Connection between nonlocal one-body and local three-body correlations of the Lieb-Liniger model

We derive a connection between the fourth coefficient of the short-distance Taylor expansion of the one-body correlation function, and the local three-body correlation function of the Lieb-Liniger model of $\delta$-interacting spinless bosons in one dimension. This connection, valid at arbitrary interaction strength, involves the fourth moment of the density of quasimomenta. Generalizing recent conjectures, we propose approximate analytical expressions for the fourth coefficient covering the whole range of repulsive interactions, validated by comparison with accurate numerics. In particular, we find that the fourth coefficient changes sign at interaction strength ${\gamma }_c\simeq 3.816$, while the first three coefficients of the Taylor expansion of the one-body correlation function retain the same sign throughout the whole range of interaction strengths.

Figure 1

Dimensionless fourth moment of the distribution of quasimomenta $e_4$ as a function of the dimensionless interaction strength $\gamma$. Analytical result from the conjecture (47) (solid blue line) is in excellent agreement with independent accurate numerics from the authors (red and black dots) for all interaction strengths. The conjecture in the weakly interacting regime, Eq. (42), with appropriate coefficients (black dashed line) reproduces numerical calculations with excellent accuracy up to intermediate interactions.

Figure 2

Derivative of the dimensionless fourth moment of the distribution of quasimomenta $e_4$ with respect to the dimensionless interaction strength $\gamma$ as a function of the latter. Using the analytical results either from the conjectures (47) (solid blue line) at strong interactions or Eq. (42) (dashed black line) at weak interactions, one finds an excellent agreement with accurate numerics (black dots) for all interaction strengths.

Figure 3

Dimensionless coefficient $c_4$ as a function of the dimensionless interaction strength $\gamma$, as predicted from the conjectures (solid blue and dashed black lines), compared to accurate numerics (red dots). A sign inversion occurs around $\gamma =3.8$.

Figure 4

Dimensionless coefficients $c_2, c_4$, and $c_3$ (black, blue, and red lines, from bottom to top, respectively) as predicted from conjectures, as functions of the dimensionless interaction strength $\gamma$.

Figure 5

Derivative $e_2^{\prime }$ of the dimensionless energy as a function of the dimensionless interaction strength $\gamma$, as predicted from the conjectures (solid blue and dashed black lines), compared to accurate numerics (red dots).

Figure 6

Dimensionless three-body local correlation function $g_3$, as a function of the dimensionless interaction strength $\gamma$ obtained numerically (dots) compared to the three accurate expressions in Ref. [23] (solid black, red, and blue lines) in a wide range of interaction strengths.