Quantum phase transition in mesoscopic superconductors

Quantum phase transitions are studied by solving the modified Bogoliubov de Gennes (BdG) model for mesoscopic metallic superconductor with large surface scatterings. It is found that the nonlocal surface scattering effect can be mapped as a spin-orbit interaction into the model Hamiltonian, which exerts an effective magnetic field of opposite orientation for opposite spin electrons. Spin chiral states can occur with appropriate spin-orbit coupling strength. The competitions between the chiral edge states and the spin-singlet superconducting states can result in a series of quantum phase transitions with the applied axial magnetic field. The phase diagram for different scattering strength is presented.