Path Integral Monte Carlo Method Research Articles

Path integral Monte Carlo calculations of helium and hydrogen–helium plasma thermodynamics and of the deuterium shock Hugoniot

In this work we calculate the thermodynamic properties of hydrogen-helium plasmas with different mass fractions of helium by the direct path integral Monte Carlo method. To avoid unphysical approximations we use the path integral representation of the density matrix. We pay special attention to the region of weak coupling and degeneracy and compare the results of simulation with a model based on the chemical picture. Further with the help of calculated deuterium isochors we compute the shock Hugoniot of deuterium. We analyze our results in comparison with recent experimental and calculated data on the deuterium Hugoniot.

Strongly correlated indirect excitons in quantum wells in high electric fields

We study the Stark effect on excitonic complexes confined in a GaAs-based single quantum well. We approach this problem using Path Integral Monte Carlo methods to compute the many-body density matrix. The developed method is applied for investigation of the electric field-dependence of energies, particle distribution and effective exciton dipole moment. Using these results as an input we apply thermodynamical Monte Carlo methods to investigate systems of several tens to thousands indirect excitons in a 2D quantum well with a lateral confinement arising from the quantum confined Stark effect. Depending on the field strength, exciton density and temperature different phases (gas, liquid and solid) of indirect excitons are predicted.

Polaron action for multimode dispersive phonon systems

Path-integral approach to the tight-binding polaron is extended to multiple optical phonon modes of arbitrary dispersion and polarization. The non-linear lattice effects are neglected. Only one electron band is considered. The electron-phonon interaction is of the density-displacement type, but can be of arbitrary spatial range and shape. Feynman's analytical integration of ion trajectories is performed by transforming the electron-ion forces to the basis in which the phonon dynamical matrix is diagonal. The resulting polaron action is derived for the periodic and shifted boundary conditions in imaginary time. The former can be used for calculating polaron thermodynamics while the latter for the polaron mass and spectrum. The developed formalism is the analytical basis for numerical analysis of such models by path-integral Monte Carlo methods.

Finite-temperature properties of quasi-two-dimensional Bose-Einstein condensates

Using the finite-temperature path integral Monte Carlo method, we investigate dilute, trapped Bose gases in a quasi-two-dimensional (quasi-2D) geometry. The quantum particles have short-range, $s$-wave interactions described by a hard-sphere potential whose core radius equals its corresponding scattering length. The effect of both the temperature and the interparticle interaction on the equilibrium properties such as the total energy, the density profile, and the superfluid fraction is discussed. We compare our accurate results with both the semiclassical approximation and the exact results of an ideal Bose gas. Our results show that for repulsive interactions (i) the minimum value of the aspect ratio, where the system starts to behave quasi-2D, increases as the two-body interaction strength increases; (ii) the superfluid fraction for a quasi-two-dimensionally Bose gas is distinctly different from that for both a quasi-1D Bose gas and a true 3D system, i.e., the superfluid fraction for a quasi-2D Bose gas decreases faster than that for a quasi-1D system and a true 3D system with increasing temperature, and shows a stronger dependence on the interaction strength; (iii) the superfluid fraction for a quasi-2D Bose gas lies well below the values calculated from the semiclassical approximation; and (iv) the Kosterlitz-Thouless transition temperature decreases as the strength of the interaction increases.

Effective charging energy for a regular granular metal array

We study the Ambegaokar-Eckern-Sch\"{o}n (AES) model for a regular array of metallic grains coupled by tunnel junctions of conductance $g$ and calculate both paramagnetic and diamagnetic terms in the Kubo formula for the conductivity. We find analytically, and confirm by numerical path integral Monte Carlo methods, that for $0<g<4$ the conductivity obeys an Arrhenius law $\sigma(T)\sim\exp[-E^{*}(g)/T]$ with an effective charging energy $E^{*} (g)$ when the temperature is sufficiently low, due to a subtle cancellation between $T^2$ inelastic-cotunneling contributions in the paramagnetic and diamagnetic terms. We present numerical results for the effective charging energy and compare the results with recent theoretical analyses. We discuss the different ways in which the experimentally observed $\sigma(T)\sim\exp[-\sqrt{T_{0}/T}]$ law could be attributed to disorder.

