Quenched fluctuation-induced force arising from polarization disorder

We investigate the zero-temperature behavior of the fluctuation force induced by the quenched disorder of electric dipoles frozen randomly into a material. Examples of such materials include relaxor ferroelectrics. In terms of the setup and geometry, we focus on a layered system comprising two coplanar semi-infinite slabs separated by a distance $\mathrm{\ell }$ as well as a system comprising a neutral atom in the vacuum located at a distance $\mathrm{\ell }$ above the surface of a semi-infinite slab. For both systems, we consider the cases where the quenched random dipolar disorder occurs inside the bulk as well as on the surface of the slabs. In all of these cases, we find that the bulk (surface) dipolar disorder-induced force grows with the mean square quenched electric dipole moment per unit volume (area). The bulk (surface) dipolar disorder-induced pressure between two semi-infinite single-layered slabs decays with ${\mathrm{\ell }}^{-3}$ (${\mathrm{\ell }}^{-4}$), whereas the bulk (surface) dipolar disorder-induced force on an atom in the vacuum near a slab containing the disorder decays with ${\mathrm{\ell }}^{-4}$ (${\mathrm{\ell }}^{-5}$). We also find that the quenched dipolar disorder-induced force between two coplanar slabs can be repulsive in a three-layered dielectric system that obeys a zero-frequency analogue of the Dzyaloshinskii-Lifshitz-Pitaevskii condition, and serve to enhance the ``nanolevitation effect" if the latter is present in the disorder-free system.