Spatially nonuniform phases in the one-dimensional $\mathrm{SU}\left(n\right)$ Hubbard model for commensurate fillings

The one-dimensional repulsive $\mathrm{SU}\left(n\right)$ 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.

Figure 1

Block entropy $s_N\left(l\right)$ of finite chains with $N=18$ and $N=16$ sites, respectively, for $n=3$ and 4 at fillings $f=1∕n$ and $f=1∕2$ for $U=10$. The solid line is our fit using Eq. (3).

Figure 2

(Color online) Finite-size dependence of $\mid \tilde{s}\left(k^{*}\right)\mid$ for various $n$ and fillings for $U=10$. The solid line is the finite-size-scaling fit.

Figure 3

Same as Fig. 1 but for (a) $n=3$, $f=2∕5$, $N=20$ and (b) $n=4$, $f=1∕3$, $N=24$.

Figure 4

(Color online) Same as Fig. 2 but for $n=5$. The solid line is the finite-size-scaling fit.

Figure 5

(Color online) Umklapp processes in the SU(3) symmetric Hubbard model. (a) Any two of the three types of fermion can be scattered in the half-filled case. (b) Fermions with the three possible spin orientations participate in the scattering process in the one-third-filled case.

Figure 6

(Color online) Finite-size dependence of $n\left(k^{*}\right)$ for various $n$ and fillings for $U=10$. The solid line is the finite-size-scaling fit.

Figure 7

(Color online) Finite-size dependence of $D_s$ for various $n$ and fillings for $U=10$. The solid line is the finite-size-scaling fit.

Figure 8

(Color online) Finite-size dependence of $T_s$ for various $n$ and fillings for $U=10$. The solid line is the finite-size-scaling fit.

Figure 9

(Color online) Finite-size dependence of $Q_s$ for various $n$ and fillings for $U=10$. The solid line is the finite-size-scaling fit.

Figure 10

(Color online) Finite-size dependence of $D_s$, $T_s$, $Q_s$, and $P_s$ for the half-, one-third-, and quarter-filled one-dimensional SU(5) Hubbard models at $U=10$.

Figure 11

(Color online) Schematic plot of the local bond strength in half-, one-third-, and quarter-filled SU(5) Hubbard chains.