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In this work we give, for the first time, the full relativistic Lagrangiandensity describing the motion of induced electric dipoles in the electricfields which induce the dipole, and the magnetic fields which generate the HMWtopological phase. We then use this relativistic Lagrangian density to derivethe complete set of conditions for producing topological phases with induceddipoles. We also give the relativistic Lagrangian density describing the motionof induced magnetic dipoles in the magnetic fields which induce the dipole, andthe electric fields which generate the AC topological phase, and derive theconditions for this AC phase to be topological. These conditions have beenincompletely discussed in previous studies. We note that, in both the AC andHMW cases, the topological phases are generated by the cross product ofelectric and magnetic fields in the form $\bm{B} \times \bm{E}$ whichreinforces the dual nature of these two topological phases.

Relativistic dipole interaction and the topological nature for induced HMW and AC phases