We present a method for approximating real-time correlation functions and quantum transport coefficients of bosonic condensed phases. The direct evaluation of quantum real-time correlation functions in the path integral formulation is impossible due to a severe dynamical sign problem. Evaluating imaginary-time correlation functions and inverting them using analytic continuation is famously ill-posed. Ring polymer molecular dynamics (RPMD) provides an alternative approach resulting in accurate approximations for real-time correlation functions of condensed phases but neglects exchange effects. In this work, we develop a bosonic RPMD method, which is exact in the harmonic, short time, and high temperature limits. A critical enabling observation is that the Kubo-transformed correlation function for bosons is real only when the correlated observables are symmetric under exchange. We benchmark the method on harmonic and anharmonic model systems and then use it to directly obtain the Lieb-Liniger gas density-density real-time correlation functions at various temperatures and momentum transfers. This work enables the direct simulation of finite-temperature, real-time dynamics of large bosonic condensed phases using RPMD for the first time.
We present a method for approximating real-time correlation functions and quantum transport coefficients of bosonic condensed phases. The direct evaluation of quantum real-time correlation functions in the path integral formulation is impossible due to a severe dynamical sign problem. Evaluating imaginary-time correlation functions and inverting them using analytic continuation is famously ill-posed. Ring polymer molecular dynamics (RPMD) provides an alternative approach resulting in accurate approximations for real-time correlation functions of condensed phases but neglects exchange effects. In this work, we develop a bosonic RPMD method, which is exact in the harmonic, short time, and high temperature limits. A critical enabling observation is that the Kubo-transformed correlation function for bosons is real only when the correlated observables are symmetric under exchange. We benchmark the method on harmonic and anharmonic model systems and then use it to directly obtain the Lieb-Liniger gas density-density real-time correlation functions at various temperatures and momentum transfers. This work enables the direct simulation of finite-temperature, real-time dynamics of large bosonic condensed phases using RPMD for the first time.
The ring polymer molecular dynamics (RPMD) method has gained popularity in recent years as a simple approximation for calculating real time quantum correlation functions in condensed media. However, the extent to which RPMD captures real dynamical quantum effects and why it fails under certain situations have not been clearly understood. Addressing this issue has been difficult in the absence of a genuine justification for the RPMD algorithm starting from the quantum Liouville equation. To this end, a new and exact path integral formalism for the calculation of real time quantum correlation functions is presented in this work, which can serve as a rigorous foundation for the analysis of the RPMD method as well as providing an alternative derivation of the well established centroid molecular dynamics method. The new formalism utilizes the cyclic symmetry of the imaginary time path integral in the most general sense and enables the expression of Kubo-transformed quantum time correlation functions as that of physical observables pre-averaged over the imaginary time path. Upon filtering with a centroid constraint function, the formulation results in the centroid dynamics formalism. Upon filtering with the position representation of the imaginary time path integral, we obtain an exact quantum dynamics formalism involving the same variables as the RPMD method. The analysis of the RPMD approximation based on this approach clarifies that an explicit quantum dynamical justification does not exist for the use of the ring polymer harmonic potential term (imaginary time kinetic energy) as implemented in the RPMD method. It is analyzed why this can cause substantial errors in nonlinear correlation functions of harmonic oscillators. Such errors can be significant for general correlation functions of anharmonic systems. We also demonstrate that the short time accuracy of the exact path integral limit of RPMD is of lower order than those for finite discretization of path. The present quantum dynamics formulation also serves as the basis for developing new quantum dynamical methods that utilize the cyclic nature of the imaginary time path integral.
The maximum entropy analytic continuation (MEAC) and ring polymer molecular dynamics (RPMD) methods provide complementary approaches to the calculation of real time quantum correlation functions. RPMD becomes exact in the high temperature limit, where the thermal time betavariant Planck's over 2pi tends to zero and the ring polymer collapses to a single classical bead. MEAC becomes most reliable at low temperatures, where betavariant Planck's over 2pi exceeds the correlation time of interest and the numerical imaginary time correlation function contains essentially all of the information that is needed to recover the real time dynamics. We show here that this situation can be exploited by combining the two methods to give an improved approximation that is better than either of its parts. In particular, the MEAC method provides an ideal way to impose exact moment (or sum rule) constraints on a prior RPMD spectrum. The resulting scheme is shown to provide a practical solution to the "nonlinear operator problem" of RPMD, and to give good agreement with recent exact results for the short-time velocity autocorrelation function of liquid parahydrogen. Moreover these improvements are obtained with little extra effort, because the imaginary time correlation function that is used in the MEAC procedure can be computed at the same time as the RPMD approximation to the real time correlation function. However, there are still some problems involving long-time dynamics for which the RPMD+MEAC combination is inadequate, as we illustrate with an example application to the collective density fluctuations in liquid orthodeuterium.
Trapped bosons exhibit fundamental physical phenomena and are at the core of emerging quantum technologies. We present a method for simulating bosons using path integral molecular dynamics. The main difficulty in performing such simulations is enumerating all ring-polymer configurations, which arise due to permutations of identical particles. We show that the potential and forces at each time step can be evaluated by using a recurrence relation which avoids enumerating all permutations, while providing the correct thermal expectation values. The resulting algorithm scales cubically with system size. The method is tested and applied to bosons in a 2-dimensional (2D) trap and agrees with analytical results and numerical diagonalization of the many-body Hamiltonian. An analysis of the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations, is also presented.
