Field theoretic approach to roughness corrections

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, σ 2 , and correlation length, ℓ, of the profile. We obtain the partition function of a massless scalar quantum field interacting with the height profile of the surface via a δ-function potential. The partition function of this model also is given by a holographic reduction 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 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, a eff , to a rough plate is measured to an equivalent plane that is displaced a distance ρ ∝ σ 2 /ℓ from the mean of its profile. This definition of the separation eliminates corrections to the free energy of order 1/a 4 eff and results in a unitary model. We derive an effective low-energy theory in the limit ℓ ≪ a. It gives the scattering matrix and equivalent planar surface of a very rough plate in terms of the single length scale ρ. The Casimir force on a rough plate is found to always weaken with decreasing correlation length ℓ. 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 Dirichlet plate with σ 0.5ℓ being weaker than on a flat plate at any separation.

The effects of surface electronic structure on Casimir interaction at short range

“…From a fundamental point of view, this means a more realistic modelling of the surface boundary that includes the short distances phenomenon, such as electron tunneling and nonlocal effects, need to be constructed and employed. Similar problems can be found when calculating the Casimir energies for the objects or surface structures with small length, a=1/ω p , scale [4,5] where the high momentum contributions dominate the interaction.…”

“…The Casimir energy density thus can be calculated as whereĀ is the plane's surface area andT is the time. Using the quantized vector potential, ( ) t A x, , in equations (4), (5) can be evaluated at the poles of the photon propagators located at the second and fourth quadrant by performing an Euclidean rotation and switching to imaginary frequency ω→iζ…”