Derivation of the Hartree dynamics for a N-particles fermionic 2D system interacting via long-range electric and magnetic two-body singular potentials in a dilute regime

TL;DR

This paper derives effective equations describing the time evolution of a large 2D fermionic system with long-range electric and magnetic interactions, showing the preservation of Slater structure in a dilute regime.

Contribution

It introduces a novel derivation of coupled Chern-Simons-Schrodinger equations for 2D fermions with singular long-range interactions, handling three-body potential diagonalization.

Findings

Wave function approximated by a Slater determinant with evolving orbitals.

Preservation of Slater structure over time in the large N limit.

Mathematical innovation in diagonalizing three-body potentials.

Abstract

We show that a 2-dimensional system of N fermions interacting through a pairwise electric and magnetic singular interactions with Slater initial data preserves its Slater structure over time when N gets large. In other words, the wave function of the system can be approximated by a Slater determinant whose orbitals evolves according to a coupled system of N Chern-Simons- Schr\"odinger type effective equations. The result holds in a dilute regime where the density is of order one on a large volume proportional to the number of particles. The pairwise magnetic feature of the system implies to deal with the diagonalization of three-body potentials which is the main mathematical innovation of the paper.