Static Local Field Correction Research Articles

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.

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.

Integral equation theory based dielectric scheme for strongly coupled electron liquids.

In a recent paper, Lucco Castello et al. (arXiv:2107.03537) provided an accurate parameterization of classical one-component plasma bridge functions that was embedded in a novel dielectric scheme for strongly coupled electron liquids. Here, this approach is rigorously formulated, its set of equations is formally derived, and its numerical algorithm is scrutinized. A systematic comparison with available and new path integral Monte Carlo simulations reveals a rather unprecedented agreement especially in terms of the interaction energy and the long wavelength limit of the static local field correction.

Local field correction to ionization potential depression of ions in warm or hot dense matter.

An analytical self-consistent approach was recently established to predict the ionization potential depression (IPD) in multicomponent dense plasmas, which is achieved by considering the self-energy of ions and electrons within the quantum statistical theory. In order to explicitly account for the exchange-correlation effect of electrons, we incorporate the effective static approximation of local field correction (LFC) within our IPD framework through the connection of dynamical structure factor. The effective static approximation poses an accurate description for the asymptotic large wave number behavior with the recently developed machine learning representation of static LFC induced from the path-integral Monte Carlo data. Our calculation shows that the introduction of static LFC through dynamical structure factor brings a nontrivial influence on IPD at warm/hot dense matter conditions. The correlation effect within static LFC could provide up to 20% correction to free-electron contribution of IPD in the strong coupling and degeneracy regime. Furthermore, a new screening factor is obtained from the density distribution of free electrons calculated within the average-atom model, with which excellent agreements are observed with other methods and experiments at warm/hot dense matter conditions.

Dust-Acoustic Wave Dispersion in Thermal Dusty Plasmas at Weak and Moderate Couplings

The dispersion of dust-acoustic waves (DAWs) in weakly and moderately coupled thermal dusty plasmas is studied in the framework of the linear density-response formalism with the static local-field correction for interdust interactions. The plasma medium composition and the charge of dust particles are simultaneously determined within a recently developed chemical model (Physical Review E, vol. 101, 063203, 2020) based on minimizing the Helmholtz free energy of the system under investigation. Stemming from the generalized Poisson-Boltzmann equation, the renormalization procedure is consistently applied to derive an interdust screened potential that takes into account the finiteness of dust grains. Within the framework of the Ornstein-Zernike relationship in the hypernetted chain approximation, the static structure factor of the dust component is evaluated to manifest the appearance of local extrema on its wavenumber dependence, thereby indicating the short-range order formation in the arrangement of dust particles with respect to one another. It is shown that the DAW dispersion law is completely governed by the static structure factor and, therefore, exhibits a nonmonotonic dependence on the wavenumber as well. In the long-wavelength limit, the acoustic-like behavior of the DAW dispersion is strictly confirmed and the corresponding phase speed, reduced in units of the dust thermal velocity, is ultimately expressed via the static structure factor at zero wavenumber.

Analytical representation of the local field correction of the uniform electron gas within the effective static approximation

The description of electronic exchange--correlation effects is of paramount importance for many applications in physics, chemistry, and beyond. In a recent Letter, Dornheim \textit{et al.} [\textit{Phys. Rev. Lett.}~\textbf{125}, 235001 (2020)] have presented the \emph{effective static approximation} (ESA) to the local field correction (LFC), which allows for the highly accurate estimation of electronic properties such as the interaction energy and the static structure factor. In the present work, we give an analytical parametrization of the LFC within ESA that is valid for any wave number, and available for the entire range of densities ($0.7\leq r_s \leq20$) and temperatures ($0\leq \theta\leq 4$) that are relevant for applications both in the ground state and in the warm dense matter regime. A short implementation in Python is provided, which can easily be incorporated into existing codes. In addition, we present an extensive analysis of the performance of ESA regarding the estimation of various quantities like the dynamic structure factor, static dielectric function, the electronically screened ion-potential, and also stopping power in electronic medium. In summary, we find that the ESA gives an excellent description of all these quantities in the warm dense matter regime, and only becomes inaccurate when the electrons start to form a strongly correlated electron liquid ($r_s\sim20$). Moreover, we note that the exact incorporation of exact asymptotic limits often leads to a superior accuracy compared to the neural-net representation of the static LFC [\textit{J.~Chem.~Phys.}~\textbf{151}, 194104 (2019)].

