Path Integral Monte Carlo Research Articles

Path-integral calculation of the fourth virial coefficient of helium isotopes.

We use the path-integral Monte Carlo (PIMC) method and state-of-the-art two-body and three-body potentials to calculate the fourth virial coefficients D(T) of 4He and 3He as functions of temperature from 2.6 K to 2000 K. We derive expressions for the contributions of exchange effects due to the bosonic or fermionic nature of the helium isotope; these effects have been omitted from previous calculations. The exchange effects are relatively insignificant for 4He at the temperatures considered, but for 3He, they are necessary for quantitative accuracy below about 4 K. Our results are consistent with previous theoretical work (also with some of the limited and scattered experimental data) for 4He; for 3He, there are no experimental values, and this work provides the first values of D(T) calculated at this level. The uncertainty of the results depends on the statistical uncertainty of the PIMC calculation, the estimated effect of omitting four-body terms in the potential energy, and the uncertainty contribution propagated from the uncertainty of the potentials. At low temperatures, the uncertainty is dominated by the statistical uncertainty of the PIMC calculations, while at high temperatures, the uncertainties related to the three-body potential and omitted higher-order contributions become dominant.

Scaling advantage over path-integral Monte Carlo in quantum simulation of geometrically frustrated magnets

The promise of quantum computing lies in harnessing programmable quantum devices for practical applications such as efficient simulation of quantum materials and condensed matter systems. One important task is the simulation of geometrically frustrated magnets in which topological phenomena can emerge from competition between quantum and thermal fluctuations. Here we report on experimental observations of equilibration in such simulations, measured on up to 1440 qubits with microsecond resolution. By initializing the system in a state with topological obstruction, we observe quantum annealing (QA) equilibration timescales in excess of one microsecond. Measurements indicate a dynamical advantage in the quantum simulation compared with spatially local update dynamics of path-integral Monte Carlo (PIMC). The advantage increases with both system size and inverse temperature, exceeding a million-fold speedup over an efficient CPU implementation. PIMC is a leading classical method for such simulations, and a scaling advantage of this type was recently shown to be impossible in certain restricted settings. This is therefore an important piece of experimental evidence that PIMC does not simulate QA dynamics even for sign-problem-free Hamiltonians, and that near-term quantum devices can be used to accelerate computational tasks of practical relevance.

Attenuating the fermion sign problem in path integral Monte Carlo simulations using the Bogoliubov inequality and thermodynamic integration.

Accurate thermodynamic simulations of correlated fermions using path integral Monte Carlo (PIMC) methods are of paramount importance for many applications such as the description of ultracold atoms, electrons in quantum dots, and warm-dense matter. The main obstacle is the fermion sign problem (FSP), which leads to an exponential increase in computation time both with an increase in the system size and with a decrease in the temperature. Very recently, Hirshberg et al. [J. Chem. Phys. 152, 171102 (2020)] have proposed to alleviate the FSP based on the Bogoliubov inequality. In the present work, we extend this approach by adding a parameter that controls the perturbation, allowing for an extrapolation to the exact result. In this way, we can also use thermodynamic integration to obtain an improved estimate of the fermionic energy. As a test system, we choose electrons in 2D and 3D quantum dots and find in some cases a speed-up exceeding 106, as compared to standard PIMC, while retaining a relative accuracy of ∼0.1%. Our approach is quite general and can readily be adapted to other simulation methods.

Quantum-Mechanical Model of Neural Network Elements

A model of a neural-network-like quantum-mechanical system is considered. The role of neurons is played by quantum-mechanical particles which evolve under the action of an external double-well potential. An instanton is used as a spike. Neurons are connected by the interaction potential $${{V}_{{{\text{int}}}}}({{\hat {q}}_{i}},{{\hat {q}}_{j}})$$ . The connection is directional owing to asymmetry of the interaction potential. Therefore, the complete Hamiltonian of the system is $$\hat {H} = \sum\nolimits_i \left( {\frac{1}{2}\hat {p}_{i}^{2} + {{V}_{0}}({{{\hat {q}}}_{i}})} \right) + \sum\nolimits_{i > j} \,{{V}_{{{\text{int}}}}}({{\hat {q}}_{i}},{{\hat {q}}_{j}})$$ . The system is investigated by the path integral Monte Carlo (PIMC) method. It is shown that with a certain choice of parameters in this system it is possible to transfer activity in long chains of neurons.

Finite-size effects in the reconstruction of dynamic properties from ab initio path integral Monte Carlo simulations.

