Symmetry breaking in the self-consistent Kohn-Sham equations

TL;DR

This paper investigates how the self-consistent Kohn-Sham equations for fermion systems exhibit symmetry breaking at zero temperature, leading to stable non-uniform solutions such as crystallized structures.

Contribution

It demonstrates the conditions under which symmetry breaking occurs in Kohn-Sham equations and provides explicit examples including fermions on a sphere crystallizing into C60-like structures.

Findings

Unique high-temperature solutions are uniform.

Stable non-uniform solutions emerge at zero temperature.

Fermions on a sphere crystallize into C60-like structures.

Abstract

The Kohn-Sham (KS) equations determine, in a self-consistent way, the particle density of an interacting fermion system at thermal equilibrium. We consider a situation when the KS equations are known to have a unique solution at high temperatures and this solution is a uniform particle density. We show that, at zero temperature, there are stable solutions that are not uniform. We provide the general principles behind this phenomenon, namely the conditions when it can be observed and how to construct these non-uniform solutions. Two concrete examples are provided, including fermions on the sphere which are shown to crystallize in a structure that resembles the C$_{60}$ molecule.