Unsupervised learning of quantum phases with topological order

Machine learning methods are emerging as powerful tools for detecting quantum phase transitions from simulated or experimental data. Most successes in this regard, have been achieved in the context of phases with a local order parameter, which can be described by the Landau theory of symmetry breaking. However, it was shown that phases that do not have such local order parameters but depend on the underlying global topology are much more difficult to capture. While there have been recent successes with unsupervised learning of topological phase transitions of classical models or non-interacting quantum systems, it remains an open question whether machine learning methods can identify interacting quantum phases of topological nature without involved feature engineering. 

In this work, we present a unsupervised machine-learning procedure, combining machine-learning-aided variational wavefunction descriptions and clustering techniques, to identify quantum phases with topological order. In particular, we consider the Kitaev Toric code model, which hosts Z2 topological order. We demonstrate that our procedure can capture both the loss of topological order at finite temperature and the transition to a polarized phase that emerges when a sufficiently strong magnetic field is applied. Our work opens the door to identifying exotic topological quantum phases without prior knowledge of the phase.