Mott transition and dimerization in the one-dimensional $\mathrm{SU}\left(n\right)$ Hubbard model

The one-dimensional $\mathrm{SU}\left(n\right)$ Hubbard model is investigated numerically for $n=2$, 3, 4, and 5 at half-filling and $1∕n$ filling using the density-matrix renormalization-group method. The energy gaps and various quantum information entropies are calculated. In the half-filled case, finite spin and charge gaps are found for arbitrary positive $U$ if $n>2$. Furthermore, it is shown that the transition to the gapped phase at $U_c=0$ is of Kosterlitz-Thouless type and is accompanied by a bond dimerization both for even and odd $n$. In the $1∕n$-filled case, the transition has similar features as the metal-insulator transition in the half-filled SU(2) Hubbard model. The charge gap opens exponentially slowly for $U>U_c=0$, the spin sector remains gapless, and the ground state is nondimerized.

Figure 1

$U$ dependence and finite-size scaling of the spin gap for the half-filled SU(2) Hubbard model. The dashed lines are a guide for the eyes; the solid lines show the result of our fit.

Figure 2

$U$ dependence and finite-size scaling of the charge gap for the half-filled SU(2) Hubbard model. The dashed lines are a guide for the eyes; the solid lines show the result of our fit.

Figure 3

One-site and two-site entropies and dimerization of the two-site entropy in the middle of the chain as a function of $U$ for the half-filled SU(2) Hubbard model. The two sets of curves for the two-site entropy correspond to $i=N∕2$ and $i=N∕2+1$. The dashed lines are a guide for the eyes and the solid line is our parabolic fit.

Figure 4

Entropy functions plotted as in Fig. 3, but for the half-filled SU(4) Hubbard model.

Figure 5

The spin gap plotted as in Fig. 1, but for the half-filled SU(3) Hubbard model.

Figure 6

The charge gap plotted as in Fig. 2, but for the half-filled SU(3) Hubbard model.

Figure 7

Entropy functions plotted as in Fig. 3, but for the half-filled SU(3) Hubbard model.

Figure 8

Finite-size scaling of the $q=\pi$ Fourier component of the half-filled SU $\left(n\right)$ Hubbard model for $n=2,3,4$ at $U=4$.

Figure 9

Spin gap plotted as in Fig. 1, but for the one-third-filled SU(3) Hubbard model.

Figure 10

Charge gap plotted as in Fig. 2, but for the one-third-filled SU(3) Hubbard model.

Figure 11

Entropy functions plotted as in Fig. 3, but for the one-third-filled SU(3) Hubbard model.

Figure 12

Block entropy, $s(l=N∕2)$, plotted as a function of $U$ for the one-third-filled SU(3) Hubbard model. The solid lines are the result of polynomial fits. The inset shows the finite-size scaling of $U_c$ determined from the maximum of the block entropy for the ground state as well as for excited states with $N+1$ or $N-1$ particles.