Interacting fermions on non-commutative spaces: exactly solvable quantum field theories in 2n+1 dimensions Original Research Article

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∞⊕gl∞ at the special points Bθ=I and Bθ=−I, where B and θ are the matrices defining the magnetic field and the star product, respectively. 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.