Derivative expansion for the electromagnetic and Neumann Casimir effects in $2+1$ dimensions with imperfect mirrors

TL;DR

This paper develops a second-order derivative expansion to approximate the Casimir energy between imperfect mirrors in 2+1 dimensions, revealing non-analytic behavior near perfect conductivity and establishing a duality with a scalar field model.

Contribution

It introduces a second-order derivative expansion for the electromagnetic Casimir effect with imperfect mirrors, including analysis of non-analyticities and duality with scalar field models.

Findings

Second-order DE provides accurate Casimir energy estimates.

Non-analytic behavior emerges near perfect conductors.

Results apply to scalar fields via duality.

Abstract

We calculate the Casimir interaction energy in $d=2$ spatial dimensions between two (zero-width) mirrors, one flat, and the other slightly curved, upon which {\em imperfect\/} conductor boundary conditions are imposed for an Electromagnetic (EM) field. Our main result is a second-order Derivative Expansion (DE) approximation for the Casimir energy, which is studied in different interesting limits. In particular, we focus on the emergence of a non-analyticity beyond the leading-order term in the DE, when approaching the limit of perfectly-conducting mirrors. We also show that the system considered is equivalent to a dual one, consisting of a massless real scalar field satisfying imperfect Neumann conditions (on the very same boundaries). Therefore, the results obtained for the EM field hold also true for the scalar field model