Excitation Spectrum of the Lieb-Liniger Model

Abstract

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.

An Introduction to Integrable Techniques for One-Dimensional Quantum Systems

“…In this way, one eventually reaches: 51) which is convergent if γ > 2 [44]. One can also approach this asymptotic expansion following [45].…”

Tan’s contact of a harmonically trapped one-dimensional Bose gas: Strong-coupling expansion and conjectural approach at arbitrary interactions

“…The study of the equation of state of an interacting 1D Bose gas has received a renewed attention [25,27]. In particular, in reference [25] we have derived a strongcoupling expansion for the ground-state energy to an unprecedented accuracy as well as a conjectural expression which is extremely close to the exact numerical solution for a wide range of interaction strengths.…”

Connection between nonlocal one-body and local three-body correlations of the Lieb-Liniger model

“…Using a basis of orthogonal polynomials to systematically find a 1/γ expansion of the moments as explained in [11] and [12], together with the conjecture Eq. (46), we find by identification: …”

“…In particular, we derive a relation, first proposed in [10], that links the fourth coefficient of the Taylor expansion of the onebody correlation function at short distances with various moments of the quasi-momentum distribution and their derivatives with respect to the coupling constant. Then, * Maxim.Olchanyi@umb.edu we use a recently-developed method [11,12], and generalize recent conjectures [12][13][14], to evaluate these quantities with excellent accuracy in a wide range of interaction strengths.…”