Predicting topological invariants and unconventional superconducting pairing from density of states and machine learning

Competition between magnetism and superconductivity can lead to unconventional and topological superconductivity. However, the experimental confirmation of the presence of Majorana edge states and unconventional pairing currently poses a major challenge. Here we consider a two-dimensional lattice model for a superconductor with spin-orbit coupling and exchange coupling to randomly distributed magnetic impurities. Depending on parameters of the model, this system may display topologically trivial or nontrivial edge states. We map out the phase diagram by computing the Bott index, a topological invariant defined in real space. We then use machine learning (ML) algorithms to predict the Bott index from the local density of states (LDOS) at zero energy, obtaining high-accuracy results. We also train ML models to predict the amplitude of odd-frequency pairing in the anomalous Green's function at zero energy. Once the ML models are trained using the LDOS, which is experimentally accessible via scanning tunneling spectroscopy, our method could be applied to predict the number of Majorana edge states and to estimate the magnitude of odd-frequency pairing in real materials.