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

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.

23 pages, LaTex

References in corpus (9)

• Noncommutative Field Theory
• Quantum Field Theory on Noncommutative Spaces
• Introduction to M(atrix) theory and noncommutative geometry, Part II
• Duality in Scalar Field Theory on Noncommutative Phase Spaces
• A Class of Exactly Solvable Pairing Models
• Integrable model for interacting electrons in metallic grains
• Superconducting correlations in metallic nanoparticles: exact solution of the BCS model by the algebraic Bethe ansatz
• Quantum Aspects of GMS Solutions of Noncommutative Field Theory and Large N Limit of Matrix Models
• Exactly solvable models for 2D correlated fermions

Cited by in corpus (11)

• Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base
• Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces
• Renormalization of Non-Commutative Phi^4_4 Field Theory in x Space
• Non-commutative Renormalization
• Exact Solution of Noncommutative Field Theory in Background Magnetic Fields
• Galilean symmetry in noncommutative field theory
• Lectures on Matrix Field Theory I
• Quantum field theories on noncommutative R^4 versus theta-expanded quantum field theories
• Exactly solvable models for 2D correlated fermions
• D-Branes in Noncommutative Field Theory
• Non-commutative geometry and exactly solvable systems