Separation of quadrupole, spin, and charge across the magnetic phases of a one-dimensional interacting spin-1 gas

We study the low-energy collective properties of a one-dimensional 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 separation of the quadrupole-spin-charge sectors, confirmed by time-matrix-product states (time-MPS) numerical simulations. 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.