Path Integral Monte Carlo Approach Research Articles

The quantum density of states and distribution functions of the helium-3: Wigner approach in path integral Monte Carlo simulations

A new path integral representation of the quantum density of states (DOS) and distribution functions of strongly correlated fermions are derived in the Wigner formulation of quantum mechanics. A new path integral Monte Carlo approach to calculate DOS and thermodynamic functions is suggested. Using 3 He as an interesting example, we calculate the DOS, internal energy distribution (IED), momentum distribution functions (MDFs), spin-resolved radial distribution functions at different densities and temperatures. The physical meaning and parameters of exchange-correlation holes, the quantum oscillations of IEDs and DOS as well as the high-momentum asymptotics (‘quantum tails’) of MDFs have been considered and explained.

Phase-space path-integral representation of the quantum density of states: Monte Carlo simulation of strongly correlated soft-sphere fermions.

The Wigner formulation of quantum mechanics is used to derive a path-integral representation of the quantum density of states (DOS) of strongly correlated fermions in the canonical ensemble. A path-integral Monte Carlo approach for the simulation of DOS and other thermodynamic functions is suggested. The derived Wigner function in the phase space resembles the Maxwell-Boltzmann distribution but allows for quantum effects. We consider a three-dimensional quantum system of strongly correlated soft-sphere fermions at different densities and temperatures. The calculated properties include the DOS, momentum distribution functions, spin-resolved radial distribution functions, potentials of mean force, and related energy levels obtained from the Bohr-Sommerfeld condition. We observe sharp peaks on DOS and momentum distribution curves, which are explained by the appearance of fermionic bound states.

Density of states of a 2D system of soft-sphere fermions by path integral Monte Carlo simulations

The Wigner formulation of quantum mechanics is used to derive a new path integral representation of quantum density of states (DOSs). A path integral Monte Carlo approach is developed for the numerical investigation of DOSs, internal energy and spin-resolved radial distribution functions for a 2D system of strongly correlated soft-sphere fermions. The peculiarities of the DOSs and internal energy distributions depending on the hardness of the soft-sphere potential and particle density are investigated and explained. In particular, at high enough densities the DOSs rapidly tends to a constant value, as for an ideal system of 2D fermions.

A wide-range semiclassical self-consistent average atom model

The discovery of material properties at extremes, which are essential for high energy density physics development, requires the most advanced experimental facilities, theories, and computations. Nowadays, it is possible to model properties of matter in such conditions using the state-of-the-art density functional theory (DFT) or path-integral Monte Carlo approaches with remarkable precision. However, fundamental and computational limitations of these methods impede their practical usage, while wide-range thermodynamic and transport models of plasma are required. As a consequence, an average atom (AA) framework is still relevant today and has been attracting more and more attention lately. The self-consistent field and electron density in an atomic cell is usually obtained using the Thomas–Fermi (TF), Hartree–Fock, Kohn–Sham approaches, or their extensions. In this study, we present the AA model, where semiclassical wave functions are used for bound states, while free electrons are approximated by the TF model with a thermodynamically consistent energy boundary. The model is compared in various regions of temperatures and pressures with the reference data: the Saha model for rarefied plasma, DFT for warm dense matter, and experimental shock Hugoniot data. It is demonstrated that a single AA model may provide a reasonable agreement with the established techniques at low computational cost and with stable convergence of the self-consistent field.

Solution of the ‘sign problem’ in the path integral Monte Carlo simulations of strongly correlated Fermi systems: thermodynamic properties of helium-3

A quantum path integral Monte Carlo approach [J. Phys. A 55, 035001 (2021)] for strongly correlated degenerate fermions has been modified to solve the ‘fermionic sign problem’. The exchange interaction of fermions was presented in the form of a positive semidefinite Gram determinant which significantly simplified the calculations. The verification of this approach has been carried out by comparison of the calculated equation of state with experimental data for quantum and degenerate unpolarised . The obtained results agree with experimental data in a wide range of reduced inverse density and temperature . The range includes the gaseous and liquid states as well as the regions of the respective phase transition and the critical point. For the first time spin-resolved pair distribution functions have been calculated to analyse the spatial arrangement of helium atoms. The calculated internal energy has been presented for the gaseous region of quantum . The simulations have been carried out for the Lennard–Jones (LJ) potential with parameters Å and K.

