Quantum Field Theory of Correlated Bose-Einstein condensates: II. Ward-Takahashi Identities and Correlation Functions

TL;DR

This paper derives Ward-Takahashi identities for correlated Bose-Einstein condensates, enabling exact results on correlation functions and opening pathways to develop superfluid Bose liquid theories similar to fermionic systems.

Contribution

It introduces Ward-Takahashi identities for correlated Bose-Einstein condensates, linking correlation functions to low-energy Green's functions and vertices, and explores their implications.

Findings

Exact relations for density and current correlation functions.

Expressions of correlation functions in terms of low-energy Green's functions.

Vertices exhibit different limits depending on frequency and wavenumber limits.

Abstract

We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the expressions of the first-order variations $(\delta\Psi,\delta G)$ due to perturbations obtained in the preceding paper [T. Kita, J. Phys. Soc. Jpn. $\bf 90$, 024001 (2021)] for the condensate wave function $\Psi$ and Green's function $G$. They enable us to obtain several exact results on the density and current correlation functions $K_{\nu\nu'}^{}$, and also express $K_{\nu\nu'}^{}$ in terms of low-energy Green's functions and vertices. The latter expressions open up the possibility of constructing theory of superfluid Bose liquids in the same way as that for fermions at low temperatures. The vertices are found to have different limits depending on which of frequency $\omega$ and wavenumber $q$ is set equal to zero first.