Unsupervised Learning of Topological Order through Diffusion on Data Sets

May 5, 2019

Fig. 1: Topological classification using sample connectivity. (Rodriguez-Nieva and Scheurer*)

Motivated by the success in many different fields of science and engineering, such as image recognition and analysis of big data in biology, machine learning has recently come to prominence in condensed matter physics โ€“ in particular, for the purpose of classification and detection of phase transitions. While symmetry-breaking phase transitions can be easily identified with machine learning, topological phase transitions are more difficult to capture; in some cases, they cannot be identified properly even with supervised learning, i.e., when the labels of the training data are provided. Such difficulty stems from the non-local nature of topological phase transitions.

Recently, two postdocs at Harvard University, Joaquin Rodriguez-Nieva and Mathias Scheurer, proposed an unsupervised machine learning approach that identifies topological phase transitions from raw data without the need of manual feature engineering. The key idea behind their approach is to construct a diffusion process on the data set to identify smooth deformations between samples (see figure, above). Using bare spin configurations as input, the approach is shown to be capable of classifying samples of the two-dimensional XY model by winding number and capture the BKT transition. They also demonstrate the success of the approach on the Ising gauge theory. In order to explain why their approach succeeds in topological classification, the authors derive a connection between the output of the machine learning method and the eigenstates of a quantum-mechanical problem.

This research is described in a recent article in Nature Physics: Joaquin F. Rodriguez-Nieva and Mathias S. Scheurer, โ€œIdentifying topological order through unsupervised machine learningโ€, Nature Physics (2019)