The correlation function is an important object in the physics of ultra-cold quantum gases. However, the study of high order correlation functions of interacting many-body systems is a notoriously difficult problem. In this talk, we first derive the explicit form of the M-body local correlation functions gM of the one-dimensional (1D) strongly interacting bosons in terms of dimensionless interaction strength and the Haldane exclusion statistical parameter ∝ = 1 − 2/ϒ at zero, low, and intermediate temperatures. We rigorously prove that such a statistical parameter ∝ solely contributes the sub-leading order to the M-body local correlation function of the gas with a finitely strong interaction regime. We also express the leading order of the short distance nonlocal correlation functions of the strongly repulsive Bose gas in terms of the wave function of M bosons at zero collision energy and zero total momentum. Finally, we further discuss the pair correlation and exact bounds between magnetism and Tan’s contact of 1D interacting fermions.
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