Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Identifying topological order through unsupervised machine learning

Nature Physics volume 15, pages 790–795 (2019)Cite this article

Subjects

Abstract

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.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

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.

Similar content being viewed by others

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.

References

  1. Nielsen, M. A. Neural Networks and Deep Learning (Determination Press, 2015).

  2. Goodfellow, I., Bengio, Y. & Courville, A. Deep Learning (MIT Press, 2016).

  3. Mehta, P. et al. A high-bias, low-variance introduction to machine learning for physicists. Preprint at https://arxiv.org/abs/1803.08823 (2018).

  4. Mehta, P. & Schwab, D. J. An exact mapping between the variational renormalization group and deep learning. Preprint at https://arxiv.org/abs/1410.3831 (2014).

  5. Carleo, G. & Troyer, M. Solving the quantum many-body problem with artificial neural networks. Science 355, 602–606 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  6. Deng, D.-L., Li, X. & Das Sarma, S. Machine learning topological states. Phys. Rev. B 96, 195145 (2017).

    Article  ADS  Google Scholar 

  7. Liu, J., Qi, Y., Meng, Z. Y. & Fu, L. Self-learning Monte Carlo method. Phys. Rev. B 95, 041101 (2017).

    Article  ADS  Google Scholar 

  8. Torlai, G. et al. Neural-network quantum state tomography. Nat. Phy. (2018).

  9. You, Y.-Z., Yang, Z. & Qi, X.-L. Machine learning spatial geometry from entanglement features. Phys. Rev. B 97, 045153 (2018).

    Article  ADS  Google Scholar 

  10. Carleo, G., Nomura, Y. & Imada, M. Constructing exact representations of quantum many-body systems with deep neural networks. Nat. Commun. 9, 5322 (2018).

    Article  ADS  Google Scholar 

  11. Spillard, S., Turner, C. J. & Meichanetzidis, K. Machine learning entanglement freedom. Int. J. Quantum Inf. 16, 1840002 (2018).

    Article  Google Scholar 

  12. Carrasquilla, J. & Melko, R. G. Machine learning phases of matter. Nat. Phys. 13, 431–434 (2017).

    Article  Google Scholar 

  13. Ch’ng, K., Carrasquilla, J., Melko, R. G. & Khatami, E. Machine learning phases of strongly correlated fermions. Phys. Rev. X 7, 031038 (2017).

    Google Scholar 

  14. Zhang, Y. & Kim, E.-A. Quantum loop topography for machine learning. Phys. Rev. Lett. 118, 216401 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  15. Zhang, Y., Melko, R. G. & Kim, E.-A. Machine learning ℤ2 quantum spin liquids with quasiparticle statistics. Phys. Rev. B 96, 245119 (2017).

    Article  ADS  Google Scholar 

  16. van Nieuwenburg, E. P. L., Liu, Y.-H. & Huber, S. D. Learning phase transitions by confusion. Nat. Phys. 13, 435–439 (2017).

    Article  Google Scholar 

  17. Ohtsuki, T. & Ohtsuki, T. Deep learning the quantum phase transitions in random two-dimensional electron systems. J. Phys. Soc. Jpn 85, 123706 (2016).

    Article  ADS  Google Scholar 

  18. Ohtsuki, T. & Ohtsuki, T. Deep learning the quantum phase transitions in random electron systems: applications to three dimensions. J. Phys.Soc. Jpn 86, 044708 (2017).

    Article  ADS  Google Scholar 

  19. Yoshioka, N., Akagi, Y. & Katsura, H. Learning disordered topological phases by statistical recovery of symmetry. Phys. Rev. B 97, 205110 (2018).

    Article  ADS  Google Scholar 

  20. Zhang, P., Shen, H. & Zhai, H. Machine learning topological invariants with neural networks. Phys. Rev. Lett. 120, 066401 (2018).

    Article  ADS  Google Scholar 

  21. Carvalho, D., García-Martínez, N. A., Lado, J. L. & Fernández-Rossier, J. Real-space mapping of topological invariants using artificial neural networks. Phys. Rev. B 97, 115453 (2018).

    Article  ADS  Google Scholar 

  22. Huembeli, P., Dauphin, A. & Wittek, P. Identifying quantum phase transitions with adversarial neural networks. Phys. Rev. B 97, 134109 (2018).

    Article  ADS  Google Scholar 

  23. Iakovlev, I. A., Sotnikov, O. M. & Mazurenko, V. V. Supervised learning approach for recognizing magnetic skyrmion phases. Phys. Rev. B 98, 174411 (2018).

    Article  ADS  Google Scholar 

  24. Vargas-Hernández, R. A., Sous, J., Berciu, M. & Krems, R. V. Extrapolating quantum observables with machine learning: inferring multiple phase transitions from properties of a single phase. Phys. Rev. Lett. 121, 255702 (2018).

    Article  ADS  Google Scholar 

  25. Beach, M. J. S., Golubeva, A. & Melko, R. G. Machine learning vortices at the Kosterlitz–Thouless transition. Phys. Rev. B 97, 045207 (2018).