Study of coulomb drag with the effect of local field correction and dielectric medium

We present a numerical calculation for the drag resistivity of charge carriers interaction in coupled layer system separated by a barrier. Analytical expressions of non-linear susceptibility function and effective interlayer Coulomb interaction have been shown for non homogeneous dielectric environment at low temperature and low frequency regime. The system is studied within the model of random phase approximation (RPA) and taking into account the static local field correction (LFC). We employ Hubbard approximations (HA) and self consistent STLS approximations based LFC for obtaining the results, where LFC includes the exchange and correlation effects. Exchange and correlation effects enhanced the drag resistivity and the net screening. Another important quantity is the local form factors (LFF), which is the function of the bare inter- and intra-layer potential of the dielectric media. LFF for a multi-layer system is derived from the Poisson equation solution.

Description of longitudinal modes in moderately coupled Yukawa systems with the static local field correction

In moderately coupled Yukawa fluids, longitudinal mode dispersion is determined by the competition between kinetic and potential effects. In a recent paper [S. Khrapak and L. Couëdel, Phys. Rev. E 102, 033207 (2020)], a semi-phenomenological dispersion relation was constructed by the ad hoc addition of the Bohm–Gross kinetic term to the generalized instantaneous excess bulk modulus, which showed very good agreement with simulations. In this paper, a nearly identical dispersion relation is derived in a rigorous manner based on a dielectric formulation with static local field corrections. At moderate coupling, this formalism is revealed to be more accurate than other successful theoretical approaches.

Ionized impurity scattering in quantum wells and quantum wires

Low-field mobility for scattering of electrons by ionized impurities is calculated in quantum wells and quantum wires. Analytical expressions for mobility are obtained on the basis of the quantum-kinetic approach using the quantum mechanical formalism of the dielectric function for non-interacting and interacting electrons. If the kinetic energy of electrons is much higher than their inter-particle potential energy, the electron system remains qualitatively similar to a non-interacting electron gas. The Lindhard dielectric function is used to calculate the mobility and screening factor of non-interacting electrons. If the potential energy of electrons is close to or prevails over their kinetic energy, the electron system displays a collective behavior similar to an electron liquid. In this case, electrons are considered interacting. To calculate the mobility of the interacting electrons, we have used the static local-field correction factor to the electron dielectric function. Analytical dependences of the electron mobility on dimensionality of the electron system, sizes of the quantum structures in the confining directions, temperature, forms of confining potential, electron density, and material parameters are obtained. The quantum confinement is modeled by triangular and rectangular confining potentials. The results of calculations are compared with the known experimental data.

Coulomb drag study in graphene/GaAs bilayer system with the effect of local field correction and dielectric medium

The drag resistivity is measured numerically for electron-electron (e-e) and electron-hole (e-h) coupled layer system composed of single layer graphene and two-dimensional (2D) GaAs layer separated by a barrier. The system is studied within the model of random phase approximation (RPA) and taking into account the static local field correction (LFC) and layer thickness effects in the Ballistic/Boltzmann regime. We have shown analytical expressions of non-linear susceptibility function and effective interlayer Coulomb interaction for non homogeneous dielectric environment at low temperature, high density and large interlayer separation limit. Exchange and correlation effects enhanced the drag resistivity by considering the static local field correction. Width of the layer and interlayer distance dependent local form factors (LFF) are obtained from the solution of the Poisson equation for a multi-layer dielectric medium. The effects of LFC and LFF are the function of bare inter- and intra-layer potential, which enhances the drag resistivity compare to simply measured RPA results. Enhanced drag resistivity also measured for e-h bilayer system where the hole layer is considered for 2D GaAs (drag layer), cause of larger effective mass of hole compare to the electron. Similarly, e-e and e-h bilayer system measured the enhanced drag resistivity due to LFC effects.

Ion energy-loss characteristics and friction in a free-electron gas at warm dense matter and nonideal dense plasma conditions.

We investigate the energy-loss characteristics of an ion in warm dense matter (WDM) and dense plasmas concentrating on the influence of electronic correlations. The basis for our analysis is a recently developed ab initio quantum Monte Carlo- (QMC) based machine learning representation of the static local field correction (LFC) [Dornheim et al., J. Chem. Phys. 151, 194104 (2019)JCPSA60021-960610.1063/1.5123013], which provides an accurate description of the dynamical density response function of the electron gas at the considered parameters. We focus on the polarization-induced stopping power due to free electrons, the friction function, and the straggling rate. In addition, we compute the friction coefficient which constitutes a key quantity for the adequate Langevin dynamics simulation of ions. Considering typical experimental WDM parameters with partially degenerate electrons, we find that the friction coefficient is of the order of γ/ω_{pi}=0.01, where ω_{pi} is the ionic plasma frequency. This analysis is performed by comparing QMC-based data to results from the random-phase approximation (RPA), the Mermin dielectric function, and the Singwi-Tosi-Land-Sjölander (STLS) approximation. It is revealed that the widely used relaxation time approximation (Mermin dielectric function) has severe limitations regarding the description of the energy loss of ions in a correlated partially degenerate electrons gas. Moreover, by comparing QMC-based data with the results obtained using STLS, we find that the ion energy-loss properties are not sensitive to the inaccuracy of the static local field correction (LFC) at large wave numbers, k/k_{F}>2 (with k_{F} being the Fermi wave number), but that a correct description of the static LFC at k/k_{F}≲1.5 is important.

