Inhomogeneous superconducting states in two weakly linked superconducting ultrathin films

A sufficiently large parallel magnetic field will generate staggered supercurrent loops and a superfluid density wave in two weakly linked superconducting (SC) ultrathin films, resulting in an inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. The SC order parameter of such an FFLO state is characterized by Bloch wave functions, called the ``Bloch SC state.'' The staggered supercurrent loops form an array of Josephson vortex-antivortex pairs, instead of the usual Josephson vortex lattice. Enclosing a unit cell of the array, the London's fluxoid is quantized as ${\mathrm{\ensuremath{\Phi}}}^{\ensuremath{'}}={\mathrm{\ensuremath{\Phi}}}₀=hc/2e$, while the net orbital magnetization caused by the staggered supercurrent is zero. Meanwhile, a small parallel magnetic field gives rise to an Fulde-Ferrell (FF) state that has uniform superfluid density. The phase transition between the Bloch SC state and the FF state belongs to the universality class of two-dimensional commensurate-incommensurate transitions. An analytical solution in terms of Jacobian elliptic functions is found to be an excellent approximation to the Bloch SC order parameter.