection between the spatial structure and the NCRI, since the former can be nearly identical for Fermi- and Bose-statistics for parameters where the superfluid fraction is diverging towards negative infinity.

Spin-resolved density response of the warm dense electron gas

We present extensive ab initio path integral Monte Carlo (PIMC) results for the spin-resolved density response of the uniform electron gas at warm dense matter conditions. This allows us to unambiguously assess the accuracy of previous theoretical approximations, thereby providing valuable insights for the future development of dielectric schemes. From a physical perspective, we observe a nontrivial manifestation of an effective electron-electron attraction that emerges in the spin-offdiagonal static density response function at strong coupling, ${r}_{s}\ensuremath{\gtrsim}5$. All PIMC results are freely available online and can be used to benchmark approximations and simulation schemes.

Effective electronic forces and potentials from ab initio path integral Monte Carlo simulations.

The rigorous description of correlated quantum many-body systems constitutes one of the most challenging tasks in contemporary physics and related disciplines. In this context, a particularly useful tool is the concept of effective pair potentials that take into account the effects of the complex many-body medium consistently. In this work, we present extensive, highly accurate ab initio path integral Monte Carlo (PIMC) results for the effective interaction and the effective force between two electrons in the presence of the uniform electron gas. This gives us a direct insight into finite-size effects, thereby, opening up the possibility for novel domain decompositions and methodological advances. In addition, we present unassailable numerical proof for an effective attraction between two electrons under moderate coupling conditions, without the mediation of an underlying ionic structure. Finally, we compare our exact PIMC results to effective potentials from linear-response theory, and we demonstrate their usefulness for the description of the dynamic structure factor. All PIMC results are made freely available online and can be used as a thorough benchmark for new developments and approximations.

Classical bridge functions in classical and quantum plasma liquids

Bridge functions, the missing link in the exact description of strong correlations, are indirectly extracted from specially designed molecular dynamics simulations of classical one-component plasma liquids and accurately parameterized. Their incorporation into an advanced integral equation theory description of Yukawa one-component plasma liquids and a novel dielectric formalism scheme for quantum one-component plasma liquids lead to an unprecedented agreement with available molecular dynamics simulations and new ab initio path integral Monte Carlo simulations, respectively.

Nonlinear interaction of external perturbations in warm dense matter

AbstractWe present extensive new ab initio path integral Monte Carlo (PIMC) results for an electron gas at warm dense matter conditions that is subject to multiple harmonic perturbations. In addition to the previously investigated nonlinear effects at the original wave number (Dornheim et al., Phys. Rev. Lett., 2020, 125, 085001) and the excitation of higher harmonics (Dornheim et al., Phys. Rev. Res., 2021, 3, 033231), the presence of multiple external potentials leads to mode‐coupling effects, which constitute the dominant nonlinear effect and lead to a substantially more complicated density response compared to linear response theory. One possibility to estimate mode‐coupling effects from a PIMC simulation of the unperturbed system is given in terms of generalized imaginary‐time correlation functions that have been recently introduced by Dornheim et al. (J. Chem. Phys., 2021, 155, 054110). In addition, we extend our previous analytical theory of the nonlinear density response of the electron gas in terms of the static local field correction (Dornheim et al., 2020, Phys. Rev. Lett., 125, 235001), which allows for a highly accurate description of the PIMC results with negligible computational cost.