Path Integral Ground State Research Articles

Path-integral Monte Carlo method for Rényi entanglement entropies.

We introduce a quantum Monte Carlo algorithm to measure the Rényi entanglement entropies in systems of interacting bosons in the continuum. This approach is based on a path-integral ground state method that can be applied to interacting itinerant bosons in any spatial dimension with direct relevance to experimental systems of quantum fluids. We demonstrate how it may be used to compute spatial mode entanglement, particle partitioned entanglement, and the entanglement of particles, providing insights into quantum correlations generated by fluctuations, indistinguishability, and interactions. We present proof-of-principle calculations and benchmark against an exactly soluble model of interacting bosons in one spatial dimension. As this algorithm retains the fundamental polynomial scaling of quantum Monte Carlo when applied to sign-problem-free models, future applications should allow for the study of entanglement entropy in large-scale many-body systems of interacting bosons.

First-principles prediction of the Raman shifts in parahydrogen clusters.

We report a first-principles prediction of the Raman shifts of parahydrogen (pH2) clusters of sizes N = 4-19 and 33, based on path integral ground-state simulations with an ab initio potential energy surface. The Raman shifts are calculated, using perturbation theory, as the average of the difference-potential energy surface between the potential energy surfaces for vibrationally excited and ground-state parahydrogen monomers. The radial distribution of the clusters is used as a weight function in this average. Very good overall agreement with experiment [G. Tejeda, J. M. Fernández, S. Montero, D. Blume, and J. P. Toennies, Phys. Rev. Lett. 92, 223401 (2004)] is achieved for p(H2)(2-8,13,33). A number of different pair potentials are employed for the calculation of the radial distribution functions. We find that the Raman shifts are sensitive to slight variations in the radial distribution functions.

Quantum Monte Carlo study of a vortex in superfluidHe4and search for a vortex state in the solid

We have performed a microscopic study of a straight quantized vortex line in three dimensions in condensed $^{4}\mathrm{He}$ at zero temperature using the shadow path integral ground state method and the fixed phase approximation. We have characterized the energy and the local density profile around the vortex axis in superfluid $^{4}\mathrm{He}$ at several densities, ranging from below the equilibrium density up to the overpressurized regime. For the Onsager-Feynman (OF) phase our results are exact and represent a benchmark for other theories. The inclusion of backflow correlations in the phase improves the description of the vortex with respect to the OF phase by a large reduction of the core energy of the topological excitation. At all densities the phase with backflow induces a partial filling of the vortex core and this filling slightly increases with density. The core size slightly decreases for increasing density and the density profile has well defined density dependent oscillations whose wave vector is closer to the wave vector of the main peak in the static density response function rather than to the roton wave vector. Our results can be applied to vortex rings of large radius $R$ and we find good agreement with the experimental value of the energy as a function of $R$ without any free parameter. We have studied also $^{4}\mathrm{He}$ above the melting density in the solid phase using the same functional form for the phase as in the liquid. We found that off-diagonal properties of the solid are not qualitatively affected by the velocity field induced by the vortex phase, both with and without backflow correlations. Therefore we find evidence that a perfect $^{4}\mathrm{He}$ crystal is not a marginally stable quantum solid in which rotation would be able to induce off-diagonal long-range coherence.

Inclusion of trial functions in the Langevin equation path integral ground state method: application to parahydrogen clusters and their isotopologues.

We developed and studied the implementation of trial wavefunctions in the newly proposed Langevin equation Path Integral Ground State (LePIGS) method [S. Constable, M. Schmidt, C. Ing, T. Zeng, and P.-N. Roy, J. Phys. Chem. A 117, 7461 (2013)]. The LePIGS method is based on the Path Integral Ground State (PIGS) formalism combined with Path Integral Molecular Dynamics sampling using a Langevin equation based sampling of the canonical distribution. This LePIGS method originally incorporated a trivial trial wavefunction, ψT, equal to unity. The present paper assesses the effectiveness of three different trial wavefunctions on three isotopes of hydrogen for cluster sizes N = 4, 8, and 13. The trial wavefunctions of interest are the unity trial wavefunction used in the original LePIGS work, a Jastrow trial wavefunction that includes correlations due to hard-core repulsions, and a normal mode trial wavefunction that includes information on the equilibrium geometry. Based on this analysis, we opt for the Jastrow wavefunction to calculate energetic and structural properties for parahydrogen, orthodeuterium, and paratritium clusters of size N = 4 - 19, 33. Energetic and structural properties are obtained and compared to earlier work based on Monte Carlo PIGS simulations to study the accuracy of the proposed approach. The new results for paratritium clusters will serve as benchmark for future studies. This paper provides a detailed, yet general method for optimizing the necessary parameters required for the study of the ground state of a large variety of systems.

