Low-energy excitations of a one-dimensional Bose gas with weak contact repulsion

We study elementary excitations of a system of one-dimensional bosons with weak contact repulsion. We show that the Gross-Pitaevskii regime, in which the excitations are the well-known Bogoliubov quasiparticles and dark solitons, does not extend to the low-energy limit. Instead, the spectra of both excitations have finite curvatures at zero momentum, in agreement with the phenomenological picture of fermionic quasiparticles. We describe analytically the crossover between the Gross-Pitaevskii and the low-energy regimes, and discuss implications of our results for the behavior of the dynamic structure factor.

Figure 1

The crossover functions $e^{\pm }\left(s\right)$ [see Eqs. (16) and (17)] describing the nonlinear in momentum corrections to the spectra of Lieb's type I and type II excitations. The inset illustrates that $e^{\pm }\left(s\right)\simeq \pm s^2$ at $s\to 0$ [see Eq. (19)].