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

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• Long-range interactions
• Spontaneous symmetry breaking
• 1-dimensional systems
• Ultracold gases
• Luttinger liquid model
• Renormalization
• Spin chains