A polymer density functional theory (P-DFT) has been extended to the case of quantum statistics within the framework of Feynman path integrals. We start with the exact P-DFT formalism for an ideal open chain and adapt its efficient numerical solution to the case of a ring. We show that, similarly, the path integral problem can, in principle, be solved exactly by making use of the two-particle pair correlation function (2p-PCF) for the ends of an open polymer, half of the original. This way the exact data for one-dimensional quantum harmonic oscillator are reproduced in a wide range of temperatures. The exact solution is not, though, reachable in three dimensions (3D) because of a vast amount of storage required for 2p-PCF. In order to treat closed paths in 3D, we introduce a so-called "open ring" approximation which proves to be rather accurate in the limit of long chains. We also employ a simple self-consistent iteration so as to correctly account for the interparticle interactions. The algorithm is speeded up by taking convolutions with the aid of fast Fourier transforms. We apply this approximate path integral DFT (PI-DFT) method to systems within spherical symmetry: 3D harmonic oscillator, atoms of hydrogen and helium, and ions of He and Li. Our results compare rather well to the known data, while the computational effort (some seconds or minutes) is about 100 times less than that with Monte Carlo simulations. Moreover, the well-known "sign problem" is expected to be considerably reduced within the reported PI-DFT, since it allows for a direct estimate of the corresponding partition functions.
We propose a method which uses centroid molecular dynamics (CMD) [J. Cao and G. A. Voth, J. Chem. Phys. 100, 5106 (1994)] real-time data in conjunction with the imaginary-time data generated using path integral Monte Carlo simulations in a numerical analytic continuation scheme based on the maximum entropy approach. We show that significant improvement is achieved by including short-time CMD data with the imaginary-time data. In particular, for a particle bilinearly coupled to a harmonic bath, these methods lead to significant improvements over previous calculations and even allow accurate determination of transport coefficients such as the diffusion coefficient and mobility for this system. In addition we show how maximum entropy method can be used to extract accurate dynamic information from short-time CMD data, and that this approach is superior to the direct Fourier transform of long-time data for systems characterized by broad, featureless spectral distributions.
The dynamic structure factor of liquid para-hydrogen and ortho-deuterium in corresponding thermodynamic states, (T = 20.0 K, n = 21.24 nm⁻³) and (T = 23.0 K, n = 24.61 nm⁻³) respectively, has been computed by both the Feynman-Kleinert linearized path-integral (FK-LPI) and Ring-Polymer Molecular Dynamics (RPMD) methods and compared with Inelastic X-ray Scattering spectra. The combined use of computational and experimental methods enables a reduction in experimental uncertainties for the determination of the true sample spectrum. Furthermore, the refined experimental spectrum of para-hydrogen and ortho-deuterium is consistently reproduced by both FK-LPI and RPMD at momentum transfers lower than 12.8nm⁻¹. At larger momentum transfers the F K - LP I results agree with experiment much better for ortho-deuterium than for para-hydrogen. More specifically we found that for k ~ 20.0 nm⁻¹ para-hydrogen provides a test case for improved approximations to quantum dynamics. We meet this demand for an improved approximate quantum dynamics method by developing two classes of quasi-classical dynamics that are shown to conserve the initial quantum ensemble when used in conjunction with the Feynman-Kleinert approximation of the density operator. As shown, both classes of dynamics are able to recover the exact classical and high temperature limits of the quantum time correlation function, while a subset is able to recover the exact harmonic limit. A comparison of the approximate quantum time correlation functions obtained from both classes of dynamics are made with the exact results for the challenging model problems of the quartic and double-well potentials. It is found that this new Feynman-Kleinert Quasi-Classical Wigner (FK-QCW) method provides a great improvement over the Feynman-Kleinert implementation of the classical Wigner approximation, also known as FK-LPI, in which purely classical dynamics are used. Furthermore, it is shown that the first class of dynamics reduces to Centroid Molecular Dynamics (CMD) when used within the framework of the classical Wigner approximation for the Kubo transformed time correlation function. Finally, we apply the Feynman-Kleinert Quasi-Classical Wigner (FK- QCW) method to the same liquid para-hydrogen and ortho-deuterium system, previously studied using FK-LPI and RPMD. When applied to this challenging system, it is shown that this new FK-QCW method consistently reproduces the experimental dynamic structure factor for all momentum transfers considered. This shows that FK-QCW provides a great improvement over FK-LPI for not only model problems, but also realistic systems. Furthermore, for small momentum transfers, where RPMD is applicable, it is shown that FK-QCW provides nearly the same results as RPMD, thus suggesting that FK-QCW provides a potentially more appealing algorithm than RPMD since one is not limited to correlation functions involving linear operators. This then suggests that the FK-QCW method is a top contender in the realm of approximate quantum dynamics methods which allow for the practical evaluation of time correlation functions.