Identifying topological order through unsupervised machine learning

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The Landau description of phase transitions relies on the identification of a local order parameter that indicates the onset of a symmetry-breaking phase. In contrast, topological phase transitions evade this paradigm and, as a result, are harder to identify. Recently, machine learning techniques have been shown to be capable of characterizing topological order in the presence of human supervision. Here, we propose an unsupervised approach based on diffusion maps that learns topological phase transitions from raw data without the need for manual feature engineering. 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 Berezinskii–Kosterlitz–Thouless transition. We also demonstrate the success of the approach on the Ising gauge theory, another paradigmatic model with topological order. In addition, a connection between the output of diffusion maps and the eigenstates of a quantum-well Hamiltonian is derived. Topological classification via diffusion maps can therefore enable fully unsupervised studies of exotic phases of matter.

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Fig. 1: Topological classification using sample connectivity.
Fig. 2: Detection of topological order in the 1D XY model.
Fig. 3: Detection of topological order in the 2D XY model.
Fig. 4: Detection of topological order in the Ising gauge theory.
Fig. 5: Illustration of the diffusion map in the continuum limit.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

Code availability

The code used to generate the results presented in this work can be found in the online version of the paper.


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The authors thank S. Chatterjee, E. Demler, P.P. Orth, H. Pichler, S. Sachdev, H. Shackleton and Y.-Z. You for useful discussions. J.F.R.-N. acknowledges support from AFOSR-MURI: Photonic Quantum Matter (award FA95501610323). M.S.S. acknowledges support from the German National Academy of Sciences Leopoldina through grant no. LPDS 2016-12 and from the National Science Foundation under grant no. DMR-1664842.

Author information

J.F.R.-N. and M.S.S. contributed equally to the design of the study, performing the simulations, analysis of the results and writing of the manuscript.

Correspondence to Joaquin F. Rodriguez-Nieva or Mathias S. Scheurer.

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