Path-integral ground state and superfluid hydrodynamics of a bosonic gas of hard spheres

We study a bosonic gas of hard spheres by using the exact zero-temperature path-integral ground-state (PIGS) Monte Carlo method and the equations of superfluid hydrodynamics. The PIGS method is implemented to calculate for the bulk system the energy per particle and the condensate fraction through a large range of the gas parameter $n{a}^{3}$ (with $n$ the number density and $a$ the $s$-wave scattering length), going from the dilute gas into the solid phase. The Maxwell construction is then adopted to determine the freezing at $n{a}^{3}=0.264\ifmmode\pm\else\textpm\fi{}0.003$ and the melting at $n{a}^{3}=0.290\ifmmode\pm\else\textpm\fi{}0.003$. In the liquid phase, where the condensate fraction is finite, the equations of superfluid hydrodynamics, based on the PIGS equation of state, are used to find other relevant quantities as a function of the gas parameter: the chemical potential, the pressure, and the sound velocity. In addition, within Feynman's approximation, from the PIGS static structure factor we determine the full excitation spectrum, which displays a maxon-roton behavior when the gas parameter is close to the freezing value. Finally, the equations of superfluid hydrodynamics with the PIGS equation of state are solved for the bosonic system under axially symmetric harmonic confinement obtaining its collective breathing modes.

Excitation spectrum in two-dimensional superfluid 4He

In this work we perform an ab-initio study of an ideal two-dimensional sample of 4He atoms, a model for 4He films adsorbed on several kinds of substrates. Starting from a realistic hamiltonian we face the microscopic study of the excitation phonon-roton spectrum of the system at zero temperature. Our approach relies on Path Integral Ground State Monte Carlo projection methods, allowing to evaluate exactly the dynamical density correlation functions in imaginary time, and this gives access to the dynamical structure factor of the system S(q,omega), containing information about the excitation spectrum E(q), resulting in sharp peaks in S(q,omega). The actual evaluation of S(q,omega) requires the inversion of the Laplace transform in ill-posed conditions, which we face via the Genetic Inversion via Falsification of Theories technique. We explore the full density range from the region of spinodal decomposition to the freezing density, i.e. 0.0321 A^-2 - 0.0658 A^-2. In particular we follow the density dependence of the excitation spectrum, focusing on the low wave--vector behavior of E(q), the roton dispersion, the strength of single quasi--particle peak, Z(q), and the static density response function, chi(q). As the density increases, the dispersion E(q) at low wave--vector changes from a super-linear (anomalous dispersion) trend to a sub-linear (normal dispersion) one, anticipating the crystallization of the system; at the same time the maxon-roton structure, which is barely visible at low density, becomes well developed at high densities and the roton wave vector has a strong density dependence. Connection is made with recent inelastic neutron scattering results from highly ordered silica nanopores partially filled with 4He.

Langevin Equation Path Integral Ground State

We propose a Langevin equation path integral ground state (LePIGS) approach for the calculation of ground state (zero temperature) properties of molecular systems. The approach is based on a modification of the finite temperature path integral Langevin equation (PILE) method (J. Chem. Phys. 2010, 133, 124104) to the case of open Feynman paths. Such open paths are necessary for a ground state formulation. We illustrate the applicability of the method using model systems and the weakly bound water-parahydrogen dimer. We show that the method can lead to converged zero point energies and structural properties.

