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Classifying snapshots of the doped Hubbard model with machine learning


Quantum gas microscopes for ultracold atoms can provide high-resolution real-space snapshots of complex many-body systems. We implement machine learning to analyse and classify such snapshots of ultracold atoms. Specifically, we compare the data from an experimental realization of the two-dimensional Fermi–Hubbard model to two theoretical approaches: a doped quantum spin liquid state of resonating valence bond type1,2, and the geometric string theory3,4, describing a state with hidden spin order. This technique considers all available information without a potential bias towards one particular theory by the choice of an observable and can therefore select the theory that is more predictive in general. Up to intermediate doping values, our algorithm tends to classify experimental snapshots as geometric-string-like, as compared to the doped spin liquid. Our results demonstrate the potential for machine learning in processing the wealth of data obtained through quantum gas microscopy for new physical insights.

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Fig. 1: Classifying quantum gas microscope snapshots of the doped Fermi–Hubbard model with CNNs.
Fig. 2: Classifying single snapshots of the many-body density matrix.
Fig. 3: Classifying experimental data.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. The raw data are available in ref. 32.

Code availability

The computer codes used to generate the results of this paper are available from the corresponding author upon reasonable request.


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We thank E. Altman, I. Bloch, J. Carrasquilla, M. Kanász-Nagy, E. Khatami, F. Pollmann, A. Rosch, S. Sachdev, R. Schmidt and D. Sels for insightful discussions and M. Kanász-Nagy in addition for his Heisenberg QMC code. We acknowledge support from Harvard-MIT CUA, NSF grant no. DMR-1308435, AFOSR-MURI Quantum Phases of Matter (grant FA9550-14-1-0035), AFOSR grant no. FA9550-16-10323, DoD NDSEG, the Gordon and Betty Moore Foundation EPIQS programme and grant no. 6791, NSF GRFP and grant nos. PHY-1506203 and PHY-1734011, ONR grant no. N00014-18-1-2863, SNSF, Studienstiftung des deutschen Volkes, and the Technical University of Munich - Institute for Advanced Study, funded by the German Excellence Initiative and the European Union FP7 under grant agreement 291763, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy–EXC-2111–390814868, the DFG grant no. KN1254/1-1, and DFG TRR80 (Project F8).

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Authors and Affiliations



A.B., F.G. and M.K. devised the method. A.B. carried out the numerical simulations and analysis. F.G. and E.D. developed the geometric string theory. C.S.C., G.J., M.X. and D.G. performed the experiments. M.G., E.D., F.G. and M.K. supervised the work. All authors contributed to the writing of the manuscript.

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Correspondence to Michael Knap.

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Supplementary text, Figs. 1–8 and references.

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Bohrdt, A., Chiu, C.S., Ji, G. et al. Classifying snapshots of the doped Hubbard model with machine learning. Nat. Phys. 15, 921–924 (2019).

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