Fluctuation force induced by quenched random polarizations

We investigate the zero-temperature behavior of the fluctuation force induced by random electric dipoles that are frozen into a material and incapable of fluctuating thermally. Examples of such materials are 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 $\ell$ as well as a system comprising a neutral atom in the vacuum located at a distance $\ell$ 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 fluctuation force grows with the mean square quenched electric dipole moment per unit volume (area). The bulk (surface) dipolar disorder-induced fluctuation pressure between two semi-infinite single-layered slabs decays with $\ell^{-3}$ ($\ell^{-4}$), whereas the bulk (surface) dipolar disorder-induced fluctuation force on an atom in the vacuum near a slab containing the disorder decays with $\ell^{-4}$ ($\ell^{-5}$). We also find that the quenched dipolar disorder-induced fluctuation force between two coplanar slabs can be repulsive and serve to enhance the ``nanolevitation effect" in a three-layered dielectric system that obeys the Dzyaloshinskii-Lifshitz-Pitaevskii condition.