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Operator Lanczos Approach enabling Neural Quantum States as Real-Frequency Impurity Solvers

Jonas B. Rigo, Markus Schmitt

Published Dec 9, 2025• 0 citations
Publisher

Abstract

Abstract To understand the intricate exchange between electrons of different bands in strongly correlated materials, it is essential to treat multi-orbital models accurately. For this purpose, dynamical mean-field theory (DMFT) provides an established framework, whose scope crucially hinges on the availability of efficient quantum impurity solvers. Here we present a real-frequency impurity solver based on neural quantum states (NQS) combined with an operator-Lanczos construction. NQS are an asymptotically unbiased variational ground-state ansatz that employs neural networks to capture long-range correlations on complicated graph structures. We leverage this ability to solve multi-orbital impurity problems using a systematically improvable Segmented Commutator Operator-Lanczos (SCOL) construction. Our benchmarks on both the single-orbital Anderson model and the multi-orbital Hubbard–Kanamori impurity Hamiltonian reveal excellent ground-state precision and the capacity to accurately resolve zero temperature spectral functions and self-energies. These results open avenues for extending DMFT to more challenging problems. Repository contents This repository contains the figures, data, and plotting script used to recreate Fig. 2 and Fig. 3 of “Operator Lanczos Approach enabling Neural Quantum States as Real Frequency Impurity Solvers”. The notebook reproduces spectral functions and self-energies for the Single Anderson Impurity Model (SIAM) and a Hubbard–Kanamori system with two and three orbitals (MO2, MO3). The files in this repsitory are: `plot.ipynb` – matplotlib script that reads the HDF5 files and generate the publication figures. `data/SIAM_plot.h5` – SIAM spectral/self-energy data grouped into three panels (`panel_left`, `panel_mid`, `panel_right`) with metadata for interaction strength `U`, segment depth `l_s`, and method `kind` (`NQS` or `NRG` -> for reference). `data/MO_plot.h5` – MO2 and MO3 results. `MO2` contains frequency `omega`, spectral function `A` and self-energy arrays (`omega_Sigma`, `Sigma_im_nqs` and NRG references). `MO3` holds NRG spectra plus frequncies `omega` and spectral function `A_nqs` for `l_s = 2,3,4`. `siam_square.pdf` – rendered SIAM figure (Fig. 2 in the paper). `mo.pdf` – rendered MO figure (Fig. 3 in the paper). Requirements Python 3 with `numpy`, `matplotlib`, and `h5py`. Jupyter (if you want to run the notebook interactively). A LaTeX installation (e.g., TeX Live) so matplotlib can render TeX labels (`text.usetex=True` is enabled). Install the Python packages locally (examples): pip install numpy matplotlib h5py jupyter Notes - The notebook uses Computer Modern fonts via LaTeX; if TeX is missing you can disable `text.usetex` inside the first code cell. - Data are already curated for plotting, no additional preprocessing is needed beyond loading the HDF5 datasets described above. - Tested versions (newer should also work): `numpy==1.26.4`, `matplotlib==3.10.0`, `h5py==3.13.0`, `jupyter==1.1.1`.