Ab initio simulation of warm dense matter

Warm dense matter (WDM)—an exotic state of highly compressed matter—has attracted increased interest in recent years in astrophysics and for dense laboratory systems. At the same time, this state is extremely difficult to treat theoretically. This is due to the simultaneous appearance of quantum degeneracy, Coulomb correlations, and thermal effects, as well as the overlap of plasma and condensed phases. Recent breakthroughs are due to the successful application of density functional theory (DFT) methods which, however, often lack the necessary accuracy and predictive capability for WDM applications. The situation has changed with the availability of the first ab initio data for the exchange–correlation free energy of the warm dense uniform electron gas (UEG) that were obtained by quantum Monte Carlo (QMC) simulations; for recent reviews, see Dornheim et al., Phys. Plasmas 24, 056303 (2017) and Phys. Rep. 744, 1–86 (2018). In the present article, we review recent further progress in QMC simulations of the warm dense UEG: namely, ab initio results for the static local field correction G(q) and for the dynamic structure factor S(q,ω). These data are of key relevance for comparison with x-ray scattering experiments at free electron laser facilities and for the improvement of theoretical models. In the second part of this paper, we discuss the simulations of WDM out of equilibrium. The theoretical approaches include Born-Oppenheimer molecular dynamics, quantum kinetic theory, time-dependent DFT, and hydrodynamics. Here, we analyze the strengths and limitations of these methods and argue that progress in WDM simulations will require a suitable combination of all methods. A particular role might be played by quantum hydrodynamics, and we concentrate on problems, recent progress, and possible improvements of this method.

Strongly coupled electron liquid: Ab initio path integral Monte Carlo simulations and dielectric theories

The strongly coupled electron liquid provides a unique opportunity to study the complex interplay of strong coupling with quantum degeneracy effects and thermal excitations. To this end, we carry out extensive \textit{ab initio} path integral Monte Carlo (PIMC) simulations to compute the static structure factor, interaction energy, density response function, and the corresponding static local field correction in the range of $20\leq r_s \leq 100$ and $0.5\leq \theta\leq 4$. We subsequently compare these data to several dielectric approximations, and find that different schemes are capable to reproduce different features of the PIMC results at certain parameters. Moreover, we provide a comprehensive data table of interaction energies and compare those to two recent parametrizations of the exchange-correlation free energy, where they are available. Finally, we briefly touch upon the possibility of a charge-density wave. The present study is complementary to previous investigations of the uniform electron gas in the warm dense matter regime and, thus, further completes our current picture of this fundamental model system at finite temperature. All PIMC data are available online.

The static local field correction of the warm dense electron gas: An ab initio path integral Monte Carlo study and machine learning representation.

The study of matter at extreme densities and temperatures as they occur in astrophysical objects and state-of-the-art experiments with high-intensity lasers is of high current interest for many applications. While no overarching theory for this regime exists, accurate data for the density response of correlated electrons to an external perturbation are of paramount importance. In this context, the key quantity is given by the local field correction (LFC), which provides a wave-vector resolved description of exchange-correlation effects. In this work, we present extensive new path integral Monte Carlo (PIMC) results for the static LFC of the uniform electron gas, which are subsequently used to train a fully connected deep neural network. This allows us to present a representation of the LFC with respect to continuous wave-vectors, densities, and temperatures covering the entire warm dense matter regime. Both the PIMC data and neural-net results are available online. Moreover, we expect the presented combination of ab initio calculations with machine-learning methods to be a promising strategy for many applications.

Ab initio path integral Monte Carlo approach to the static and dynamic density response of the uniform electron gas

In a recent Letter [T. Dornheim et al., Phys. Rev. Lett. 121, 255001 (2018)] we have presented ab initio results for the dynamic structure factor $S(\mathbf{q},\ensuremath{\omega})$ of the uniform electron gas for conditions ranging from the warm dense matter regime to the strongly correlated electron liquid. This was achieved on the basis of exact path integral Monte Carlo data by stochastically sampling the dynamic local field correction $G(\mathbf{q},\ensuremath{\omega})$. In this paper we introduce in detail this reconstruction method and provide several practical demonstrations. Moreover, we thoroughly investigate the associated imaginary-time density-density correlation function $F(\mathbf{q},\ensuremath{\tau})$. The latter also gives us access to the static density-response function $\ensuremath{\chi}(\mathbf{q})$ and static local field correction $G(\mathbf{q})$, which are compared to standard dielectric theories like the widespread random phase approximation. In addition, we study the high-frequency limit of $G(\mathbf{q},\ensuremath{\omega})$ and provide extensive results for the dynamic structure factor for different densities and temperatures. Finally, we discuss the implications of our findings for warm dense matter research and the interpretation of experiments.

