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Machine Learning in Electronic Quantum Matter Imaging Experiments
Yi Zhang, A. Mesaros, K. Fujita, S. D. Edkins, M. H. Hamidian, K. Ch'ng, H. Eisaki, S. Uchida, J. C. Séamus Davis, E. Khatami, Eun-Ah Kim
TL;DR
Large-scale EQM experiments produce complex image data that can defy effective human analysis. This paper develops ANN-based strategies to analyze experimentally derived images and discovers a commensurate, four-unit-cell periodic, translational-symmetry-breaking state with coincident nematicity.
Problem
Complex experimental EQM image structures can defy effective analysis, motivating machine-learning strategies for scientific discovery.
Method
The paper develops and demonstrates an ANN-based protocol, including atomic-mask processing, for analyzing experimentally derived EQM images.
Results
The ANNs identify a commensurate, 4a0-periodic, translational-symmetry-breaking electronic structure modulation and unidirectional, nematic behavior in EQM images.
Takeaways & Limitations
The observations are congruent with strong-coupling theories of electronic liquid crystals.
Takeaways & Limitations
Extreme disorder in cuprate EQM complicates traditional analysis, and the Fourier cutoff was based on human visual inspection.
Abstract
from arXiv · showhide
Essentials of the scientific discovery process have remained largely unchanged for centuries: systematic human observation of natural phenomena is used to form hypotheses that, when validated through experimentation, are generalized into established scientific theory. Today, however, we face major challenges because automated instrumentation and large-scale data acquisition are generating data sets of such volume and complexity as to defy human analysis. Radically different scientific approaches are needed, with machine learning (ML) showing great promise, not least for materials science research. Hence, given recent advances in ML analysis of synthetic data representing electronic quantum matter (EQM), the next challenge is for ML to engage equivalently with experimental data. For example, atomic-scale visualization of EQM yields arrays of complex electronic structure images, that frequently elude effective analyses. Here we report development and training of an array of artificial neural networks (ANN) designed to recognize different types of hypothesized order hidden in EQM image-arrays. These ANNs are used to analyze an experimentally-derived EQM image archive from carrier-doped cuprate Mott insulators. Throughout these noisy and complex data, the ANNs discover the existence of a lattice-commensurate, four-unit-cell periodic, translational-symmetry-breaking EQM state. Further, the ANNs find these phenomena to be unidirectional, revealing a coincident nematic EQM state. Strong-coupling theories of electronic liquid crystals are congruent with all these observations.
arrays. These ANNs are used to analyze an experimentally-derived EQM image
The ANNs analyze an experimentally derived EQM image archive from carrier-doped cuprate Mott insulators, despite the complexity of the data.
- The archive contains complex electronic-structure data that challenge effective analysis.
- The ANNs analyze an experimentally derived EQM image archive from carrier-doped cuprate Mott insulators.
cell periodic, translational-symmetry-breaking EQM state. Further, the ANNs find
The ANNs find the observed phenomena to be unidirectional, revealing a coincident nematic EQM state.
- The phenomena are unidirectional, revealing a coincident nematic EQM state.
Strong-coupling theories of electronic liquid crystals18,19 are congruent with all these
ANN analysis of noisy experimental EQM images identifies a lattice-commensurate, unidirectional 4a₀-periodic electronic structure modulation in carrier-doped CuO₂. The modulation breaks rotational symmetry near the pseudogap energy, revealing a coincident nematic state consistent with strong-coupling electronic-liquid-crystal theories.
- ANNs identify a predominant translational-symmetry breaking with wavelength λ=4a₀ in CuO₂ images.The signal is found for electron densities 0.06≤p≤0.14.
- The 4a₀-periodic modulation persists for energies exceeding 66meV despite intense masking by quasiparticle-interference phenomena.ANNs recognize the commensurate density-wave modulation across this energy range.
- The modulation preferentially breaks C4 rotational symmetry and occurs primarily along the x-axis of the CuO₂ plane.This unidirectionality produces the observed nematic character.
- A nematic state emerges near the pseudogap energy scale Δ₀≈80meV from highly disordered yet unidirectional 4a₀-periodic modulations.The authors identify this as a vestigial nematic state whose characteristic energy gap is the pseudogap.
