Field theoretic approach to roughness corrections

Abstract

We develop a systematic field theoretic description of roughness corrections to the Casimir free energy of a massless scalar field in the presence of parallel plates with mean separation $a$. Roughness is modeled by specifying a generating functional for correlation functions of the height profile. The two-point correlation function being characterized by its variance, ${\ensuremath{\sigma}}^{2}$, and correlation length, $\ensuremath{\ell}$. We obtain the partition function of a massless scalar quantum field interacting with the height profile of the surface via a $\ensuremath{\delta}$-function potential. The partition function is given by a holographic reduction of this model to three coupled scalar fields on a two-dimensional plane. The original three-dimensional space with a flat parallel plate at a distance $a$ from the rough plate is encoded in the nonlocal propagators of the surface fields on its boundary. Feynman rules for this equivalent $2+1$-dimensional model are derived and its counterterms constructed. The two-loop contribution to the free energy of this model gives the leading roughness correction. The effective separation, ${a}_{\mathrm{eff}}$, to a rough plate is measured to a plane that is displaced a distance $\ensuremath{\rho}\ensuremath{\propto}{\ensuremath{\sigma}}^{2}/\ensuremath{\ell}$ from the mean of its profile. This definition of the separation eliminates corrections to the free energy of order $1/{a}^{4}$ and results in unitary scattering matrices. We obtain an effective low-energy model in the limit $\ensuremath{\ell}\ensuremath{\ll}a$. It determines the scattering matrix and equivalent planar scattering surface of a very rough plate in terms of the single length scale $\ensuremath{\rho}$. The Casimir force on a rough plate is found to always weaken with decreasing correlation length $\ensuremath{\ell}$. The two-loop approximation to the free energy interpolates between the free energy of the effective low-energy model and that of the proximity force approximation -- the force on a very rough plate with $\ensuremath{\sigma}\ensuremath{\gtrsim}0.5\ensuremath{\ell}$ being weaker than on a planar Dirichlet surface at any separation.