Topological phases of the Bogoliubov de Gennes Hamiltonian

Abstract

We investigate a two-dimensional superconducting system with a smoothly and periodically varying order parameter. The order parameter is modulated along one direction while remaining uniform in the perpendicular direction, leading to a spatially periodic superconducting phase. We show that the periodicity of the order parameter determines the winding number of the eigenfunctions, which serves as a topological characterization of the system. A topological invariant is identified that links the winding number directly with the Bloch vector. By solving the Bogoliubov-de Gennes equation, we obtain both plane-wave solutions describing bulk states and exponentially localized solutions that correspond to edge modes. The analytic bulk-edge connection is employed to identify the conditions under which the edge states emerge from the bulk spectrum. We find that the winding numbers depend on the boundary conditions, which differ between the plane-wave and exponential solutions. These results establish a direct connection between the spatial modulation of the order parameter, the topological structure of the eigenstates, and the emergence of edge modes in periodically modulated superconducting order parameters.