TWO DIMENSIONAL AND NOVEL QUASI TWO DIMENSIONAL QUANTUM LIQUIDS

In this thesis we have used Quantum Monte Carlo techniques to study two systems that can be regarded as the archetype for neutral strongly interacting systems: 4He, and its fermionic counterpart 3He.More specifically, we have used the Path Integral Ground State and the Path Integral Monte Carlo methods to study a system of two dimensional 3He (2d-3He) and a system of 4He adsorbed on Graphene-Fluoride (GF) and Graphane (GH) at both zero and finite temperature. The purpose of the study of 4He on GF (GH) was the research of new physical phenomena, whereas in the case of 2d-3He it was the application of novel methodologies for the ab-initio study of static and dynamic properties of Fermi systems. In the case of 2d-3He we have computed the spin susceptibility as function of density which turned out to be in very good agreement with experimental data; we have also obtained the first ab-initio evaluation of the zero-sound mode and the dynamic structure factor of 2d-3He that is in remarkably good agreement with experiments. In the case of 4He adsorbed on GF (GH), we determined the zero temperature equilibrium density of the first monolayer of 4He showing also that the commensurate sqrt(3) x sqrt(3) R30 phase is unstable on both substrates; at equilibrium density we found that 4He on GF (GH) is a modulated superfluid with an anisotropic phono-rotonic spectrum; at high coverages we found an incommensurate triangular solid and, on both GF and GH, a commensurate phase at filling factor x= 2/7 that is locally stable or at least metastable. Remarkably, in this commensurate solid phase and for both GF and GH, our computations show preliminary evidence of the presence of a superfluid fraction.

Novel substrates for Helium adsorption: Graphane and Graphene—Fluoride

The discovery of fullerenes has stimulated extensive exploration of the resulting behavior of adsorbed films. Our study addresses the planar substrates graphene—fluoride (GF) and graphane (GH) in comparison to graphene. We present initial results concerning the potential energy, energy bands and low density behavior of 4He and 3He films on such different surfaces. For example, while graphene presents an adsorption potential that is qualitatively similar to that on graphite, GF and GH yield potentials with different symmetry, a number of adsorption sites double that on graphene/graphite and a larger corrugation for the adatom. In the case of GF, the lowest energy band width is similar to that on graphite but the He atom has a significantly larger effective mass and the adsorption energy is about three time that on graphite. Implications concerning the monolayer phase diagram of 4He are explored with the exact path integral ground state method. A commensurate ordered state similar to the R30° state on graphite is found the be unstable both on GF and on GH. The ground states of submonolayer 4He on both GF and GH are superfluids with a Bose Einstein condensate fraction of about 10%.

Quantized vortices in two dimensional solid 4He

Diagonal and off-diagonal properties of 2D solid 4He systems doped with a quantized vortex have been investigated via the Shadow Path Integral Ground State method using the fixed-phase approach. The chosen approximate phase induces the standard Onsager-Feynman flow field. In this approximation the vortex acts as a static external potential and the resulting Hamiltonian can be treated exactly with Quantum Monte Carlo methods. The vortex core is found to sit in an interstitial site and a very weak relaxation of the lattice positions away from the vortex core position has been observed. Also other properties like Bragg peaks in the static structure factor or the behavior of vacancies are very little affected by the presence of the vortex. We have computed also the one-body density matrix in perfect and defected 4He crystals finding that the vortex has no sensible effect on the off-diagonal long range tail of the density matrix. Within the assumed Onsager Feynman phase, we find that a quantized vortex cannot auto-sustain itself unless a condensate is already present like when dislocations are present. It remains to be investigated if backflow can change this conclusion.

Excitations and Stripe Phase Formation in a Two-Dimensional Dipolar Bose Gas with Tilted Polarization

We present calculations of the ground state and excitations of an anisotropic dipolar Bose gas in two dimensions, realized by a nonperpendicular polarization with respect to the system plane. For sufficiently high density, an increase of the polarization angle leads to a density instability of the gas phase in the direction where the anisotropic interaction is strongest. Using a dynamic many-body theory, we calculate the dynamic structure function in the gas phase which shows the anisotropic dispersion of the excitations. We find that the energy of roton excitations in the strongly interacting direction decreases with increasing polarization angle and almost vanishes close to the instability. Exact path integral ground state MonteCarlo simulations show that this instability is indeed a quantum phase transition to a stripe phase, characterized by long-range order in the strongly interacting direction.

Population size bias in diffusion Monte Carlo

The size of the population of random walkers required to obtain converged estimates in diffusion Monte Carlo (DMC) increases dramatically with system size. We illustrate this by comparing ground state energies of small clusters of parahydrogen (up to 48 molecules) computed by DMC and path integral ground state (PIGS) techniques. We contend that the bias associated with a finite population of walkers is the most likely cause of quantitative numerical discrepancies between PIGS and DMC energy estimates reported in the literature, for this few-body Bose system. We discuss the viability of DMC as a general-purpose ground state technique, and argue that PIGS, and even finite temperature methods, enjoy more favorable scaling, and are therefore a superior option for systems of large size.

