[Paper Review] Prediction of the atomistic Hubbard U interaction from moiré system STM-images using image recognition

Nachiket Anil Tanksale, Tobias Stauber|arXiv (Cornell University)|Feb 21, 2026

Graphene research and applications0 citations

TL;DR

The paper develops CNN-based regression to infer the on-site Hubbard U from simulated STM FT-LDOS images of twisted bilayer graphene in the flat-band regime, achieving high accuracy and revealing a weak weak-to-strong coupling crossover around U/t ≈ 1.

ABSTRACT

The atomistic Hubbard interaction U, representing the on-site Coulomb repulsion, serves as a pivotal parameter in theoretical models describing of correlated systems, yet its precise experimental determination especially in moiré systems remains challenging. Scanning Tunneling Microscopy(STM) provides real-space images of the local density of states (LDOS), offering rich data sets that reflect the unique electronic structure of the material. Here, we introduce a systematic methodology for extracting the Hubbard U parameter directly from these LDOS images through the application of machine learning (ML) in the case of twisted bilayer graphene in the flat-band regime. The regression of U is highly accurate even though the image-similarity is greater than 99.98%. Subsequent data-analysis further suggest a weak crossover between the weak and strong coupling regime at Uc/t 1

Motivation & Objective

• Infer the effective on-site Hubbard interaction U directly from STM-like FT-LDOS images of twisted bilayer graphene in the flat-band regime.
• Demonstrate that CNNs can accurately regress U from FT-LDOS data despite high image similarity across U values.
• Explore model interpretability to identify momentum-space features driving the U regression.
• Assess interpolation and extrapolation capabilities, including fractional U values, within and beyond the training range.
• Provide a data-driven framework for Hamiltonian learning in correlated moiré materials.

Proposed method

• Generate a dataset by solving a Hartree–Fock model of twisted bilayer graphene under hydrostatic pressure to realize a flat-band regime at θ ≈ 3.5°, for U in [0,6] eV with ε = 10.
• Compute LDOS and its Fourier transform (FT-LDOS) to mimic STM images at εF and generate 256×256 grayscale inputs after normalization.
• Train two CNN regressors (a custom 4-block CNN and a ResNet-18 variant) to map FT-LDOS images to U using mean squared error loss.
• Evaluate performance with MAE, RMSE, and R² on test sets and held-out fractional U values to probe interpolation and extrapolation.
• Apply Grad-CAM and guided backpropagation to localize the momentum-space regions and pixels driving predictions.
• Compare CNN performance to a PCA+ridge baseline to highlight nonlinear feature utilization.

Figure 1 : (A) Hartree-Fock bands at filling $\nu=3$ setting the Fermi energy $\epsilon_{F}$ to zero with $\epsilon=10$ obeying all symmetries, for $U=0$ – $5$ eV. (B) Density of states of the upper two flat bands for the same parameters.

Experimental results

Research questions

• RQ1 Can machine learning regress the on-site Hubbard interaction U directly from FT-LDOS STM-like images in a moiré system?
• RQ2 What FT-LDOS features in momentum space correlate with changes in U, and can interpretability tools reveal these features?
• RQ3 How well can CNNs interpolate within a discretely trained U grid and extrapolate to fractional U values not in the training set?
• RQ4 Is there a detectable crossover between weak and strong coupling regimes as U varies, and where does it occur?
• RQ5 How does the CNN-based approach compare to linear baselines in extracting U from high-dimensional STM data?

Key findings

• CNNs achieve highly accurate regression of U from FT-LDOS images with R² ≈ 0.984–0.985 on the standard test sets.
• Held-out performance for fractional U values drops (R² ≈ 0.87–0.88), indicating limited extrapolation beyond trained U ranges.
• PCA reveals a strongly low-dimensional data structure (PC1 56.8%, PC2 26.8%, PC3 6.1%), with PC1 showing monotonic U dependence.
• Grad-CAM and guided backpropagation show the regression relies on momentum-space regions near Bragg peaks and their contrasts, evolving with U.
• There is an inferred crossover around U/t ≈ 1–2.7 eV, supported by Grad-CAM patterns and PCA trends.
• Compared to ridge regression on leading PCs, CNNs provide notably better regression performance (MAE ~0.12–0.14 eV vs ~0.91 R² baseline).

Figure 2 : Fourier-transformed LDOS (FT-LDOS) for $\nu=-2$ and $\epsilon=12$ in the valley-coherence symmetry broken state, as discussed in Ref. Sánchez Sánchez et al. , 2024 . The principal Bragg peaks (black arrows) are located at the reciprocal lattice vectors of the two rotated graphene layers,