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Path Integral studies of quantum systems at finite temperatures

Sergei D. Ivanov

Published Jan 1, 2005• 1 citation
OpenAlex

Abstract

This thesis presents and develops the path integral simulation techniques in application to small quantum systems at finite temperatures. The first goal is to obtain exact thermodynamic expressions for systems of noninteracting The rest and the major part of the thesis is dedicated to the development and testing of Bead-Fourier path integral molecular dynamics. Although, path integral molecular dynamics as well as path integral Monte Carlo are well First, molecular dynamics under Bead-Fourier scheme was developed and tested on the examples of quantum harmonic oscillator and Hydrogen atom. The main attention was paid to ergodicity problems. Then we addressed the question, Later, the formalism for identical particles was developed. Finally, the question of molecular dynamics efficacy was raised. It was shown, that formalisms for identical and distinguishable particles, both, can be reformulated into a more efficient ones, providing all dynamical variables

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This thesis presents and develops the path integral simulation techniques in application to small quantum systems at finite temperatures. The first goal is to obtain exact thermodynamic expressions for systems of noninteracting The rest and the major part of the thesis is dedicated to the development and testing of Bead-Fourier path integral molecular dynamics. Although, path integral molecular dynamics as well as path integral Monte Carlo are well First, molecular dynamics under Bead-Fourier scheme was developed and tested on the examples of quantum harmonic oscillator and Hydrogen atom. The main attention was paid to ergodicity problems. Then we addressed the question, Later, the formalism for identical particles was developed. Finally, the question of molecular dynamics efficacy was raised. It was shown, that formalisms for identical and distinguishable particles, both, can be reformulated into a more efficient ones, providing all dynamical variables
Using imaginary time, quantum statistical mechanics of a finite-temperature system in thermodynamical equilibrium turns into a problem of classical ring polymers made of particle paths. We need to consider all possible ring polymers at a given temperature, which again is where Monte Carlo methods are effective, especially using a so-called worm algorithm. We show how paths can be sampled using Monte Carlo, and how the physical properties, such as energy and radial distribution, are computed from the paths. Identical particles can have their paths exchanged at will, and we see how boson superfluidity comes out of this picture. Finally, the stochastic series expansion method is introduced to show finite temperature Hubbard model can be simulated.
This thesis describes several novel approaches in quantum molecular dynamics for obtaining properties of molecular systems in different regimes. We investigate ground state properties of chains of linear rotors with dipole–dipole interactions via the density matrix renormalization group (DMRG), by deriving the appropriate form of the interaction operator and implementing it in ITensor. This provides us with further evidence of a quantum phase transition in this system. We also improve the sampling of Gaussian mixture distributions for finite temperature path integral Monte Carlo (PIMC) of vibronic Hamiltonians. To do this, we replace random sampling by quasi-random sampling, and improve sampling distributions by optimizing their parameters. Finally, we introduce estimators and integrators for constrained free energy simulations in path integral molecular dynamics (PIMD). This method is applied to the study of a water dimer, for which we obtain a quantum potential of mean force.
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.
A quantum transfer matrix method is proposed and examined. To obtain finite temperature properties, a small number of Monte Carlo samples for the trace summation is taken without the Monte Carlo sampling of the path integral. We introduce the method of a random orthonormal base in the Monte Carlo sampling. This makes it possible to investigate larger size systems than the exact diagonalization. An advantage of this method is that it does not have negative sign difficulty in the path integral as contrast to the usual quantum Monte Carlo method. A quantum molecular dynamics method is also proposed to investigate dynamical correlation functions.
In this article we develop a general method to numerically calculate physical properties for a system of anyons with path integral molecular dynamics. We provide a unified method to calculate the thermodynamics of identical bosons, fermions, and anyons. Our method is tested and applied to systems of anyons, bosons, and fermions in a two-dimensional harmonic trap. We also consider a method to calculate the energy for fermions as an application of the path integral molecular dynamics to simulate the anyon model.
By generalizing the recently developed path integral molecular dynamics for identical bosons and fermions, we consider the finite-temperature thermodynamic properties of fictitious identical particles with a real parameter ξ interpolating continuously between bosons (ξ = 1) and fermions (ξ = -1). Through general analysis and numerical experiments, we find that the average energy may have good analytical properties as a function of this real parameter ξ, which provides the chance to calculate the thermodynamical properties of identical fermions by extrapolation with a simple polynomial function after accurately calculating the thermodynamic properties of the fictitious particles for ξ ≥ 0. Using several examples, it is shown that our method can efficiently give accurate energy values for finite-temperature fermionic systems. Our work provides a chance to circumvent the fermion sign problem for some quantum systems.