Reconstructing the Hamiltonian from the local density of states using neural networks

Reconstructing a quantum system's Hamiltonian from limited yet experimentally observable information is interesting both as a practical task and from a fundamental standpoint. We investigate the inverse problem of reconstructing a Hamiltonian from a spatial map of the local density of states (LDOS) near a fixed energy. We demonstrate high-quality recovery of Hamiltonians from the LDOS using supervised learning. In particular, we generate synthetic data from single-particle Hamiltonians in 1D and 2D, train convolutional neural networks, and obtain models that solve the inverse problem with remarkably high accuracy. Moreover, we are able to generalize beyond the training distribution and develop models with strong robustness to noise. Finally, we comment on possible experimental applications to scanning tunneling microscopy, where we propose that maps of the electronic LDOS might be used to reveal a sample's unknown underlying energy landscape.