We study the Hartree-Fock-Bogoliubov mean-field theory as applied to a
two-dimensional finite trapped Bose gas at low temperatures and find that, in
the Hartree-Fock approximation, the system can be described either with or
without the presence of a condensate; this is true in the thermodynamic limit
as well. Of the two solutions, the one that includes a condensate has a lower
free energy at all temperatures. However, the Hartree-Fock scheme neglects the
presence of phonons within the system, and when we allow for the possibility of
phonons we are unable to find condensed solutions; the uncondensed solutions,
on the other hand, are valid also in the latter, more general scheme. Our
results confirm that low-energy phonons destabilize the two-dimensional
condensate.Comment: 8 pages, 3 figures, REVTeX 4. To appear in J. Low Temp. Phys.
Corrected a mistake in a calculation and changed the conclusions accordingl
