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Path Integral Monte Carlo and Bose-Einstein condensation in quantum fluids and solids

R. Rota

Published Dec 20, 2011• 5 citations
Publisher

Abstract

Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.

DOI
10.5821/dissertation-2117-94513

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Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.
Bose-Einstein condensation has been experimentally found to take place in finite trapped systems when one of the confining frequencies is increased until the gas becomes effectively two-dimensional (2D). We confirm the plausibility of this result by performing path-integral Monte Carlo (PIMC) simulations of trapped Bose gases of increasing anisotropy and comparing them to the predictions of finite-temperature many-body theory. PIMC simulations provide an essentially exact description of these systems; they yield the density profile directly and provide two different estimates for the condensate fraction. For the ideal gas, we find that the PIMC column density of the squeezed gas corresponds quite accurately to that of the exact analytic solution and, moreover, is well mimicked by the density of a 2D gas at the same temperature; the two estimates for the condensate fraction bracket the exact result. For the interacting case, we find 2D Hartree-Fock solutions whose density profiles coincide quite well with the PIMC column densities and whose predictions for the condensate fraction are again bracketed by the PIMC estimates.
Quantum Monte Carlo methods are a powerful suite of techniques for solving the quantum many-body problem. By using random numbers to stochastically sample quantum properties, QMC methods are capable of studying low-temperature quantum systems well beyond the reach of conventional deterministic techniques. QMC techniques have likewise been indispensible tools for augmenting our current knowledge of superfluidity and superconductivity. In this thesis, I present two new quantum Monte Carlo techniques, the Monte Carlo Power Method and Bose-Fermi Auxiliary-Field Quantum Monte Carlo, and apply previously developed Path Integral Monte Carlo methods to explore two new phases of quantum hard spheres and hydrogen. I lay the foundation for a subsequent description of my research by first reviewing the physics of quantum liquids in Chapter One and the mathematics behind Quantum Monte Carlo algorithms in Chapter Two. I then discuss the Monte Carlo Power Method, a stochastic way of computing the first several extremal eigenvalues of a matrix too memory-intensive to be stored and therefore diagonalized. As an illustration of the technique, I demonstrate how it can be used to determine the second eigenvalues of the transition matrices of several popular Monte Carlo algorithms. This information may be used to quantify how rapidly a Monte Carlo algorithm is converging to the equilibrium probability distribution it is sampling. I next present the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm. This algorithm generalizes the well-known Auxiliary-Field Quantum Monte Carlo algorithm for fermions to bosons and Bose-Fermi mixtures. Despite some shortcomings, the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm represents the first exact technique capable of studying Bose-Fermi mixtures of any size in any dimension. In Chapter Six, I describe a new Constant Stress Path Integral Monte Carlo algorithm for the study of quantum mechanical systems under high pressures. While the eventual hope is to apply this algorithm to the exploration of yet unidentified high-pressure, low-temperature phases of hydrogen, I employ this algorithm to determine whether or not quantum hard spheres can form a low-temperature bcc solid if exchange is not taken into account. In the final chapter of this thesis, I use Path Integral Monte Carlo once again to explore whether glassy para-hydrogen exhibits superfluidity. Physicists have long searched for ways to coax hydrogen into becoming a superfluid. I present evidence that, while glassy hydrogen does not crystallize at the temperatures at which hydrogen might become a superfluid, it nevertheless does not exhibit superfluidity. This is because the average binding energy per p-H2 molecule poses a severe barrier to exchange regardless of whether the system is crystalline. All in all, this work extends the reach of Quantum Monte Carlo methods to new systems and brings the power of existing methods to bear on new problems.
