Conserving and gapless approximations for the composite bosons in terms of the constituent fermions

TL;DR

This paper demonstrates that for composite bosons formed from fermion pairs, conserving approximations inherently lead to a gapless excitation spectrum, resolving a long-standing dichotomy in many-body theory.

Contribution

It shows that in the strong-coupling limit, conserving approximations for constituent fermions produce gapless bosonic excitations when solved self-consistently.

Findings

Conserving approximations yield gapless spectra for composite bosons.

The dichotomy between conserving and gapless approximations does not exist for composite bosons.

Self-consistent solutions for Green's functions ensure both conservation laws and gapless excitations.

Abstract

A long-standing problem with the many-body approximations for interacting condensed bosons has been the dichotomy between the ``conserving'' and ``gapless'' approximations, which either obey the conservations laws or satisfy the Hugenholtz-Pines condition for a gapless excitation spectrum, in the order. It is here shown that such a dichotomy does not exist for a system of composite bosons, which form as bound-fermion pairs in the strong-coupling limit of the fermionic attraction. By starting from the constituent fermions, for which conserving approximations can be constructed for any value of the mutual attraction according to the Baym-Kadanoff prescriptions, it is shown that these approximations also result in a gapless excitation spectrum for the boson-like propagators in the broken-symmetry phase. This holds provided the corresponding equations for the fermionic single- and two-particle Green's functions are solved self-consistently.