Unsupervised learning of topological phase diagram using topological data analysis

Topology and machine learning are two actively researched topics not only in condensed matter physics, but also in data science. Here, we propose the use of topological data analysis in unsupervised learning of the topological phase diagrams. This is possible because the quantum distance can capture the shape of the space formed by the Bloch wave functions as we sweep over the Brillouin zone. Therefore, if we minimize the volume of the space formed by the wave function through a continuous deformation, the wave functions will end up forming distinct spaces which depend on the topology of the wave functions. Combining this observation with the topological data analysis, which provides tools such as the persistence diagram to capture the topology of the space formed by the wave functions, we can cluster together Hamiltonians that give rise to similar persistence diagrams after the deformation. By examining these clusters as well as representative persistence diagrams in the clusters, we can draw the phase diagram as well as distinguish between topologically trivial and nontrivial phases. Our proposal to minimize the volume can be interpreted as finding geodesics in a one-dimensional (1D) Brillouin zone and minimal surfaces in 2D and higher-dimensional Brillouin zones. Using this interpretation, we can guarantee the convergence of the minimization under certain conditions, which is an outstanding feature of our algorithm. We demonstrate the working principles of our machine learning algorithm using various models.