The density–temperature range of exchange–correlation exciton existence by the fermionic path integral Monte Carlo method

A recently developed fermionic path integral Monte Carlo approach [Filinov et al., Phys. Rev. E 102, 033203 (2020) and Filinov et al., J. Phys. A 55, 035001 (2021)] has been applied for the estimation of the density–temperature range of exchange–correlation exciton existence in a strongly coupled degenerate uniform electron gas. The approach allows us to reduce the “fermionic sign problem” taking into account the interference effects of the Coulomb and exchange interaction of electrons in the basic Monte Carlo cell and its periodic images. Our results for radial distribution functions demonstrate the formation and decay of a short-range quantum ordering of electrons associated with exchange–correlation excitons in the literature. Such excitons have never been observed earlier in standard path integral Monte Carlo simulations.

Path integral Monte Carlo approach to the structural properties and collective excitations of liquid ^3{text {He}} without fixed nodes

Due to its nature as a strongly correlated quantum liquid, ultracold helium is characterized by the nontrivial interplay of different physical effects. Bosonic ^4{text {He}} exhibits superfluidity and Bose-Einstein condensation. Its physical properties have been accurately determined on the basis of ab initio path integral Monte Carlo (PIMC) simulations. In contrast, the corresponding theoretical description of fermionic ^3{text {He}} is severely hampered by the notorious fermion sign problem, and previous PIMC results have been derived by introducing the uncontrolled fixed-node approximation. In this work, we present extensive new PIMC simulations of normal liquid ^3{text {He}} without any nodal constraints. This allows us to to unambiguously quantify the impact of Fermi statistics and to study the effects of temperature on different physical properties like the static structure factor S({mathbf {q}}), the momentum distribution n({mathbf {q}}), and the static density response function chi ({mathbf {q}}). In addition, the dynamic structure factor S({mathbf {q}},omega ) is rigorously reconstructed from imaginary-time PIMC data. From simulations of ^3{text {He}}, we derived the familiar phonon–maxon–roton dispersion function that is well-known for ^4{text {He}} and has been reported previously for two-dimensional ^3{text {He}} films (Nature 483:576–579 (2012)). The comparison of our new results for both S({mathbf {q}}) and S({mathbf {q}},omega ) with neutron scattering measurements reveals an excellent agreement between theory and experiment.

Monte Carlo study of transport in low-dimensional quantum disorder systems at finite temperature

Using the quantum generalized Langevin equation and the path integral Monte Carlo approach, we study the transport dynamics of low-dimensional quantum disorder systems at finite temperature. Motivated by the nature of the classical-to-quantum transformation in fluctuations in the time domain, we extend the treatment to the spatial domain and propose a quantum random-correlated potential, describing specifically quantum disorder. For understanding the Anderson localization from the particle transport perspective, we present an intuitive treatment using a classical analogy in which the particle moves through a flat periodic crystal lattice corrugated by classical or quantum disorder. We emphasize an effective classical disorder potential in studying the quantum effects on the transport dynamics. Compared with the classical case, we find that the quantum escape rate from a disordered metastable potential is larger. Moreover, the diffusion enhancement of a quantum system moving in a weak, biased, periodic disorder potential is more significant compared with the classical case; for an effective rock-ratcheted disorder potential, quantum effects increase the directed current with decreasing temperature. For the classical case, we explore surface diffusion on a two-dimensional biased disorder potential at finite temperature; surprisingly, the optimal angle of the external bias force is found to enhance diffusion in the biased disorder surface. Furthermore, to explain the quantum transport dynamics in a disorder potential, we adopt the barrier-crossing mechanism and the mean first passage time theory to establish the probability distribution function.

Nonlinear density response from imaginary-time correlation functions: Ab initio path integral Monte Carlo simulations of the warm dense electron gas.

The 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 (ITCFs). For example, the well-known density-density ITCF F(q, τ) allows one to estimate the linear response of a given system for all wave vectors q from a single simulation of the unperturbed system. Moreover, it constitutes the basis for the reconstruction of the dynamic structure factor S(q, ω)-a key quantity in state-of-the-art scattering experiments. In this work, we present analogous relations between the nonlinear density response in the 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(q1, ω1; q2, ω2), which evokes the enticing possibility to study dynamic three-body effects on an ab initio level.

