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We introduce a new term into the Dirac equation based on the Lorentz symmetryviolation background in order to make a theoretical description of therelativistic quantum dynamics of a spin-half neutral particle, where the wavefunction of the neutral particle acquires a relativistic Abelian quantum phasegiven by the interaction between a fixed time-like 4-vector background andcrossed electric and magnetic fields, which is analogous to the geometric phaseobtained 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 inducedelectric dipole moment. We also discuss the flux dependence of energy levels ofbound states analogous to the Aharonov-Bohm effect for bound states.

Abelian geometric phase for a Dirac neutral particle in a Lorentz symmetry violation environment