# The reduced Hartree-Fock model with self-generated magnetic fields

## TL;DR

This paper analyzes the stability of the reduced Hartree-Fock model with self-generated magnetic fields, identifying a critical fine structure constant that determines system stability and establishing conditions for the existence of minimizers.

## Contribution

It introduces a critical value for the fine structure constant in the model and characterizes it via a minimization problem, extending understanding of stability in quantum systems with magnetic fields.

## Key findings

- Existence of a critical  for stability
- Explicit characterization of  as a minimization problem
- Existence of minimizers for neutral or positively charged systems

## Abstract

We study the well-posedness of the reduced Hartree-Fock model for molecules and perfect crystals when taking into account a self-generated magnetic field. We exhibit a critical value $\alpha_c > 0$ such that, if the fine structure constant $\alpha$ is smaller than $\alpha_c$, then the corresponding system is stable, whereas if $\alpha$ is greater than $\alpha_c$, it is unstable. We give an explicit characterisation of $\alpha_c$ as a minimisation problem over the set of zero-modes, and we prove that the critical values for the molecular case and the periodic case coincide. Finally, we prove the existence of minimisers when the system is neutral or positively charged.