Exact Diagonalization of $\mathrm{SU}(N)$ Fermi-Hubbard Models

We show how to perform exact diagonalizations of $\mathrm{SU}(N)$ Fermi-Hubbard models on $L$-site clusters separately in each irreducible representation (irrep) of $\mathrm{SU}(N)$. Using the representation theory of the unitary group $\mathrm{U}(L)$, we demonstrate that a convenient orthonormal basis, on which matrix elements of the Hamiltonian are very simple, is given by the set of semistandard Young tableaux (or, equivalently, the Gelfand-Tsetlin patterns) corresponding to the targeted irrep. As an application of this color factorization, we study the robustness of some $\mathrm{SU}(N)$ phases predicted in the Heisenberg limit upon decreasing the on-site interaction $U$ on various lattices of size $L\le 12$ and for $2\le N\le 6$. In particular, we show that a long-range color ordered phase emerges for intermediate $U$ for $N=4$ at filling $1/4$ on the triangular lattice.

Figure 1

Examples of $\mathrm{SU}(N=2)$ irrep $\alpha$ for a $L=4$-site cluster and $M=6$ fermions in (a) and $M=4$ fermions in (b). The Young tableaux are $\alpha =[3,3]$ (respectively, $\alpha =[3,1]$), representing singlets (respectively, spin 1) states. We associate the transpose YT, flipping the shape: $\alpha =[3,3]\to \overline{\alpha }=[2,2,2]$ for (a) and $\alpha =[3,1]\to \overline{\alpha }=[2,1,1]$ for (b). We fill up $\overline{\alpha }$ to get the highest weight state to which we associate $d_{N=2}^{\alpha }$ physical states, each of which generates an independent $d_L^{\overline{\alpha }}$-dimensional sector invariant under the application of the operators $E_{ij}$. See text for details.

Figure 2

Correlation patterns of the ground states of the FHM at filling $1/N$ for $U=10$ in (a),(c),(d) and for $U=12$ in (b). They are defined as $⟨P_{1j}⟩-1/N$, where $P_{1j}\equiv -1+E_{1j}{E}_{j1}$, with the reference site 1 in black, and $j$ the site indices being blue (red) for positive (negative) correlation, with area proportional to its absolute value. At the top, triangular lattice with $L=12$ sites, for SU(3) (a) compatible with the three-sublattice Néel order [35, 75, 76] and for SU(4) (b) compatible with the four-sublattice Néel order [32]. (c) SU(5) on the cluster $\sqrt{10}\times \sqrt{10}$, pattern compatible with the (chess) knight move LRO [59]; (d) SU(6) on the $3\times 4$ cluster, compatible with the SU(6) plaquette state [42].

Figure 3

Energy spectra of the FHM in Eq. (1) (with $t_{ij}\equiv 1$ at filling $1/N$) as a function of the quadratic Casimir $C_2$ for various values of $U$ and $N$ for the $L=12$ periodic triangular lattice [(a)–(d)] and for the $L=4\times 3$ periodic square lattice in (e), where we focused on $N=6$. Note that the constant $LU/2$ has been withdrawn. (c) The Atos is reminiscent of the one revealing the four-sublattice order in the HM with nnn couplings [32]. Inset of (e): spin (singlet) gap in blue (red).

Figure 4

(a) Charge gaps ${\mathrm{\Delta }}_{\mathrm{c}}$ for the FHM on the $L=12$ sites triangular lattice for $N=2$, 3, and 4 at filling $1/N$. The dashed lines (cf. the Supplemental Material [63] for the fitting procedure) cross the $x$ axis at $U\simeq 8.6$ for $N=2$, $U\simeq 9.8$ for $N=3$, and $U\simeq 11.2$ for $N=4$ separating the small $U$ metallic phase from the Mott insulators. (b) $\mathrm{SU}(N)$ spin gaps ${\mathrm{\Delta }}_{\mathrm{s}}$.