Understanding the behavior of quantum systems subject to magnetic fields is of fundamental importance and underpins quantum technologies. However, modeling these systems is a complex task, because of many-body interactions and because many-body approaches such as density-functional theory get complicated by the presence of a vector potential into the system Hamiltonian. We use the metric-space approach to quantum mechanics to study the effects of varying the magnetic vector potential on quantum systems. The application of this technique to model systems in the ground state provides insight into the fundamental mapping at the core of current density-functional theory, which relates the many-body wave function, particle density, and paramagnetic current density. We show that the role of the paramagnetic current density in this relationship becomes crucial when considering states with different magnetic quantum numbers $m$. Additionally, varying the magnetic field uncovers a richer complexity for the ``band structure'' present in ground-state metric spaces, as compared to previous studies varying scalar potentials. The robust nature of the metric-space approach is strengthened by demonstrating the gauge invariance of the related metric for the paramagnetic current density. We go beyond ground-state properties and apply this approach to excited states. The results suggest that, under specific conditions, a universal behavior may exist for the relationships between the physical quantities defining the system.
Understanding the behavior of quantum systems subject to magnetic fields is of fundamental importance and underpins quantum technologies. However, modeling these systems is a complex task, because of many-body interactions and because many-body approaches such as density-functional theory get complicated by the presence of a vector potential into the system Hamiltonian. We use the metric-space approach to quantum mechanics to study the effects of varying the magnetic vector potential on quantum systems. The application of this technique to model systems in the ground state provides insight into the fundamental mapping at the core of current density-functional theory, which relates the many-body wave function, particle density, and paramagnetic current density. We show that the role of the paramagnetic current density in this relationship becomes crucial when considering states with different magnetic quantum numbers m. Additionally, varying the magnetic field uncovers a richer complexity for the ``band structure'' present in ground-state metric spaces, as compared to previous studies varying scalar potentials. The robust nature of the metric-space approach is strengthened by demonstrating the gauge invariance of the related metric for the paramagnetic current density. We go beyond ground-state properties and apply this approach to excited states. The results suggest that, under specific conditions, a universal behavior may exist for the relationships between the physical quantities defining the system.
The metric space approach to quantum mechanics is a new, powerful method for deriving metrics for sets of quantum mechanical functions from conservation laws. We develop this approach to show that, from a standard form of conservation law, a universal method exists to generate a metric for the physical functions connected to that conservation law. All of these metric spaces have an "onion-shell" geometry consisting of concentric spheres, with functions conserved to the same value lying on the same sphere. We apply this approach to generate metrics for wavefunctions, particle densities, paramagnetic current densities, and external scalar potentials. In addition, we demonstrate the extensions to our approach that ensure that the metrics for wavefunctions and paramagnetic current densities are gauge invariant. \n \nWe use our metric space approach to explore the unique relationship between ground-state wavefunctions, particle densities and paramagnetic current densities in Current Density Functional Theory (CDFT). We study how this relationship is affected by variations in the external scalar potential, pairwise electronic interaction strength, and magnetic field strength. We find that all of the metric spaces exhibit a "band structure", consisting of "bands" of points characterised by the value of the angular momentum quantum number, m. These "bands" were found to either be separated by "gaps" of forbidden distances, or be "overlapping". We also extend this analysis beyond CDFT to explore excited states. \n \nWe apply our metrics in order to gain new insight into the Hohenberg-Kohn theorem and the Kohn-Sham scheme of Density Functional Theory. For the Hohenberg-Kohn theorem, we find that the relationship between potential and wavefunction metrics, and between potential and density metrics, is monotonic and includes a linear region. Comparing Kohn-Sham quantities to many-body quantities, we find that the distance between them increases as the electron interaction dominates over the external potential.
External potentials play a crucial role in modeling quantum systems, since, for a given interparticle interaction, they define the system Hamiltonian. We use the metric-space approach to quantum mechanics to derive, from the energy conservation law, two natural metrics for potentials. We show that these metrics are well defined for physical potentials, regardless of whether the system is in an eigenstate or if the potential is bounded. In addition, we discuss the gauge freedom of potentials and how to ensure that the metrics preserve physical relevance. Our metrics for potentials, together with the metrics for wave functions and densities from I. D'Amico et al. [Phys. Rev. Lett. 106, 050401 (2011)] paves the way for a comprehensive study of the two fundamental theorems of density-functional theory. We explore these by analyzing two many-body systems for which the related exact Kohn-Sham systems can be derived. First we consider the information provided by each of the metrics, and we find that the density metric performs best in distinguishing two many-body systems. Next we study for the systems at hand the one-to-one relationships among potentials, ground-state wave functions, and ground-state densities defined by the Hohenberg-Kohn theorem as relationships in metric spaces. We find that, in metric space, these relationships are monotonic and incorporate regions of linearity, at least for the systems considered. Finally, we use the metrics for wave functions and potentials in order to assess quantitatively how close the many-body and Kohn-Sham systems are: We show that, at least for the systems analyzed, both metrics provide a consistent picture, and for large regions of the parameter space the error in approximating the many-body wave function with the Kohn-Sham wave function lies under a threshold of 10%.
We show that conservation laws in quantum mechanics naturally lead to metric spaces for the set of related physical quantities. All such metric spaces have an ``onion-shell'' geometry. We demonstrate the power of this approach by considering many-body systems immersed in a magnetic field, with a finite ground state current. In the associated metric spaces we find regions of allowed and forbidden distances, a ``band structure'' in metric space directly arising from the conservation of the z component of the angular momentum.
