Long-range order in one-dimensional spinless Fermi gas with attractive dipole-dipole interaction

One-dimensional spinless Fermi gas with attractive dipole-dipole interaction is investigated. Results obtained show that when the interaction is weak, the excitation spectrum is linear and the superconducting correlation function decays as a power law, indicating the validity of the Tomonaga-Luttinger (TL) liquid picture. However, when the interaction reaches a critical value, the excitation spectrum is nonlinear and the superconducting correlation function remains finite for infinity separation, indicating that real long-range order has been established and the breakdown of the TL liquid picture. We prove that the existence of long-range order is not in contradiction with the Hohenberg theorem and show that this system is related to the Kitaev toy model, therefore, it has potential applications for future topological quantum computation.