Superfluid State of 4He on Graphane and Graphene–Fluoride: Anisotropic Roton States

We explore the phase behavior of Helium films on two variants of graphene: graphane (graphene coated with H, denoted GH) and graphene–fluoride (GF). A semiempirical interaction with these substrates is used in T=0 K Path Integral Ground State and finite temperature Path Integral Monte Carlo simulations. We predict that 4He forms anisotropic fluid states at low coverage. This behavior differs qualitatively from that on graphite because of the different surface composition, symmetry and spacing of the adsorption sites. The 4He ground state on both substrates is thus a self-bound anisotropic superfluid with a superfluid fraction ρs/ρ lower than 1 due to the corrugation of the adsorption potential. In the case of GF such corrugation is so large that ρs/ρ=0.6 at T=0 K and the superfluid is essentially restricted to move in a multiconnected space, along the bonds of a honeycomb lattice. We predict a superfluid transition temperature T≃ 0.25 (1.1) K for 4He on GF (GH). We have studied the elementary excitation spectrum of 4He on GF at equilibrium density finding a phonon–maxon–roton dispersion relation that is strongly anisotropic in the roton region. We conclude that these new platforms for adsorption studies offer the possibility of studying novel superfluid phases of quantum condensed matter.

Microscopic characterization of overpressurized superfluid4He

We have studied static and dynamical properties of superfluid ${}^{4}$He at $T=0$ K in the pressure range from $\ensuremath{-}6$ up to 87 atm well above freezing into the metastable region. Zero temperature properties have been obtained with the exact shadow path integral ground state (SPIGS) method. Information about dynamic structure factors at different pressures have been obtained from imaginary time correlation functions via the genetic inversion via falsification of theories (GIFT) method. In the full pressure range sharp roton excitations are always present in the spectral functions. The roton energy decreases at higher pressures in good agreement with experimental data also in the metastable region. The roton energies have essentially a linear trend with pressure, going from about 7.4 K near freezing to about 4.3 K at about 87 atm. The pressure at which the linear trend would extrapolate to a zero roton energy turns out to be about 170 atm. At $T=0$ K, no sign of metastable glass phase has been found; the disordered systems studied at pressures above about 87 atm readily start homogeneous nucleation processes. Our results in the metastable phase for the condensate fractions and roton gaps differ remarkably from previous ones obtained via a diffusion Monte Carlo study.

Condensate Fraction in Liquid 4He at Zero Temperature

We present results of the one-body density matrix (OBDM) and the condensate fraction n_0 of liquid 4He calculated at zero temperature by means of the Path Integral Ground State Monte Carlo method. This technique allows to generate a highly accurate approximation for the ground state wave function Psi_0 in a totally model-independent way, that depends only on the Hamiltonian of the system and on the symmetry properties of Psi_0. With this unbiased estimation of the OBDM, we obtain precise results for the condensate fraction n_0 and the kinetic energy K of the system. The dependence of n_0 with the pressure shows an excellent agreement of our results with recent experimental measurements. Above the melting pressure, overpressurized liquid 4He shows a small condensate fraction that has dropped to 0.8% at the highest pressure of p = 87 bar.

A Ground State Monte Carlo Approach for Studies of Dipolar Systems with Rotational Degrees of Freedom

We have developed a path integral ground state Monte Carlo (PIGSMC) algorithm for quantum simulations of rotating dipolar molecules, using a highly accurate sixth-order algorithm. The method allows us to calculate unbiased estimates of ground state properties of dipolar molecules in a variety of geometries, with or without an external electric field. To demonstrate the capability of the approach, we calculate the orientational phase diagram of a one dimensional lattice system of rotating point dipoles in the absence of any external electric fields. We find that for finite lattice size, this system exhibits an order–disorder transition at finite dipolar interaction strength in contrast to the well-known orientational disorder of the corresponding one dimensional O(3) quantum rotor models. Comparison of the quantum Monte Carlo results with a self-consistent field estimate of the phase transition shows the emergence of an ordered phase at non-zero dipolar strength, confirming the symmetry breaking role of the anisotropic dipole–dipole interaction.

