Reconstructing the Hamiltonian from the local density of states using neural networks

Technical Summary: Reconstructing the Hamiltonian from the Local Density of States using Neural Networks

Problem Statement
This work addresses an inverse problem at the intersection of quantum condensed matter physics and machine learning: reconstructing a single-particle Hamiltonian from the spatial map of the local density of states (LDOS) at a fixed energy. While the "forward problem"—calculating the LDOS $\rho_E(\mathbf{r})$ from a known potential $V(\mathbf{r})$ and Hamiltonian $H$—is straightforward, the reverse is generally ill-posed. Distinct Hamiltonians can theoretically yield identical LDOS maps within a specific energy window (e.g., if the energy lies within spectral gaps). The authors investigate whether approximate solutions to this inverse problem are feasible using supervised learning, specifically focusing on recovering the potential landscape $V(\mathbf{r})$ from a single energy slice of the LDOS.

Methodology
The authors formulate the problem within the context of tight-binding models on 1D and 2D lattices, defined by the Hamiltonian:
$$H = -t \sum_{\langle \mathbf{r}\mathbf{r}' \rangle} c^\dagger_\mathbf{r} c_{\mathbf{r}'} + \sum_\mathbf{r} V(\mathbf{r}) c^\dagger_\mathbf{r} c_\mathbf{r}$$
where $V(\mathbf{r})$ represents a random disorder potential.

• Data Generation: Synthetic datasets are generated by sampling potentials $V(\mathbf{r})$ from Gaussian random fields with specific correlation lengths ($\xi$) and amplitudes ($V_0$). The corresponding LDOS is computed numerically for a fixed energy $E$.
• Preprocessing: Both the input LDOS and the target potential are normalized (zero mean, unit standard deviation) to facilitate training.
• Model Architecture: The core approach utilizes Image-to-Image Convolutional Neural Networks (CNNs). Unlike standard classification CNNs that map images to class labels, these models map an input image (LDOS) directly to an output image (reconstructed potential).
  • The 1D model uses convolutional layers with ReLU activations.
  • The 2D model incorporates residual (skip) connections every two layers to improve gradient flow and performance.
  • The networks do not use pooling or fully connected layers, preserving spatial resolution.
• Training Objective: The models are trained to minimize the Mean Squared Error (MSE) between the normalized predicted potential and the ground truth potential.
• Baselines: Performance is compared against a single-parameter fit ($\tilde{V} \propto \tilde{\rho}^n$) and a translation-augmented $k$-nearest-neighbors ($k$-nn) approach.

Key Results
The study demonstrates that supervised learning can achieve high-fidelity reconstruction of the potential landscape from LDOS data in both 1D and 2D systems.

• Reconstruction Accuracy:
  • 1D Case: The CNN achieves a test set MSE loss ($L$) of approximately 0.016, significantly outperforming the single-parameter fit ($L \approx 0.995$) and $k$-nn methods ($L \approx 0.707$ for $k=10$). Visual inspection shows the predicted potential is nearly indistinguishable from the true potential in best and median cases.
  • 2D Case: The CNN achieves a test set MSE loss of approximately $4.6 \times 10^{-3}$, again vastly superior to baselines ($L \approx 0.623$ for $k$-nn).
• Robustness to Noise:
  • The models exhibit inherent tolerance to noise; even without specific regularization, the 2D model maintains low error ($L < 0.1$) with up to 20% correlated test noise.
  • Noise Regularization: Intentionally adding synthetic correlated Gaussian noise to the training data (training noise) acts as a regularizer. A model trained with Signal-to-Noise Ratio (SNR) = 1 noise maintains $L < 0.05$ even when tested with noise levels approaching the signal amplitude ($\sigma/\sigma_\rho \approx 1$).
• Generalization (Out-of-Distribution):
  • The models generalize to parameter shifts not seen during training. When tested on potentials with disorder amplitudes ($V_0$) and correlation lengths ($\xi$) slightly different from the training distribution, the models maintain low error ($L \lesssim 0.1$) within a reasonable neighborhood of the training parameters.
• Learnability Analysis: The reconstruction quality is not strictly correlated with spectral density or eigenstate localization length. Performance degrades near the edges and middle of the energy spectrum, suggesting specific spectral features influence learnability.

Significance and Claims
The paper claims that reconstructing a Hamiltonian from limited, experimentally observable LDOS data is a feasible and tractable inverse problem when approached via supervised machine learning.

• Practical Utility: The authors propose that this method could be applied to Scanning Tunneling Microscopy (STM) data. Since STM measures the differential conductance (proportional to LDOS) with sub-angstrom resolution, the trained models could theoretically invert these images to reveal the underlying unknown energy landscape (potential $V(\mathbf{r})$) of a sample.
• Applications: This reconstruction enables downstream calculations of physical quantities (e.g., transport coefficients, magnetic responses) that require knowledge of the potential. The authors specifically note potential applications in studying surface reconstructions (e.g., complex atomic patterns on Si(111) or Au(111)) where the energy landscape is complicated and aperiodic.
• Data Efficiency: The models achieve high accuracy with relatively small datasets (1,200 samples for 2D), suggesting the approach may be viable even when generating large amounts of training data is computationally expensive (e.g., for systems with electron-electron interactions).
• Limitations: The authors modestly note that for highly regular surfaces with low defect densities, the problem space is "low-dimensional," and simple regression or human inspection might suffice, limiting the advantage of ML in those specific cases. The current work is confined to non-interacting, single-orbital tight-binding models.

In conclusion, the work establishes that approximate solutions to the Hamiltonian reconstruction problem exist for specific distributions and demonstrates that image-to-image CNNs can solve this inverse problem with high accuracy and robustness, bridging the gap between experimental LDOS maps and theoretical Hamiltonian descriptions.