Unsupervised learning of topological order

Machine learning algorithms have proven useful in exploring phases and phase transitions in condensed matter Hamiltonians. Here, we show that a combination of unsupervised machine learning algorithms can be used to identify topological order in classical Z₂ and Z₃ lattice gauge theories. In particular, we apply dimensional reduction algorithms such as principal component analysis, clustering algorithms such as k-means, and the internal cluster performance metric known as silhouette analysis directly to raw spin configurations sampled from Monte Carlo simulations. Our method takes advantage of locality to define a patch of spins to be used as units of contrast instead of the individual spins to generate pairwise distance matrices based on the Hamming distance metric. We show that this modification enables us to identify non-local topological order in lattice gauge theories while maintaining performance for systems with local order parameters such as the Ising model. We also demonstrate non-trivial scaling for internal and external performance metrics for varying patch and system sizes.