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A neural network can reconstruct a quantum lattice's disorder potential from a single-energy image of the local density of states.

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2026-08-04 18:44 UTC pith:NXWOXXHH

load-bearing objection Solid empirical demonstration of LDOS-to-potential inversion; the normalization ambiguity is resolved by the appendix, and the paper deserves peer review. the 1 major comments →

arxiv 2509.09604 v1 pith:NXWOXXHH submitted 2025-09-11 cond-mat.dis-nn quant-ph

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

Nisarga Paul , Andrew Ma , Kevin P. Nuckolls This is my paper
classification cond-mat.dis-nn quant-ph
keywords Hamiltonian reconstructionlocal density of statesscanning tunneling microscopyconvolutional neural networkinverse problemAnderson modeldisordersupervised learning
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether a quantum system's Hamiltonian can be recovered from a single spatial map of its local density of states (LDOS) at one energy. The authors train convolutional neural networks on synthetic data from 1D and 2D tight-binding models with random disorder, and find that the networks reconstruct the disorder potential with high accuracy: test MSE of 0.016 in 1D and 0.005 in 2D, far below linear-fit and nearest-neighbor baselines. The networks also tolerate moderate noise and generalize to nearby disorder strengths and correlation lengths. If the result extends to experiments, scanning tunneling microscopy maps of the LDOS could become a direct probe of a sample's unknown energy landscape.

Core claim

The paper's central claim is that the inverse problem—recovering the potential V(r) from a single-energy LDOS image ρ_E(r)—is practically solvable by supervised learning for a class of single-particle lattice Hamiltonians. Using an image-to-image CNN trained on normalized pairs (ρ_E, V), the authors achieve normalized mean-squared errors of 0.016 (1D, L=128) and 0.005 (2D, L=64), outperforming baseline approaches by more than an order of magnitude. The reconstruction quality degrades gracefully under noise when the network is trained with synthetic noise regularization, and remains low for disorder amplitudes V0 between 0.3 and 0.6 and correlation lengths ξ between 2 and 4.5. The authors con

What carries the argument

The key instrument is an image-to-image convolutional neural network (CNN): a convolutional network that maps one image to another image, rather than to a class label. It takes the normalized LDOS field at one energy as input and outputs the normalized disorder potential field. The network uses convolutional layers with ReLU activations and, in 2D, residual connections, with circular padding to respect periodic boundary conditions. Training data is generated by exact diagonalization of the tight-binding Hamiltonian, followed by a windowed sum of eigenstate densities, and the loss is the mean-squared error between predicted and true normalized potentials. The argument works because the LDOS i

Load-bearing premise

The load-bearing premise is that the single-particle, nearest-neighbor tight-binding model with a Gaussian random-field site potential used to generate all training data is an adequate description of the real system; if electron-electron interactions or multi-orbital physics are significant, the reconstructed 'potential' may not correspond to the physical energy landscape.

What would settle it

Apply the trained network to an LDOS image, taken from a scanning tunneling microscopy experiment on a surface whose actual potential landscape is independently known (for example, a quantum corral built by atom manipulation). If the network's predicted potential disagrees with the known landscape beyond the reported accuracy, the assumption that the simple tight-binding forward model captures the essential physics of the experiment would be falsified.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • A single STM conductance image could be used to estimate the local potential energy landscape of a surface, enabling downstream calculations of transport and other observables.
  • The low data requirement (about 1000 samples in 2D) suggests the method may be feasible when synthetic data generation is expensive, such as for interacting models.
  • Noise-regularized training (injecting synthetic noise) maintains reconstruction accuracy even when test noise reaches the LDOS signal amplitude, indicating robustness to realistic experimental conditions.
  • The network retains accuracy for disorder strengths and correlation lengths outside its training range (e.g., V0 ∈ [0.3,0.6], ξ ∈ [2,4.5]), implying that exact knowledge of the disorder prior may not be necessary.
  • Because the model uses only a single energy slice, experiments need not perform full spectroscopic mapping to extract the Hamiltonian's potential term.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If validated on experimental STM data, the same image-to-image paradigm could be extended to reconstruct other spatially varying Hamiltonian parameters, such as hopping modulations or local interaction strengths, from LDOS maps at multiple energies.
  • The observed failure modes—charging rings near potential minima and plateaus near maxima—suggest that incorporating physical priors (like the sign of the potential or spectral bounds) as additional loss terms could sharpen reconstruction in extreme regions.
  • A multi-energy input (stacking LDOS slices at several bias voltages) could resolve ambiguities that a single slice leaves hidden, an extension the authors mention but do not test; this is a natural next experiment within their framework.
  • The method's reliance on a specific forward model means its success on real materials will depend on how faithfully a tight-binding model captures the sample; surfaces with strong electron-electron correlations may require the network to be retrained on data from a more complete model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 8 minor

Summary. The paper studies the inverse problem of recovering a site-diagonal disorder potential V(r) of a single-particle nearest-neighbor tight-binding Hamiltonian (Eq. 1) from a single-energy spatial map of the LDOS (Eq. 2). The authors generate synthetic training pairs from correlated Gaussian random-field disorder in 1D and 2D, normalize both fields, and train convolutional image-to-image networks to map the normalized LDOS to the normalized potential. They report test MSE L≈0.016 (1D, L=128) and L≈4.6×10^-3 (2D, L=64), far below a linear fit and translation-augmented k-NN baselines (Table I). They also study robustness to added noise with noise-injected training and show modest out-of-distribution generalization around the training values V0=0.5, ξ=3. The paper frames the result as a step toward STM-based Hamiltonian reconstruction and explicitly limits the setting to noninteracting, single-orbital tight-binding with fixed hopping.

