  We use large-scale fermion quantum Monte Carlo simulations to study metallic
and deconfined quantum phase transitions in a bilayer honeycomb model, focusing
on their quantum critical and finite-temperature properties. At weak
interaction, a fully symmetric Dirac semimetal state is realized. At
intermediate and strong interaction, respectively, two long-range-ordered
phases that break different symmetries are stabilized. The ordered phases
feature partial and full, respectively, gap openings in the fermion spectrum.
We clarify the symmetries of the different zero-temperature phases and the
symmetry breaking patterns across the two quantum phase transitions between
them. The first transition between the disordered and long-range-ordered
semimetallic phases has previously been argued to be described by the
$(2+1)$-dimensional Gross-Neveu-SO(3) field theory. By performing simulations
with an improved symmetric Trotter decomposition, we further substantiate this
claim by computing the critical exponents $1/\nu$, $\eta_\phi$, and
$\eta_\psi$, which turn out to be consistent with the field-theoretical
expectation within numerical and analytical uncertainties. The second
transition between the two long-range-ordered phases has previously been
proposed as a possible instance of a metallic deconfined quantum critical
point. We further develop this scenario by analyzing the spectral functions in
the single-particle, particle-hole, and particle-particle channels. Our results
indicate gapless excitations with a unique velocity, supporting the emergence
of Lorentz symmetry at criticality. We also determine the finite-temperature
phase boundaries above the fully gapped state at large interaction, which
smoothly vanish near the putative metallic deconfined quantum critical point,
consistent with a continuous or weakly first-order transition.