Quantum Simulation of a Hydrated Noradrenaline Analog with the Torsional Path Integral Method

An extended version of the torsional path integral Monte Carlo (TPIMC) method is presented and shown to be useful for studying the conformation of flexible molecules in solvated clusters. The new technique is applied to the hydrated clusters of the 2-amino-1-phenyl-ethanol (APE) molecule. APE + nH2O clusters with n = 0-4 are studied at 100 and 300 K using both classical and quantum simulations. Only at the lower temperature is the hydration number n found to impact the conformational distribution of the APE molecule. This is shown to be a result of the temperature-dependent balance between the internal energy and entropy contributions to the relative conformer free energies. Furthermore, at 100 K, large quantum effects are observed in the calculated conformer populations. A particularly large quantum shift of 30% of the total population is calculated for the APE + 2H2O cluster, which is explained in terms of the relative zero point energy of the lowest-energy hydrated structures for this cluster. Finally, qualitative agreement is found between the reported calculations and recent spectroscopy experiments on the hydrated clusters of APE, including an entropically driven preference for the formation of AG-type hydrated structures and the formation of a water "droplet" in the APE + 4H2O cluster.

Superfluidity of Mesoscopic Bose Gases under Varying Confinements

The superfluid fraction Ns/N of 27 bosons under varying confinement is investigated at finite temperature using well-known properties of the harmonic oscillator and the microscopic path integral Monte Carlo method. We find that Ns/N (i) is essentially independent of the interaction strength for all temperatures considered, (ii) changes profoundly as the effective dimensionality is varied from three to one dimensional, (iii) is approximately equal to the condensate fraction N0/N for spherical Bose gases, and (iv) deviates dramatically from N0/N for highly elongated Bose gases.

Quantum rotation of carbonyl sulfide molecules in superfluid helium clusters: a pathintegral hybrid Monte Carlo study

Carbonyl sulfide (OCS) molecules in superfluid helium-4 clusters have been studied by pathintegral hybrid Monte Carlo methods. A new technique was developed to treat thequantum rotational degree of freedom of the molecules in hybrid Monte Carlo methods;this is referred to as a ‘Legendre potential technique’. Then, our method was applied toOCS-doped helium clusters. It was found that although the molecule is solvated inside thecluster, the calculated orientational correlation function exhibits free-rotor-type behaviour.The estimated effective rotational constant was in good agreement with the experimentalvalue.

Superfluidity of DenseHe4in Vycor

We calculate properties of a model of $^{4}\mathrm{He}$ in Vycor using the path integral Monte Carlo method. We find that $^{4}\mathrm{He}$ forms a distinct layered structure with a highly localized first layer, a disordered second layer with some atoms delocalized and able to give rise to the observed superfluid response, and higher layers of nearly perfect crystals. The addition of a single $^{3}\mathrm{He}$ atom was enough to bring down the total superfluidity by blocking the exchange in the second layer. Our results are consistent with the persistent liquid-layer model to explain the observations. Such a model may be relevant to the experiments on bulk solid $^{4}\mathrm{He}$, if there is a fine network of grain boundaries in those systems.

High-Precision Quantum Thermochemistry on Nonquasiharmonic Potentials: Converged Path-Integral Free Energies and a Systematically Convergent Family of Generalized Pitzer−Gwinn Approximations

Accurate quantum mechanical (QM) vibrational-rotational partition functions for HOOD, D(2)O(2), H(18)OOH, H(2)(18)O(2), D(18)OOH, and H(18)OOD are determined using a realistic potential energy surface for temperatures ranging from 300 to 2400 K by using the TT-FPI-ESPE path-integral Monte Carlo method. These data, together with our prior results for H(2)O(2), provide benchmarks for testing approximate methods of estimating isotope effects for systems with torsional motions. Harmonic approximations yield poor accuracy for these systems, and although the well-known Pitzer-Gwinn (PG) approximation provides better results for absolute partition functions, it yields the same results as the harmonic approximation for isotope effects because these are intrinsically quantal phenomena. We present QM generalizations of the PG approximation that can provide high accuracy for both isotope effects and absolute partition functions. These approximations can be systematically improved until they approach the accurate result and converge rapidly. These methods can also be used to obtain affordable estimates of zero-point energies from accurate partition functions-even those at relatively high temperatures.

