Spontaneous breaking of Lorentz symmetry in $(2+ϵ)$-dimensional QED

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 $ϵ$ 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.