Ground state energy of the $\delta$-Bose and Fermi gas at weak coupling from double extrapolation

TL;DR

This paper accurately computes higher-order terms in the weak coupling expansion of the ground state energy for delta Bose and Fermi gases using double extrapolation of numerical solutions, confirming and extending previous theoretical results.

Contribution

It introduces a numerical double extrapolation method to determine higher-order coefficients in the weak coupling expansion of the Lieb-Liniger and related Fermi gases, improving precision and confirming conjectured values.

Findings

Coefficient of γ^{5/2} in bosonic expansion approximately -0.0015877

Coefficient of γ^{3} in fermionic expansion approximately -ζ(3)/π^{4}

Numerical results support and refine previous theoretical conjectures.

Abstract

We consider the ground state energy of the Lieb-Liniger gas with $\delta$ interaction in the weak coupling regime $\gamma\to0$. For bosons with repulsive interaction, previous studies gave the expansion $e_{\text{B}}(\gamma)\simeq\gamma-4\gamma^{3/2}/3\pi+(1/6-1/\pi^{2})\gamma^{2}$. Using a numerical solution of the Lieb-Liniger integral equation discretized with $M$ points and finite strength $\gamma$ of the interaction, we obtain very accurate numerics for the next orders after extrapolation on $M$ and $\gamma$. The coefficient of $\gamma^{5/2}$ in the expansion is found approximately equal to $-0.00158769986550594498929$, 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 $\gamma$. It was identified as $(3\zeta(3)/8-1/2)/\pi^{3}$ by G. Lang. The next two coefficients are also guessed from numerics. For balanced spin $1/2$ fermions with attractive interaction, the best result so far for the ground state energy was $e_{\text{F}}(\gamma)\simeq\pi^{2}/12-\gamma/2+\gamma^{2}/6$. An analogue double extrapolation scheme leads to the value $-\zeta(3)/\pi^{4}$ for the coefficient of $\gamma^{3}$.