  The machine learning technique of persistent homology classifies complex
systems or datasets by computing their topological features over a range of
characteristic scales. There is growing interest in applying persistent
homology to characterize physical systems such as spin models and multiqubit
entangled states. Here we propose persistent homology as a tool for
characterizing and optimizing band structures of periodic photonic media. Using
the honeycomb photonic lattice Haldane model as an example, we show how
persistent homology is able to reliably classify a variety of band structures
falling outside the usual paradigms of topological band theory, including "moat
band" and multi-valley dispersion relations, and thereby control the properties
of quantum emitters embedded in the lattice. The method is promising for the
automated design of more complex systems such as photonic crystals and Moire
superlattices.
