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.
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.
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 study metric representations in finite-dimensional non-Hermitian quantum mechanics. The main purpose of this work is to emphasize that the physical description of a non-Hermitian system may be formulated in different, but isomorphic, Hilbert spaces. In particular, within the Krein-space formalism, we show that the vector space endowed with an indefinite Krein metric can be explicitly related to the standard Hilbert space through a suitable isomorphism. This observation is essential for a consistent description of non-Hermitian Hamiltonians. Physical states, metrics, and operators must be transported through the corresponding Hilbert-space isomorphism. In this way, equivalent representations of the same system can be used without changing the physical content of the theory. We illustrate these theoretical aspects by studying a two-level non-Hermitian spin model. We use the Robertson uncertainty relation as a consistency test. Apparent violations can arise when operators and states are kept fixed while the metric is changed, and therefore reflect a mismatch of representations rather than a failure of the uncertainty principle.
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.
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.
By acting the projector to the filled lattice eigenstates on a specific position, or applying 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.