Molecular Hamiltonian learning from setpoint-dependent scanning tunneling spectroscopy

Abstract

Molecular quantum magnets adsorbed on surfaces exhibit rich spin and orbital excitations that can be probed by scanning tunneling microscopy with inelastic electron tunneling spectroscopy (STM-IETS). However, the quantitative extraction of the underlying multiorbital Hamiltonian from experimental spectra remains a fundamental challenge. Here, we introduce molecular Hamiltonian learning, a machine learning strategy that infers the microscopic Hamiltonian parameters of a single adsorbed molecule directly from the setpoint-dependence of STM-IETS data. The method leverages the systematic evolution of spectral features as the STM tip tunes the local electrostatic environment for different tip-sample distances. We demonstrate this approach on iron phthalocyanine on ferroelectric SnTe, training our algorithm on theory spectra from a realistic multiorbital model, including spin-orbit coupling, electrostatic interactions, local crystal field, and substrate effects. The algorithm, trained solely on theoretical many-body simulations, allows reconstructing Hamiltonian parameters directly from experimental spectra. Our manuscript establishes a flexible and automated strategy for Hamiltonian reconstruction from STM-IETS, transforming setpoint-dependent spectroscopy into quantitative characterization of quantum materials at the atomic scale.

Figures (4)

• Figure 1: (a) STM measurement principle: varying the tip–molecule distance $z$ via the setpoint current tunes the local electrostatic environment and orbital energies. (b) Evolution of simulated d$I$/d$V$ spectra with decreasing $z$ (c) Scheme of Hamiltonian learning methodology.
• Figure 2: (a) Schematic side view: a FePc molecule adsorbed on SnTe. (b) Top view of FePc and crystal field splitting of the $3d$ orbitals. The ferroelectric substrate polarization $\mathbf{P}$ induces a rectangular distortion of the ligand-field symmetry.
• Figure 3: Representative theoretical spectrum: d$^2I$/d$V^2$ (a) and its cumulative integral d$I$/d$V$ (b), with the central bias window ($\pm 3$ meV) set to zero. The integrated dynamical correlator serves as input to the ML model. (c) Test set d$I$/d$V$ spectrum (gray) with added Gaussian noise ($W=5\%$, colored) used as network input. (d) Reconstructed d$I$/d$V$ obtained from the Hamiltonian parameters predicted by the network for the noisy input in (c); predicted values are indicated. (e) The fidelity $\mathcal{F}$ for Hamiltonian parameters ($\tau$, $\lambda_{\mathrm{SOC}}$, $\varepsilon_{\mathrm{min}}$, $\varepsilon_{\mathrm{max}}$) as a function of noise $W$.
• Figure 4: (a) Experimental $d^2I/dV^2$ spectrum of a single FePc molecule on SnTe. The derivative is obtained numerically from the measured d$I$/d$V$ signal. (b) Reconstructed dynamical correlator from the Hamiltonian parameters predicted by the model from the input in (a). (c) Orbital-conserving spin-flip and (d) orbital cotunneling channel contributions to the total reconstructed excitation spectrum.