Casimir-Foucault interaction: Free energy and entropy at low temperature

It was recently found that thermodynamic anomalies which arise in the Casimir effect between metals described by the Drude model can be attributed to the interaction of fluctuating Foucault (or eddy) currents [F. Intravaia and C. Henkel, Phys. Rev. Lett. 103, 130405 (2009).] We focus on the transverse electric (TE) polarization, where the anomalies occur, and show explicitly that the two leading terms of the low-temperature correction to the Casimir free energy of interaction between two plates are identical to those pertaining to the Foucault current interaction alone, up to a correction which is very small for good metals. Moreover, a mode density along real frequencies is introduced, showing that the TE contribution to the Casimir free energy, as given by the Lifshitz theory, separates in a natural manner into contributions from eddy currents and propagating cavity modes, respectively. The latter have long been known to be of little importance to the low-temperature Casimir anomalies. This convincingly demonstrates that eddy current modes are responsible for the large temperature correction to the Casimir effect between Drude metals, predicted by the Lifshitz theory, but not observed in experiments.

Figure 1

(a) Complex eigenfrequencies in the parallel plate geometry, for a fixed wave vector $k$ (not to scale). The thick dotted lines show branch cuts (three-dimensional mode continuum) and crosses mark poles (discrete frequencies). The circular dot is a pole of the first factor in Eq. (2.18). The pole structure is symmetric with respect to the imaginary axis, according to Eq. (2.19). The frequency ${\omega }_{\mathrm{B}}(k)$ marks the transition from discrete cavity modes to a continuum of bulk modes (see Ref. [27] for details). (b) Integration path: $C_D$ around the eddy current continuum, $C_{\omega }$ around the pole at $z=\omega +i0$, $C_{+}$ around cavity and propagating modes. The corresponding paths in the left half plane are denoted $C_{-\omega }$ and $C_{-}$, respectively. (c) Integration path that encircles all modes, as relevant for the Lifshitz theory. Closing this contour in the upper half plane, one gets the residue from the pole $z=\omega +i0$ in the upper half plane.