Spectrum of Elementary Excitations in Galilean-Invariant Integrable Models

The spectrum of elementary excitations in one-dimensional quantum liquids is generically linear at low momenta. It is characterized by the sound velocity that can be related to the ground-state energy. Here we study the spectrum at higher momenta in Galilean-invariant integrable models. Somewhat surprisingly, we show that the spectrum at arbitrary momentum is fully determined by the properties of the ground state. We find general exact relations for the coefficients of several terms in the expansion of the excitation energy at low momenta and arbitrary interaction and express them in terms of the Luttinger liquid parameter. We apply the obtained formulas to the Lieb-Liniger model and obtain several new results.

Figure 1

Plot of the coefficient $\lambda$ (in units where $\hslash =n=m=1$) as a function of $\gamma$. The dots represent the numerically exact result, while the two lines are our analytical results given by Eqs. (17) and (20).

Figure 2

Plot of the coefficient $\nu$ (in units where $\hslash =n=m=1$) as a function of $\gamma$. The dots represent the numerically exact result, while the two lines are our analytical results given by Eqs. (18) and (21).