Exact results for the spectra of interacting bosons and fermions on the lowest Landau level

TL;DR

This paper derives exact matrix elements for interacting bosons and fermions in the lowest Landau level, enabling precise energy calculations for large N and revealing rational low-energy values for small systems.

Contribution

It provides generic expressions for interaction matrix elements in the lowest Landau level and exact energies for large fermionic systems, advancing understanding of quantum many-body spectra.

Findings

Exact energy calculations for N up to 1000 fermions.

Identification of rational low-lying energy values for N=3 with Coulomb interaction.

General expressions for interaction matrix elements in the lowest Landau level.

Abstract

A system of N interacting bosons or fermions in a two-dimensional harmonic potential (or, equivalently, magnetic field) whose states are projected onto the lowest Landau level is considered. Generic expressions are derived for matrix elements of any interaction, in the basis of angular momentum eigenstates. For the fermion "ground state" (N=1 Laughlin state), this makes it possible to exactly calculate its energy all the way up to the mesoscopic regime N ~ 1000. It is also shown that for N = 3 and Coulomb interaction, several rational low-lying values of energy exist, for bosons and fermions alike.