Thermodynamic properties of the finite-temperature electron gas by the fermionic path integral Monte Carlo method

The recently developed quantum path integral Monte Carlo approach [Filinov et al., Phys. Rev. E 102(3), 033203 (2020)] has been applied for the entropy difference calculations for the strongly coupled degenerated uniform electron gas (UEG), a well-known model of simple metals. The used approach includes the Coulomb and exchange interaction of fermions in the basic Monte Carlo cell and its periodic images as well as the change of variables in the path integral measure and exchange determinant. Calculations have been carried out at finite temperature in the canonical ensemble. Presented results include the pair distribution functions, the isochors, and isotherms of pressure, internal energy, and the entropy change in the strongly coupled and degenerate UEG. The obtained data may be useful for density functional theory.

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.

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

Precise knowledge of the static density response function (SDRF) of the uniform electron gas serves as key input for numerous applications, most importantly for density functional theory beyond generalized gradient approximations. Here we extend the configuration path integral Monte Carlo (CPIMC) formalism that was previously applied to the spatially uniform electron gas to the case of an inhomogeneous electron gas by adding a spatially periodic external potential. This procedure has recently been successfully used in permutation blocking path integral Monte Carlo simulations (PB-PIMC) of the warm dense electron gas [T. Dornheim et al., Phys. Rev. E 96, 023203 (2017)], but this method is restricted to low and moderate densities. Implementing this procedure into CPIMC allows us to obtain highly accurate finite temperature results for the SDRF of the electron gas at high to moderate densities closing the gap left open by the PB-PIMC data. In this paper, we demonstrate how the CPIMC formalism can be efficiently extended to the spatially inhomogeneous electron gas and present the first data points. Finally, we discuss finite size errors involved in the quantum Monte Carlo results for the SDRF in detail and present a solution how to remove them that is based on a generalization of ground state techniques.

Path-integral Monte Carlo study of particles obeying quantum mechanics and classical statistics

Ultracold atomic systems have been of great research interest in the past, with more recent attention being paid to systems of mixed species. In this work, we carry out nonperturbative path-integral Monte Carlo (PIMC) simulations of $N$ distinguishable particles at finite temperature, which can be thought of as an ultracold atomic system containing $N$ distinct species. We use the PIMC approach to calculate thermodynamic properties of particles interacting via hard-sphere and hard-cavity potentials. The first problem we study is a two-particle system interacting via a hard-sphere and hard-cavity interaction in order to test the effectiveness of two approximations for the thermal density matrix corresponding to these potentials. We then apply the PIMC method to a system of many hard-sphere particles under periodic boundary conditions at varying temperature in order to calculate the energy per particle, pressure, and specific heat of the system. We examine how finite-size effects impact the results of PIMC simulations of hard-sphere particles and when the thermodynamic limit has been reached. Our results provide microscopic benchmarks for a system containing distinguishable particles, which can be thought of as a limiting case for ultracold atomic systems of mixed species.

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.

Quantum virial coefficients of molecular nitrogen

ABSTRACTWe report virial coefficients up to third order in density for molecular nitrogen, investigating 103 temperatures in the range (15 K, 3000 K). All calculations are based on an ab initio-based potential taken from the literature. Path-integral Monte Carlo (PIMC) is applied to account for nuclear quantum effects, and these results are compared to a more approximate but faster semiclassical treatment. Additionally, we examine a PIMC approach that employs semiclassical beads for the path-integral images, but find that it offers marginal advantage. A recently developed orientation sampling algorithm is used in conjunction with Mayer sampling to compute precise virial coefficients. We find that, within the precision of our calculations of the second-order coefficient (B2), semiclassical methods are adequate for temperatures greater than 250 K, and are needed to correct classical behaviour for temperatures as high as 800 K. For the third-order coefficient (B3), the semiclassical methods are adequate above 150 K, and are required up to the highest temperature examined (3000 K) in order to correct the classical treatment within the precision of the calculations. However, three-body contributions to the potential are much more significant than nuclear quantum effects for the evaluation of B3.

Dynamics of non-Markovian open quantum systems

Open quantum systems (OQSs) cannot always be described with the Markov approximation, which requires a large separation of system and environment time scales. An overview is given of some of the most important techniques available to tackle the dynamics of an OQS beyond the Markov approximation. Some of these techniques, such as master equations, Heisenberg equations, and stochastic methods, are based on solving the reduced OQS dynamics, while others, such as path integral Monte Carlo or chain mapping approaches, are based on solving the dynamics of the full system. The physical interpretation and derivation of the various approaches are emphasized, how they are connected is explored, and how different methods may be suitable for solving different problems is examined.

