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

Abstract

We consider the ground state energy of the Lieb-Liniger gas with $\delta$ interaction in the weak coupling regime $\gamma \to 0$. 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 ${\gamma }^3$.

arXiv comment11 pages, 2 figures, 3 tables

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