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Fractional topological excitations and quantum phase transition in a bilayer 2DEG adjacent to a superconductor film

We study a bilayer two-dimension-electron-gas (2DEG) adjacent to a type-II superconductor thin film with a pinned vortex lattice. We find that with increasing interlayer tunneling, the system of half filling presents three phases: gapped phase-I (topological insulator), gapless critical phase-II (metal), and gapped phase-III (band insulator). The Hall conductance for phase-I/III is 2/0 $e^{2}/h$, and has non-quantized values in phase-II. The excitation (response to topological defect, a local vortex defect) in these three phases shows different behaviors due to the topological property of the system, including fractional charge $e/2$ for each layer in phase-I. While in the case of quarter filling, the system undergoes a quantum phase transition from metallic phase to topological insulator phase (with excitation of fractional charge $e/4$).