Fermion condensates induced by axial interactions

We study the formation of fermion condensates induced by axial interactions. The starting point is a Dirac field with a quartic interaction written in terms of the axial current. This interaction has the form of a spin-spin coupling. It can arise, for example, in gravitational theories with spacetime torsion. The main problem is to understand whether this interaction can modify the vacuum of the theory. In particular, we ask if the vacuum can acquire a non-zero expectation value of the axial current. If this happens, the vacuum is no longer the free one. It becomes a condensate of fermion-antifermion pairs. Our aim is to describe this new vacuum and to derive the conditions for its existence. We also want to show that the condensate can generate a new source of axial current. This source is purely quantum. It is not imposed by an external field. We use a mean-field approximation. This approximation replaces the quartic interaction with an effective quadratic Hamiltonian. Then we quantize the Dirac field. We diagonalize the Hamiltonian through a Bogoliubov transformation. The diagonalization defines a new vacuum state. Finally, we impose self-consistency conditions on the axial current. The result is that a non-trivial condensate can appear for a spin-aligning interaction. It also breaks Lorentz symmetry spontaneously. For this reason, the mechanism may have implications for the early universe, CPT violation and cosmological models with torsion.