Hamiltonian Learning via STM-IETS Techniques

• Hamiltonian learning involves extracting microscopic parameters from STM-IETS data using model-based and machine learning approaches.
• The methodology is applicable to diverse systems, such as molecular magnets and topological superconductors, through spectroscopic data analysis.
• Techniques leverage impurities and tip-induced modifications to analyze local spectral signatures and derive Hamiltonian parameters.

Hamiltonian learning from scanning tunneling microscopy inelastic electron tunneling spectroscopy (STM-IETS) refers to the set of techniques that quantitatively extract the microscopic Hamiltonian parameters of a quantum system—such as molecules, engineered spin lattices, or correlated materials—directly from spectroscopic data acquired via position-resolved and inelastic tunneling processes. This methodology leverages the local spectral signatures induced by impurities or tip-molecule coupling and typically combines model-based fitting, analytic inversion, and modern ML approaches. Recent developments demonstrate the applicability of these methods to a wide range of physical systems, including topological moiré superconductors, molecular magnets, and quantum spin models realized on surfaces (Khosravian et al., 2023, Ferri-Cortés et al., 6 Mar 2025, Lupi et al., 27 Jan 2026).

1. Model Hamiltonians and STM-IETS Observable Foundations

STM-IETS experiments resolve changes in conductance at the atomic scale through differential ($dI/dV$) and second-derivative ($d^2I/dV^2$) spectroscopy, probing not only single-particle density of states (LDOS) but also local inelastic excitations such as spin flips or vibrational modes. The parameters of interest are those of the system’s effective Hamiltonian $\mathcal{H}$, which may include bilinear and higher-order interactions in spin systems, spatially modulated exchange and pairing in topological materials, or multiorbital effects in molecular magnets. These Hamiltonians are typically expressed in second-quantized form, incorporating on-site energies, (tip-induced) crystal field terms, Coulomb repulsion, spin-orbit coupling, electron–boson couplings, and substrate hybridization (Lupi et al., 27 Jan 2026).

In STM-IETS, the observable spectral steps or peaks correspond to transitions between discrete many-body or quasi-particle states. The voltage position and intensity of these inelastic features encode the energy differences and matrix elements set by $\mathcal{H}$. Thus, a direct mapping between STM-IETS features and Hamiltonian parameters is possible, provided the relevant selection rules, occupation kinetics, and coupling mechanisms are properly accounted for.

2. Analytic Inversion in Localized Spin and Edge-State Systems

For well-controlled finite systems—e.g., spin dimers or edge states in AKLT-type fragments—the link between Hamiltonian parameters and observable IETS spectral features can be derived analytically. For a $S=3/2$ dimer with bilinear ($J$) and biquadratic ($K$) exchange, the eigenenergies are

$$E(S_T) = \frac{J}{2}A(S_T) + \frac{K}{4}A(S_T)^2, \quad A(S_T) = S_T(S_T+1) - \frac{15}{2}$$

with $S_T=0,1,2,3$ (Ferri-Cortés et al., 6 Mar 2025). Steps or peaks in $dI/dV$ at bias voltages $d^2I/dV^2$0 occur at excitation energies $d^2I/dV^2$1, corresponding to spin multiplet transitions that obey selection rules $d^2I/dV^2$2. The analytic dependence of $d^2I/dV^2$3 on $d^2I/dV^2$4 and $d^2I/dV^2$5 enables (overconstrained) algebraic inversion of experimental step energies to obtain the exchange parameters. For boundary “dangling” $d^2I/dV^2$6 pairs in AKLT-model edge states, energy splittings are directly proportional to the effective coupling $d^2I/dV^2$7 and can be measured by the position of inelastic steps, thus allowing for the mapping of the BLBQBC (bilinear–biquadratic–bicubic) parameter space onto observable spectroscopic data.

3. Real-Space Impurity Tomography in Topological and Correlated Materials

In correlated and topologically non-trivial 2D materials, Hamiltonian learning adopts a different paradigm. Rather than solving for all eigenstates, an impurity is introduced to locally probe the electronic structure via LDOS modulations. For example, in moiré topological superconductors, Hamiltonian parameters such as exchange and superconducting modulation amplitudes ($d^2I/dV^2$8, $d^2I/dV^2$9) are spatially varying and manifest in the properties of impurity-bound states (Khosravian et al., 2023).

The “impurity tomography” approach relies on a calculated embedding Green function at the impurity site to generate local spectral maps,

$\mathcal{H}$0

where $\mathcal{H}$1 is the supercell Hamiltonian with impurity and $\mathcal{H}$2 is the embedding self-energy. The resulting LDOS maps, when expanded in harmonic or radial basis (e.g., $\mathcal{H}$3 for $\mathcal{H}$4 angular and $\mathcal{H}$5 radial components), capture the salient features of in-gap states across the moiré pattern.

