Hamiltonian Reconstruction via STM-IETS

• The paper demonstrates that STM-IETS spectra, combined with many-body simulations and machine learning, quantitatively extract complex Hamiltonian parameters in quantum materials.
• Inversion methodologies rely on statistical distance minimization and neural networks to reconstruct multiorbital and lattice-model Hamiltonians from experimental dI/dV maps.
• Case studies on Fe-phthalocyanine/SnTe and AKLT chains illustrate the method’s accuracy and potential for mapping magnetic, orbital, and exchange interactions.

Hamiltonian reconstruction from scanning tunneling microscopy with inelastic electron tunneling spectroscopy (STM-IETS) enables direct, quantitative extraction of microscopic interaction parameters in quantum materials by leveraging the spectral fingerprints arising from many-body excitations. STM-IETS spectra encode rich information on local spin, orbital, and vibrational degrees of freedom. Recent advances combine realistic many-body simulations, statistical inversion methodologies, and machine learning to extract full multiorbital or lattice-model Hamiltonians from experiment, transforming atomic-scale measurements into precise characterizations of magnetic, orbital, and exchange interactions.

1. Multiorbital and Lattice-Model Hamiltonians in STM-IETS

The microscopic Hamiltonian for STM-IETS studies is system-dependent. For adsorbed molecular quantum magnets (e.g., iron phthalocyanine on SnTe), the multiorbital Hamiltonian $\mathcal{H}_{\rm model}(z)$ includes four essential components: intrinsic crystal-field splitting ($\mathcal{H}^0_{\rm CF}$), tip-induced perturbation ($\mathcal{H}^{\rm tip}_{\rm CF}(z)$, e.g., Stark shift), full four-index Coulomb repulsion among $d$ orbitals ($\mathcal{H}_{\rm Coulomb}$), and atomic spin–orbit coupling ($\mathcal{H}_{\rm SOC}$), with explicit $z$-dependent terms to model the STM setpoint (Lupi et al., 27 Jan 2026).

For lattice systems, the canonical form is a classical Ising-type Hamiltonian on an $N \times N$ lattice, with nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) exchange couplings, and possible quenched bond disorder:

$\mathcal{H}^0_{\rm CF}$0

This minimal lattice model allows exploration of symmetry-breaking, frustration, and competing interactions that produce complex phase diagrams and rich local motif statistics (Valleti et al., 2020).

In quantum spin systems, e.g. the AKLT ($\mathcal{H}^0_{\rm CF}$1) chain, the relevant Hamiltonian is a bilinear-biquadratic form:

$\mathcal{H}^0_{\rm CF}$2

with analytic eigenenergies for each total spin multiplet $\mathcal{H}^0_{\rm CF}$3 (Ferri-Cortés et al., 6 Mar 2025).

2. Physical Mechanisms and Setpoint Dependence

The STM setpoint (current $\mathcal{H}^0_{\rm CF}$4 and resulting tip–sample distance $\mathcal{H}^0_{\rm CF}$5) is a controllable knob that exponentially tunes the orbital overlap and local electrostatic field at the molecule or substrate. In practice, the tip-induced Stark shift $\mathcal{H}^0_{\rm CF}$6 linearly depends on $\mathcal{H}^0_{\rm CF}$7 across the relevant experimental range, allowing the on-site energy of specific orbitals (e.g., $\mathcal{H}^0_{\rm CF}$8) to shift by up to $\mathcal{H}^0_{\rm CF}$9 meV (Lupi et al., 27 Jan 2026). This tuning reveals the fingerprints of Hamiltonian parameters in the evolution of conductance spectra.

Setpoint-dependent STM-IETS spectra show discrete steps or peaks in $\mathcal{H}^{\rm tip}_{\rm CF}(z)$0 and $\mathcal{H}^{\rm tip}_{\rm CF}(z)$1 whenever $\mathcal{H}^{\rm tip}_{\rm CF}(z)$2 crosses a many-body excitation, with substrate effects (e.g. ferroelectric polarization) symmetry-breaking orbital pairs and leading to characteristic splitting and hybridization patterns. The bias window defines the accessible excitation energies.

