  The phase diagram of massless quantum electrodynamics in three space-time
dimensions as a function of fermion flavor number $N$ exhibits two well-known
phases: at large $N > N_c^{conf}$ the system is in a conformal gapless state,
while for small $N < N_c^{\chi SB}$ the fermions are expected to develop a
dynamical mass due to spontaneous chiral symmetry breaking. Using $\epsilon$
expansion near the lower critical dimension of 2, as well as the recent results
on the generalization of the $F$ theorem to continuous dimension, we show that
$N_c^{conf} > N_c^{\chi SB}$. There is therefore an intermediate range of
values of $N$ at which a third phase is stabilized. We demonstrate that this
phase is characterized by spontaneous breaking of Lorentz symmetry, in which a
composite vector boson field acquires a vacuum expectation value with the
fermions and the photon remaining massless.
