Spatially nonuniform phases in the one-dimensional SU(n) Hubbard model for commensurate fillings

The one-dimensional repulsive SU(n) Hubbard model is investigated analytically by bosonization approach and numerically using the density-matrix renormalization-group method for n=3, 4, and 5 for commensurate fillings f=p/q, where p and q are relatively primes. It is shown that the behavior of the system is drastically different depending on whether q > n, q=n, or q < n. When q > n, the umklapp processes are irrelevant and the model is equivalent to an n-component Luttinger liquid with central charge c=n. When q=n, the charge and spin modes are decoupled, the umklapp processes open a charge gap for finite U > 0, whereas the spin modes remain gapless and the central charge c=n - 1. The translational symmetry is not broken in the ground state for any n. On the other hand, when q < n, the charge and spin modes are coupled, the umklapp processes open gaps in all excitation branches, and a spatially nonuniform ground state develops. Bond-ordered dimerized, trimerized, or tetramerized phases are found depending on the filling.