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 (DMRG) method for $n=3,4$, and 5 for commensurate fillings $f=p/q$ where $p$ and $q$ are relatively prime. 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, 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.