Hamiltonian parameter inference from resonant inelastic x-ray scattering with active learning

TL;DR

The paper tackles the challenge of extracting reliable low-energy Hamiltonians from information-dense RIXS data, where inverse scattering is typically underconstrained. It introduces a Bayesian-optimization workflow that couples RIXS spectral simulations within the KH formalism (via EDRIXS) to a Gaussian-process surrogate of a spectral-distance metric, enabling efficient, automated inference of multi-parameter atomic models. Applied to NiPS3, NiCl2, Fe2O3, and Ca3LiOsO6, the approach achieves parameter estimates that reproduce hand-fitted spectra and provides the first atomic-model parameters for Fe2O3 and Ca3LiOsO6, while exposing multiple near-minima and symmetry-based distinctions among solutions. Overall, the method automates inverse scattering in quantum materials, offering a scalable path toward high-throughput, model-agnostic parameter inference and paving the way for more sophisticated cluster or impurity models and automated experimental design.

Abstract

Identifying model Hamiltonians is a vital step toward creating predictive models of materials. Here, we combine Bayesian optimization with the EDRIXS numerical package to infer Hamiltonian parameters from resonant inelastic X-ray scattering (RIXS) spectra within the single atom approximation. To evaluate the efficacy of our method, we test it on experimental RIXS spectra of NiPS3, NiCl2, Ca3LiOsO6, and Fe2O3, and demonstrate that it can reproduce results obtained from hand-fitted parameters to a precision similar to expert human analysis while providing a more systematic mapping of parameter space. Our work provides a key first step toward solving the inverse scattering problem to extract effective multi-orbital models from information-dense RIXS measurements, which can be applied to a host of quantum materials. We also propose atomic model parameter sets for two materials, Ca3LiOsO6 and Fe2O3, that were previously missing from the literature.

Figures (11)

• Figure 1: Tanabe-Sugano-style plots for $d$-shell electronic (initial) Hamiltonians of the form $x_1 U_\text{coul}^{F^2}+(1-x_1)V_\text{cf} + x_2 U_\text{coul}^{F^4}-(1-x_2)U_\text{coul}^{F^2}$ with occupation numbers 2 or 8. On the left subplot, $x_2=0$ and $x_1$ is varied ($x_1$ subplot). On the right subplot, $x_1=1$ and $x_2$ is varied. Dotted lines correspond to spin singlets, dashed lines denote triplets. States transforming according to different irreducible representations of the octahedral group are shown in different colors: $A_1$ -- blue, $A_2$ -- green, $E$ -- red, $T_1$ -- teal, $T_2$ -- purple. Ground state energies are subtracted so that occupations $2$ and $8$ yield the same figure. This plot gives an overview of the spectrum of $H_i$ as a function of its three most important parameters $F^2$, $F^4$, and $10Dq$, with other parameters fixed to zero.
• Figure 2: Flowchart of the Bayesian optimization algorithm used to solve the inverse scattering problem for RIXS. A. The preprocessor imports and processes a spectrum image from a RIXS experiment. B. The GPR sampler builds a set of $N_\text{run}$ models of the distance function via an active learning protocol. Each run consists of $N_\text{iter}$ GPR iterations, followed by $N_\text{init}$ initial evaluations at random points. C. Parameters obtained from the $N_\text{greedy}$ smallest queried distances are refined by a subsequent greedy optimizer. D. Finally, the results are presented by the data analyzer.
• Figure 3: Experimental and simulated spectra for NiPS$_{3}$. (a) experimental data from Ref. He2024MagneticallyPH, (b) hand-fitted (reference), (c) result of GPR, (d) result after greedy fine-tuning (Powell's method). The corresponding $L_1$ sum distances for the models shown in panels b-d are $\chi^2_{L_1} = 0.692$, $0.694$, and $0.480$, respectively.
• Figure 4: Experimental and simulated spectra for NiCl$_{2}$. (a) experimental data from Ref. Occhialini2024, (b) hand-fitted (reference), (c) result of GPR, (d) result after greedy fine-tuning (Powell's method). The corresponding L1 sum distances for panels (b)-(d) are $\chi^2_{L_1}=0.496$, $0.406$, and $0.373$, respectively.
• Figure 5: Experimental and simulated spectra for Fe$_{2}$O$_{3}$. (a) experimental data from Ref. JieminLi2023Fe2O3, (b) result of GPR, (c) result after greedy fine-tuning (Powell's method). $L_1$ max distances for panels (b)-(c) are $\chi^2_{L^\prime_1}=158.8$ and $116.8$, respectively.