Persistent homology of topological band structures

Persistent homology is a powerful machine learning technique for classifying complex systems or data sets, based on computing topological features over a range of spatial or energy scales. There is growing interest in applying persistent homology to characterize complex condensed matter systems, with recent applications including the classification of multiqubit entangled states and identification of hidden order in interacting spin models. In this work, we propose to use persistent homology to characterize band structures of periodic lattices. Using the Haldane model as an example, we show that persistent homology is able to reliably identify novel band structures including degenerate ``moat'' band minima and transitions between trivial and non-trivial Chern insulating phases. Our method is promising for the characterization of more complex wave systems with many internal degrees of freedom, such as Moire superlattices and superconducting circuits.