# Spin symmetry breaking in the translation-invariant Hartree-Fock electron gas

## TL;DR

This paper investigates spin symmetry breaking in a translation-invariant Hartree-Fock electron gas, revealing phase transitions between ferromagnetic and paramagnetic states at zero and positive temperatures, with numerical phase diagrams for Coulomb interactions.

## Contribution

It provides a rigorous proof of a first order phase transition in the Coulomb case and explores multiple transitions for other potentials, including numerical phase diagrams at positive temperature.

## Key findings

- Existence of a unique first order transition at zero temperature for Coulomb interaction.
- Identification of multiple phase transitions for other interaction potentials.
- Numerical phase diagram showing paramagnetic and ferromagnetic regions at positive temperature.

## Abstract

We study the breaking of spin symmetry for the nonlinear Hartree-Fock model describing an infinite translation-invariant interacting quantum gas (fluid phase). At zero temperature and for the Coulomb interaction in three space dimensions, we can prove the existence of a unique first order transition between a pure ferromagnetic phase at low density and a paramagnetic phase at high density. Multiple first or second order transitions can happen for other interaction potentials, as we illustrate on some examples. At positive temperature $T>0$ we compute numerically the phase diagram in the Coulomb case. We find the paramagnetic phase at high temperature or high density and a region where the system is ferromagnetic. We prove that the equilibrium state is unique and paramagnetic at high temperature or high density.