We conceptually study the existence of four distinct quantum phases: the Luttinger liquid, the Wigner crystal, the Coulomb plasma and the molecular crystal in one-dimensional many-body systems with long-range interaction. We show that the anomaly of the ground state wavefunction indicates a quantum Kosterlitz–Thouless transition at zero temperature, which separates into two regimes of the Luttinger liquid and the Wigner crystal. We also postulate a scaling theory which discriminate the four phases using the Kosterlitz–Thouless scaling theory.