Particle distribution tail and related energy formula

We present a simple derivation of the energy formula found by Tan, relative to the single-channel Hamiltonian relevant for ultracold Fermi gases. This derivation is generalized to particles with different masses, to arbitrary mixtures, and to two-dimensional space. We show how in a field-theoretic approach, the $1/k^4$ tail in the momentum distribution and the energy formula arise in a natural way. As a specific example, we consider quantitative calculations of the energy from different formulas within the ladder diagram approximation in the normal state. The comparison of the results provides an indication on the quality of the approximation.

Figure 1

Energy of the gas at unitarity as a function of reduced temperature, obtained from Eq. (45) (dashed line), Eq. (25) (dashed-dotted line), and from the scaling relation (full line) $E=-\left(3/2\right)\Omega$, with the grand potential $\Omega$ obtained by integrating the particle density. The inset shows the results with an extended scale.