4. Machine Learning Architectures and Training Protocols

For high-dimensional or many-body Hamiltonians, direct analytic inversion is impractical. In these cases, supervised machine learning, particularly neural networks (NNs), can be trained to regress Hamiltonian parameters from simulated or experimental spectral data. The pipeline consists of:

• Generating a labeled dataset: Theory spectra are computed by diagonalizing $\mathcal{H}$6 over a sampled parameter grid (e.g., $\mathcal{H}$7, or $\mathcal{H}$8 for molecular systems) and extracting simulated observables such as LDOS or $\mathcal{H}$9 maps (Lupi et al., 27 Jan 2026, Khosravian et al., 2023).
• Preprocessing the input: Harmonic/radial expansion (impurity tomography) or direct 2D CNN input (spectral maps as images).
• Training: Fully connected or convolutional neural networks fit with mean squared error (MSE) loss and optimized via Adam optimizer, with dropout and batch processing for regularization and performance improvement.
• Parameter estimation: For experimental data, preprocessing is matched to the training procedure, and the trained network yields Hamiltonian parameter estimates, with uncertainty quantification via model ensembles or Bayesian dropout (Lupi et al., 27 Jan 2026).

In extensive benchmarks, physics-informed harmonic-expansion networks outperform standard CNNs in resolving subtle order parameters such as $\mathcal{H}$0 in superconducting states, while both approaches are comparably effective for stronger signals like exchange modulation $\mathcal{H}$1 (Khosravian et al., 2023).

5. STM-IETS-Specific Protocols and Their Generalization

STM-IETS implementation for Hamiltonian learning requires carefully controlled measurement conditions (e.g., low temperatures $\mathcal{H}$2, lock-in detection for $\mathcal{H}$3 with small AC modulation, high conductance regimes for non-equilibrium state population). For setpoint-dependent STM-IETS, the systematic variation of the tip-sample distance alters the local electrostatic environment, inducing Stark shifts and modifying excited state energies in a controlled manner (Lupi et al., 27 Jan 2026). Recording full $\mathcal{H}$4 “spectroscopic maps” enables a more robust inference of multi-parameter Hamiltonians, as tip-induced symmetry breaking and level shifts differentiate otherwise overlapping features.

Advances have extended machine learning pipelines to accept these multidimensional data as inputs, yielding parameter prediction for realistic models encompassing local fields, multi-orbital interactions, spin–orbit coupling, tip-induced crystal field, and substrate hybridization. The mapping from STM-IETS spectra to Hamiltonian is thus encoded as a nonlinear regression problem, made tractable by large-scale synthetic datasets and modern neural network architectures.

A critical consideration is the inclusion of noise and non-ideal effects: training and validation protocols systematically inject synthetic noise (e.g., up to $\mathcal{H}$5 amplitude for FePc/SnTe), and Bayesian model ensembles yield realistic confidence intervals for predicted parameters, as reported by Lupi et al. (Lupi et al., 27 Jan 2026).

6. Extensions, Limitations, and Prospects

The current Hamiltonian-learning framework from STM-IETS is robust for a range of systems but has explicit limitations and open directions:

• Substrate-induced hybridizations, higher-order tip–orbital couplings, and nonequilibrium occupation effects can introduce discrepancies between simple model Hamiltonians and real experimental conditions (Lupi et al., 27 Jan 2026).
• Zero-bias (Kondo, phonon) features are typically masked or omitted, although more complete nonequilibrium or quantum master-equation approaches are under development.
• For strongly correlated or truly many-body systems, exact diagonalization becomes computationally intensive, and density-matrix renormalization group (DMRG) or tensor network approaches are required for generating reference spectra.
• Future progress includes model generalization (e.g., ab initio DFT+Wannier models), probabilistic learning (e.g., Gaussian processes, Bayesian NNs), and active learning strategies for optimal STM setpoint selection.

A central advance is the recognition that setpoint-dependent STM-IETS is not merely a complication but a functional lever for Hamiltonian learning—by tuning the tip-environment, one can enhance parameter distinguishability and enable quantitative, model-agnostic spectroscopy at the atomic scale (Lupi et al., 27 Jan 2026). The paradigm of “impurity tomography + ML” positions STM-IETS as a universal quantum materials diagnosis tool, capable of extracting both conventional (exchange, pairing) and emergent (fractional edge-spin) couplings from real-space spectral data.