  We introduce a new term into the Dirac equation based on the Lorentz symmetry
violation background in order to make a theoretical description of the
relativistic quantum dynamics of a spin-half neutral particle, where the wave
function of the neutral particle acquires a relativistic Abelian quantum phase
given by the interaction between a fixed time-like 4-vector background and
crossed electric and magnetic fields, which is analogous to the geometric phase
obtained by Wei \textit{et al} [H. Wei, R. Han and X. Wei, Phys. Rev. Lett.
\textbf{75}, 2071 (1995)] for a spinless neutral particle with an induced
electric dipole moment. We also discuss the flux dependence of energy levels of
bound states analogous to the Aharonov-Bohm effect for bound states.
