  Recently, the zero-pairing limit of Hartree-Fock-Bogoliubov (HFB) mean-field
theory was studied in detail in arXiv:2006.02871. It was shown that such a
limit is always well-defined for any particle number A, but the resulting
many-body description differs qualitatively depending on whether the system is
of closed-(sub)shell or open-(sub)shell nature. Here, we extend the discussion
to the more general framework of Finite-Temperature HFB (FTHFB) which deals
with statistical density operators, instead of pure many-body states. We
scrutinize in detail the zero-temperature and zero-pairing limits of such a
description, and in particular the combination of both limits. For closed-shell
systems, we find that the FTHFB formulism reduces to the (zero-temperature)
Hartree-Fock formulism, i.e. we recover the textbook solution. For open-shell
systems, however, the resulting description depends on the order in which both
limits are taken: if the zero-temperature limit is performed first, the FTHFB
density operator demotes to a pure state which is a linear combination of a
finite number of Slater determinants, i.e. the case of arXiv:2006.02871. If the
zero-pairing limit is performed first, the FTHFB density operator remains a
mixture of a finite number of Slater determinants with non-zero entropy, even
as the temperature vanishes. These analytical findings are illustrated
numerically for a series of Oxygen isotopes.
