Ultracold Fermions and the $\mathrm{SU}(N)$ Hubbard Model

We investigate the fermionic $\mathrm{SU}(N)$ Hubbard model on the two-dimensional square lattice for weak to moderate interactions using renormalization group and mean-field methods. For the repulsive case $U>0$ at half filling and small $N$ the dominant tendency is towards breaking of the $\mathrm{SU}(N)$ symmetry. For $N>6$ staggered flux order takes over as the dominant instability, in agreement with the large-$N$ limit. Away from half filling for $N=3$ two flavors remain half filled by cannibalizing the third flavor. For $U<0$ and odd $N$ a full Fermi surface coexists with a superconductor. These results may be relevant to future experiments with cold fermionic atoms in optical lattices.

Figure 1

(a) AF spin-density wave state for $N=2$. Spin-up and spin-down particles occupy the two sublattices with different probabilities (here idealized to 0 and 1). (b) Flavor-density wave state for $N=3$. Flavors 1 and 2 prefer one sublattice, flavor 3 the other. (c) Staggered flux state for $N>6$: the arrows indicate the particle currents.

Figure 2

Left plots: RG results for SU(3) at half filling and $U=4t$. Upper left: flow of ${\overline{V}}_s^s$ in the SU(3)-breaking channel (dashed line) and ${\overline{V}}_c^d$ in the SF channel, averaged around the FS. Lower left: $V_s(\overrightarrow{k},{\overrightarrow{k}}^{\prime },\overrightarrow{Q})$ (dashed line) and $V_c(\overrightarrow{k},{\overrightarrow{k}}^{\prime },\overrightarrow{Q})$ with $\overrightarrow{k}$ fixed at $(\pi ,0)$ and ${\overrightarrow{k}}^{\prime }$ moving around the FS. Right plots: the same for SU(8) at half filling.

Figure 3

Left: uniform densities of the $N=3$ flavors (solid line: flavors 1 and 2; crosses: flavor 3) and staggered densities (squares) vs total density for $U=1.6t$. Right: difference in uniform density between the two majority flavors and the minority flavor vs $U$ and total density $⟨n⟩$ per site. The scale bar indicates the density difference per site.