Short Range Scaling Laws of Quantum Gases With Contact Interactions

Abstract

The probability amplitude for $N$ particles in a quantum gas with negligible range of interparticle interaction potentials to come to a small region of size $r$ scales like $r^{\gamma }$. It is shown that $\gamma$ is quantitatively related to the ground state energy of these $N$ fermions in the unitarity limit, confined by an isotropic harmonic potential. For large $N$, the short range density distribution of these $N$ particles is predominantly the same as the Thomas-Fermi profile of the gas in the unitarity limit confined by such a harmonic potential. These results may shed light on strongly interacting ultracold atomic Fermi gases, in a trap or on an optical lattice.

4 pages, 1 figure. 2nd version: the last sentence of the abstract changed; acknowledgements completed

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