For AKLT or similar quantum spin chains, non-equilibrium regimes with increased STM junction conductance allow populations of excited states, so transitions among excited multiplets become visible as overshoot peaks (Ferri-Cortés et al., 6 Mar 2025).

3. Generation of Theoretical Spectra and Descriptor Extraction

Exact diagonalization in the full Fock space of the local shell (e.g., Fe $\mathcal{H}^{\rm tip}_{\rm CF}(z)$3, $\mathcal{H}^{\rm tip}_{\rm CF}(z)$4 orbitals with appropriate filling) yields eigenstates and energies. Dynamical correlators compute excitation spectra in both spin-flip and orbital cotunneling channels:

• Spin-flip:

$\mathcal{H}^{\rm tip}_{\rm CF}(z)$5

• Orbital cotunneling:

$\mathcal{H}^{\rm tip}_{\rm CF}(z)$6

• The integrated inelastic conductance:

$\mathcal{H}^{\rm tip}_{\rm CF}(z)$7

Experimental STM-IETS spectra are compressed into local descriptors—peak positions, step heights, local motif labels—such that each tip position maps to a discrete set of “motifs” or cluster configurations. Relative motif frequencies form an $\mathcal{H}^{\rm tip}_{\rm CF}(z)$8-dimensional histogram $\mathcal{H}^{\rm tip}_{\rm CF}(z)$9, serving as the input for statistical inversion. This approach is generalizable to local vibrational or spin excitations in lattice systems (Valleti et al., 2020).

4. Inversion Methodologies: Statistical and Machine Learning Approaches

Hamiltonian reconstruction proceeds by inverting the parameter-to-spectrum mapping using supervised learning, statistical distance minimization, or least-squares fitting.

For multiorbital models, a feed-forward neural network with three hidden layers and ReLU activations is trained on $d$010,000 simulated spectra spanning the physically relevant ranges of symmetry-breaking ($d$1), spin–orbit coupling ($d$2), and Stark shifts ($d$3). The 2D normalized conductance map $d$4 is the input, and the output is the best-fit Hamiltonian parameter set (Lupi et al., 27 Jan 2026).

For lattice models, the motif histogram from experiment is compared to that generated by Monte Carlo simulation at candidate $d$5 values. The Wootters statistical distance

$d$6

quantifies the mismatch. Maximum a-posteriori estimates $d$7 are obtained by minimizing $d$8 (with optional priors), using quasi-Newton or derivative-free optimizers. Parameter uncertainties are derived from the Hessian of the log-posterior (Valleti et al., 2020).

For AKLT dimers, analytic formulas relate the measured STM-IETS peak voltages $d$9 to exchange couplings $\mathcal{H}_{\rm Coulomb}$0 and $\mathcal{H}_{\rm Coulomb}$1, allowing least-squares fitting or direct linear inversion:

$\mathcal{H}_{\rm Coulomb}$2

This closed-form inversion is robust given multiple resolvable thresholds (Ferri-Cortés et al., 6 Mar 2025).

5. Case Studies and Representative Results

In Fe-phthalocyanine/SnTe (multiorbital case), ML inversion yields symmetry-breaking hybridization $\mathcal{H}_{\rm Coulomb}$3 eV, enhanced spin–orbit coupling $\mathcal{H}_{\rm Coulomb}$4 eV, and Stark-shift window $\mathcal{H}_{\rm Coulomb}$5 meV. The reconstructed conductance maps accurately reproduce all observed excitation branches, two from spin-flip and one from orbital cotunneling, matching experimental peak positions across all setpoint values (Lupi et al., 27 Jan 2026).

