Casimir force between vortex matter in anisotropic and layered superconductors

We present an approach to calculate the attractive long-range vortex-vortex interaction of the van der Waals type present in anisotropic and layered superconductors. The mapping of the statistical mechanics of vortex lines onto the imaginary time quantum mechanics of two-dimensional charged bosons allows us to define a two-dimensional (2D) Casimir problem: Two half spaces of (dilute) vortex matter separated by a gap of width R are mapped to two dielectric half planes of charged bosons interacting via a massive gauge field. We determine the attractive Casimir force between the two half planes and show that it agrees with the pairwise summation of the van der Waals force between vortices previously found by Blatter and Geshkenbein [Phys. Rev. Lett. 77, 4958 (1996)].