Exactly solvable models for 2D correlated fermions

TL;DR

This paper introduces exactly solvable 2D fermion models, including a quantum Hall system, using group theory and dynamical symmetries, enabling explicit solutions for energy states and phase diagrams.

Contribution

It presents a novel class of exactly solvable 2D fermion models based on group theory, extending mean field approaches to include non-trivial correlations.

Findings

Exact solutions for energy eigenvalues and eigenfunctions.

Identification of a non-trivial zero temperature phase diagram.

Development of simplified models capturing correlations beyond mean field.

Abstract

I discuss many-body models for interacting fermions in two space dimensions which can be solved exactly using group theory. The simplest example is a model of a quantum Hall system: 2D fermions in a constant magnetic field and a particular non-local 4-point interaction. It is exactly solvable due to a dynamical symmetry corresponding to the Lie algebra $\gl_\infty\oplus \gl_\infty$. There is an algorithm to construct all energy eigenvalues and eigenfunctions of this model. The latter are, in general, many-body states with spatial correlations. The model also has a non-trivial zero temperature phase diagram. I point out that this QH model can be obtained from a more realistic one using a truncation procedure generalizing a similar one leading to mean field theory. Applying this truncation procedure to other 2D fermion models I obtain various simplified models of increasing complexity which generalize mean field theory by taking into account non-trivial correlations but nevertheless are treatable by exact methods.