Ferromagnetism in the $\text{SU}\left(n\right)$ Hubbard model with a nearly flat band

We present rigorous results for the $\mathrm{SU}\left(n\right)$ Fermi-Hubbard model on the railroad-trestle lattice. We first study the model with a flat band at the bottom of the single-particle spectrum and prove that the ground states exhibit $\mathrm{SU}\left(n\right)$ ferromagnetism when the total fermion number is the same as the number of unit cells. We then perturb the model by adding extra hopping terms and make the flat band dispersive. Under the same filling condition, it is proved that the ground states of the perturbed model remain $\mathrm{SU}\left(n\right)$ ferromagnetic when the bottom band is nearly flat. This is the first rigorous example of the ferromagnetism in nonsingular $\mathrm{SU}\left(n\right)$ Hubbard models in which both the single-particle density of states and the on-site repulsive interaction are finite.

Figure 1

(a) The railroad-trestle lattice with hopping amplitudes $t_1=\nu t$ and $t_2={\nu }^{2}t$. Odd (black) and even (white) sites have on-site potentials $t$ and $2{\nu }^{2}t$, respectively. The shaded region indicates the unit cell. (b) The energy bands for $t=1,\nu =1/\sqrt{2}$. The lowest band at zero energy is completely flat.

Figure 2

(a) The lattice geometry of $H_{\mathrm{hop}}^{\prime }$. The hopping amplitudes are given by $t_1=\nu (t+s),t_2={\nu }^{2}t$, and $t_2^{\prime }=-{\nu }^{2}s$. Odd (black) and even (white) sites have on-site potentials $t-2{\nu }^{2}s$ and $-s+2{\nu }^{2}t$, respectively. The corresponding energy bands are shown in panel (b) for $t=1,\nu =1/\sqrt{2},s=1/10$.

Figure 3

The positive semidefiniteness of $h_x(n=4)$ holds in the shaded region for $\nu =1/\sqrt{2},\kappa =0$. The plot is obtained by diagonalizing $h_x$ numerically. Lemma 1 says that the ground states of the full Hamiltonian $H_2$ are fully polarized states in the shaded region. For example, the ferromagnetism is established if $t/s\ge 4.5$ when $U/s=25$.