Bose and Fermi Gases with Lennard–Jones Interactions

We study a model for cold Bose and Fermi gases based on the Lennard–Jones interaction, using the optimized (Fermi-)hypernetted-chain ((F)HNC-EL) method. For comparison, we also have carried out path integral ground state Monte Carlo (PIGSMC) simulations in the Bose case. By varying the density and the coupling strength for the Lennard–Jones potential, we cover the whole range of dilute, weakly interacting gases up to the dense, strongly interacting case of liquid 3He and 4He. Below about 20 percent helium equilibrium density, the simplest version of the (F)HNC-EL theory is accurate within better than 1 percent.

Theoretical Analysis of the Anomalous Spectral Splitting of Tetracene in 4He Droplets

We present a theoretical analysis of the electronic absorption spectra of tetracene in (4)He droplets based on many-body quantum simulations. Using the path integral ground state approach, we calculate one- and two-body reduced density matrices of the most strongly localized He atoms near the molecule surface and use these to investigate the helium ground-state quantum coherence and correlations when tetracene is in its electronic ground and excited states. We identify a trio of quasi-one-dimensional, strongly localized atoms adsorbed along the long axis of the molecule that show some quantum coherence among themselves but far less with the remaining solvating helium. We evaluate the single-particle natural orbitals of the localized He atoms by diagonalization of the one-body density matrix and use these to construct single- and many-particle solvating helium basis states with which the zero-phonon spectral features of the tetracene-(4)He(N) absorption spectrum are then calculated. The absorption spectrum resulting from the three-body density matrix for the strongly bound trio of helium atoms is in very good agreement with the experimental data, accounting quantitatively for the anomalous splitting of the zero-phonon line [Hartmann, M.; Lindinger, A.; Toennies, J. P.; Vilesov, A. F. Chem. Phys. 1998, 239, 139; Krasnokutski, S.; Rouillé, G.; Huisken, F. Chem. Phys. Lett. 2005, 406, 386]. Our results indicate that the combination of strong localization and the quasi-one-dimensional nature of trios of helium atoms adsorbed along the long axis of tetracene leads to a quantum coherent, yet highly correlated ground state for the helium density closest to the molecule. The spectroscopic analysis shows that this feature accounts quantitatively for the anomalous splittings and hitherto unexplained fine structure observed in the absorption spectra of tetracene and suggests that it may be responsible for the corresponding zero-phonon splittings in other quasi-one-dimensional planar aromatic molecules.

Quantum dislocations: the fate of multiple vacancies in two-dimensional solid 4He

Defects are believed to play a fundamental role in the supersolid state of 4He. We havestudied solid 4He in two dimensions (2D) as a function of the number of vacanciesnv, up to 30, inserted in the initial configuration atρ = 0.0765 Å − 2, close to the melting density, with the exact zero-temperature shadow path integral groundstate method. The crystalline order is found to be stable also in the presenceof many vacancies and we observe two completely different regimes. For smallnv,up to about 6, vacancies form a bound state and cause a decrease of the crystalline order. At largernv, the formation energy of an extra vacancy at fixed density decreases by one order ofmagnitude to about 0.6 K. It is no longer possible to recognize vacancies in the equilibratedstate because they mainly transform into quantum dislocations and crystalline order isfound almost independently of how many vacancies have been inserted in the initialconfiguration. The one-body density matrix in this latter regime shows a non-decayinglarge distance tail: dislocations, that in 2D are point defects, turn out to be mobile, theirnumber is fluctuating, and they are able to induce exchanges of particles across the systemmainly triggered by the dislocation cores. These results indicate that the notionof the incommensurate versus the commensurate state loses meaning for solid 4He in 2D, because the number of lattice sites becomes ill defined when the system is notcommensurate. Crystalline order is found to be stable also in 3D in the presence of up to100 vacancies.

High-order time expansion path integral ground state

The feasibility of path integral Monte Carlo ground state calculations with very few beads using a high-order short-time Green's function expansion is discussed. An explicit expression of the evolution operator which provides dramatic enhancements in the quality of ground-state wave functions is examined. The efficiency of the method makes possible to remove the trial wave function and thus obtain completely model-independent results still with a very small number of beads. If a single iteration of the method is used to improve a given model wave function, the result is invariably a shadow-type wave function, whose precise content is provided by the high-order algorithm employed.