A gapless time-reversal symmetry violating superconductor

We consider a layered superconductor with a complex order parameter whose phase switches sign from one layer to the next. This system is shown to exhibit gapless superconductivity for sufficiently large interlayer pairing or interlayer hopping. In addition, this description is consistent with experiments finding signals of time reversal symmetry breaking in high-temperature superconductors only at the surface and not in the sample bulk.