Thermal conductivity and impurity scattering in the accreting neutron star crust

We calculate the thermal conductivity of electrons for the strongly correlated multi-component ion plasma expected in the outer layers of neutron star's crust employing a Path Integral Monte Carlo (PIMC) approach. This allows us to isolate the low energy response of the ions and use it to calculate the electron scattering rate and the electron thermal conductivity. We find that the scattering rate is enhanced by a factor 2-4 compared to earlier calculations based on the simpler electron-impurity scattering formalism. This findings directly impacts the interpretation of thermal relaxation observed in transiently accreting neutron stars and has implications for the composition and nuclear reactions in the crust that occur during accretion.

Path-Integral Monte Carlo Determination of the Fourth-Order Virial Coefficient for a Unitary Two-Component Fermi Gas with Zero-Range Interactions

The unitary equal-mass Fermi gas with zero-range interactions constitutes a paradigmatic model system that is relevant to atomic, condensed matter, nuclear, particle, and astrophysics. This work determines the fourth-order virial coefficient b_{4} of such a strongly interacting Fermi gas using a customized abinitio path-integral MonteCarlo (PIMC) algorithm. In contrast to earlier theoretical results, which disagreed on the sign and magnitude of b_{4}, our b_{4} agrees within error bars with the experimentally determined value, thereby resolving an ongoing literature debate. Utilizing a trap regulator, our PIMC approach determines the fourth-order virial coefficient by directly sampling the partition function. An on-the-fly antisymmetrization avoids the Thomas collapse and, combined with the use of the exact two-body zero-range propagator, establishes an efficient general means to treat small Fermi systems with zero-range interactions.

Ab initioquantum Monte Carlo simulations of the uniform electron gas without fixed nodes

The uniform electron gas (UEG) at finite temperature is of key relevance for many applications in the warm dense matter regime, e.g. dense plasmas and laser excited solids. Also, the quality of density functional theory calculations crucially relies on the availability of accurate data for the exchange-correlation energy. Recently, new benchmark results for the N = 33 spin-polarized electrons at high density, r_s = r/a_B <= 4 and low temperature, have been obtained with the configuration path integral Monte Carlo (CPIMC) method [T. Schoof et al., Phys. Rev. Lett. 115, 130402 (2015)]. To achieve these results, the original CPIMC algorithm [T. Schoof et al., Contrib. Plasma Phys. 51, 687 (2011)] had to be further optimized to cope with the fermion sign problem (FSP). It is the purpose of this paper to give detailed information on the manifestation of the FSP in CPIMC simulations of the UEG and to demonstrate how it can be turned into a controllable convergence problem. In addition, we present new thermodynamic results for higher temperatures. Finally, to overcome the limitations of CPIMC towards strong coupling, we invoke an independent method|the recently developed permutation blocking path integral Monte Carlo approach [T. Dornheim et al., accepted for publication in J. Chem Phys., arXiv:1508.03221]. The combination of both approaches is able to yield ab initio data for the UEG over the entire density range, above a temperature of about one half of the Fermi temperature. Comparison with restricted path integral Monte Carlo data [E. W. Brown et al., Phys. Rev. Lett. 110, 146405 (2013)] allows us to quantify the systematic error arising from the free particle nodes.

Exact modeling of finite temperature and quantum delocalization effects on reliability of quantum-dot cellular automata

A thorough simulation study is carried out on thermal and quantum delocalization effects on the feasibility of a quantum-dot cellular automata (QCA) cell. The occupation correlation of two electrons is modeled with a simple four-site array of harmonic quantum dots (QD). QD sizes range from 20 nm to 40 nm with site separations from 20 nm to 100 nm, relevant for state-of-the-art GaAs/InAs semiconductor technology. The choice of parameters introduces QD overlap, which is only simulated properly with exact treatment of strong Coulombic correlation and thermal equilibrium quantum statistics. These are taken into account with path integral Monte Carlo approach. Thus, we demonstrate novel joint effects of quantum delocalization and decoherence in QCA, but also highly sophisticated quantitative evidence supporting the traditional relations in pragmatic QCA design. Moreover, we show the effects of dimensionality and spin state, and point out the parameter space conditions, where the ‘classical’ treatment becomes invalid.