Ferromagnetic ground state of the SU(3) Hubbard model on the Lieb lattice

We investigate the magnetic properties of a repulsive fermionic $\mathrm{SU}\left(3\right)$ Hubbard model on the Lieb lattice from weak to strong interaction by means of the mean-field approximation. To validate the method we employed, we first discuss the $\mathrm{SU}(2$) Hubbard model at the mean-field level, and find that our results are consistent with known rigorous theorems. We then extend the calculation to the case of $\mathrm{SU}\left(3\right)$ symmetry. We find that, at $4/9$ filling, the $\mathrm{SU}\left(3\right)$ symmetry spontaneously breaks into the $\mathrm{SU}\left(2\right)\times \mathrm{U}\left(1\right)$ symmetry in the ground state, leading to a staggered ferromagnetic state for any repulsive $U$ at zero temperature. We then investigate the stability of the ferromagnetic state by relaxing the filling away from $4/9$, and conclude that the ferromagnetic state is sensitive but robust to fillings, as it can persist within a certain filling regime. We also apply the mean-field approximation to finite temperature to calculate the critical temperature and the critical entropy of the ferromagnetic state. As the resulting critical entropy per particle is significantly greater than what can be realized in experiments, we expect some quasi-long-range-ordered features of such a ferromagnetic state can be realized and observed with fermionic alkaline-earth-metal(-like) atoms loaded into optical lattices.

Figure 1

(a) Lattice structure of the Lieb lattice. The dotted square shows one unit cell, containing three lattice sites with indices A, B, and C. (b) The band structure of the tight-binding model on the Lieb lattice. (c) The band structure along high symmetric lines in the first Brillouin zone (BZ), where the inset shows the first BZ and high symmetric points.

Figure 2

The properties of the magnetic order of the SU(2) Hubbard model at half filling. (a) The magnetization orders on each site $m_{\mu }^z$ and total magnetization in one unit cell $m_{\mathrm{tot}}^z$ as functions of $U/t$, where $\mu =\mathrm{A},\mathrm{B},\mathrm{C}$. The result of $m_{\mu }^{x,y}=0$ is not shown. (b) The particle numbers of each spin component on each site $\langle n_{\mu ,\alpha }\rangle$ as functions of $U/t$, where $\alpha =\uparrow ,\downarrow$. (c) The correlation functions $m\left(0\right)$ and $m\left(Q\right)$ as functions of $U/t$. (d) A schematic figure showing the staggered ferromagnetic long-range order in the large-$U$ limit.

Figure 3

The zero-temperature band structure of the SU(2) model at half filling with different interactions: (a) $U/t=0.1$, (b) $U/t=0.5$, (c) $U/t=1.5$, and (d) $U/t=5.0$. The band filled by spin-up (spin-down) particles is labeled by thick solid black (dashed red) line. The remaining three bands labeled by thin dotted black lines are empty.

Figure 4

The total magnetization in one unit cell, $m_{\mathrm{tot}}^z$, for the SU(2) model with $U/t=0.0505$. Similar to the half-filling case, the magnetization orders are collinear: ${\mathit{m}}_{\mathrm{B}}$ is parallel with ${\mathit{m}}_{\mathrm{C}}$ and antiparallel with ${\mathit{m}}_{\mathrm{A}}$, where ${\mathit{m}}_{\mathrm{B}}={\mathit{m}}_{\mathrm{C}}$. Therefore, we only plot total magnetization in one unit cell, $m_{\mathrm{tot}}^z$ ($m_{\mathrm{tot}}^{x,y}=0$ are not shown).

Figure 5

The properties of the magnetic order for the SU(3) Hubbard model at 4/9 filling. (a) Results of $m_{\mathrm{A},\mathrm{B}\left(\mathrm{C}\right)}^8$ and $m_{\mathrm{tot}}^8$ in one unit cell as functions of $U/t$. (b) The particle numbers on each site with different flavors $\langle n_{\mu ,\alpha }\rangle$. (c) The correlation functions $m{\left(0\right)}_{I,V,U}$ and $m{\left(Q\right)}_{I,V,U}$ as functions of $U/t$. (d) A schematic figure showing the flavor-density distribution and the staggered ferromagnetic long-range order in the large-$U$ limit.

Figure 6

The equivalent ferromagnetic ground states can be shown with the aid of a weights diagram of (a) the $\mathrm{SU}\left(2\right)$ Lie algebra and (b) the $\mathrm{SU}\left(3\right)$ Lie algebra. The coordinates of the weights are labeled in each diagram.

Figure 7

The zero-temperature band structure of the SU(3) model at 4/9 filling with different interactions (a) $U/t=0.1$, (b) $U/t=0.5$, (c) $U/t=1.5$, and (d) $U/t=5.0$. The bands filled by flavor-1 or flavor-2 particles are labeled by dashed red lines, while the bands filled by flavor-3 particles are labeled by solid black lines. The empty bands are labeled by thin dotted black lines.

Figure 8

The magnetization for the SU(3) model at finite temperature with different interactions. (a) The critical temperature $T_c/t=0.0192$ and the critical entropy per particle, $S_c=0.457$, when $U/t=0.11$; (b) $T_c/t=0.31$ and $S_c=0.818$ when $U/t=1.11$; (c) $T_c/t=1.08$ and $S_c=1.406$ when $U/t=3.01$; and (d) $T_c/t=1.80$ and $S_c=1.452$ when $U/t=5.01$.

Figure 9

(a) The norm of the total magnetization in one unit cell, $|{\mathit{m}}_{\mathrm{tot}}|$, for the SU(3) model; (b) the norms of magnetization on each sublattice site, $|{\mathit{m}}_{\mu }|$, for the SU(3) model, where $\mu =\mathrm{A},\mathrm{B},\mathrm{C}$. The interaction strength used in this plot is within the weak-interacting limit with $U/t=0.0505$.

Figure 10

The norm of total magnetization for the SU(3) model in one unit cell, $|{\mathit{m}}_{\mathrm{tot}}|$, in the plane of $(h/t,{\mu }_0/t)$ with $U/t=0.0505$.