Abstract
The recently discovered universal thermodynamic behavior of dilute, strongly interacting Fermi gases also implies a universal structure in the many-body pair-correlation function at short distances, as quantified by the contact 
. Here, we theoretically calculate the temperature dependence of this universal contact for a Fermi gas in free space and in a harmonic trap. At high temperatures above the Fermi degeneracy temperature, T≳TF, we obtain a reliable non-perturbative quantum virial expansion up to third order. At low temperatures, we compare different approximate strong-coupling theories. These make different predictions, which need to be tested either by future experiments or by advanced quantum Monte Carlo simulations. We conjecture that in the universal unitarity limit, the contact or correlation decreases monotonically with increasing temperature, unless the temperature is significantly lower than the critical temperature, T≪Tc∼0.2TF. We also discuss briefly how to measure the universal contact in either homogeneous or harmonically trapped Fermi gases.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. Ultracold atoms, cooled to below a micro-Kelvin, provide exceptionally well-controlled models for the behaviour of strongly interacting fermions. This is one of the most challenging problems in present-day physics. A better understanding of strongly interacting fermions has wide-ranging implications for systems ranging from quark matter in neutron stars to high-temperature superconductors. The recent realization of ultracold molecule formation resonances has provided a means for verifying fermionic universality: all strongly interacting, dilute Fermi gases should behave identically, depending only on a scaling factor. An important universal property is the 'contact', which gives the short-distance correlations.
Main results. In this paper, we investigate the temperature dependence of the contact for a universal Fermi gas. Firstly, we obtain a reliable high-temperature quantum virial expansion study of the contact up to third order. At lower temperatures, as there is no controllable small interaction parameter, a comparative study using different strong coupling theories is necessary. We find that the theoretical descriptions of the finite-temperature contact from different strong coupling theories show considerable discrepancies near the critical temperature Tc~0.2TF for the onset of superfluidity.
Wider implications. This theoretical discrepancy remains to be resolved by future experiments and Monte Carlo simulations, exploring the critical temperature regime. We have proposed several experimental measurements. All of these are within reach of current techniques. In the larger picture, obtaining a reliable theory for strongly interacting fermions at finite temperature is also crucial to many other fields of physics.
Figure. Universal contact of a homogeneous Fermi gas in the unitarity limit. The predictions from different strong coupling theories are compared to one another. At high temperatures, they are also compared to the virial expansion results. The symbols at low temperature indicate the predictions calculated by using the experimental equation of state (star), ground state quantum Monte Carlo results (circle) and finite temperature lattice quantum Monte Carlo simulations (square).