  We revisit the physics of electron gas bilayers in the quantum Hall regime
[Nature, 432 (2004) 691; Science, 305 (2004) 950], where transport and
tunneling measurements provided evidence of a superfluid phase being present in
the system. Previously, this behavior was explained by the possible formation
of a BEC of excitons in the half-filled electron bilayers, where empty states
play the role of holes. We discuss the fundamental difficulties with this
scenario, and propose an alternative approach based on a treatment of the
system as a pseudospin magnet. We show that the experimentally observed
tunneling peak can be linked to the XY ferromagnet (FM) to Ising
antiferromagnet (AFM) phase transition of the S=1/2 XXZ pseudospin model,
driven by the change in total electron density. This transition is accompanied
by a qualitative change in the nature of the low energy spin wave dispersion
from a gapless linear mode in the XY-FM phase to a gapped, quadratic mode in
the Ising-AFM phase.