We systematically investigate finite-size effects in the dynamic structure factor S(q,ω) of the uniform electron gas obtained via the analytic continuation of ab initio path integral Monte Carlo data for the imaginary-time density-density correlation function F(q,τ). Using the recent scheme by Dornheim etal. [Phys. Rev. Lett. 121, 255001 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.255001], we find that the reconstructed spectra are not afflicted with any finite-size effects for as few as N=14 electrons both at warm dense matter (WDM) conditions and at the margins of the strongly correlated electron liquid regime. Our results further corroborate the high quality of our current description of the dynamic density response of correlated electrons, which is of high importance for many applications in WDM theory and beyond.

Dynamic properties of the warm dense electron gas based on ab initio path integral Monte Carlo simulations

There is growing interest in warm dense matter (WDM) -- an exotic state on the border between condensed matter and plasmas. Due to the simultaneous importance of quantum and correlation effects WDM is complicated to treat theoretically. A key role has been played by \textit{ab initio} path integral Monte Carlo (PIMC) simulations, and recently extensive results for thermodynamic quantities have been obtained. The first extension of PIMC simulations to the dynamic structure factor of the uniform electron gas were reported by Dornheim \textit{et al.} [Phys. Rev. Lett. \textbf{121}, 255001 (2018)]. This was based on an accurate reconstruction of the dynamic local field correction. Here we extend this concept to other dynamical quantities of the warm dense electron gas including the dynamic susceptibility, the dielectric function and the conductivity.

Nonlinear Electronic Density Response in Warm Dense Matter.

Warm dense matter (WDM)-an extreme state with high temperatures and densities that occurs, e.g., in astrophysical objects-constitutes one of the most active fields in plasma physics and materials science. These conditions can be realized in the lab by shock compression or laser excitation, and the most accurate experimental diagnostics is achieved with lasers and free electron lasers which is theoretically modeled using linear response theory. Here, we present first abinitio path integral MonteCarlo results for the nonlinear density response of correlated electrons in WDM and show that for many situations of experimental relevance nonlinear effects cannot be neglected.

Path-integral Monte Carlo simulations of quantum dipole systems in traps: Superfluidity, quantum statistics, and structural properties

We present extensive \textit{ab initio} path integral Monte Carlo (PIMC) simulations of two-dimensional quantum dipole systems in a harmonic confinement, taking into account both Bose- and Fermi-statistics. This allows us to study the nonclassical rotational inertia, which can lead to a negative superfluid fraction in the case of fermions [Phys. Rev. Lett. \textbf{112}, 235301 (2014)]. Moreover, we study in detail the structural characteristics of such systems, and are able to clearly resolve the impact of quantum statistics on density profiles and the respective shell structure. Further, we present results for a more advanced center-two particle correlation function [Phys. Rev. E \textbf{91}, 043104 (2015)], which allows to detect differences between Fermi- and Bose-systems that do not manifest in other observables like the density. Overall, we find that bosonic systems sensitively react to even small values of the dipole--dipole coupling strength, whereas such a weak interaction is effectively masked for fermions by the Pauli exclusion principle. In addition, the abnormal superfluid fraction for fermions is not reflected by the structural properties of the system, which are equal to the bosonic case even though the moments of inertia diverge from each other. Lastly, we have demonstrated that fermionic PIMC simulations of quantum dipole systems are feasible despite the notorious fermion sign problem, which opens up new avenues for future investigations in this field.

Ab initio path integral monte carlo simulation of the uniform electron gas in the high energy density regime

The response of the uniform electron gas (UEG) to an external perturbation is of paramount importance for many applications. Recently, highly accurate results for the static density response function and the corresponding local field correction have been provided both for warm dense matter [2019 J. Chem. Phys. 151 194104] and strongly coupled electron liquid [2020 Phys. Rev. B 101 045129] conditions based on exact ab initio path integral Monte Carlo (PIMC) simulations. In the present work, we further complete our current description of the UEG by exploring the high energy density regime, which is relevant for, e.g. astrophysical applications and inertial confinement fusion experiments. To this end, we present extensive new PIMC results for the static density response in the range of 0.05 ≤ rs≤ 0.5 and 0.85 ≤ θ ≤ 8. These data are subsequently used to benchmark the accuracy of the widely used random phase approximation and the dielectric theory by Singwi, Tosi, Land, and Sjölander (STLS). Moreover, we compare our results to configuration PIMC data where they are available and find perfect agreement with a relative accuracy of 0.001 − 0.01%. All PIMC data are available online.

Path integral molecular dynamics for fermions: Alleviating the sign problem with the Bogoliubov inequality.

