One-dimensional Bose-Hubbard model with long-range hopping

Interacting one-dimensional bosons with long range hopping decaying as a power law $r^{-α}$ with distance $r$ are considered with the renormalization group and the self-consistent harmonic approximation. For $α\ge 3$, the ground state is always a Tomonaga-Luttinger liquid, whereas for $α<3$, a ground state with long range order breaking the continuous global gauge symmetry becomes possible for sufficiently weak repulsion. At positive temperature, continuous symmetry breaking becomes restricted to $α<2$, and for $2<α<3$, a Tomonaga-Luttinger liquid with the Tomonaga-Luttinger exponent diverging at low temperature is found.