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=+/-1, 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 held as the state whose fermion and spin numbers are equal to zero.