Rigorous Results for the Ground States of the Spin-2 Bose-Hubbard Model

We present rigorous and universal results for the ground states of the $f=2$ spinor Bose-Hubbard model. The model includes three two-body on site interaction terms, two of which are spin dependent while the other one is spin independent. We prove that, depending only on the coefficients of the two spin dependent terms, the ground state exhibits maximum or minimum total spin or SU(5) ferromagnetism. Exact ground-state degeneracies and the forms of ground-state wave function are also determined in each case. All these results are valid regardless of dimension, lattice structure, or particle number. Our approach takes advantage of the symmetry of the Hamiltonian and employs mathematical tools including the Perron-Frobenius theorem and the Lie algebra $\mathfrak{so}(5)$.