Quadrupole-spin-charge separation and magnetic phases of a 1D interacting spin-1 gas

We study the low-energy collective properties of a 1D spin-1 Bose gas using bosonization. After giving an overview of the technique, emphasizing the physical aspects, we apply it to the S=1 Bose-Hubbard Hamiltonian and find a novel separation of the quadrupole-spin-charge sectors. Additionally, through the single particle spectrum, we show the existence of the superfluid-Mott insulator transition and the point at which the physics are described by a Heisenberg-like Hamiltonian. The magnetic phase diagrams are found for both the superfluid and insulating regimes; the latter is determined by decomposing the complete Heisenberg bilinear-biquadratic Hamiltonian, which describes the Mott insulator, into simpler, effective Hamiltonians. This allows us to keep our methods flexible and transferable to other interesting interacting condensed matter systems.