Metric-space analysis of systems immersed in a magnetic field

Abstract

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 wavefunction, 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.

Metric-space approach to potentials and its relevance to density-functional theory

“…In order to derive a metric for external potentials, we use the metric space approach to quantum mechanics [4][5][6], which allows us to derive metrics from conservation laws of the form…”

“…In this work, we apply the metric space approach to quantum mechanics [4][5][6] to potentials in order to gain insight into the two fundamental theorems of DFT. First, we use the general procedure from Ref.…”

Metric-Space Approach for Distinguishing Quantum Phase Transitions in Spin-Imbalanced Systems

“…In metric space one can assign distances between wave functions, densities and external potentials of two systems, which quantify the closeness between the systems. Recently natural distances have been proposed for physical systems [8][9][10] and applied in several contexts [11][12][13][14][15][16][17][18].…”

Metrics for Two Electron Random Potential Systems

“…Although this is a key concept for DFT, the relationship is not fully understood. Here we follow on from previous research [4][5][6][7], using these metrics to further understanding of this oneto-one mapping of density and wavefunction.…”