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

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