- The identified modulation is lattice-commensurate, unidirectional, and has a d-symmetry form factor.Its signature is reported across the 0.06≤p≤0.14 density range.
- ANNs reliably identify broken symmetries in complex, non-synthetic experimental EQM image arrays, supporting further ML-driven discovery in EQM.The approach achieves generally greater than 99% accuracy on validation images.
17 Fujita, K. et al, Spectroscopic Imaging STM: Atomic-Scale Visualization of Electronic
This section lists prior work on cuprate electronic structure, charge order, nematicity, electronic liquid-crystal phases, Fourier analysis, and neural networks.
- The bibliography also covers electronic liquid-crystal phases, nematic Fermi fluids, and intertwined orders in high-temperature superconductors.
- The references include studies of atomic-scale electronic structure and d-symmetry form-factor density-wave states in cuprates.
- Several cited works address charge-order patterns, including stripes, checkerboards, and commensurate 4a0-period modulations.
METHODS
The methods construct synthetic EQM images representing candidate ordered states and train fully connected ANNs to classify them, then test robustness across disorder and network variations.
- Fourier-transform comparison: The ANN analyzes all real-space image data without the information loss associated with conventional Fourier analysis.
- Training image set generation: Synthetic training images combine d-wave and s-wave form-factor components with amplitude and phase fluctuations, noise, and DFF dislocations.
- Training image set generation: Atomic masks place s-wave intensity at Cu sites and d-wave intensity with opposite signs on the two oxygen sites.
- ANN configuration: The fully connected ANN uses weighted nonlinear neurons and a softmax output for probabilistic classification across candidate categories.
- ANN configuration: Training uses 90% of images, with 10% reserved for validation, while stochastic gradient descent minimizes cross-entropy cost with L2 regularization.
- Robustness checks: 81 independently initialized ANNs produced robust, quantitatively consistent outputs, and the reported results average across them.
- Robustness checks: Across disorder models, the analysis robustly favors commensurate period 4a0 for 0.06<p<0.14 but becomes confused among categories at p=0.2.
6 Discommensurations and Maximum Intensity Wavevector. The Fourier
The section explains why Fourier-based analyses can misidentify disordered or defect-rich density-wave patterns, and presents ANN analysis as a faster, less manually constrained alternative.
- Fourier limitations: Fourier methods perform poorly when density-wave patterns contain randomly placed disordered patches and topological defects.Their linear wavevector-basis representation is most useful for sharp features.
- Fourier limitations: A commensurate 4a0 modulation with discommensurations can be incorrectly identified as having an apparent period Q=0.3*2π/a0.The incorrect result arises from residue minimization or Fourier-amplitude maximization.
- Fourier limitations: The earlier DR approach depended on human inspection to identify commensurate patches and select the Fourier cutoff Λ.It also averaged over dislocations, ignoring their role.
- ANN alternative: The ANN approach avoids arbitrary Fourier cutoffs and manually selected regions of interest because it is nonlinear and basis-independent.It analyzes complete image arrays without ad-hoc Fourier filtering or selection.
- ANN alternative: Once trained, the ANNs assess new data sets in minutes, enabling high-throughput analysis of complete electron-density and electron-energy dependence.This efficiency enabled discovery of a connection between nematic and commensurate density-wave states at the pseudogap energy scale.
Methods References
The methods and extended-data passages describe ANN training, experimental image inputs, benchmark categorizations, and evidence for commensurate, unidirectional electronic motifs.
- Experimental results: The power spectral density for a p~0.08 sample has maximum intensity at <Q>=0.28.The measurement is obtained from a Fourier transform of the electronic-structure data.
- Training and categorization: The ANN training set defines four categories using d-wave form-factor modulations with wavelengths 4.348a0, 4a0, 3.704a0, and 3.448a0.The CuO2 unit-cell size is set to a0=6 pixels.
- Training and categorization: Finalized ANNs achieve generally over 99% accuracy on synthesized data.Training evaluates accuracy and cross-entropy across activation functions and hidden-layer neuron counts.
- Experimental inputs: The same image and its 90-degree rotation are supplied to obtain ANN outputs for modulation orientations X and Y.This orientation comparison supports assessment of unidirectionality.
- Robustness: ANN categorizations remain robust when training-set parameters change, including analyses of doping and electron-energy evolution.The comparison includes outputs from a single ANN trained with a different training set.