Interacting fermions on noncommutative spaces: Exactly solvable quantum field theories in 2n+1 dimensions

TL;DR

This paper introduces a new class of exactly solvable quantum field theories describing non-relativistic fermions on noncommutative spaces with magnetic fields, revealing a special dynamical symmetry and providing explicit eigenstates.

Contribution

It constructs and solves a novel class of noncommutative fermionic quantum field theories with exact eigenstates at special parameter points, highlighting their correlated nature.

Findings

Models exhibit a $ ext{gl}_ infty imes ext{gl}_ infty$ symmetry at specific parameters.

Explicit eigenvalues and eigenstates are constructed for these models.

The solutions reveal strongly correlated fermionic behavior not accessible by mean-field methods.

Abstract

I present a novel class of exactly solvable quantum field theories. They describe non-relativistic fermions on even dimensional flat space, coupled to a constant external magnetic field and a four point interaction defined with the Groenewold-Moyal star product. Using Hamiltonian quantization and a suitable regularization, I show that these models have a dynamical symmetry corresponding to $\gl_\infty\oplus \gl_\infty$ at the special points where the magnetic field $B$ is related to the matrix $\theta$ defining the star product as $B\theta=\pm I$. I construct all eigenvalues and eigenstates of the many-body Hamiltonian at these special points. I argue that this solution cannot be obtained by any mean-field theory, i.e. the models describe correlated fermions. I also mention other possible interpretations of these models in solid state physics.