Permutation-blocking path-integral Monte Carlo approach to the static density response of the warm dense electron gas.

The static density response of the uniform electron gas is of fundamental importance for numerous applications. Here we employ the recently developed ab initio permutation blocking path integral Monte Carlo (PB-PIMC) technique [T. Dornheim et al., New J. Phys. 17, 073017 (2015)10.1088/1367-2630/17/7/073017] to carry out extensive simulations of the harmonically perturbed electron gas at warm dense matter conditions. In particular, we investigate in detail the validity of linear response theory and demonstrate that PB-PIMC allows us to obtain highly accurate results for the static density response function and, thus, the static local field correction. A comparison with dielectric approximations to our new ab initio data reveals the need for an exact treatment of correlations. Finally, we consider a superposition of multiple perturbations and discuss the implications for the calculation of the static response function.

Elastic Properties of Zr<sub>50</sub>Cu<sub>43</sub>Ag<sub>7</sub> Bulk Metallic Glass Using Pseudopotential Theory

Using Hubbard-Beeby approach for phonon dynamics, in conjunction with our recently proposed model pseudopotential; phonon frequencies for longitudinal and transverse modes are computed and associated elastic properties of technologically important Zr-based Zr50Cu43Ag7 bulk metallic glass (BMG) are evaluated. Five different forms of the static local field correction functions, viz., Hartree etal. (H), Taylor et al. (T), Ichimaru and Utsumi et al. (IU), Farid et al. (F) and Sarkar et al. (S) are employed to investigate the influence of the screening effect on the vibrational dynamics of Zr50Cu43Ag7 BMG. Results for bulk modulus, modulus of rigidity, Poisson's ratio, Young modulus, propagation velocity of elastic waves and dispersion curves are studied. The theoretical computations are found to be in good agreement with the available experimental results, which confirms the use of our model pseudopotential to study elastic properties of such a glassy system.

Effect of pressure on some physical properties of gallium based semiconductors

The gallium based semiconductor compounds are very useful materials for optical spectroscopy and optoelectronic applications, we have studied the effect of pressure on various physical properties like total energy, static bulk modulus, energy band gap at the point X on the Jones-zone face, pressure derivative of bulk modulus and equation of state of gallium based binary compounds GaSb, GaAs, GaP and GaN using pseudopotential theory beyond second order with our well established single parametric model potential. We have incorporated Nagy's static local field correction function to include exchange and correlation effects. The results are compared with those obtained using few other local field correction functions. The present results agree satisfactorily with available experimental and other such theoretical data confirming the application.

Spin-polarized electron–electron quantum bilayers: Finite width effect

We study the effect of finite width on the ground-state of a spin-polarized electron–electron quantum bilayers (EEBL) system at temperature T=0. Correlations among carriers are treated beyond the static mean-field theories by using the quantum or dynamical version of Singwi, Tosi, Land and Sjölander (qSTLS) theory. Numerical results are presented for the pair-correlation function, the ground-state energy, the static density susceptibility, and the static local-field correction factor as a function of density parameter r sl and interlayer spacing d. Interestingly, we find that the inclusion of finite width lowered the critical density, for the onset of Wigner crystal (WC) ground-state, as compared to the similar recent study of spin-polarized EEBL system without finite width effect. Further, spin-polarization effect is seen to introduce a marked change in the ground-state energy of the EEBL system as compared to the results of unpolarized EEBL system with finite width. Results of ground-state energies are also compared with the recent diffusion Monte Carlo (DMC) and variational Monte Carlo (VMC) simulation studies of spin-polarized EEBL system with zero width.

Ground-State Properties of Electron-Electron Quantum Bilayers

The ground-state behavior of electron-electron quantum bilayer systems within the neutralizing background and zero magnetic field is studied by including unequal density of layers. The quantum self-consistent mean-field approximation of Singwi, Tosi, Land and Sjolander (qSTLS) is used to study the intra- and inter layer properties, such as the static pair-correlation functions, the static structure factors, the static density susceptibility and the static local-field correction factors over a wide range of layer density parameter rsι and layer spacing d. We find that the inclusion of unequal density brings a phase transition from charge density wave (CDW) to coupled Wigner crystal (WC) ground state at the close proximity of the layers. We also find that inclusion of unequal density markedly reduce critical rsι for the onset of Wigner crystallization. The results are compared with recent findings of the equal density effects in electron-electron quantum bilayer systems.