Hamiltonian learning with real-space impurity tomography in topological moiré superconductors

Extracting Hamiltonian parameters from available experimental data is a challenge in quantum materials. In particular, real-space spectroscopy methods such as scanning tunneling spectroscopy allow probing electronic states with atomic resolution, yet even in those instances extracting the effective Hamiltonian is an open challenge. Here we show that impurity states in modulated systems provide a promising approach to extracting non-trivial Hamiltonian parameters of a quantum material. We show that by combining the real-space spectroscopy of different impurity locations in a moiré topological superconductor, modulations of exchange and superconducting parameters can be inferred via machine learning. We demonstrate our strategy with a physically-inspired harmonic expansion combined with a fully-connected neural network that we benchmark against a conventional convolutional architecture. We show that while both approaches allow extracting exchange modulations, only the former approach allows inferring the features of the superconducting order. Our results demonstrate the potential of machine learning methods to extract Hamiltonian parameters by real-space impurity spectroscopy as local probes of a topological state.