Hilbert space combines the properties of two different types of mathematical spaces: vector space and metric space. While the vector-space aspects are widely used, the metric-space aspects are much less exploited. Here we show that a suitable metric stratifies Fock space into concentric spheres on which maximum and minimum distances between states can be defined and geometrically interpreted. Unlike the usual Hilbert-space analysis, our results apply also to the reduced space of only ground states and to that of particle densities, which are metric, but not Hilbert, spaces. The Hohenberg-Kohn mapping between densities and ground states, which is highly complex and nonlocal in coordinate description, is found, for three different model systems, to be simple in metric space, where it becomes a monotonic and nearly linear mapping of vicinities onto vicinities.
Magnetism is a subject that has fascinated mankind for countless generations. With the development of quantum mechanics around a century ago a fundamental understanding of many of the underlying causes of magnetism were obtained. However, the long range interaction combined with the complex systems magnetism appears in makes it very hard to investigate it. In our days this research area is more active than ever, mainly due to future demand of the electronic industry for components engineered down to the atomic level. Quantum effects gets more important at short scales, especially in lower dimensional materials like sheets and wires. The rapidly increasing amount of computational power available makes the numerical techniques a more important part of the research effort. Many of these are especially well suited to analyze lower dimensional quantum problems. Three of the most important techniques, (classical) Monte Carlo, Exact Diagonalization (ED) and matrix product states (MPS) based techniques, like density renormalization group (DMRG) and time evolving block decimation (TEBD) will be described in some detail and put to use later in this dissertation.With the rapid development of experimental techniques for ultracold gases in optical traps a new approach to investigate magnetic properties that are hard achieve or control in the solid state has emerged. We first study the ground-state phase diagram of a spin-1 condensate trapped in an optical trap when the magnetic dipole interaction between the atoms is taken into account along with confinement and spin precession. The boundaries between the regions of ferromagnetic and polar phases move as the dipole strength is varied and the ferromagnetic phases can be modulated. The magnetization of the ferromagnetic phase perpendicular to the field becomes modulated as a helix winding around the magnetic field direction, with a wavelength inversely proportional to the dipole strength. This modulation should be observable for current experimental parameters in 87Rb. Hence the much-sought supersolid state, with broken continuous translation invariance in one direction and broken global U(1) invariance, occurs generically as a metastable state in this system as a result of dipole interaction. The ferromagnetic state parallel to the applied magnetic field becomes striped in a finite system at strong dipolar coupling.The development of artificial gauge fields, that can mimic magnetic fields, in ultracold gases suggests that atomic realization of fractional quantum Hall physics will become experimentally practical in the near future. While it is known that bosons on lattices can support quantum Hall states, the universal edge excitations that provide the most likely experimental probe of the topological order have not been obtained. We find that the edge excitations of an interacting boson lattice model are surprisingly sensitive to interedge hybridization and edge-bulk mixing for some confining potentials. With properly chosen potentials and fluxes, the edge spectrum is surprisingly clear even for small systems with strong lattice effects such as bandwidth. Various fractional quantum Hall phases for bosons can be obtained, and the phases ν=1/2 and ν=2/3 have the edge spectra predicted by the chiral Luttinger liquid theory.Also, some of the traditional experimental techniques for detecting magnetic order and dynamics in solid state materials, like neutron scattering has had somewhat of a renaissance lately. In a recent experiment on CoNb2O6, Coldea et. al. found for the first time experimental evidence of the exceptional Lie algebra E8. The emergence of this symmetry was theoretically predicted long ago for the transverse quantum Ising chain in the presence of a weak longitudinal field. We consider an accurate microscopic model of CoNb2O6 incorporating additional couplings and calculate numerically the dynamical structure function using a recently developed matrix-product-state method. The excitation spectra show bound states characteristic of the weakly broken textrmE8 symmetry. We compare the observed bound state signatures in this model to those found in the transverse Ising chain in a longitudinal field and to experimental data.Finally, we investigate the ground state phase diagram of a related quantum spin chain, the S=2 XXZ chain with single-ion anisotropy. The interest in this system comes mainly from connecting the highly quantum mechanical spin-1 phase diagram with the classical S=∞ phase diagram. While most of these questions where believed to have been satisfactorily answered mainly with DMRG, some recent studies have questioned some of the conclusions. We use several of the recent advances within DMRG and perform a detailed analysis of the whole phase diagram. We extend the phase diagram by considering different types of single ion anisotropies which help us to answer two important questions: First we show that one can adiabatically move from the isotropic Heisenberg point to the so-called large-D phase with a continuous change of the Hamiltonian. Second, we can tune the model into a predicted intermediate phase which is equivalent to the topologically non-trivial spin-1 Haldane phase. Furthermore, we study the spin-3 XXZ chain to help explaining the development of the classical phase diagram.
By applying the projector to the filled lattice eigenstates on a specific position, or applying the local electron annihilation operator on the many-body ground state, one can construct a quantum state localized around a specific position in a solid. The overlap of two such local states at slightly different positions defines a quantum metric in real space, which manifests even in systems as simple as particles in a box. For continuous systems like electron gas, this metric weighted by the density gives the momentum variance of electrons, which is readily measurable by ARPES. The presence of disorder curves the real space manifold and gives rise to various differential geometrical quantities like Riemann tensor and Ricci scalar, indicating the possibility of engineering differential geometrical properties by disorder, as demonstrated by lattice models of 2D metals and topological insulators.