Haldane Phase in Spin-1 Bose-Hubbard Model with Flat Band

The Haldane phase has been widely discussed in spin chains. A famous rigorous example is the AKLT model. However, in itinerant spin systems which have both unfrozen spin and charge degrees of freedom, the Haldane phase has been rarely investigated, to say nothing of any rigorous results. Here we study the spin-1 Bose-Hubbard model on a sawtooth lattice with flat band. At half-unit filling with fine-tuned interaction strength, we prove that the ground state can be exactly written down in the form of a matrix product state (MPS) similar to that of the AKLT model. The ground state correlation lengths, edge states, string order parameters, etc. are analytically given for both spin and charge degrees of freedom. The entanglement spectrum also analytically shows perfect two-fold degeneracy. Moreover, with general interaction strength, when a parameter (related to hopping and on-site potential of the lattice) is sufficiently large, perturbation theory yields the spin-1 bilinear-biquadratic model as the effective spin model, which can be either symmetry-protected-topological (Haldane) or trivial, depending on the interaction strength.