A convolutional hamming distance metric for unsupervised learning of topological order

Machine learning algorithms have proven to be effective tools for the exploration of phases and phase transitions in many-body systems. Much prior work has focused on applying supervised and unsupervised machine learning algorithms to distinguish phases and identify phase transitions and crossovers in systems with local order parameters as well as those with topological phases that lack a local order parameter. In this work we study how the performance of unsupervised machine learning algorithms depend on the choice of a distance metric in configuration space. We introduce a metric based on the Hamming distance and a convolution of local patches of spins, which we call the convolutional hamming distance. We show that this distance metric allows us to identify topological order in classical Z₂ and Z₃ lattice gauge theories. Additionally we demonstrate how the choice of topologically non-trivial patches can distinguish the topological sectors of the ground state of Z₂ gauge theory. We study various performance metrics as a function of patch and system size using a combination of unsupervised machine learning algorithms. The method introduced in this work has the potential to reduce the amount of physical input required for using unsupervised machine learning to identify phases and phase transitions from raw experimental or simulation data.