Exact SU($N$) symmetry resolution of bosonic Hubbard models: general boson-fermion mappings and flat-band singlet-cluster states

We show how to perform exact diagonalization of SU($N$)-symmetric Bose-Hubbard models separately in each irreducible representation of SU($N$), by exploiting the Howe duality between U($L$) and U($N$). Three structural consequences follow for any lattice and any Hamiltonian built from the U($L$) generators: (i) the spectrum in a given orbital sector is independent of $N$ and of the particle statistics, (ii) the canonical partition functions for bosons and fermions obey the exact identity $Z_{\mathrm{b}}(-N)=(-1{)}^{N_{\mathrm{b}}}{Z}_{\mathrm{f}}(N)$, (iii) with unfrustrated hopping the groundstate is always a color ferromagnet, so that the color-resolved method we introduce is useful to study excited sectors, frustrated lattices and finite temperatures. As applications, we first derive the finite-temperature phase diagram of attractive SU($N$) bosons on few-site rings at large $N$ for arbitrary filling. We then show the emergence of singlet-cluster states from frustrated hopping, the counterpart of fermionic ferromagnetism, with the example of the sawtooth and kagome lattice.