After describing these advancements in methodology, we apply our new technology to fluid sodium near its liquid-vapor critical point. In particular, we explore the microscopic mechanisms which drive the continuous change from a dense metallic liquid to an expanded insulating vapor above the critical temperature. We show that the dynamic aggregation and dissociation of clusters of atoms play a significant role in determining the conductivity and that the formation of these clusters is highly density and temperature dependent. Finally, we suggest several avenues for research to further improve our simulations.
After describing these advancements in methodology, we apply our new technology to fluid sodium near its liquid-vapor critical point. In particular, we explore the microscopic mechanisms which drive the continuous change from a dense metallic liquid to an expanded insulating vapor above the critical temperature. We show that the dynamic aggregation and dissociation of clusters of atoms play a significant role in determining the conductivity and that the formation of these clusters is highly density and temperature dependent. Finally, we suggest several avenues for research to further improve our simulations.
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.
Metallic quantum critical phenomena are believed to play a key role in many strongly correlated materials, including high-temperature superconductors. Theoretically, the problem of quantum criticality in the presence of a Fermi surface has proven to be highly challenging. However, it has recently been realized that many models used to describe such systems are amenable to numerically exact solution by quantum Monte Carlo (QMC) techniques, without suffering from the fermion sign problem. In this review, we examine the status of the understanding of metallic quantum criticality and the recent progress made by QMC simulations. We focus on the cases of spin-density wave and Ising nematic criticality. We describe the results obtained so far and their implications for superconductivity, non-Fermi liquid behavior, and transport near metallic quantum critical points. Some of the outstanding puzzles and future directions are highlighted.
We present a new quantum molecular dynamics (MD) method where the electronic structure and atomic forces are solved by a real-space dynamical mean-field theory (DMFT). Contrary to most quantum MD methods that are based on effective single-particle wave functions, the DMFT approach is able to describe correlation-induced Mott metal-insulator transitions and the associated incoherent electronic excitations in an atomic liquid. We apply the DMFT-MD method to study Mott transitions in an atomic liquid model which can be viewed as the liquid-state generalization of the Hubbard model. The half-filled Hubbard liquids also provide a minimum model for alkali fluid metals. Our simulations uncover two distinct types of Mott transition depending on the atomic bonding and short-range structures in the electronically delocalized phase. In the first scenario where atoms tend to form dimers, increasing the Hubbard repulsion gives rise to a transition from a molecular insulator to an atomic insulator with a small window of enhanced metallicity in the vicinity of the localization transition. On the other hand, for Hubbard liquids with atoms forming large conducting clusters, the localization of electrons leads to the fragmentation of clusters and is intimately related to the liquid-gas transition of atoms. Implications of our results for metal-insulator transitions in fluid alkali metals are discussed.
We determine the critical temperature of a 3D homogeneous system of hard-sphere bosons by path-integral Monte Carlo simulations and finite-size scaling. In the low density limit, we find that the critical temperature is increased by the repulsive interactions, as $\ensuremath{\Delta}{T}_{C}{/T}_{0}\ensuremath{\sim}({\mathrm{na}}^{3}{)}^{\ensuremath{\gamma}}$, where $\ensuremath{\gamma}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.34\ifmmode\pm\else\textpm\fi{}0.03$. At high densities the result for liquid helium, namely, a lower critical temperature than in the noninteracting case, is recovered. We give a microscopic explanation for the observed behavior.
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.