Berry phase for a spin-1/2 particle moving in a flat space-time with torsio
n is investigated in the context of the Einstein-Cartan-Dirac model. It is
shown that if the torsion is due to a dense polarized background, then ther
e is a Berry phase only if the fermion is massless and its momentum is perp
endicular to the direction of the background polarization. The order of mag
nitude of this Berry phase is discussed in other theoretical frameworks.