  By starting from the modified Maxwell theory coupled to gravity, the arising
of geometric quantum phases in the relativistic and nonrelativistic quantum
dynamics of a Dirac neutral particle from the effects of the violation of the
Lorentz symmetry in the cosmic string spacetime is investigated. It is shown
that the Dirac equation can be written in terms of an effective metric and a
relativistic geometric phase stems from the topology of the cosmic string
spacetime and the Lorentz symmetry breaking effects. Further, the
nonrelativistic limit of the Dirac equation is discussed and it is shown that
both Lorentz symmetry breaking effects and the topology of the defect yields a
phase shift in the wave function of the nonrelativistic spin-1/2 particle.
