A $(2+1)$-Dimensional Gauge Model with Electrically Charged Fermions

We consider a $(2+1)$-dimensional gauge theory with a nonzero fermion density and an initial Chern–Simons topological term, whose Lorentz invariance is spontaneously broken in a certain Lorentz reference frame by the generation of a constant homogenous magnetic field. We propose interpreting the number $\eta=\pm1$, which characterizes the two nonequivalent representations of Dirac matrices in $2+1$ dimensions, as a quantum number that explicitly describes the spin of the fermion. In particular, this interpretation allows determining the vacuum state of the model in a constant homogenous magnetic field as the state whose fermion and spin numbers are equal to zero.