    Article  ADS  Google Scholar 

  26. Wang, C. & Zhai, H. Machine learning of frustrated classical spin models (II): Kernel principal component analysis. Front. Phys. 13, 130507 (2018).

    Article  Google Scholar 

  27. Hu, W., Singh, R. R. P. & Scalettar, R. T. Discovering phases, phase transitions and crossovers through unsupervised machine learning: a critical examination. Phys. Rev. E 95, 062122 (2017).

    Article  ADS  Google Scholar 

  28. Wetzel, S. J. Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders. Phys. Rev. E 96, 022140 (2017).

    Article  ADS  Google Scholar 

  29. Wang, C. & Zhai, H. Machine learning of frustrated classical spin models. I. Principal component analysis. Phys. Rev. B 96, 144432 (2017).

    Article  ADS  Google Scholar 

  30. Cristoforetti, M., Jurman, G., Nardelli, A. I. & Furlanello, C. Towards meaningful physics from generative models. Preprint at https://arxiv.org/abs/1705.09524 (2017).

  31. Broecker, P., Assaad, F. F. & Trebst, S. Quantum phase recognition via unsupervised machine learning. Preprint at https://arxiv.org/abs/1707.00663 (2017).

  32. Zhang, W., Liu, J. & Wei, T.-C. Machine learning of phase transitions in the percolation and XY models. Preprint at https://arxiv.org/abs/1804.02709(2018).

  33. Suchsland, P. & Wessel, S. Parameter diagnostics of phases and phase transition learning by neural networks. Phys. Rev. B 97, 174435 (2018).

    Article  ADS  Google Scholar 

  34. Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C 6, 1181 (1973).

    Article  ADS  Google Scholar 

  35. Sachdev, S. Topological order, emergent gauge fields, and Fermi surface reconstruction. Rep. Prog. Phys. 82, 014001 (2019).

    Article  ADS  MathSciNet  Google Scholar 

  36. Coifman, R. R. et al. Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc. Natl Acad. Sci USA 102, 7426–7431 (2005).

    Article  ADS  Google Scholar 

  37. Nadler, B., Lafon, S., Kevrekidis, I. & Coifman, R. R. Diffusion maps, spectral clustering and eigenfunctions of Fokker–Planck operators. In Advances in Neural Information Processing Systems Vol. 18 (eds Weiss, Y., Scholkopf, B. & Platt, J.) 955–962 (MIT Press, 2006).

  38. Coifman, R. R. & Lafon, S. Diffusion maps. Appl. Comput. Harmon, Anal. 21, 5–30 (2006).

    Article  MathSciNet  Google Scholar 

  39. Michalevsky, Y., Talmon, R. & Cohen, I. Speaker identification using diffusion maps. In 19th European Signal Processing Conference, 2011, 1299–1302 (IEEE, 2011).

  40. Barkan, O., Weill, J., Wolf, L. & Aronowitz, H. Fast high dimensional vector multiplication face recognition. In Proceedings of the IEEE International Conference on Computer Vision, 1960–1967 (IEEE, 2013).

  41. Berezinskii, V. Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems. Sov. Phys. JETP 32, 493–500 (1971).

    ADS  MathSciNet  Google Scholar 

  42. Berezinskii, V. Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems. Sov. J. Exp. Theor. Phys. 34, 610 (1972).

    ADS  Google Scholar 

  43. Kosterlitz, J. M. The critical properties of the two-dimensional XY model. J. Phys. C 7, 1046 (1974).

    Article  ADS  Google Scholar 

  44. Wegner, F. J. Duality in generalized Ising models and phase transitions without local order parameters. J. Math. Phys. 12, 2259–2272 (1971).

    Article  ADS  MathSciNet  Google Scholar 

  45. Kogut, J. B. An introduction to lattice gauge theory and spin systems. Rev. Mod. Phys. 51, 659–713 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  46. Su, W. P., Schrieffer, J. R. & Heeger, A. J. Solitons in polyacetylene. Phys. Rev. Lett. 42, 1698–1701 (1979).

    Article  ADS  Google Scholar 

  47. Kitaev, A. Y. Unpaired Majorana fermions in quantum wires. Phys.-Uspekhi 44, 131 (2001).

    Article  ADS  Google Scholar 

  48. Komura, Y. & Okabe, Y. Large-scale Monte Carlo simulation of two-dimensional classical XY model using multiple GPUs. J. Phys. Soc. Jpn 81, 113001 (2012).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

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

Authors and Affiliations

  1. Department of Physics, Harvard University, Cambridge, MA, USA

    Joaquin F. Rodriguez-Nieva & Mathias S. Scheurer

Authors
  1. Joaquin F. Rodriguez-Nieva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mathias S. Scheurer
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

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.

Corresponding authors

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

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Eight figures, three references.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rodriguez-Nieva, J.F., Scheurer, M.S. Identifying topological order through unsupervised machine learning. Nat. Phys. 15, 790–795 (2019). https://doi.org/10.1038/s41567-019-0512-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-019-0512-x

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing