Metric space analysis of systems immersed in a magnetic field

TL;DR

This paper applies the metric space approach to quantum mechanics to analyze how magnetic vector potentials influence many-body quantum systems, revealing new insights into density functional theory and the role of current densities.

Contribution

It introduces a metric space framework to study magnetic effects in quantum systems, highlighting the importance of paramagnetic current density and gauge invariance in the fundamental mapping.

Findings

Paramagnetic current density is crucial for states with different magnetic quantum numbers.

Varying magnetic fields reveals complex band structures in metric spaces.

The metric space approach is gauge-invariant and applicable to excited states.

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.