Torsional Anharmonicity in the Conformational Analysis of β-d-Galactose

Schemes to include a treatment of torsional anharmonicity in the conformational analysis of biological molecules are introduced. The approaches combine ab initio electronic energies and harmonic frequencies with anharmonic torsional partition functions calculated using the torsional path integral Monte Carlo method on affordable potential energy surfaces. The schemes are applied to the conformational study of the monosaccharide beta-d-galactose in the gas phase. The global minimum structure is almost exclusively populated at 100 K, but a large number of conformers are present at ambient and higher temperatures. Both quantum mechanical and anharmonic effects in the torsional modes have little effect on the populations at all temperatures considered, and it is, therefore, expected that standard harmonic treatments are satisfactory for the conformational study of monosaccharides.

Bose-Einstein condensation of trapped atoms with dipole interactions

The path integral Monte Carlo method is used to simulate dilute trapped Bose gases and to investigate the equilibrium properties at finite temperatures. The quantum particles have a long-range dipole-dipole interaction and a short-range s-wave interaction. Using an anisotropic pseudopotential for the long-range dipolar interaction and a hard-sphere potential for the short-range s-wave interaction, we calculate the energetics and structural properties as a function of temperature and the number of particles. Also, in order to determine the effects of dipole-dipole forces and the influence of the trapping field on the dipolar condensate, we use two cylindrically symmetric harmonic confinements (a cigar-shaped trap and a disk-shaped trap). We find that the net effect of dipole-dipole interactions is governed by the trapping geometry. For a cigar-shaped trap, the net contribution of dipolar interactions is attractive and the shrinking of the density profiles is observed. For a disk-shaped trap, the net effect of long-range dipolar forces is repulsive and the density profiles expand.

Simulation of excited states and the sign problem in the path integral Monte Carlo method

An approach is presented to compute properties of excited states in path integral Monte Carlo simulations of quantum systems. The approach is based on the introduction of several images of the studied system which have the total wavefunction antisymmetric over permutations of these images, and a simulation of the whole system at low enough temperature. The success of the approach relies, however, on the solution of the sign problem for the corresponding system. In the cases when the sign problem may be resolved (as for fermions in one dimension), properties of excited states can be computed with high precision. This was demonstrated by a simulation of excited states of a single particle in one- and two-dimensional harmonic oscillators. An example of the simulation of excited states of several interacting electrons in one dimension is also given.

Hybrid Monte Carlo implementation of the Fourier path integral algorithm

This paper formulates a hybrid Monte Carlo implementation of the Fourier path integral (FPI-HMC) approach with partial averaging. Such a hybrid Monte Carlo approach allows one to generate collective moves through configuration space using molecular dynamics while retaining the computational advantages associated with the Fourier path integral Monte Carlo method. In comparison with the earlier Metropolis Monte Carlo implementations of the FPI algorithm, the present HMC method is shown to be significantly more efficient for quantum Lennard-Jones solids and suggests that such algorithms may prove useful for efficient simulations of a range of atomic and molecular systems.

Coupled Electron–Ion Monte Carlo calculations of atomic hydrogen

We present a new Monte Carlo method which couples Path Integral for finite temperature protons with Quantum Monte Carlo for ground state electrons, and we apply it to metallic hydrogen for pressures beyond molecular dissociation. This method fills the gap between high temperature electron–proton Path Integral and ground state Diffusion Monte Carlo methods. Our data exhibit more structure and higher melting temperatures of the proton crystal than Car–Parrinello Molecular Dynamics results using LDA. We further discuss the quantum motion of the protons and the zero temperature limit.