Path integral Monte Carlo approach to the structural properties and collective excitations of liquid 3He^3{\text {He}}3He 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], the momentum distribution [Formula: see text], and the static density response function [Formula: see text]. In addition, the dynamic structure factor [Formula: see text] is rigorously reconstructed from imaginary-time PIMC data. From simulations of [Formula: see text], we derived the familiar phonon-maxon-roton dispersion function that is well-known for [Formula: see text] and has been reported previously for two-dimensional [Formula: see text] films (Nature 483:576-579 (2012)). The comparison of our new results for both [Formula: see text] and [Formula: see text] with neutron scattering measurements reveals an excellent agreement between theory and experiment.
We evaluate the Bose-Einstein condensate density and the superfluid fraction of bosons in a periodic external potential using path-integral Monte Carlo methods. The periodic lattice consists of a cubic cell containing a potential well that is replicated along one dimension using periodic boundary conditions. The aim is to describe bosons in a one-dimensional optical lattice or helium confined in a periodic porous medium. The one-body density matrix is evaluated and diagonalized numerically to obtain the single boson natural orbitals and the occupation of these orbitals. The condensate fraction is obtained as the fraction of bosons in the orbital that has the highest occupation. The superfluid density is obtained from the winding number. From the condensate orbital and superfluid fraction, we investigate (1) the impact of the periodic external potential on the spatial distribution of the condensate, and (2) the correlation of localizing the condensate into separated parts and the loss of superflow along the lattice. For high-density systems, as the well depth increases, the condensate becomes depleted in the wells and confined to the plateaus between successive wells, as in pores between necks in a porous medium. For low-density systems, as the well depth increases the BEC is localized at the center of the wells (tight binding) and depleted between the wells. In both cases, the localization of the condensate suppresses superflow leading to a superfluid-insulator crossover. The impact of the external potential on the temperature dependence of the superfluidity is also investigated. The external potential suppresses the superfluid fraction at all temperatures, with a superfluid fraction significantly less than one at low temperature. The addition of an external potential does not, however, significantly reduce the transition temperature.
We present path integral Monte Carlo (PIMC) calculations of the superfluid fraction, ${\ensuremath{\rho}}_{S}/\ensuremath{\rho}$, and the one-body density matrix (OBDM) [Bose-Einstein condensation (BEC)] of liquid 4He^{4}\mathrm{He}4He confined in nanopores. Liquid 4He^{4}\mathrm{He}4He in nanopores represents a dense Bose liquid at reduced dimension and in disorder. The goal is to determine the effective dimensions of the liquid in the pores. It is to test whether observed properties, such as a very low onset temperature for superflow, Tc{T}_{c}Tc​, can be predicted by a standard, static PIMC ${\ensuremath{\rho}}_{S}/\ensuremath{\rho}$. We simulate a cylinder of liquid of diameter dL{d}_{L}dL​ surrounded by 5 \AA{} of inert solid 4He^{4}\mathrm{He}4He in a nanopore of diameter ddd; d=dL+10A˚d={d}_{L}+10\phantom{\rule{0.28em}{0ex}}\AA{}d=dL​+10A˚. We find a PIMC ${\ensuremath{\rho}}_{S}(T)/\ensuremath{\rho}$ and OBDM that scales as a 1D fluid Luttinger liquid at extremely small liquid pore diameters only, dL=6A˚{d}_{L}=6\phantom{\rule{0.28em}{0ex}}\AA{}dL​=6A˚. At this dL{d}_{L}dL​, the liquid fills the pore in a 1D line at the center of the pore and there is no PIMC superflow. In the range $8\ensuremath{\le}{d}_{L}\ensuremath{\le}22\phantom{\rule{0.28em}{0ex}}\AA{}$ the PIMC ${\ensuremath{\rho}}_{S}(T)/\ensuremath{\rho}$ scales as a 2D liquid. In this dL{d}_{L}dL​ range the liquid fills the pores in 2D-like cylindrical layers. The crossover from no superflow at d=16A˚d=16\phantom{\rule{0.28em}{0ex}}\AA{}d=16A˚ to superflow at $d\ensuremath{\ge}18\phantom{\rule{0.28em}{0ex}}\AA{}$ agrees with experiment. There is a crossover to 3D scaling at larger ${d}_{L}\ensuremath{\simeq}22\phantom{\rule{0.28em}{0ex}}\AA{}$. In the range $8\ensuremath{\le}{d}_{L}\ensuremath{\le}22\phantom{\rule{0.28em}{0ex}}\AA{}$, the Tc{T}_{c}Tc​ predicted using the Kosterlitz-Thouless 2D scaling criterion of the OBDM agrees well with that obtained from ${\ensuremath{\rho}}_{S}(T)/\ensuremath{\rho}$. These results suggest that the superflow observed in small pore media is standard static superflow with the low Tc{T}_{c}Tc​ arising from its 2D character. An operational onset temperature, TBEC{T}_{\mathrm{BEC}}TBEC​, for BEC can be defined as the temperature at which there is a crossover from exponential to algebraic decay in the OBDM. This definition leads to a ${T}_{\mathrm{BEC}}\ensuremath{\ge}{T}_{c}$ as observed in larger pore media.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.