This thesis considers (1) novel manifestations and applications of quantum interference in complex systems and (2) development of new approaches to study complex quantum systems. First, I examine a general model of particles with long-range hopping amplitudes. For at least 30 years, it has been widely accepted that these particles do not undergo Anderson localization in 3D lattices. We show that these particles do undergo Anderson localization in 3D lattices if their hopping amplitudes are isotropic. In contrast, particles with anisotropic long-range hopping appear to follow the widely held expectations. We show these results by demonstrating that cooperative shielding extends to 3D cubic lattices with isotropic long-range hopping, but not with anisotropic long-range hopping and by computing the scaling behaviour of participation ratios and energy level statistics. Secondly, I develop a fully quantum mechanical model of molecular surface spin-echo experiments, which study surface properties and dynamics by scattering molecules off the sample surface. This model, based on the transfer matrix method, incorporates molecular hyperfine degrees of freedom, allows for the efficient calculation of the experimental signal given a molecule-surface scattering matrix, and permits us to begin addressing the inverse scattering problem. This fully quantum model is required to properly describe these experiments as the semi-classical methods used to describe experiments using helium-3 atoms do not take magnetic-field induced momentum changes into account. We apply our method to the case of ortho-hydrogen and then apply Bayesian optimization to determine the molecule-surface scattering matrix elements from a calculated signal, for a scattering matrix defined by three parameters. Our work sets the stage for Bayesian optimization to solve the inverse scattering problem for these experiments. Finally, I propose using Bayesian model calibration to improve the convergence of Monte Carlo calculations in regions where the sign problem or critical slowing down are an issue. Specifically, Bayesian model calibration would correct poorly converged Monte Carlo calculations with the information from a small number of well-converged Monte Carlo calculations. As a simple proof of principle demonstration, we apply Bayesian model calibration to a diagrammatic Monte Carlo calculation of the scattering length of a spherical potential barrier.
This thesis considers (1) novel manifestations and applications of quantum interference in complex systems and (2) development of new approaches to study complex quantum systems. First, I examine a general model of particles with long-range hopping amplitudes. For at least 30 years, it has been widely accepted that these particles do not undergo Anderson localization in 3D lattices. We show that these particles do undergo Anderson localization in 3D lattices if their hopping amplitudes are isotropic. In contrast, particles with anisotropic long-range hopping appear to follow the widely held expectations. We show these results by demonstrating that cooperative shielding extends to 3D cubic lattices with isotropic long-range hopping, but not with anisotropic long-range hopping and by computing the scaling behaviour of participation ratios and energy level statistics. Secondly, I develop a fully quantum mechanical model of molecular surface spin-echo experiments, which study surface properties and dynamics by scattering molecules off the sample surface. This model, based on the transfer matrix method, incorporates molecular hyperfine degrees of freedom, allows for the efficient calculation of the experimental signal given a molecule-surface scattering matrix, and permits us to begin addressing the inverse scattering problem. This fully quantum model is required to properly describe these experiments as the semi-classical methods used to describe experiments using helium-3 atoms do not take magnetic-field induced momentum changes into account. We apply our method to the case of ortho-hydrogen and then apply Bayesian optimization to determine the molecule-surface scattering matrix elements from a calculated signal, for a scattering matrix defined by three parameters. Our work sets the stage for Bayesian optimization to solve the inverse scattering problem for these experiments. Finally, I propose using Bayesian model calibration to improve the convergence of Monte Carlo calculations in regions where the sign problem or critical slowing down are an issue. Specifically, Bayesian model calibration would correct poorly converged Monte Carlo calculations with the information from a small number of well-converged Monte Carlo calculations. As a simple proof of principle demonstration, we apply Bayesian model calibration to a diagrammatic Monte Carlo calculation of the scattering length of a spherical potential barrier.
This thesis illustrates the use of machine learning algorithms and exact numerical methods to study quantum observables for different systems. The first part of this thesis depicts how to construct accurate potential energy surfaces (PESs) using supervised learning algorithms such as Gaussian Process (GP) regression. PESs have a leading part in quantum chemistry since they are used to study chemical reaction dynamics. Constructing the PES from quantum reactive scattering calculations, as the reaction probability, is known as the inverse scattering problem. Here, we illustrate a possible solution to the inverse scattering problem with a two-tiered GP model one GP model interpolates the PES and the second in Bayesian optimization (BO) algorithm. The end result is an accurate PES constructed with a GP with fewer points than with standard methods previously used for PES. BO is an optimization algorithm for black-box functions that use GP regression as an approximation of the interrogative function. We applied BO to find the optimal parameters of hybrid-density functionals. Quantum observables can differ between phases of matter. GP models with kernel combinations can extrapolate quantum observables such as the polaron dispersion energy between different phases and discover phases of matter. The same algorithm can predict quantum observables where standard numerical techniques lack convergence. In the second half of the dissertation, we studied the evolution of quantum walks in various graphs with Hamiltonians permitting particle number changes. We showed that particle number-changing interactions accelerate quantum walks for any of the graph considered. Quantum simulators to study many-body physics is an active research field. We proposed the use of magnetic atoms trapped in optical lattices to experimentally mimic Bose-Hubbard type models by preparing atoms in different Zeeman states.
