A Field Theoretic Approach to Roughness Corrections

TL;DR

This paper introduces a field theoretic framework to accurately compute roughness corrections to the Casimir free energy between parallel plates, accounting for surface height correlations and their impact on the force.

Contribution

It develops a systematic field theoretic and holographic model for roughness corrections, including a two-loop calculation and an effective low-energy theory, advancing the understanding of surface roughness effects.

Findings

Roughness weakens the Casimir force as correlation length decreases.

The effective separation accounts for surface roughness, removing certain correction orders.

The two-loop approximation bridges proximity force approximation and low-energy models.

Abstract

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