  In bilayer systems, the additional layer pseudospin enables the emergence of
interlayer coherence (IC), which is a direct consequence of the interlayer
Coulomb interaction. This study presents a comprehensive HF investigation of IC
in 2D bilayers, uncovering ground-state behaviors and temperature-dependent
phase transitions that are distinct from single-layer 2DEG. This IC signals a
spontaneous breaking of the U(1) symmetry in pseudospin. We explore the
zero-temperature phase diagrams as a function of the electron density and
interlayer separation within the HF formalism. We also calculate the critical
temperature ($T_c$) of the interlayer coherence onset by self-consistently
solving the HF gap-like equation. We contrast this IC phase in e-e bilayers
with the closely related excitonic superfluid phase in e-h bilayers. Though
both e-e and e-h bilayers spontaneously break the pseudospin U(1) symmetry, e-h
bilayers produce BCS-BEC crossover intrinsic to the excitons acting as
effective bosons or Cooper pairs, whereas the symmetry-broken phase in e-e
bilayers is akin to the XY or easy-plane pseudospin ferromagnetism. Using the
same system parameters and a similar theoretical framework, we find that $T_c$
of the interlayer coherent phase in e-e bilayers is about one-third of that in
exciton condensates, suggesting a weaker IC in e-e bilayers. In addition, we
examine the effect of a weak interlayer tunneling on the IC order parameter,
drawing parallels with the influence of an effective in-plane magnetic field on
the XY pseudospin ferromagnetism. Our findings provide a comparative
theoretical framework that bridges the gap between the IC physics in e-e and
e-h bilayers, contributing to a unified understanding of phase transitions in
low-dimensional electron/hole systems and establishing in particular the same
universality class for interlayer phase coherence in both e-e and e-h bilayers.
