Periodic vortex lattices for the Lawrence-Doniach model of layered superconductors in a parallel field

We consider the Lawrence-Doniach model for layered superconductors, in which stacks of parallel superconducting planes are coupled via the Josephson effect. We assume that the superconductor is placed in an external magnetic field oriented parallel to the superconducting planes and study periodic lattice configurations in the limit as the Josephson coupling parameter r tends to zero. This limit leads to the "transparent state" discussed in the physics literature, which is observed in very anisotropic high-temperature superconductors at sufficiently high applied fields and below a critical temperature. We use a Lyapunov-Schmidt reduction to prove that energy minimization uniquely determines the geometry of the optimal vortex lattice: a period-2 (in the layers) array proposed by Bulaevskii & Clem. Finally, we discuss the apparent conflict with previous results for finite-width samples, in which the minimizer in the small coupling regime takes the form of "vortex planes" (introduced by Theodorakis and Kuplevakhsky.)