Landau Theory for States with local translation symmetry and the Phenomenological Potential with minimal electron-phonon interactions

In the paper, the inhomogeneous states with local translational symmetry are under investigation in the framework of the Landau theory. The phonon potential is defined as a compensating field of local representation of the subgroup of translations. The constructed phenomenological Ginzburg-Landau potential takes into account both electromagnetic and electron-phonon interactions explicitly. The suggested phenomenological model does not require doubling of charge to get the correct expression of current density in the Londons equations and is in agreement with the BCS model. A phenomenological description of sound based on the phonon potential is also proposed.