Charge and Current Sum Rules in Quantum Media Coupled to Radiation II

TL;DR

This paper extends previous sum rules for charge and current density correlations in quantum media to inhomogeneous interface geometries, revealing new sum rules and interface effects in fluctuating media.

Contribution

It generalizes charge-charge sum rules to interface geometries and introduces new charge-current and current-current sum rules for quantum media.

Findings

Charge-charge sum rules relate to interface dipole moments.

New current-current sum rules show breaking of directional invariance.

Explicit expressions provided for classical and jellium models.

Abstract

This paper is a continuation of the previous study [\v{S}amaj, L.: J. Stat. Phys. {\bf 137}, 1-17 (2009)], where a sequence of sum rules for the equilibrium charge and current density correlation functions in an infinite (bulk) quantum media coupled to the radiation was derived by using Rytov's fluctuational electrodynamics. Here, we extend the previous results to inhomogeneous situations, in particular to the three-dimensional interface geometry of two joint semi-infinite media. The sum rules derived for the charge-charge density correlations represent a generalization of the previous ones, related to the interface dipole moment and to the long-ranged tail of the surface charge density correlation function along the interface of a conductor in contact with an inert (not fluctuating) dielectric wall, to two fluctuating semi-infinite media of any kind. The charge-current and current-current sum rules obtained here are, to our knowledge, new. The current-current sum rules indicate a breaking of the directional invariance of the diagonal current-current correlations by the interface. The sum rules are expressed explicitly in the classical high-temperature limit (the static case) and for the jellium model (the time-dependent case).