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

Source abstract

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 e F ( γ ) ≃ π 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 .