  We analyze the many-flavor phase diagram of quantum electrodynamics (QED) in
2+1 (Euclidean) space-time dimensions. We compute the critical flavor number
above which the theory is in the quasi-conformal massless phase. For this, we
study the renormalization group fixed-point structure in the space of gauge
interactions and pointlike fermionic self-interactions, the latter of which are
induced dynamically by fermion-photon interactions. We find that a reliable
estimate of the critical flavor number crucially relies on a careful treatment
of the Fierz ambiguity in the fermionic sector. Using a Fierz-complete basis,
our results indicate that the phase transition towards a chirally-broken phase
occurring at small flavor numbers could be separated from the quasi-conformal
phase at larger flavor numbers, allowing for an intermediate phase which is
dominated by fluctuations in a vector channel. If these interactions approach
criticality, the intermediate phase could be characterized by a
Lorentz-breaking vector condensate.
