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

Nachiket Tanksale, Tobias Stauber

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

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

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
Paper Structure (16 sections, 7 equations, 8 figures, 1 table)

This paper contains 16 sections, 7 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tight-binding Model
  3. Hartree-Fock Solution
  4. Local Density of States
  5. Methodology for image-recognition of FT-LDOS data
  6. Dataset Generation from Hartree--Fock Theory
  7. FT-LDOS Preprocessing
  8. Neural-Network Architectures and Training Procedure
  9. Model Interpretability and Auxiliary Analyses
  10. Results and Discussion
  11. Evolution of FT-LDOS with Interaction Strength
  12. Regression of Interaction Strength from FT-LDOS
  13. CNNs as Detectors of Interaction-Driven Spectral Redistribution
  14. Crossover between weak and strong Hubbard interaction
  15. Summary
  16. ...and 1 more sections

Figures (8)

  • 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.
  • Figure 2: Fourier-transformed LDOS (FT-LDOS) for $\nu=-2$ and $\epsilon=12$ in the valley-coherence symmetry broken state, as discussed in Ref. sanchez_sanchez_nematic_2024. The principal Bragg peaks (black arrows) are located at the reciprocal lattice vectors of the two rotated graphene layers, while the red arrows indicate the reciprocal vectors associated with the Kekulé superlattice composed of three graphene unit cells.
  • Figure 3: Local density of states (LDOS) for $\nu=3$ and $\epsilon=10$ and its Fourier transformed local density of states (FT-LDOS).
  • Figure 4: Fourier-transformed LDOS (FT-LDOS) for $\nu=3$ and $\epsilon=10$ in the symmetric state for there random values of the Hubbard interaction $U=0.59711$, $U=1.56952$, and $U=3.08776$. No difference is seen by the bare eye which is also confirmed by cosine similarity, shown in Fig. \ref{['fig:cosine_sim']}.
  • Figure 5: Spectral Similarity: Cosine similarity between the FT-LDOS magnitude maps across the surveyed range of $U$ (exceeding $99.98\%$)
  • ...and 3 more figures