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

We show how to perform exact diagonalizations of $\mathrm{S}\mathrm{U}(N)$ Fermi-Hubbard models on $L$-site clusters separately in each irreducible representation ({irrep}) of $\mathrm{S}\mathrm{U}(N)$. 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 {\it 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 $\mathrm{S}\mathrm{U}(N)$ 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 $1\mathrm{/}4$ on the triangular lattice.