Particle distribution tail and related energy formula

TL;DR

This paper derives and generalizes the energy formula for ultracold Fermi gases, explaining the momentum distribution tail and comparing approximation methods for energy calculations.

Contribution

It provides a simple, generalized derivation of Tan's energy formula for various particle types and dimensions, linking it to the momentum distribution tail in a field theoretical framework.

Findings

Derivation of a generalized energy formula for different particle masses and dimensions.

Explanation of the $1/k^4$ tail in momentum distribution within a field theoretical approach.

Comparison of energy calculation methods within ladder diagrams approximation.

Abstract

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 theoretical 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 diagrams approximation in the normal state. The comparison of the results provides an indication on the quality of the approximation.