Abnormal Meissner state in a superconducting bilayer

An ultrathin superconducting bilayer can admit a monotonic magnetic field when one layer has a slit even in a Meissner state. The rotation of the superconducting quantum phase by a vector potential for the magnetic field is compensated by a phase shift accompanying an inter-layer phase difference mode which grows from the slit. This phase difference mode may grow to form a fractional vortex, and it may show a close relationship with experimentally observed fractional vortices.