Ground-state energy and excitation spectrum of the Lieb-Liniger model : accurate analytical results and conjectures about the exact solution

Abstract

We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in [1] and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasimomenta, the ground-state energy, the local two-body correlation function and Tan's contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.

Strongly Correlated Quantum Droplets in Quasi-1D Dipolar Bose Gas

“…In the limit of a very weak contact interactions g → 0 we retrieve the GPE, while for g → ∞ we restore the TGGPE. It should be emphasized that we use a simplified energy density functional for the ground state, which is a rough approximation of the full Lieb-Liniger expression, see for instance [36,37] and references therein. We aim to solve the Eq.…”

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

“…In the strongly-interacting regime, we generalize a recent conjecture on e 2 [12], by stating that the asymptotic expansion in 1/γ is partially resummed in a natural way as…”

“…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.…”

Conjectures about the structure of strong- and weak-coupling expansions of a few ground-state observables in the Lieb-Liniger and Yang-Gaudin models

“…We have pushed this procedure up to order n = 20 in [55], the explicit result is given here in appendix A. More generally, at order n the expression contains 1 + ⌊(n + 1)/2⌋⌊1 + n/2⌋ different terms, where ⌊.…”