Structure Function Sum rules for Systems with Large Scattering Lengths

TL;DR

This paper derives model-independent sum rules for the dynamic structure functions of systems with large scattering lengths using dispersion relations and the operator product expansion, applicable to both fermions and bosons.

Contribution

It introduces a general method to obtain sum rules for response functions in systems with large scattering lengths, including corrections at finite scattering lengths.

Findings

Sum rules relate structure functions to the contact parameter.

Borel transform emphasizes low-energy data.

Applicable to both fermionic and bosonic systems.

Abstract

We use a dispersion relation in conjunction with the operator product expansion (OPE) to derive model independent sum rules for the dynamic structure functions of systems with large scattering lengths. We present an explicit sum rule for the structure functions that control the density and spin response of the many-body ground state. Our methods are general, and apply to either fermions or bosons which interact through two-body contact interactions with large scattering lengths. By employing a Borel transform of the OPE, the relevant integrals are weighted towards infrared frequencies, thus allowing for greater overlap low energy data. Similar sum rules can be derived for other response functions. The sum rules can be used to extract the contact parameter introduced by Tan, including universality violating corrections at finite scattering lengths.