Excitation Spectrum of the Lieb-Liniger Model

We study the integrable model of one-dimensional bosons with contact repulsion. In the limit of weak interaction, we use the microscopic hydrodynamic theory to obtain the excitation spectrum. The statistics of quasiparticles changes with the increase of momentum. At lowest momenta good quasiparticles are fermions, while at higher momenta they are Bogoliubov bosons, in accordance with recent studies. In the limit of strong interaction, we analyze the exact solution and find exact results for the spectrum in terms of the asymptotic series. Those results undoubtedly suggest that fermionic quasiparticle excitations actually exist at all momenta for moderate and strong interaction and also at lowest momenta for arbitrary interaction. Moreover, at strong interaction we find highly accurate analytical results for several relevant quantities of the Lieb-Liniger model.

Figure 1

The dispersion curve for the Lieb-Liniger model at $\gamma =10$ is represented by solid line, see Eqs. (22) and (23). Dashed lines are the two limiting cases, Eqs. (17) and (24). Doted line is given for comparison, and represents the limiting spectrum at $\gamma \to \infty$, Eq. (11). In the units used for the plot, $p_0^{*}=\lambda$.

Figure 2

The spectrum of elementary excitations in the Lieb-Liniger model has its characteristic form in five regions of the momentum-interaction plane. The number in parentheses denotes the number of equation where the limiting form of the spectrum is given. In regions I, II, and III, elementary excitations are fermionic quasiparticles, while IV and V they are bosonic quasiparticles. The crossover momenta are $p^{*}\sim {\gamma }^{3/4}$, $p_0\sim {\gamma }^{1/2}$, and $p_0^{*}\sim \gamma$, see the main text. As we neglected the pure numbers in all the asymptotes, ${\gamma }_c=1$, corresponding to $p_c=\hslash n$.