Non-commutative Gross-Neveu model at large N

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

“…It corresponds to taking the limit N → ∞ with Λ 2n = N −1 det(B/4π) fixed and it produces an ultra-local propagator 6) where the parameter Λ now has the interpretation of an infrared cutoff. As remarked above, however, there is an alternative interpretation of both of these limits in which Λ may be interpreted simply as the mass scale set by the background magnetic field, i.e.…”

“…Heref is the Fourier transform of the field f with the conventioñ 6) and for any two momentum space vectors k and q we have defined their skew-product by…”

“…The ground state wavefunction φ 0 (x) ≡ φ 1,1 (x) can be computed explicitly by solving the differential equations a φ 0 = 0 and b φ 0 = 0 using (3.2) to get 6) while in general they can be computed by introducing the generating function…”

“…In fact, this mixing is known to render some noncommutative field theories non-renormalizable, even if their commutative counterparts are renormalizable. A class of such models are asymptotically-free noncommutative field theories [5,6], wherein the combination of asymptotic freedom and UV/IR mixing forces the couplings to run to their trivial Gaussian fixed points in the ultraviolet scaling limit. Such theories possess non-trivial interactions only when they are defined with a finite ultraviolet cutoff.…”

Renormalization group flow for noncommutative Fermi liquids

“…Recently some non-relativistic systems in noncommutative spaces have been discussed in references [36,37,38,39,40]. The study of the renormalization group flow in the noncommutative Gross-Neveu model of interacting fermions has been carried out in [41]. Moreover, a superconducting vortex liquid system in the lowest Landau level approximation was studied from the view point of the noncommutative field theory in [42].…”