Significance. If the reported accuracy holds, the paper is a useful empirical demonstration that a CNN can approximate the inverse of the LDOS map within a disordered noninteracting tight-binding model, using modest data and compute (<1 GPU hour). Strengths include a clean train/validation/test split, hyperparameter tuning on validation, comparison against two baselines, honest worst-case examples, and an explicit statement of model limitations in Section IV. The appendix's global-normalization statement preserves the disorder amplitude in the target, so the reported MSE is not merely shape recovery. The paper does not overclaim uniqueness: it acknowledges that the inverse problem is ill-posed in general and frames the result as distribution-dependent approximate recovery. Within its stated scope, the contribution is solid, although it is an empirical benchmark rather than a new analytical method.

major comments (1)
  1. [§II, Eqs. (3)–(4), and Appendix A1] The normalization equations in the main text are ambiguous and, if read as per-sample standardization, would make the headline MSE (0.016/0.005) and the V0 out-of-distribution study misleading. Appendix A1 explicitly states that a global training-set normalization was used and that no sample-wise normalization was attempted, which resolves the substantive concern: the CNN target retains the V0 scale. However, the main text should be amended to refer to the global scheme, since as written a reader could reasonably interpret Eqs. (3)–(4) as per-sample and thereby misread the central numerical claim. This is a clarity issue in a load-bearing location, not a flaw in the implementation.
minor comments (8)
  1. [§III.A vs Appendix A2] The 1D sample count is inconsistent: Section III.A says 'we generated 12000 data points', while Appendix A2 says 'The total number of data points was N=15000'; the baseline paragraph also says 12000 points. Please reconcile and specify the exact split used.
  2. [§III.A] The text says 'We show the results of NN-1D on four test LDOS profiles in Fig. 2', but Fig. 2 shows three profiles (best, median, worst). Please correct.
  3. [Eq. (2), §II] The notation 'µ|E−E′|' should be 'µ(|E−E′|)' or the weighting function should be defined with explicit arguments, to avoid confusion with multiplication.
  4. [§II.C and Table II] The superscript n in 'ρ~_E^n' appears before noise is introduced in the main text. Clarify whether it denotes a noised LDOS or is a typo.
  5. [Fig. 2 caption] The caption says the LDOS input is shown '(right)' and the predicted potential is also '(right)'; one of these should be '(left)'.
  6. [Appendix B] The text refers to 'Fig. 7b and Fig. 7d', but Fig. 7 appears to have two panels (uncorrelated and correlated noise). Please renumber or relabel the subfigures consistently.
  7. [Global reproducibility] No code, data, or random seeds are made available. Given the ML methodology, releasing the data-generation and training code would substantially improve reproducibility.
  8. [Fig. 5b] The out-of-distribution averages use 10 realizations per grid point but no error bars or standard errors are shown. Adding uncertainty estimates would make the claimed L≲0.1 region more persuasive.

Circularity Check

0 steps flagged

No circularity: the CNN is trained and evaluated on synthetic pairs from an independent forward model; the reported inverse mapping is a learned empirical fit, not an input-equivalent construction.

full rationale

The paper's central claim is that a CNN can approximately invert the LDOS-to-potential map of a specified tight-binding model. The forward model (Eq. 1) and LDOS definition (Eq. 2 / A1) generate synthetic training pairs; the CNN is a generic image-to-image regressor trained by MSE loss (Eqs. 5-6) and evaluated on held-out samples from the same distribution. This is standard supervised learning: the target V is not defined in terms of the input rho, and the network weights are free parameters fit to the data. The baselines (single-parameter fit, k-NN) are independent comparators and are explicitly reported as such, not presented as predictions. No step reduces a claimed result to its own input by construction. The only internal inconsistency is the normalization definition: Sec. II (Eqs. 3-4) writes per-sample mean/std, while Appendix A2 states a global training-set normalization and explicitly says 'We did not attempt sample-wise normalization.' This is a correctness/ambiguity concern, not circularity: it affects what the reported MSE measures (absolute vs. shape recovery) but does not make the CNN's output equal to its input by definition. There are no load-bearing self-citations: the only author self-citation is a contextual review reference (Ref. [22]) used to motivate STM imaging, not to justify the inverse result. The paper explicitly acknowledges limitations (single-orbital, noninteracting model) in Sec. IV, which further supports that no universal physical law is being claimed. Thus the derivation chain is self-contained and non-circular.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central claim rests on the training distribution and the CNN hyperparameters, both of which are either chosen by hand or fitted to validation data. No new physical entities are introduced.

