We study the quantum dynamics of a generic model fluid with internal quantum states and classical translational degrees of freedom in two spatial dimensions. The path integral Monte Carlo data for the imaginary time correlation functions are presented and analyzed by the maximum entropy method. A comparison of the frequency distribution with those of a mean field approximation and virial expansion shows good agreement at high and low densities, respectively. \textcopyright{} 1996 The American Physical Society.
We study the quantum dynamics of a generic model fluid with internal quantum states and classical translational degrees of freedom in two spatial dimensions. The path integral Monte Carlo data for the imaginary time correlation functions are presented and analyzed by the maximum entropy method. A comparison of the frequency distribution with those of a mean field approximation and virial expansion shows good agreement at high and low densities, respectively. \textcopyright{} 1996 The American Physical Society.
A new numerical procedure for the study of finite temperature quantum dynamics is developed. The method is based on the observation that the real and imaginary time dynamical data contain complementary types of information. Maximum entropy methods, based on a combination of real and imaginary time input data, are used to calculate the spectral densities associated with real time correlation functions. Model studies demonstrate that the inclusion of even modest amounts of short-time real time data significantly improves the quality of the resulting spectral densities over that achievable using either real time data or imaginary time data separately.
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.
We present a way of combining real-time path integral Monte Carlo simulations with a maximum entropy numerical analytic continuation scheme in a new approach for calculating time correlation functions for finite temperature many body quantum systems. The real-time dynamics is expressed in the form of the symmetrized time correlation function, which is suitable for Monte Carlo methods, and several simulation techniques are presented for evaluating this function accurately up to moderate values of time. The symmetrized time correlation function is then analytically continued in combination with imaginary time data to obtain the real-time correlation function. We test this approach on several exactly solvable problems, including two one-dimensional systems, as well two cases of vibrational relaxation of a system coupled to a dissipative environment. The computed time correlation functions are in good agreement with exact results over several multiples of the thermal time βℏ, and exhibit a significant improvement over analytic continuation of imaginary time correlation functions. Moreover, we show how the method can be systematically improved.
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.
We formulate a new quantum many body simulation method for a general quantum fluid at any given temperature. Unlike path integral Monte Carlo our method evolves in imaginary time the density matrix from its initial delta function condition to its final thermal form at time time equal to the inverse temperature. It does this with a molecular dynamics scheme on a classical Hamiltonian that has the same functional form as the one of the quantum mechanical Hamiltonian according to the properties of the continuous representation of John R. Klauder. We then end up with the thermal density matrix that can be used to extract thermal averages of observables using the Monte Carlo method equally well in any statistics.
The main objective of the thesis is to study static and/or dynamic properties of a set of quantum fluids by means of quantum Monte Carlo techniques, mainly using the path integral formalism to obtain results both at zero temperature and finite temperature. First, we present briefly some of the more important quantum Monte Carlo methods, and introduce the Path Integral Monte Carlo (PIMC) method, which has been used during all this thesis, as well as the Path Integral Ground State (PIGS), which is an extension of the first at ground state. After introducing the basic formalism, we comment on the approximations needed and provide a comparison between different actions. We also comment on parallelization schemes and advanced sampling techniques. The first results shown in this thesis are for the phase diagram of a one-dimensional Coulomb gas, which have been obtained using the PIMC method. The phase diagram have been constructed mainly by calculating energetic and structural properties. The obtained results extend previous knowledge of different phases in the one-dimensional Coulomb gas at zero temperature. Our results show the existence of a quantum Wigner crystal regime and a Ideal Fermi gas regime at low temperatures. As temperature increases, we reach a classic Wigner crystal regime and a classical gas. In the following chapter we show the results of a quasi-one-dimensional para-H2. The aim of this work is to see how the quasi-one-dimensionality affects the Luttinger parameter when comparing it with the pure one-dimensional case. This is done at zero temperature using PIGS. As para-hydrogen is an important candidate to superfluidity, the main idea behind study a quasi-one-dimensional system is to reduce dimensionality in order to soften intermolecular interaction. For that, we try different external potentials to control the opening of the system in two dimensions. Despite an increase in the Luttinger parameter in the various quasi-one-dimensional cases, it still