Two-dimensional Cooper pairs in a periodic landscape

We study the phenomenology of a two-dimensional (2D) superconductor subjected to long-wavelength periodic modulation.This effective landscape interacts with the Cooper pairs, disrupting the homogeneity of the superconducting state. Near criticality, we represent the order parameter as a superposition of zero-th Landau levels and analytically minimize the Ginzburg-Landau free energy. We find that a 1D modulation hinders the formation of the Abrikosov vortex lattice and, in its place, promotes the localization of the Cooper pairs into an array of disconnected 1D filaments, effectively reducing the dimensionality of the superconductor over a finite range of magnetic fields. We support our phenomenological results with a microscopic analysis and highlight the significant experimental transport signatures associated to this novel superconducting phase transition.