Low-energy excitations of the one-dimensional half-filled SU(4) Hubbard   model with an attractive on-site interaction: Density-matrix   renormalization-group calculations and perturbation theory
  We investigate low-energy excitations of the one-dimensional half-filled SU(4) Hubbard model with an attractive on-site interaction U < 0 using the density matrix renormalization group method as well as a perturbation theory. We find that the ground state is a charge density wave state with a long range order. The ground state is completely incompressible since all the excitations are gapful. The charge gap which is the same as the four-particle excitation gap is a non-monotonic function of U, while the spin gap and others increase with increasing |U| and have linear asymptotic behaviors. 
Jize Zhao, Kazuo Ueda and Xiaoqun Wang