Disorder effects inevitably exist in realistic samples, manifesting in various physical properties. In this paper, we review the recent progress in understanding the disorder effects on quantum transport and quantum phase transition properties in low-dimensional superconducting and topological systems. As a consequence of the pronounced quantum fluctuation in low-dimensional systems, rare events drastically change the physical characteristics and underlying microscopic transport process in these systems, which are beyond the traditional paradigms. Associating with recent experimental observations, we emphasize the microscopic mechanism for disordered Ising superconductivity, the quantum Griffiths singularity of superconductor metal transition and the discrete scale invariance in topological materials.
We solve the quasi-bound state-energy spectra and wavefunctions of an NPN-type graphene quantum dot under a perpendicular magnetic field. The evolution of the quasi-bound state spectra under the magnetic field is investigated using a Wentzel–Kramers–Brillouin approximation. In numerical calculations, we also show that the twofold energy degeneracy of the opposite angular momenta breaks under a weak magnetic field. As the magnetic field strengthens, this phenomenon produces an observable splitting of the energy spectrum. Our results demonstrate the relation between the quasi-bound state-energy spectrum in graphene quantum dots and magnetic field strength, which is relevant to recent measurements in scanning tunneling microscopy.
Electron in gapless bilayer graphene can form quasi-bound states when a circular symmetric potential is created in bilayer graphene. These quasi-bound states can be adjusted by tuning the radius and strength of the potential barrier. We investigate the evolution of quasi-bound states spectra in the circular n–p junction of bilayer graphene under the magnetic field numerically. The energy levels of opposite angular momentum split and the splitting increases with the magnetic field. Moreover, weak magnetic fields can slightly shift the energy levels of quasi-bound states. While strong magnetic fields induce additional resonances in the local density states, which originates from Landau levels. We demonstrate that these numerical results are consistent with the semiclassical analysis based on Wentzel–Kramers–Brillouin approximation. Our results can be verified experimentally via scanning tunneling microscopy measurements.
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