  We study a complex Dirac field in the chiral representation minimally coupled
to gravity in 3+1 dimensions in the context of Einstein-Cartan theory.
Generically the matter content gravitates in two different ways: On the one
hand, the energy-momentum induces spacetime curvature; on the other hand, the
presence of spin acts as a source for the spacetime torsion, which does not
propagate. In this setup we consider the most general stationary spherically
symmetric ansatz and we find an analytic black hole solution that supports a
nontrivial spinor configuration. The spinor field affects the geometry by
inducing spacetime torsion, though, remarkably, it does not alter the black
hole metric, which retains its Schwarzschild form. We find solutions both in
asymptotically flat and asymptotically (Anti) de Sitter spaces. Additionally,
we consider how the solution gets deformed when the so-called Holst term is
included in the gravity action. We discuss possible observational effects due
to the coupling of fermions in the nonvanishing torsion background.
Additionally we include a Maxwell field into the problem and find that it
forces the spinor field to be zero. Finally, nonminimal couplings of the spinor
with spacetime curvature are considered, obtaining any effect on the
spherically symmetric solutions.
