Exact Relations for a Strongly-interacting Fermi Gas from the Operator Product Expansion

TL;DR

This paper derives exact relations connecting the high-momentum tail of a strongly-interacting Fermi gas to local pair density using the operator product expansion, providing a simpler theoretical framework for understanding universal properties.

Contribution

It introduces a straightforward derivation of Tan's relations employing the operator product expansion, clarifying the connection between momentum tails and local pair densities.

Findings

Derived exact relations for Fermi gas properties

Identified the tail coefficient as a local pair density integral

Simplified the theoretical understanding of universal relations

Abstract

The momentum distribution in a Fermi gas with two spin states and a large scattering length has a tail that falls off like 1/k^4 at large momentum k, as pointed out by Shina Tan. He used novel methods to derive exact relations between the coefficient of the tail in the momentum distribution and various other properties of the system. We present simple derivations of these relations using the operator product expansion for quantum fields. We identify the coefficient as the integral over space of the expectation value of a local operator that measures the density of pairs.