Multilevel algorithm for quantum-impurity models

A continuous-time path integral quantum Monte Carlo method using the directed-loop algorithm is developed to simulate the Anderson single-impurity model in the occupation number basis. Although the method suffers from a sign problem at low temperatures, the new algorithm has many advantages over conventional algorithms. For example, the model can be easily simulated in the Kondo limit without time discretization errors. Furthermore, many observables including the impurity susceptibility and a variety of fermionic observables can be calculated efficiently. Finally the new approach allows us to explore a general technique, called the multilevel algorithm, to solve the sign problem. We find that the multilevel algorithm is able to generate an exponentially large number of configurations with an effort that grows as a polynomial in inverse temperature such that configurations with a positive sign dominate over those with negative signs. Our algorithm can be easily generalized to other multi-impurity problems.

Magnetic droplets in a metal close to a ferromagnetic quantum critical point

Using analytical and path integral Monte Carlo methods, we study the susceptibility ${\ensuremath{\chi}}_{\mathrm{dc}}(T)$ of a spin-$S$ impurity with $XY$ rotational symmetry embedded in a metal. Close to a ferromagnetic quantum critical point, the impurity polarizes conduction electrons in its vicinity and forms a large magnetic droplet with moment $M⪢S$. At not too low temperatures, the strongly damping paramagnon modes of the conduction electrons suppress large quantum fluctuations (or spin flips) of this droplet. We show that the susceptibility follows the law ${\ensuremath{\chi}}_{\mathrm{dc}}(T)=({M}^{2}∕T)[1\ensuremath{-}{(\ensuremath{\pi}g)}^{\ensuremath{-}1}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}(g{E}_{0}∕T)]$, where the parameter $g⪢1$ describes the strong damping by conduction electrons, and ${E}_{0}$ is the bandwidth of paramagnon modes. At exponentially low temperatures $T⪡{T}_{*}\ensuremath{\sim}{E}_{0}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}g∕2)$ we show that spin flips cannot be ignored. In this regime we find that ${\ensuremath{\chi}}_{\mathrm{dc}}(T)\ensuremath{\approx}{\ensuremath{\chi}}_{\mathrm{dc}}(0)[1\ensuremath{-}(2∕3){(T∕{T}_{*})}^{2}]$, where ${\ensuremath{\chi}}_{\mathrm{dc}}(0)\ensuremath{\sim}{M}^{2}∕{T}_{*}$ is finite and exponentially large in $g$. We also discuss these effects in the context of the multichannel Kondo impurity model.

Anharmonic corrections to the electrostatic energy of a Coulomb crystal

Model calculations of the electrostatic energy of body-centered-cubic and face-centered-cubic Coulomb crystals are presented. The electrostatic energy is calculated using a pair distribution function of ions in the harmonic approximation. This distribution function is shown to be in a good agreement with the function derived by the Monte Carlo method in the high-temperature classical limit ( T ⪢ ℏ ω p , ω p being the ion plasma frequency). It is also in a good agreement with the distribution function obtained by the path integral Monte Carlo method in the low-temperature quantum limit ( T ⪡ ℏ ω p ). First- and second-order anharmonic corrections to the electrostatic energy are analyzed in a wide temperature range, from the high-temperature classical regime to the low-temperature quantum regime. For typical conditions in the envelopes of neutron stars and the cores of white dwarfs the anharmonic corrections to the electrostatic energy do not exceed a few percent of the electrostatic energy of harmonic lattice.

The Hamiltonian limit of (3+1)D SU(3) lattice gauge theory

The extreme anisotropic limit of Euclidean SU(3) lattice gauge theory is examined to extract the Hamiltonian limit, using standard path integral Monte Carlo (PIMC) methods. We examine the mean plaquette and string tension and compare them to results obtained within the Hamiltonian framework of Kogut and Susskind. The results are a significant improvement upon previous Hamiltonian estimates, despite the extrapolation procedure necessary to extract observables. We conclude that the PIMC method is a reliable method of obtaining results for the Hamiltonian version of the theory. Our results also clearly demonstrate the universality between the Hamiltonian and Euclidean formulations of lattice gauge theory.

Calculation of the Shock Hugoniot of Deuterium at Pressures above 1 Mbar by the Path-Integral Monte Carlo Method

The shock Hugoniot of deuterium at pressures above 1 Mbar is calculated by the path-integral Monte Carlo method without introducing additional physical assumptions and approximations. The results obtained are compared to calculations by other authors, various theoretical models, and experimental data.