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Register a new userNonlinear density response from imaginary-time correlation functions: Ab initio path integral Monte Carlo simulations of the warm dense electron gas
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The emph{ab initio} path integral Monte Carlo (PIMC) approach is one of the most successful methods in quantum many-body theory. A particular strength of this method is its straightforward access to imaginary-time correlation functions (ITCF). For example, the well-known density-density ITCF $F(mathbf{q},tau)$ allows one to estimate the linear response of a given system for all wave vectors $mathbf{q}$ from a single simulation of the unperturbed system. Moreover, it constitutes the basis for the reconstruction of the dynamic structure factor $S(mathbf{q},omega)$ -- a key quantity in state-of-the-art scattering experiments. In this work, we present analogous relations between the nonlinear density response in quadratic and cubic order of the perturbation strength and generalized ITCFs measuring correlations between up to four imaginary-time arguments. As a practical demonstration of our new approach, we carry out simulations of the warm dense electron gas and find excellent agreement with previous PIMC results that had been obtained with substantially larger computational effort. In addition, we give a relation between a cubic ITCF and the triple dynamic structure factor $S(mathbf{q}_1,omega_1;mathbf{q}_2,omega_2)$, which evokes the enticing possibility to study dynamic three-body effects on an emph{ab initio} level.
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The accurate description of electrons at extreme density and temperature is of paramount importance for, e.g., the understanding of astrophysical objects and inertial confinement fusion. In this context, the dynamic structure factor $S(mathbf{q},omega)$ constitutes a key quantity as it is directly measured in X-ray Thomson (XRTS) scattering experiments and governs transport properties like the dynamic conductivity. In this work, we present the first textit{ab initio} results for $S(mathbf{q},omega)$ by carrying out extensive path integral Monte Carlo simulations and developing a new method for the required analytic continuation, which is based on the stochastic sampling of the dynamic local field correction $G(mathbf{q},omega)$. In addition, we find that the so-called static approximation constitutes a promising opportunity to obtain high-quality data for $S(mathbf{q},omega)$ over substantial parts of the warm dense matter regime.
Fractional derivatives are nonlocal differential operators of real order that often appear in models of anomalous diffusion and a variety of nonlocal phenomena. Recently, a version of the Schrodinger Equation containing a fractional Laplacian has been proposed. In this work, we develop a Fractional Path Integral Monte Carlo algorithm that can be used to study the finite temperature behavior of the time-independent Fractional Schrodinger Equation for a variety of potentials. In so doing, we derive an analytic form for the finite temperature fractional free particle density matrix and demonstrate how it can be sampled to acquire new sets of particle positions. We employ this algorithm to simulate both the free particle and $^{4}$He (Aziz) Hamiltonians. We find that the fractional Laplacian strongly encourages particle delocalization, even in the presence of interactions, suggesting that fractional Hamiltonians may manifest atypical forms of condensation. Our work opens the door to studying fractional Hamiltonians with arbitrarily complex potentials that escape analytical solutions.
The problem of calculating real-time correlation functions is formulated in terms of an imaginary-time partial differential equation. The latter is solved analytically for the perturbed harmonic oscillator and compared with the known exact result. The first order approximation for the short-time propagator is derived and used for numerical solution of the equation by a Monte Carlo integration. In general, the method provides a reformulation of the dynamic sign problem, and is applicable to any two-time correlation function including single-particle, density-density, current-current, spin-spin, and others. The prospects of extending the technique onto multi-dimensional problems are discussed.
Warm dense matter (WDM) -- an exotic state of highly compressed matter -- has attracted high 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 textit{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 textit{et al.}, Phys. Plasmas textbf{24}, 056303 (2017) and Phys. Rep. textbf{744}, 1-86 (2018). In the present article we review recent further progress in QMC simulations of the warm dense UEG: namely, textit{ab initio} results for the static local field correction $G(q)$ and for the dynamic structure factor $S(q,omega)$. These data are of key relevance for the 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 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 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.
In a classical plasma the momentum distribution, $n(k)$, decays exponentially, for large $k$, and the same is observed for an ideal Fermi gas. However, when quantum and correlation effects are relevant simultaneously, an algebraic decay, $n_infty(k)sim k^{-8}$ has been predicted. This is of relevance for cross sections and threshold processes in dense plasmas that depend on the number of energetic particles. Here we present extensive textit{ab initio} results for the momentum distribution of the nonideal uniform electron gas at warm dense matter conditions. Our results are based on first principle fermionic path integral Monte Carlo (CPIMC) simulations and clearly confirm the $k^{-8}$ asymptotic. This asymptotic behavior is directly linked to short-range correlations which are analyzed via the on-top pair distribution function (on-top PDF), i.e. the PDF of electrons with opposite spin. We present extensive results for the density and temperature dependence of the on-top PDF and for the momentum distribution in the entire momentum range.
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