Unsupervised Learning to Recognize Quantum Phases of Matter

Drawing the quantum phase diagram of a many-body system in the parameter space of its Hamiltonian can be seen as a learning problem, which implies labelling the corresponding ground states according to some classification criterium that defines the phases. In this work we adopt unsupervised learning, where the algorithm has no access to any priorly labeled states, as a tool for determining quantum phase diagrams of many-body systems. The algorithm directly works with quantum states: given the ground-state configurations for different values of the Hamiltonian parameters, the process uncovers the most significant way of grouping them based on a similarity criterion that refers to the fidelity between quantum states, that can be easily estimated, even experimentally. We benchmark our method with two specific spin-frac{1}{2} chains, with states determined via tensor network techniques. We find that unsupervised learning algorithms based on spectral clustering, combined with "silhouette" and "elbow" methods for determining the optimal number of phases, can accurately reproduce the phase diagrams. Our results show how unsupervised learning can autonomously recognize and possibly unveil novel phases of quantum matter.