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Classifying Snapshots of the Doped Hubbard Model with Machine Learning

Annabelle Bohrdt, Christie S. Chiu, Geoffrey Ji, Muqing Xu, Daniel Greif, Markus Greiner, Eugene Demler, Fabian Grusdt, Michael Knap

arXiv:1811.12425v1cond-mat.quant-gascond-mat.dis-nncond-mat.str-el

TL;DR

Quantum gas microscope snapshots contain information beyond conventional observables, while the finite-doping Fermi-Hubbard model lacks consensus on its theoretical description. This paper uses a CNN to classify experimental snapshots against theoretical categories and finds that the preferred category varies with doping.

  • Problem

    At finite doping, there is no consensus on the theoretical description of the Fermi-Hubbard model, and microscope snapshots contain more information than conventional observables capture.

    Source: p. 1

  • Method

    A CNN classifies experimental Fermi-Hubbard snapshots into theoretical categories to assess which theory describes the system best microscopically.

    Source: p. 2

  • Results

    Experimental snapshots more closely resemble geometric string theory at low doping, while the π-flux RVB data resembles them more closely beyond a qualitative change between 15% and 20% doping.

    Source: p. 4

  • Takeaways & Limitations

    The analysis demonstrates that CNNs can classify individual quantum-gas-microscope snapshots against theoretical predictions and identify which theory fits better within the studied doping regime.

    Source: pp. 2, 4

    The main-text geometric-string snapshots use experimental half-filling data, creating a potential bias relative to the purely theoretical π-flux data.

    Source: Geometric string theory based on… · p. 6

Abstract

from arXiv · show

Quantum gas microscopes for ultracold atoms can provide high-resolution real-space snapshots of complex many-body systems. We implement machine learning to analyze 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 type, and the geometric string theory, describing a state with hidden spin order. This approach considers all available information without a potential bias towards one particular theory by the choice of an observable and can therefore select the theory which 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.

Data

The dataset includes experimental snapshots across doping and temperature conditions, and the CNN’s sorted classes have similar staggered magnetization values. Snapshots were post-selected by staggered magnetization, with results reported to be independent of the post-selection percentage.

Data

  • Both CNN-sorted experimental classes have comparably close average staggered magnetization values, indicating classification uses more involved patterns.The π-flux-classified set has a slightly lower average than the geometric-string-classified set.Source: Data · p. 5, Figure S1 · p. 5
  • Table S1 reports experimental snapshot counts by doping at T = 0.6J ± 0.1J and by temperature at half-filling.The supplied caption gives the table’s scope but not its cell values.Source: Table S1 · p. 5
  • About 30% of snapshots with the highest absolute staggered magnetization are used to avoid images whose Néel ordering vector is misaligned with the measurement axis.The analysis was reported to be independent of the post-selection percentage.Source: Data · p. 5

Errors

Training and sorting are repeated across different test-data choices to average fluctuations, and reported error bars represent standard deviations across runs. Network parameters are listed for the figures covered by this analysis.

Errors

  • Training, test-data selection, and experimental sorting are repeated 10 times to average out fluctuations.Each test set contains 400 snapshots held out from training.Source: Errors · p. 5
  • Error bars for accuracy and the percentage of experimental snapshots assigned to either theory represent standard deviations across runs.The passage defines N as the number of runs, x_i as the quantity in run i, and x̄ as its average over runs.Source: Errors · pp. 5–6
  • Table S2 lists network parameters for Fig. 2, Fig. 3, and Fig. S5.These are the figures identified in the supplied passages.Source: Table S2 · p. 6, Errors · p. 6

Network

The CNN’s qualitative result is robust to architecture changes, although optimal architectures vary across tasks and available training data.

Network

  • The qualitative result remains robust across tested network architectures despite quantitative fluctuations.Different tasks and training-data amounts for Fig. 2, Fig. 3, and Fig. S5 yield slightly different architectures with the best overall test accuracy.Source: Network · p. 6

Geometric string theory based on Heisenberg QMC

To reduce a potential bias from using experimental half-filling data for geometric string snapshots, the authors also construct those snapshots from Heisenberg QMC. The resulting classification trend is qualitatively similar to the main-text result.

Geometric string theory based on Heisenberg QMC

  • Using experimental half-filling snapshots for geometric string data creates a potential bias because π-flux data is purely theoretical apart from a fitted energy scale.The authors describe this bias as to some extent intrinsic because geometric string theory specifies how holes change the state, not the half-filled state itself.Source: Geometric string theory based on… · p. 6
  • With geometric-string snapshots based on Heisenberg QMC, the qualitative classification result is very similar to the main-text result.The network is trained on these snapshots and π-flux data, then used to sort experimental data into the two categories.Source: Geometric string theory based on… · p. 6, Figure S2 · p. 6

Distinguishing geometric string theory from random images

The CNN distinguishes geometric-string snapshots from random images, with training precision close to 100% below 10% doping. The experiment tests how it classifies experimental images at matching doping values.

Distinguishing geometric string theory from random images

  • For doping below 10%, CNN training precision is close to 100%, showing it can learn to distinguish the theory classes almost perfectly.The figure then applies the trained network to experimental images at the corresponding doping values.Source: Distinguishing geometric string theory… · p. 6
  • Across the classification tasks reported so far, CNN training precision is around 75%.Source: Distinguishing geometric string theory… · p. 6
  • Figure S3 averages experimental-image classifications by doping and reports held-out precision for the trained classes after ten repetitions.Source: Figure S3 · p. 7

Classifying sets of snapshots

Grouping four rotated snapshots into one classification improves precision compared with the single-snapshot result. Below 14% doping, 65–85% of experimental images are classified as string theory.

Classifying sets of snapshots

Comparing finite doping to high temperature

The CNN classifies experimental images below approximately 15% doping as geometric-string-like, but its declining precision makes classifications around 15% unreliable. The comparison uses high-temperature, half-filled experimental images with holes added to match the doping.

Comparing finite doping to high temperature

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