does not reach the values displaying superfluidity. The next work shown in the thesis is our extensive study of the dynamic structure factor for the 4He. Using Path Integral Monte Carlo, we compute the intermediate scattering function at different temperatures and perform an inversion in order to gain access at the dynamics of the system. Despite the ill-posed problem of this inversion, we obtain results in a qualitative agreement with the experiments and prove that our method of inversion, despite having to yield with inversion problems achieves to obtain better numerical results for 4He at finite temperature than the ones previously reported. In this sense, we provide comparisons with the Maximum Entropy method and with experimental results. The study at different temperatures shows us the dissappearance of the roton peak when we cross T=2.17K from the superfluid regime to the normal fluid. We also observe a kink in the momentum distribution at the superfluid regime that dissappears at higher temperatures, for which does not exist an explanation in the theory. In the final chapter of the thesis we provide a method to sample complex-time correlation functions whose aim is to obtain better dynamic structure factor functions than the ones obtained via pure imaginary-time correlation functions. This model has already been tested for single-particle systems. Our aim is to test it for multi-particle systems, and to see if we can still recover good results at a reasonable high complex-time when the number of particles is closer to the typical simulation values of real systems. We tested it with particles interacting with an harmonic potential. Despite an increased variance compared with the one-particle case, we obtain good results that allow us to obtain the dynamic structure factor. Comparing the results with ones obtained at pure-imaginary time, we show how the complex-time inversion is superior and provides results closer to the exact ones. L'objectiu principal d'aquesta tesi es l'estudi de propietats estàtiques i dinàmiques de diferents fluids quàntics utilitzant tècniques de Monte Carlo quàntiques, principalment emprant el formalisme de path integrals per obtenir resultats tan a temperatura zero com a temperatura finita. Primer de tot, presentem els mètodes de Monte Carlo quàntics més importants, i introduïm el mètode de Path Integral Monte Carlo (PIMC), que fem servir al llarg de tota la tesi, i el mètode de Path Integral Ground State (PIGS), que es una extensió del primer però a temperatura zero. Després d'introduir el formalisme bàsic, comentem les diferents aproximacions necessàries i aportem una comparació entre elles. També expliquem un possible mètode de paral·lelització i tècniques de mostreig avançat. Els primers resultats que mostrem en aquesta tesi son pel diagrama de fases d'un gas de Coulomb unidimensional, que hem obtingut emprant PIMC. Hem construït el diagrama de fases mitjançant el càlcul de propietats energètiques i estructurals. Els nostres resultats amplien estudis previs que s'havien realitzat pel mateix sistema a temperatura zero. Els nostres resultats mostren l'existència d'un règim de cristall de Wigner quàntic i un d'un gas de Fermi ideal a temperatures baixes. Incrementant la temperatura obtenim un cristall de Wigner clàssic i un gas clàssic. En el següent capítol ensenyem els resultats per un sistema quasi-unidimensional de parahidrogen. L'objectiu d'aquest estudi es veure si la quasi-unidimensionalitat afecta al paràmetre de Luttinger quan el comparem pel cas purament unidimensional. Això ho fem a temperatura zero utilitzant PIGS. Sent el parahidrogen un fort candidat a ser superfluid, la idea principal es veure si reduint la dimensionalitat del sistema podem alleugerir suficient la interacció intermolecular. Per fer-ho, provem diferents potencials externs per controlar l'obertura del sistema en dues de les dimensions. Tot i l'increment del paràmetre de Luttinger respecte al cas unidimensional, aquest no arriba als valors esperats per mostrar superfluïdesa. El següents resultats són del nostre estudi sobre el factor d'estructura dinàmic per 4He. Utilitzant PIMC, calculem la funció de dispersió a diferents temperatures i fem una inversió per tal d'accedir a les propietats dinàmiques del sistema. Tot i la naturalesa de problema mal posat d'aquesta inversió, obtenim resultats qualitativament bons en comparació amb els experimentals, i provem que el nostre mètode d'inversió obté resultats superiors per 4He a temperatura finita que els obtinguts prèviament utilitzant altres mètodes. En aquest sentit, aportem una comparació amb el mètode de màxima entropia i amb resultats experimentals. L'estudi a diferents temperatures ens deixa veure la desaparició del pic del rotó quan creuem T=2.17K des de el règim superfluid al fluid normal. També observem una curvatura estranya en la distribució de moments en el règim de superfluïdesa que desapareix a temperatures més elevades, i pel qual no existeix cap explicació teòrica. Finalment, mostrem un mètode per calcular funcions de correlació en temps complex, l'objectiu del qual es obtenir factors d'estructura dinàmic superiors als obtinguts en temps purament imaginari. Aquest model ha sigut provat amb èxit en sistemes d'una sola partícula. El nostre objectiu es veure si obtenim resultats bons en sistemes amb més partícules, i si el temps complex màxim al que podem accedir no es redueix amb aquest increment. Tot i l'increment en la variança, obtenim bons resultats pel factor dinàmic i, comparant-los amb els obtinguts amb temps imaginari, podem veure com el temps complex ofereix resultats més pròxims als exactes.