Hofstadter's Butterfly and Phase Transition of Checkerboard Superconducting Network in a Magnetic Field

Abstract

We study the magnetic effect of the checkerboard superconducting wire network. Based on the de Gennes–Alexader theory, we obtain difference equations for superconducting order parameter in the wire network. Through solving these difference equations, we obtain the eigenvalues, linked to the coherence length, as a function of magnetic field. The diagram of eigenvalues shows a fractal structure, being so-called Hofstadter's butterfly. We also calculate and discuss the dependence of the transition temperature of the checkerboard superconducting wire network on the applied magnetic field, which is related to up-edge of the Hofstadter's butterfly spectrum.