This thesis comprises a trio of separate yet thematically-linked investigations within the field of electronic structure that address challenging problems to which quantum Monte Carlo, the most accurate electronic structure method capable of treating large extended systems, has been applied only rarely or indeed never before. The first is the study of polarization and localization in many body systems. Density functional and variational quantum Monte Carlo methods are used to examine the behaviour of the many-electron localization length near band insulator to metal transitions in various oneand two-dimensional model systems. We examine when the localization length, known to be infinite in metals and finite in insulators and assumed to diverge at a metal-insulator transition, serves as a useful diagnostic of the transition. We also present a novel derivation of the formulae for polarization and localization in many body systems that expresses the connection of these two quantities to the concept of Many Body Wannier Functions. The second is a study of the surface energy of the electron gas. The electron gas, or ‘jellium’, is an important benchmark system for electronic structure methods and serves as a simple model for metals. We apply diffusion Monte Carlo (DMC), and a new method for determining surface energies without an unreliable comparison of bulk and slab results, to the calculation of the surface energy. Our method improves on previous calculations in a number of ways and removes many sources of error that plagued previous calculations. The final study applies DMC to the properties, specifically the formation energy, of point defects in Alumina. Alumina is an important technological material, and there are many important unanswered questions relating to its properties that stem from the behaviour and concentration of its point defects. We address many of the problems associated with the calculation of formation energies with electronic structure calculations, and present the first calculations of formation energies of charged point defects in DMC.
Ultracold atomic and molecular systems provide a useful platform for understanding quantum many-body physics. Recent progresses in AMO experiments enable access to systems exhibiting long-range interactions, opening a window for exploring the interplay between long-range interactions and dissipation. In this thesis, I develop theoretical approaches to study non-equilibrium dynamics in systems where such interplay is crucial.I first focus on a system of KRb molecules, where dipolar interactions and fast chemical reactions coexist. Using a classical kinetic theory and Monte Carlo methods, I study the evaporative cooling in a quasi-two-dimensional trap, and develop a protocol to reach quantum degeneracy. I also study the case where molecules are loaded into an optical lattice, and show that the strong dissipation induces a quantum Zeno effect, which suppresses the molecule loss. The analysis requires including multiple bands to explain recent experimental measurements, and can be used to determine the molecular filling fraction.I also investigate a system of radiating atoms, which experience long-range elastic and dissipative interactions. I explore the collective behavior of atoms and the role of atomic motion. The model is validated by comparison with a recent light scattering experiment using Sr atoms. I also show that incoherently pumped dipoles can undergo a dynamical phase transition to synchronization, and study its signature in the quantum regime.
In this thesis, the numerical simulation of strongly-interacting many-body quantum-mechanical systems using matrix product states (MPS) is considered. Compared to classical systems, quantum many-body systems possess an exponentially enlarged number of degrees of freedom, significantly complicating a simulation on a classical computer. Matrix-Product-States are a novel representation of arbitrary quantum many-body states. Using quantum information theory, it is possible to show that Matrix-Product-States provide a polynomial-sized representation of one-dimensional quantum systems, thus allowing an efficient simulation of one-dimensional quantum system on classical computers. Matrix-Product-States form the conceptual framework of the density-matrix renormalization group (DMRG). Based upon this connection, deeper understanding of the density matrix renormalization group can be obtained. After a general introduction in the first chapter of this thesis, the second chapter deals with Matrix-Product-States, focusing on the development of fast and stable algorithms. It is possible to extend the Matrix-Product-States approach to be able to represent arbitrary operators, the so-called Matrix-Product-Operators, which allows a fast and flexible calculation of arbitrary expectation values. To obtain algorithms to efficiently calculate groundstates, the density-matrix renormalization group is reformulated using the Matrix-Product-States framework. Further, time-dependent problems are considered. Two different algorithms are presented, one based on a Trotter decomposition of the time-evolution operator, the other one on Krylov subspaces. Finally, the evaluation of dynamical spectral functions is discussed, and a correction vector-based method is presented. In the following chapters, the methods presented in the second chapter, are applied to a number of different physical problems. The third chapter deals with the existence of chiral phases in isotropic one-dimensional quantum spin systems. A preceding analytical study based on a mean-field approach indicated the possible existence of those phases in an isotopic Heisenberg model with a frustrating zig-zag interaction and a magnetic field. In this thesis, the existence of the chiral phases will be shown numerically by using Matrix-Product-States-based algorithms. A key effect of interacting one-dimensional quantum-mechanical many-body systems is the spin-charge separation. However, up to now only signs of the spin-charge separation have been observed in experiments. In the fourth chapter, we propose an experiment using ultracold atomic gases in optical lattices, which allows a well controlled observation of the spin-charge separation (of different hyperfine states of the ultracold atoms) with current state of the art experimental techniques. Ultracold atoms in optical lattices are well described by (Bose)-Hubbard models. In order to support this proposal, we present numerical results for realistic system parameters. Matrix-Product-States are an excellent tool for the simulation of one-dimensional quantum systems, however, they are not well suited for the simulation of higher dimensional systems. For strongly-correlated systems, for instance cuprates-based high-temperature superconductors, quantum fluctuations play an essential role. Classical mean-field theories neglect any kind of fluctuations, thus they are not suitable to describe strongly-correlated systems. The dynamical mean-field theory (DMFT) fully takes local quantum fluctuations into account but neglects any kind of spatial fluctuations. The many-body problem on the lattice is mapped onto an impurity problem, which needs to be solved self-consistently. In the last chapter of this thesis, Matrix-Product-States-based algorithms are used to solve the impurity problem of the dynamical mean-field theory. We present results for a Hubbard model on a one-dimensional lattice and on a Bethe lattice obtained by the dynamical mean-field and compare them with exact results.