We present a method for performing path integral molecular dynamics (PIMD) simulations for fermions and address its sign problem. PIMD simulations are widely used for studying many-body quantum systems at thermal equilibrium. However, they assume that the particles are distinguishable and neglect bosonic and fermionic exchange effects. Interacting fermions play a key role in many chemical and physical systems, such as electrons in quantum dots and ultracold trapped atoms. A direct sampling of the fermionic partition function is impossible using PIMD since its integrand is not positive definite. We show that PIMD simulations for fermions are feasible by employing our recently developed method for bosonic PIMD and reweighting the results to obtain fermionic expectation values. The approach is tested against path integral Monte Carlo (PIMC) simulations for up to seven electrons in a two-dimensional quantum dot for a range of interaction strengths. However, like PIMC, the method suffers from the sign problem at low temperatures. We propose a simple approach for alleviating it by simulating an auxiliary system with a larger average sign and obtaining an upper bound to the energy of the original system using the Bogoliubov inequality. This allows fermions to be studied at temperatures lower than would otherwise have been feasible using PIMD, as demonstrated in the case of a three-electron quantum dot. Our results extend the boundaries of PIMD simulations of fermions and will hopefully stimulate the development of new approaches for tackling the sign problem.

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.

Quantum Monte Carlo study of strongly interacting bosonic one-dimensional systems in periodic potentials

We present diffusion Monte Carlo (DMC) and path-integral Monte Carlo (PIMC) calculations of a one-dimensional Bose system with realistic interparticle interactions in a periodic external potential. Our main aim is to test the predictions of the Luttinger liquid (LL) theory, in particular with respect to the superfluid-Mott insulator transition at both zero and finite temperatures, in the predicted robust and fragile superfluid regimes. For that purpose, we present our results of the superfluid fraction $\rho_s/\rho_0$, the one-body density matrix, the two-body correlation functions, and the static structure factor. The DMC and PIMC results in the limit of very low temperature for $\rho_s/\rho_0$ agree, but the LL model for scaling $\rho_s/\rho_0$ does not fit the data well. The critical depth of the periodic potential is close to the values obtained for ultracold gases with different models of interaction, but with the same value of the bare LL parameter, demonstrating the universality of LL description. Algebraic decay of correlation functions is observed in the superfluid regime and exponential decay in the Mott-insulator one, as well as in all regimes at finite temperature for distances larger than a characteristic length.

Path integral Monte Carlo and density functional molecular dynamics simulations of warm dense MgSiO3

In order to provide a comprehensive theoretical description of MgSiO$_3$ at extreme conditions, we combine results from path integral Monte Carlo (PIMC) and density functional molecular dynamics simulations (DFT-MD) and generate a consistent equation of state for this material. We consider a wide range of temperature and density conditions from 10$^4$ to 10$^8$ K and from 0.321 to 64.2 g$\,$cm$^{-3}$ (0.1- to 20-fold the ambient density). We study how the L and K shell electrons are ionized with increasing temperature and pressure. We derive the shock Hugoniot curve and compare with experimental results. Our Hugoniot curve is in good agreement with the experiments, and we predict a broad compression maximum that is dominated by the K shell ionization of all three nuclei while the peak compression ratio of 4.70 is obtained when the Si and Mg nuclei are ionized. Finally we analyze the heat capacity and structural properties of the liquid.

Lefschetz Thimbles and Wigner Functions

The major difficulty for the Feynman Path Integral Monte Carlo (PIMC) simulations of the quantum systems of particles is the so called “sign problem”, arising due to the fast oscillations of the path integral integrand depending on the complex-valued action. Our aim is to find universal techniques being able to solve this problem. The new method combines the basic ideas of the Metropolis and Hasting algorithms and is based on the Picard-Lefschetz theory and complex-valued version of Morse theory. The basic idea is to choose the Lefschetz thimbles as manifolds approaching the saddle point of the integrand. On this thimble the imaginary part of the complex-valued action remains constant. As a result the integrand on each thimble does not oscillate, so the “sign problem” disappears and the integral can be calculated much more effectively. The developed approach allows also finding saddle points in the complexified space of path integral integration. Some simple test calculations and comparisons with available analytical results have been carried out.

Equation of state and first principles prediction of the vibrational matrix shift of solid parahydrogen.

We generate the equation of state (EOS) of solid parahydrogen (para-H2) using a path-integral Monte Carlo (PIMC) simulation based on a highly accurate first-principles adiabatic hindered rotor potential energy curve for the para-H2 dimer. The EOS curves for the fcc and hcp structures of solid para-H2 near the equilibrium density show that the hcp structure is the more stable of the two, in agreement with experiment. To accurately reproduce the structural and energy properties of solid para-H2, we eliminated by extrapolation the systematic errors associated with the choice of simulation parameters used in the PIMC calculation. We also investigate the temperature dependence of the EOS curves, and the invariance of the equilibrium density with temperature is satisfyingly reproduced. The pressure as a function of density and the compressibility as a function of pressure are both calculated using the obtained EOS and are compared with previous simulation results and experiments. We also report the first ever a priori prediction of a vibrational matrix shift from first-principles two-body potential functions, and its result for the equilibrium state agrees well with experiment.

