One-dimensional interacting systems of quantum particles are qualitatively different from their higher-dimensional counterparts. The reason is the strong effect of quantum fluctuations, which renders the ground state in one dimension to be a liquid. At lowest energies, such a quantum state is well described by the phenomenological Luttinger liquid theory. It is characterized by two parameters : one is the excitation velocity, and the other is so-called Luttinger liquid parameter that controls, e.g., the long distance decay of the single particle correlation function. The elementary excitations in a Luttinger liquid have a linear spectrum. To access the real spectrum of a liquid that is generically nonlinear, one must invoke the approaches that account for the deviations from the Luttinger liquid theory.
We studied the spectrum at higher momenta in Galilean invariant integrable models. Somewhat surprisingly, we showed that the spectrum at arbitrary momentum is fully determined by the properties of the ground state. We found general exact relations for the coefficients of several terms in the expansion of the excitation energy at low momenta and arbitrary interaction and expressed them in terms of the Luttinger liquid parameter. We applied the obtained formulas to the Lieb-Liniger model and obtained several new results.
Reference : Aleksandra Petkovi? and Zoran Ristivojevic Phys. Rev. Lett. 120, 165302 (2018)