# Non-Universal and Universal Aspects of The Large Scattering Length Limit

## TL;DR

This paper investigates the momentum density in interacting Fermionic systems, revealing the limits of zero-range models and establishing conditions for universality based on scattering length and effective range.

## Contribution

It demonstrates the non-universality of the momentum density at high momenta and derives a universal behavior when including finite effective range effects.

## Key findings

- Zero-range models fail for momenta above a certain threshold.
- Universal behavior of $n(k)$ when including finite effective range.
- Derived a relation between Fermi-gas energy and density integral.

## Abstract

The momentum density, $n(k)$ of interacting many-body Fermionic systems is studied (for $k>k_F)$ using examples of several well-known two-body interaction models. This work shows that $n(k)$ can not be approximated by a zero-range model for momenta $k$ greater than about $1/(a r_e^2)^{1/3}$, where $a$ is the scattering length, and $r_e$ the effective range. However, if the scattering length is large and one includes the effects of a fixed value of $r_e\ne0$, $n(k)$ is universal for momenta $k$ up to about $2/r_e$. An accurate relation between the energy of a two component Fermi-gas and an integral involving the density is derived. The short separation distance, $s$, behavior of the pair density is shown to vary as $s^6$.

---
Source: https://tomesphere.com/paper/1702.03370