Path Integral Ground State Method Research Articles

Properties of fermionic systems with the path-integral ground state method

We investigate strongly correlated many-body systems composed of bosons and fermions with a fully quantum treatment using the path-integral ground state method, PIGS. To account for the Fermi-Dirac statistics, we implement the fixed-node approximation into PIGS, which we then call FN-PIGS. In great detail, we discuss the pair density matrices we use to construct the full density operator in coordinate representation, a vital ingredient of the method. We consider the harmonic oscillator as a proof-of-concept and, as a platform representing quantum many-body systems, we explore helium atoms. Pure ^44He systems demonstrate most of the features of the method. Complementarily, for pure ^33He, the fixed-node approximation resolves the ubiquitous sign problem stemming from anti-symmetric wave functions. Finally, we investigate ^33He-^44He mixtures, demonstrating the method’s robustness. One of the main features of FN-PIGS is its ability to estimate any property at temperature T = 0T=0 without any additional bias apart from the FN approximation; biases from long simulations are also excluded. In particular, we calculate the correlation function of pairs of equal and opposite spins and precise values of the ^33He kinetic energy in the mixture.

On the accuracy and efficiency of different methods to calculate Raman vibrational shifts of parahydrogen clusters.

The Raman vibrational frequency shifts of pure parahydrogen and orthodeuterium clusters of sizes N = 4-9 are calculated using the Langevin equation path integral ground state method. The shifts are calculated using three different methods; the results obtained from each are compared to experiment and variance properties are assessed. The first method requires the direct calculation of energies from two simulations: one when the cluster is in the v = 0 vibrational state and one when the cluster has v = 1 total quantum of vibration. The shift is directly calculated from the difference in those two energies. The second method requires only a v = 0 simulation to be performed. The ground state energy is calculated as usual and the excited state energy is calculated by using the distribution of the v = 0 simulation and the ratio of the density matrices between the v = 1 state and the v = 0 state. The shift is calculated from the difference in those two energies. These first two are both exact methods. The final method is based on perturbation theory where the shift is calculated by averaging the pairwise difference potential over the pair distribution function. However, this is an approximate approach. It is found that for large enough system sizes, despite the approximations, the perturbation theory method has the strongest balance between accuracy and precision when weighing against computational cost.

Ground state chemical potential of parahydrogen clusters of size N = 21-40.

We report the ground state chemical potential of parahydrogen clusters between N = 21-40 calculated using the Langevin equation Path Integral Ground State method. There has been much debate in the past whether the chemical potential size evolution in this region is jagged (indicating magic number cluster sizes) or if it is smooth (indicating some quantum melting below 1K). We compare to previous diffusion Monte Carlo and Path Integral Ground State (PIGS) results, including very recent Variational Path Integral Molecular Dynamics (VPIMD) calculations [S. Miura, J. Chem. Phys. 148, 102333 (2018)]. We find that the ground state chemical potential is not a smooth curve and that magic number clusters are present, consistent with VPIMD and PIGS Monte Carlo results.

Phase diagram of dipolar bosons in two dimensions with tilted polarization

We analyze the ground state of a system of dipolar bosons moving in the $XY$ plane and such that their dipolar moments are all aligned in a fixed direction in space. We focus on the general case where the polarization field forms a generic angle $\alpha$ with respect to the $Z$ axis. We use the Path Integral Ground State method to analyze the static properties of the system as both $\alpha$ and the density $n$ vary over a wide range were the system is stable. We use the maximum of the static structure function as an order parameter to characterize the different phases and the transition lines among them. We find that aside of a superfluid gas and a solid phase, the system reaches a stripe phase at large tilting angles that is entirely induced by the anisotropic character of the interaction. We also show that the quantum phase transition from the gas to the stripe phase is of second order, and report approximate values for the critical exponents.

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.

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.

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.

Weakly bound complexes trapped in quantum matrices: Structure, energetics, and isomer coexistence in (para-H2)N(ortho-D2)3 clusters

The ground state of mixed (para-H(2))(N)(ortho-D(2))(3) clusters of sizes ranging from N=8 to 37 is examined by means of the path integral ground state method. The chemical potential is calculated and reveals that magic numbers are consistent with those found in pure para-H(2) and ortho-D(2) clusters. The structural features of the mixed clusters are examined by analyzing density profiles, one-dimensional Pekeris distribution functions of the (ortho-D(2))(3) subsystem, and by direct visualization of density isosurfaces of the systems. The heavier (ortho-D(2))(3) complex resides in the center of the cluster for the various sizes under consideration. It is found that certain cluster sizes favor either equilateral, or near-linear isosceles (ortho-D(2))(3) configurations, while others show a coexistence between those two triangular geometries.

BOSE–EINSTEIN CONDENSATION IN BULK AND CONFINED SOLID HELIUM

We address the question if the ground state of solid 4 He has the number of lattice sites equal to the number of atoms (commensurate state) or if it is different (incommensurate state). We point out that energy computation from simulation as performed by now cannot be used to decide this question and that the presently best variational wave function, a shadow wave function, gives an incommensurate state. We have extended the calculation of the one–body density matrix ρ1 to the exact Shadow Path Integral Ground State method. Calculation of ρ1 at ρ = 0.031 Å-3 shows that Vacancy–Interstitial pair processes are present also in the exact computation but the simulated system size is too small to infer the presence of off–diagonal long range order. Variational simulations of 4 He confined in a narrow cylindrical pore are also discussed.

Path integral ground state study of finite-size systems: Application to small (parahydrogen)N (N=2–20) clusters

We use the path integral ground state method to study the energetic and structural properties of small para-H2 clusters of sizes ranging from 2 to 20 molecules. A fourth order formula is used to approximate the short imaginary-time propagator and two interaction potentials are considered. Our results are compared to those of exact basis set calculations and other quantum Monte Carlo methods when available. We find that for all cluster sizes considered, our results show a lower ground state energy than literature values obtained by diffusion Monte Carlo and variational Monte Carlo. For the dimer and trimer, ground state energies are in good agreement with exact results obtained using the discrete variable representation. Structural properties are found to be insensitive to the choice of interaction potential. We explore the use of Pekeris coordinates to analyze the importance of linear arrangement in trimers and for trimers within clusters of larger size.

A path integral ground state method

Ground state expectation values are obtained by using a path integral ground state Monte Carlo method. The method allows calculations of ground state expectation values without the extrapolations often used with Green’s function and diffusion Monte Carlo methods. We compare our results with those of Green’s function Monte Carlo by calculating some ground state properties of the van der Waals complex He2Cl2 as well as the infinite systems liquid and solid He4. Advantages and disadvantages of the present method with respect to previous ones are discussed.