Tan's contact as an indicator of completeness and self-consistency of a theory

TL;DR

This paper discusses how Tan's contact can serve as a consistency check for theories of Bose and Fermi systems with contact interactions, highlighting the Hartree-Fock-Bogoliubov approach's strengths and limitations.

Contribution

It introduces a method to verify the self-consistency of theories using Tan's contact and evaluates the HFB approach's effectiveness for dilute Bose gases.

Findings

HFB approach is the most self consistent among MFT methods.

HFB matches experimental data at low gas parameters.

HFB fails at large gas parameters, indicating need for higher-order fluctuations.

Abstract

It is well known that, Tan's contact could be calculated by using any of following three methods: by the asymptotic behavior of momentum distribution; by Tan's adiabatic sweep theorem; or by the operator product expansion as an expectation value of the interaction term. We argue that, if a theory describing Bose (or Fermi) system with the only contact interaction is self consistent, then it should lead to the same result in all three cases. As an example we considered MFT based approaches and established that among existing approximations of MFT, the Hartree - Fock - Bogoliubov (HFB) approach is the most self consistent. Actually, HFB is able to describe existing experimental data on Tan's contact for dilute Bose gas, but fails to predict its expected behavior at large gas parameter $(\gamma > 0.015)$. So, for appropriate description of properties of a Bose gas even at zero temperature, this approximation needs to be expanded by taking into account fluctuations in higher order then the second one.