Berezinskii-Kosterlitz-Thouless transition in two-dimensional dipolar stripes

A two-dimensional quantum system of dipoles, with a polarization angle not perpendicular to the plane, shows a transition from a gas to a stripe phase. We have studied the thermal properties of these two phases using the path integral Monte Carlo (PIMC) method. By simulating the thermal density matrix, PIMC provides exact results for magnitudes of interest such as the superfluid fraction and the one-body density matrix. As it is well known, in two dimensions the superfluid-to-normal phase transition follows the Berezinskii-Kosterlitz-Thouless (BKT) scenario. Our results show that both the anisotropic gas and the stripe phases follow the BKT scaling laws. At fixed density and increasing the tilting angle, the transition temperature decreases in going from the gas to the stripe phase. Superfluidity in the perpendicular direction to the stripes is rather small close to the critical temperature but it becomes larger at lower temperatures, mainly close to the transition to the gas. Our results are in qualitative agreement with the supersolidity observed recently in a quasi-one-dimensional array of dipolar droplets.

Magnesium oxide at extreme temperatures and pressures studied with first-principles simulations.

We combine two first-principles computer simulation techniques, path integral Monte Carlo and density functional theory molecular dynamics, to determine the equation of state of magnesium oxide in the regime of warm dense matter, with densities ranging from 0.35 to 71 g cm-3 and temperatures ranging from 10 000 K to 5 × 108 K. These conditions are relevant for the interiors of giant planets and stars as well as for shock wave compression measurements and inertial confinement fusion experiments. We study the electronic structure of MgO and the ionization mechanisms as a function of density and temperature. We show that the L-shell orbitals of magnesium and oxygen hybridize at high density. This results in a gradual ionization of the L-shell with increasing density and temperature. In this regard, MgO behaves differently from pure oxygen, which is reflected in the shape of the MgO principal shock Hugoniot curve. The curve of oxygen shows two compression maxima, while that of MgO shows only one. We predict a maximum compression ratio of 4.66 to occur for a temperature of 6.73 × 107 K. Finally, we study how multiple shocks and ramp waves can be used to cover a large range of densities and temperatures.

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.

Monte Carlo simulation of cosmologies with dust

The quantum theory of the Friedmann cosmological model with dust and cosmological constant () is not exactly solvable analytically. We apply path integral Monte Carlo (PIMC) techniques to study its quantum dynamics using the physical Hamiltonian corresponding to the dust field as a clock. We study (i) quantum fluctuations around classical paths and (ii) formulate the analogues of the no-boundary and tunnelling proposals and simulate the ground state wave functions. For a unique ground state wave function exists. For the physical Hamiltonian is not bounded below, but the path integral for the propagator is convergent over a range of Euclidean time T. We investigate the properties of the convergent propagator. The path integral can be made convergent for all values of T by restricting the integral over paths with action greater than equal to zero. We explore the consequences of such a choice.

Simplification of the electron-ion many-body problem: N -representability of pair densities obtained via a classical map for the electrons

The classical map hypernetted-chain (CHNC) method for interacting electrons uses a kinetic energy functional in the form of a classical-fluid temperature. Here, we show that the CHNC generated two-body densities and pair-distribution functions (PDFs) correspond to $N$-representable densities. Comparisons of results from CHNC with quantum Monte Carlo (QMC) and path-integral Monte Carlo (PIMC) are used to validate the CHNC results. Since the PDFs are sufficient to obtain the equation of state or linear response properties of electron-ion systems, we apply the CHNC method for fully classical calculations of electron-ion systems in the quantum regime, using hydrogen at 4000 K and 350 times the solid density as an example since QMC comparisons are available. We also present neutral pseudoatom (NPA) calculations which use rigorous density functional theory (DFT) to reduce the many nuclear problem to an effective one-ion problem. The CHNC PDFs and NPA results agree well with the ion-ion, electron-ion, and electron-electron PDFs from QMC, PIMC, or DFT coupled to molecular dynamics simulations where available. The PDFs of a two-dimensional (2D) electron-hole system at 5 K are given as an example of 2D ``warm dense'' matter where the electrons and the counterparticles (holes) are all in the quantum regime. Basic methods such as QMC, PIMC, or even DFT become prohibitive while CHNC methods, being independent of the number of particles or the temperature, prove to be easily deployable.