Conjectures about the ground-state energy of the Lieb-Liniger model at weak repulsion

Abstract

In this paper we develop an alternative description to solve the problem of ground state energy of the Lieb-Liniger model that describes one-dimensional bosons with contact repulsion. For this integrable model we express the Lieb integral equation in the representation of Chebyshev polynomials. The latter form is convenient to efficiently obtain very precise numerical results in the singular limit of weak interaction. Such highly precise data enables us to use the integer relation algorithm to discover the analytical form of the coefficients in the expansion of the ground state energy for small interaction parameter. We obtained the first nine terms of the expansion using quite moderate numerical efforts. The detailed knowledge of behavior of the ground state energy on the interaction immediately leads to the exact perturbative results for the excitation spectrum.Since its introduction in 1963, the Lieb-Liniger model of one-dimensional bosons with contact interaction [1, 2] continues to fascinate the scientific community. This model is remarkable in many respects and, moreover, describes the physics of realistic systems. The attractive case has deep connections with classical two-dimensional systems, in particular with a surface growth described by the Kardar-Parisi-Zhang equation [3,4]. It could be nowadays directly realized in experiments with cold gases [5][6][7], enabling us to better understand the correlation effects in one-dimensional many-body systems. Rather importantly, the Lieb-Liniger model is integrable and admits an exact solution in terms of Bethe ansatz [1,2]. It thus serves as a benchmark for the effective theories which unavoidably contain various levels of approximations [8,9]. The known results for the above model form a cornerstone for quantum one-dimensional physics of interacting particles [10,11].The explicit analytical expressions for various physical quantities of interest in integrable models are often difficult to extract from the exact solution and one is typically restricted to study special cases. The relevant information about the system's wave function and the corresponding energy of the Lieb-Liniger model is encoded into, so called, the Lieb inte-gral equation [1]. Despite its simple form, the ground state energy is only known in the limiting cases. Using the systematic procedure of Ref. [12], at strong repulsion one can generate a power series expansion to an arbitrary order in the inverse coupling strength. The other limit of weak coupling, on the other hand, is more difficult to treat since it is singular [1]. However, the first three terms were known analytically in this limit for a long time [13][14][15], until a recent work [16] (see also Ref. [17]) which contains conjectures for the analytical form of the first six terms in the ground state energy and the general structure of the following ones based on the double extrapolation of numerical solution of the discrete Bethe ansatz equations.In this paper we study the Lieb-Liniger model at weak repulsion. We develop an algorit...

Variational Bethe ansatz approach for dipolar one-dimensional bosons

“…The expression (B4) was obtained 50 by fitting the ground state energy to a polynomial expression in √ γ for γ < 15. An exact expansion has been conjectured 51,52 . We have checked that in the range 0 < γ < 8 the relative difference between the two expressions was under 2 × 10 −3 , while the relative difference between the derivatives was under 10 −2 .…”

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

“…Very recently, further coefficients have been explicitly obtained using two independent methods. On the one hand, an efficient algorithmic expansion at low γ, combined to an integer coefficients algorithm to find the explicit form of the latter has been used in [49] and allowed to identify coefficients up to order 8. On the other hand, a systematic algorithmic procedure that yields directly the exact coefficients has been developed in [50], and has yielded their value up to order 34, though they have been published up to eighth order only due to their lengthy expressions.…”

“…Although the exact ground-state energy can be obtained, in principle, from the exact Bethe Ansatz equations, only weak-and strong-coupling expansions are accessible to date [1, 40,[44][45][46][47][48][49][50][51]. As far as the Lieb-Liniger model is concerned, recently-developed algorithms allow to obtain these expansions to any order [40,49,50]. An important step forward would be to understand them well enough to predict their coefficients at arbitrary order without evaluating them algorithmically.…”

Report on scipost_201908_00007v1

“…Very recently, further coecients have been explicitly obtained using two independent methods. On the one hand, an ecient algorithmic expansion at low γ, combined to an integer coecients algorithm to nd the explicit form of the latter has been used in [49] and allowed to identify coecients up to order 8. On the other hand, a systematic algorithmic procedure that yields directly the exact coecients has been developed in [50], and has yielded their value up to order 34, though they have been published up to eighth order only due to their lengthy expressions.…”

“…Although the exact ground-state energy can be obtained, in principle, from the exact Bethe Ansatz equations, only weak-and strong-coupling expansions are accessible to date [1, 40,4451]. As far as the Lieb-Liniger model is concerned, recently-developed algorithms allow to obtain these expansions to any order [40,49,50]. An important step forward would be to understand them well enough to predict their coecients at arbitrary order without evaluating them algorithmically.…”