Many strongly correlated quantum systems are difficult to study numerically because of the exponential growth of the Hilbert space combined with the failure of numerical methods in important parameter regimes. This includes the long-standing problems of high-temperature superconductivity and frustrated magnetism. A promising solution to this dilemma are quantum simulators, well-controlled (synthetic) quantum systems, emulating model Hamiltonians not only of condensed matter systems, where parameters can be tuned at will and model complexity can be added step by step. This dissertation presents numerical simulation studies that are motivated by quantum simulators. Using large-scale quantum Monte Carlo computer simulations and mean-field theory the semiquantitative ground state phase diagram of a Penning ion trap quantum simulator, realizing a two-dimensional model of quantum magnetism with long-range interactions, is obtained. We study also thermal and quantum phase transitions in one-dimensional transverse-field Ising models with long-range interactions, which exhibit a Kosterlitz-Thouless transition and universality classes with continuously varying critical exponents. The second part of this thesis is concerned with the Fermi-Hubbard model, which has been realized recently with ultracold atoms in optical lattices, both for repulsive and attractive interactions. Equipped with a quantum gas microscope, which allows single-site single-atom detection, these experimental setups can collect histograms of the particle number from repeated projective measurements. This dissertation presents a method to compute such full counting statistics for quadratic operators such as particle number or magnetization in the framework of determinantal quantum Monte Carlo. From an analysis of the full counting statistics of the particle number for different subsystem sizes the size of preformed pairs or Cooper pairs in the attractive Hubbard model is inferred. Furthermore, excellent agreement of the numerically computed probability distribution of the staggered magnetization with a recent quantum gas experiment demonstrates that current experiments are capable of resolving the differences between the Hubbard model and its limiting Heisenberg model. An extensive method section reviews the stochastic series expansion quantum Monte Carlo method for Ising models in a transverse field, which is free of the sign problem for arbitrary interactions, with special emphasis on the efficient simulation of long-range interactions. Also a self-contained introduction to the determinantal quantum Monte Carlo method is provided.
This thesis details four research projects related to zero temperature quantum Monte Carlo. Chapters 2-4 focus on continuum quantum Monte Carlo, and specifically its application to molecular systems; whereas Chapter 5 focuses on quantum Monte Carlo in a discrete space. Chapter 2 focuses on improving upon the single-particle basis functions employed in quantum Monte Carlo calculations for molecular systems. For calculations requiring non-diverging pseudopotentials, a class of functions is introduced that is capable of producing the short- and long-range asymptotic behavior of the exact wavefunction. It is demonstrated that this form of basis function produces superior accuracy and efficiency when compared to the basis sets typically employed in quantum Monte Carlo. Although the basis functions introduced in Chapter 2 are capable of producing superior results, it is necessary that the parameters of the functional form are near-optimal for the full potential of the functions to be realized. Chapter 3 introduces a simple yet general method for constructing basis sets of a desired functional form appropriate for molecular electronic structure calculations. A standard basis set is created for each of the elements from hydrogen to argon. Chapter 4 explores the effect of different aspects of the trial wavefunction on the accuracy of quantum Monte Carlo. By systematically testing the effect of the basis size, orbital quality, and determinant expansion quality, this work offers guidance to quantum Monte Carlo practitioners for achieving results to within chemical accuracy of experiment. In Chapter 5, semistochastic projection, a hybrid of deterministic and stochastic projection, is introduced for finding the dominant eigenvalue and eigenvector of a matrix. This method, like stochastic projection, is applicable to matrices well beyond the size that can be handled by deterministic methods. Semistochastic projection improves over stochastic projection by significantly reducing the computational time required to obtain the eigenvalue within a specified statistical uncertainty. After the semistochastic projection method is introduced, it is applied to determine the ground state energy of the Hamiltonian in a discrete basis. This special case of semistochastic projection, dubbed semistochastic quantum Monte Carlo, is shown to be orders of magnitude more efficient than stochastic quantum Monte Carlo.