free parameters (4)
  • NN-1D hyperparameters = Nc=112, NL=6, k=5, lr=1.03e-3
    Selected via Optuna on the validation set; the reported accuracy depends on these choices.
  • NN-2D hyperparameters = Nc=32, NB=3, k=3, lr=7.21e-4
    Selected via Optuna on the validation set; the reported accuracy depends on these choices.
  • Global normalization statistics = mean/std of V and rho over training set
    Eq. A2 normalizes inputs and outputs using training-set statistics; the MSE is computed on these normalized fields.
  • Problem settings (V0, xi, E, deltaE, L) = V0=0.5, xi=3, E=-1.5 (1D)/-1.0 (2D), deltaE=0.25, L=128/64
    Chosen by hand to define the data distribution; results could change for other settings.
axioms (5)
  • domain assumption The system is described by a single-particle tight-binding Hamiltonian with nearest-neighbor hopping and site-diagonal potential (Eq. 1).
    Underlies all data generation; real materials can have multiple orbitals and interactions.
  • domain assumption LDOS is defined as the energy-windowed sum of eigenstate densities (Eq. 2/A1).
    This is the simulated proxy for STM differential conductance.
  • domain assumption Potentials are drawn from a Gaussian random field with exponential correlation (V0, xi).
    Defines the training distribution D_V; the inverse map is trained for this prior specifically.
  • standard math CNN with circular padding preserves the periodic boundary conditions of the lattice.
    Architectural choice matching the simulated data; standard practice.
  • domain assumption MSE loss on globally normalized fields is a meaningful measure of reconstruction quality.
    Normalization removes trivial mean/scale; the reported L values are in normalized units.

pith-pipeline@v1.3.0-alltime-deepseek · 13201 in / 12685 out tokens · 128716 ms · 2026-08-04T18:44:20.622110+00:00 · methodology

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Cite this review

Pith. "Pith review of Reconstructing the Hamiltonian from the local density of states using neural networks." pith.science (2026). https://pith.science/paper/NXWOXXHH

@misc{pith2026250909604,
  author       = {Pith},
  title        = {Pith review of: Reconstructing the Hamiltonian from the local density of states using neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NXWOXXHH}},
  note         = {Machine review of arXiv:2509.09604}
}
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read the original abstract

Reconstructing a quantum system's Hamiltonian from limited yet experimentally observable information is interesting both as a practical task and from a fundamental standpoint. We pose and investigate the inverse problem of reconstructing a Hamiltonian from a spatial map of the local density of states (LDOS) near a fixed energy. We demonstrate high-quality recovery of Hamiltonians from the LDOS using supervised learning. In particular, we generate synthetic data from single-particle Hamiltonians in 1D and 2D, train convolutional neural networks, and obtain models that solve the inverse problem with remarkably high accuracy. Moreover, we are able to generalize beyond the training distribution and develop models with strong robustness to noise. Finally, we comment on possible experimental applications to scanning tunneling microscopy, where we propose that maps of the electronic local density of states might be used to reveal a sample's unknown underlying energy landscape.

Figures

Figures reproduced from arXiv: 2509.09604 by Andrew Ma, Kevin P. Nuckolls, Nisarga Paul.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: c,d. For this analysis, the main goal was to capture how each parameter generally affects the reconstruction, and we did not optimize for low losses or tune any hyperparameters. We can observe that the model’s worst performance occurs near the middle and edges of the spectrum at any fixed V0. This indicates that the performance is not strictly correlated with either the spectral density (or sparsity) or th… view at source ↗
Figure 2
Figure 2. Figure 2: In the absence of a potential, the wavefunctions are plane waves with wavevector k at energy E as determined by the dispersion E(k) = −t cos(k). As the potential is turned on, the wavefunctions Anderson localize with a localization length that increases with increasing V0 and |E|. We illustrate the 0.2 0.4 0.6 0.8 1.0 V0/t 2 1 0 1 2 E/t 0.0 0.4 0.8 Architecture 2 L L 0.2 0.4 0.6 0.8 1.0 V0/t 2 1 0 1 2 E/t … view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: , along with the the corresponding true and predicted potentials. Our results indicate that our basic approach is fairly robust against moderate test noise levels, especially if regularization (achieved via adding training noise in our work) is applied. That said, real STM measurements can exhibit additional compli￾cations like sample-tip distance fluctuations, drift-induced distortions, and temperature ef… view at source ↗

discussion (0)

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    Baselines We compared our neural network approach against two baseline methods for the inverse mapping from normalized local density of states ˜ρE (r)to normalized disorder potential ˜V(r). The single-parameter linear fit was the first baseline. It assumes a direct proportionality between (mean-subtracted) LDOS and potential: ˜V pred(r) =α˜ρE (r)whereαis ...

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    learnability

    Figure 3 and out-of-distribution study For Fig. 3c-d, we performed a “learnability” study of the 1D Anderson model. We chose system sizeL=64, correlation lengthξ=3, and disorder amplitudesV 0/t∈[0.05,1.0]. For each disorder amplitudeV 0, we generated 500 random disorder realizations, diagonalized each Hamiltonian, and stored results for reuse across diffe...