For the classical Ising lattice, motif histogram inversion enables unbiased recovery of $\mathcal{H}_{\rm Coulomb}$6 above $\mathcal{H}_{\rm Coulomb}$7, as well as in frustrated regimes with competing interactions ($\mathcal{H}_{\rm Coulomb}$8), even when global order is absent. Bond disorder degrades precision only mildly up to $\mathcal{H}_{\rm Coulomb}$9 of $\mathcal{H}_{\rm SOC}$0 (Valleti et al., 2020).

Edge-state STM-IETS on finite AKLT $\mathcal{H}_{\rm SOC}$1 chains (hexamers) enables direct observation of fractional $\mathcal{H}_{\rm SOC}$2 spins at boundaries and effective Hamiltonian reconstruction via in-gap excitation spectroscopy. The mapping of threshold voltages to the spectrum of a bilinear chain model $\mathcal{H}_{\rm SOC}$3 quantifies three neighbor-couplings, confirming AKLT physics and ground-state manifold degeneracy (Ferri-Cortés et al., 6 Mar 2025).

6. Limitations, Uncertainties, and Generalizations

Multiple sources of uncertainty and model simplification affect reconstruction fidelity:

• Neglect of dynamic substrate hybridization (only static symmetry-breaking $\mathcal{H}_{\rm SOC}$4 included)
• Single-orbital tip coupling and absence of Kondo or non-equilibrium effects at short $\mathcal{H}_{\rm SOC}$5 (Lupi et al., 27 Jan 2026)
• Parameter uncertainty in mapping $\mathcal{H}_{\rm SOC}$6 from mechanical tip deformations (Lupi et al., 27 Jan 2026)
• Experimental noise and tip variability, with machine learning retaining fidelity $\mathcal{H}_{\rm SOC}$7 up to realistic $\mathcal{H}_{\rm SOC}$8 noise (Lupi et al., 27 Jan 2026)
• Quantum fluctuations, strong electron–phonon coupling, and finite-size effects requiring adapted motif classification and simulations (Valleti et al., 2020)
• For AKLT extraction, overlapping thresholds or thermal broadening can obscure multiplet resolution; large enough tip spin-polarization and low temperature are necessary (Ferri-Cortés et al., 6 Mar 2025)

A plausible implication is that reconstruction protocols are robust across a broad parameter regime but rely on comprehensive forward modeling and realistic training data. By extending the Hamiltonian—e.g., including more orbitals, explicit tip and substrate orbitals via Wannierization—and augmenting training data, the workflows generalize to other metallo-porphyrins, lanthanide complexes, and ligand-field-engineered systems (Lupi et al., 27 Jan 2026).

7. Ancillary Machine Learning and Phase Diagram Mapping

Histogram clustering (e.g., $\mathcal{H}_{\rm SOC}$9-means, hierarchical) partitions motif histograms $z$0 into distinct clusters, compressing $z$1 site variables to $z$2 motif dimensions. This enables rapid mapping of phase boundaries—paramagnetic, ferromagnetic, antiferromagnetic, frustrated—without exhaustive thermodynamic scans. Principal component projection and dendrogram analysis help visualize phase space topology (Valleti et al., 2020).

This suggests that STM-IETS motif clustering combined with inversion algorithms provides rigorous estimates of competing interaction parameters and accelerates discovery of complex ground-state and high-temperature phase behavior, especially in systems that resist macroscopic thermodynamic characterization.

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In summary, STM-IETS coupled with advanced inversion workflows enables robust atomic-scale reconstruction of multiorbital molecular, lattice, and quantum-spin Hamiltonians. The synergistic use of setpoint-dependent spectroscopy, exact many-body simulation, statistical inversion, and machine learning transforms experimental $z$3 maps into quantitative models of strong correlation, symmetry-breaking, and topological excitations (Lupi et al., 27 Jan 2026, Valleti et al., 2020, Ferri-Cortés et al., 6 Mar 2025).