Atoms with bosonic "electrons" in strong magnetic fields

TL;DR

This paper analyzes the ground state properties of bosonic atoms in strong magnetic fields, identifying three asymptotic regimes and introducing a magnetic Hartree functional to describe the system's behavior.

Contribution

It introduces a magnetic Hartree functional for bosonic atoms in strong magnetic fields and characterizes three asymptotic regimes based on the magnetic field strength.

Findings

Region 1: Standard Hartree theory applies.

Region 3: Ground state described by Hyper-Strong functional, independent of statistics.

Region 2: Magnetic Hartree functional captures the system's properties.

Abstract

We study the ground state properties of an atom with nuclear charge $Z$ and $N$ bosonic ``electrons'' in the presence of a homogeneous magnetic field $B$. We investigate the mean field limit $N\to\infty$ with $N/Z$ fixed, and identify three different asymptotic regions, according to $B\ll Z^2$, $B\sim Z^2$, and $B\gg Z^2$. In Region 1 standard Hartree theory is applicable. Region 3 is described by a one-dimensional functional, which is identical to the so-called Hyper-Strong functional introduced by Lieb, Solovej and Yngvason for atoms with fermionic electrons in the region $B\gg Z^3$; i.e., for very strong magnetic fields the ground state properties of atoms are independent of statistics. For Region 2 we introduce a general {\it magnetic Hartree functional}, which is studied in detail. It is shown that in the special case of an atom it can be restricted to the subspace of zero angular momentum parallel to the magnetic field, which simplifies the theory considerably. The functional reproduces the energy and the one-particle reduced density matrix for the full $N$-particle ground state to leading order in $N$, and it implies the description of the other regions as limiting cases.