Real-Space Topological Invariant for Disordered Semiconductor Majorana Wire

We discuss a new topological invariant that serves as the real-space analog of the winding number invariant for the disordered semiconductor Majorana system. Our study focuses on a one-dimensional semiconductor-superconductor (SM-SC) heterostructure with Rashba spin-orbit coupling and a parallel Zeeman field, accounting for disorder from random charged impurities. We demonstrate that, as the disorder strength increases, the winding number invariant works better than the topological visibility, used in recent literature, to distinguish between topologically trivial and non-trivial states. Additionally, we show that a limited number of large-wavelength Fourier components of the disorder profile, obtainable from the zero-bias conductance data using machine learning, can adequately capture the potential topological phases in the parameter space using the winding number invariant.