Time-dependent mean-field theory of the superfluid-insulator phase transition

We develop a time-dependent mean-field approach, within the time-dependent variational principle, to describe the superfluid-insulator quantum phase transition. We construct the zero-temperature phase diagram both of the Bose-Hubbard model (BHM), and of a spin-$S$ Heisenberg model (SHM) with the $\mathrm{XXZ}$ anisotropy. The phase diagram of the BHM indicates a phase transition from a Mott insulator to a compressibile superfluid phase, and shows the expected lobelike structure. The SHM phase diagram displays a quantum phase transition between a paramagnetic and a canted phases showing as well a lobelike structure. We show how the BHM and the quantum phase model (QPM) can be rigorously derived from the SHM. Based on such results, the phase boundaries of the SHM are mapped to the BHM ones, while the phase diagram of the QPM is related to that of the SHM. The QPM’s phase diagram obtained through the application of our approach to the SHM, describes the known onset of the macroscopic phase coherence from the Coulomb blockade regime for increasing Josephson coupling constant. The BHM and the QPM phase diagrams are in good agreement with quantum Monte Carlo results, and with the third-order strong-coupling perturbative expansion.