Higher-order local and non-local correlations for 1D strongly interacting Bose gas

The correlation function is an important quantity in the physics of ultracold quantum gases because it provides information about the quantum many-body wave function beyond the simple density profile. In this paper we first study the M-body local correlation functions, gM, of the one-dimensional (1D) strongly repulsive Bose gas within the Lieb-Liniger model using the analytical method proposed by Gangardt and Shlyapnikov [1,2]. In the strong repulsion regime the 1D Bose gas at low temperatures is equivalent to a gas of ideal particles obeying the non-mutual generalized exclusion statistics (GES) with a statistical parameter α=1−2/γ, i.e. the quasimomenta of N strongly interacting bosons map to the momenta of N free fermions via ki≈αkFi with i=1,…,N. Here γ is the dimensionless interaction strength within the Lieb-Liniger model. We rigorously prove that such a statistical parameter α solely determines the sub-leading order contribution to the M-body local correlation function of the gas at strong but finite interaction strengths. We explicitly calculate the correlation functions gM in terms of γ and α at zero, low, and intermediate temperatures. For M=2 and 3 our results reproduce the known expressions for g2 and g3 with sub-leading terms (see for instance [3-5]). We also express the leading order of the short distance \emph{non-local} correlation functions ⟨Ψ†(x1)⋯Ψ†(xM)Ψ(yM)⋯Ψ(y1)⟩ of the strongly repulsive Bose gas in terms of the wave function of M bosons at zero collision energy and zero total momentum. Here Ψ(x) is the boson annihilation operator. These general formulas of the higher-order local and non-local correlation functions of the 1D Bose gas provide new insights into the many-body physics