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, g M , of the one-dimensional (1D) strongly repulsive Bose gas within the Lieb-Liniger model using the analytical method proposed by Gangardt and Shlyapnikov (2003 Phys. Rev. Lett. 90 010401; 2003 New J. Phys. 5 79). 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 with a statistical parameter a g = -1 2 , i.e. the quasimomenta of N strongly interacting bosons map to the momenta of N free fermions via a » k kHere 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 g M in terms of γ and α at zero, low, and intermediate temperatures. For M = 2 and 3 our results reproduce the known expressions for g 2 and g 3 with sub-leading terms (see for instance (Vadim et al 2006 Phys. Rev. A 73 051604(R); Kormos et al 2009 Phys. Rev. Lett. 103 210404; Wang et al 2013 Phys. Rev. A 87 043634). We also express the leading order of the short distance non-local correlation functions1 of the strongly repulsive Bose gas in terms of the wave function of M bosons at zero collision energy and zero total momentum. Here ( ) Y x is the boson annihilation operator. These general formulas of the higherorder local and non-local correlation functions of the 1D Bose gas provide new insights into the manybody physics.

The Lieb-Liniger model is a prototypical integrable model and has been turned into the benchmark physics in theoretical and numerical investigations of low dimensional quantum systems. In this note, we present various methods for calculating local and nonlocal M -particle correlation functions, momentum distribution and static structure factor. In particular, using the Bethe ansatz wave function of the strong coupling Lieb-Liniger model, we analytically calculate twopoint correlation function, the large moment tail of momentum distribution and static structure factor of the model in terms of the fractional statistical parameter α = 1 − 2/γ, where γ is the dimensionless interaction strength. We also discuss the Tan's adiabatic relation and other universal relations for the strongly repulsive Lieb-Liniger model in term of the fractional statistical parameter.

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