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arXiv:1907.04410 · SciPost Physics

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

Guillaume Lang

Published Oct 23, 2019• 6 citations
arXivHTMLPDF

Abstract

In this paper, we apply experimental number theory to two integrable quantum models in one dimension, the Lieb-Liniger Bose gas and the Yang-Gaudin Fermi gas with contact interactions. We identify patterns in weak- and strong-coupling series expansions of the ground-state energy, local correlation functions and pressure. Based on the most accurate data available in the literature, we make a few conjectures about their mathematical structure and extrapolate to higher orders

DOI
10.21468/scipostphys.7.4.055

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In this paper, we apply experimental number theory to two integrable quantum models in one dimension, the Lieb-Liniger Bose gas and the Yang-Gaudin Fermi gas with contact interactions. We identify patterns in weak- and strong-coupling series expansions of the ground-state energy, local correlation functions and pressure. Based on the most accurate data available in the literature, we make a few conjectures about their mathematical structure and extrapolate to higher orders
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 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 quasi-momenta, 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.
Proposed is a method of solving the Lieb–Liniger integral equation for a one-dimensional δ-function bose gas at zero temperature. The integral equation reduces to a set of algebraic equations for the coefficients of the series expansion. For strong coupling case, a few coefficients are enough to have a good approximation. On the contrary, all the coefficients are in the same order for weak coupling case. The expansion in λ= c / K , where c is the coupling constant and K is the cutoff momentum, is a formal one in the sense that the coefficients may include the divergent sums. However, the physical quantities such as the ground state energy are shown to be finite in γ= c / ρ, where ρ is the number density, which agree with the results of the Bogoliubov theory.
We develop an alternative description to solve the problem of the 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 enable 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 values of the 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 exact perturbative results for the excitation spectrum.
We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb– Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem. PACS numbers: 05.30.−d, 67.40.Db, 05.30.Jp (Some figures in this article are in colour only in the electronic version) 1.
Abstract We consider the ground state energy of the Lieb–Liniger gas with δ interaction in the weak coupling regime γ → 0 . For bosons with repulsive interaction, previous studies gave the expansion e B ( γ ) ≃ γ − 4 γ 3 / 2 / 3 π + ( 1 / 6 − 1 / π 2 ) γ 2 . Using a numerical solution of the Lieb–Liniger integral equation discretized with M points and finite strength γ of the interaction, we obtain very accurate numerics for the next orders after extrapolation on M and γ . The coefficient of γ 5 / 2 in the expansion is found to be approximately equal to − 0.001 587 699 865 505 944 989 29 , accurate within all digits shown. This value is supported by a numerical solution of the Bethe equations with N particles, followed by extrapolation on N and γ . It was identified as ( 3 ζ ( 3 ) / 8 − 1 / 2 ) / π 3 by G Lang. The next two coefficients are also guessed from the numerics. For balanced spin 1/2 fermions with attractive interaction, the best result so far for the ground state energy has been
We consider the integrable one-dimensional δ-function interacting Bose gas in a hard wall box which is exactly solved via the coordinate Bethe ansatz. The ground-state energy, including the surface energy, is derived from the Lieb–Liniger-type integral equations. The leading and correction terms are obtained in the weak and strong coupling regimes from both the discrete Bethe equations and the integral equations. This allows the investigation of both finite-size and boundary effects in the integrable model. We also study the Luttinger liquid behaviour by calculating Luttinger parameters and correlations. The hard wall boundary conditions are seen to have a strong effect on the ground-state energy and phase correlations in the weak coupling regime. Enhancement of the local two-body correlations is shown by application of the Hellmann–Feynman theorem.