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

Abstract

We consider the ground state energy of the Lieb-Liniger gas with $δ$ interaction in the weak coupling regime $γ\to0$. For bosons with repulsive interaction, previous studies gave the expansion $e_{\text{B}}(γ)\simeqγ-4γ^{3/2}/3π+(1/6-1/π^{2})γ^{2}$. 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 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 $γ$. It was identified as $(3ζ(3)/8-1/2)/π^{3}$ 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 $e_{\text{F}}(γ)\simeqπ^{2}/12-γ/2+γ^{2}/6$. An analogue double extrapolation scheme leads to the value $-ζ(3)/π^{4}$ for the coefficient of $γ^{3}$.