DOI
10.5281/zenodo.17865206

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Abstract To understand the intricate exchange between electrons of different bands in strongly correlated materials, it is essential to treat multi-orbital models accurately. For this purpose, dynamical mean-field theory (DMFT) provides an established framework, whose scope crucially hinges on the availability of efficient quantum impurity solvers. Here we present a real-frequency impurity solver based on neural quantum states (NQS) combined with an operator-Lanczos construction. NQS are an asymptotically unbiased variational ground-state ansatz that employs neural networks to capture long-range correlations on complicated graph structures. We leverage this ability to solve multi-orbital impurity problems using a systematically improvable Segmented Commutator Operator-Lanczos (SCOL) construction. Our benchmarks on both the single-orbital Anderson model and the multi-orbital Hubbard–Kanamori impurity Hamiltonian reveal excellent ground-state precision and the capacity to accurately resolve zero temperature spectral functions and self-energies. These results open avenues for extending DMFT to more challenging problems. Repository contents This repository contains the figures, data, and plotting script used to recreate Fig. 2 and Fig. 3 of “Operator Lanczos Approach enabling Neural Quantum States as Real Frequency Impurity Solvers”. The notebook reproduces spectral functions and self-energies for the Single Anderson Impurity Model (SIAM) and a Hubbard–Kanamori system with two and three orbitals (MO2, MO3). The files in this repsitory are: plot.ipynb – matplotlib script that reads the HDF5 files and generate the publication figures. data/SIAM_plot.h5 – SIAM spectral/self-energy data grouped into three panels (panel_left, panel_mid, panel_right) with metadata for interaction strength U, segment depth l_s, and method kind (NQS or NRG -> for reference). data/MO_plot.h5 – MO2 and MO3 results. MO2 contains frequency omega, spectral function A and self-energy arrays (omega_Sigma, Sigma_im_nqs and NRG references). MO3 holds NRG spectra plus frequncies omega and spectral function A_nqs for l_s = 2,3,4. siam_square.pdf – rendered SIAM figure (Fig. 2 in the paper). mo.pdf – rendered MO figure (Fig. 3 in the paper). Requirements Python 3 with numpy, matplotlib, and h5py. Jupyter (if you want to run the notebook interactively). A LaTeX installation (e.g., TeX Live) so matplotlib can render TeX labels (text.usetex=True is enabled). Install the Python packages locally (examples): pip install numpy matplotlib h5py jupyter Notes - The notebook uses Computer Modern fonts via LaTeX; if TeX is missing you can disable text.usetex inside the first code cell. - Data are already curated for plotting, no additional preprocessing is needed beyond loading the HDF5 datasets described above. - Tested versions (newer should also work): numpy==1.26.4, matplotlib==3.10.0, h5py==3.13.0, jupyter==1.1.1.
To understand the intricate exchange between electrons of different bands in strongly correlated materials, it is essential to treat multi-orbital models accurately. For this purpose, dynamical mean-field theory (DMFT) provides an established framework, whose scope crucially hinges on the availability of efficient quantum impurity solvers. Here we present a real-frequency impurity solver based on neural quantum states (NQS) combined with an operator-Lanczos construction. NQS are an asymptotically unbiased variational ground-state ansatz that employs neural networks to capture long-range correlations on complicated graph structures. We leverage this ability to solve multi-orbital impurity problems using a systematically improvable Segmented Commutator Operator-Lanczos (SCOL) construction. Our benchmarks on both the single-orbital Anderson model and the multi-orbital Hubbard-Kanamori impurity Hamiltonian reveal excellent ground-state precision and the capacity to accurately resolve zero temperature spectral functions and self-energies. These results open avenues for extending DMFT to more challenging problems.
Abstract Dynamical correlation functions are essential for revealing emergent phenomena in quantum matter, since they measure the frequency-resolved response of a system at equilibrium to a perturbation and thereby probe its full spectrum. While neural quantum states provide an accurate and flexible description of correlated ground states, accessing real-frequency dynamical correlators from them remains a major challenge. Here we introduce a general and systematically improvable framework to compute real-frequency linear-response functions directly from neural quantum state ground states. Our method performs the Lanczos subspace construction through an operator recursion and controls operator growth with a suited truncation scheme. We benchmark it on demanding quantum impurity models, where high precision is required close to the Fermi energy, and show that rich spectral structure and the self-energy can be resolved over several orders of magnitude. The approach generalizes straightforwardly, for example providing a direct path toward more broadly applicable dynamical mean-field theory solvers.
Quantum impurity solvers are the computational bottleneck of quantum embedding approaches to correlated materials, such as dynamical mean-field theory (DMFT). We show that neural networks trained on synthetic, material-agnostic data learn the impurity mapping from hybridization functions and local interactions to Green's functions with quantitative accuracy for both model systems and real materials, providing fast solvers for single- and multi-orbital models. Benchmarks against numerically controlled quantum Monte Carlo show that the method reproduces the Mott transition, multi-orbital phase diagrams of Hubbard-Kanamori models, and the electronic properties of SrVO3_33​ and SrMnO3_33​. The learned solvers achieve orders-of-magnitude speedup and can initialize controlled calculations, dramatically accelerating DMFT while preserving accuracy.
We introduce a neural network impurity solver for real-frequency DMFT that employs a multihead cross-attention mechanism to map hybridization functions to spectral functions, conditioned on impurity parameters. Trained on high-quality MPS data from complex contour time evolution and incorporating derivative constraints with respect to the complex-time angle, our model achieves smooth generalization to the real-frequency axis. Benchmarking on the single-band Hubbard model for the Bethe lattice demonstrates quantitative accuracy across metallic, strongly correlated, and insulating regimes.
Predicting the physical properties of strongly correlated electron systems (SCES) requires the combined Density Functional Theory and Dynamical Mean-Field Theory (DFT+DMFT) framework. However, the computational bottle-neck lies in the impurity solver. This work focuses on rigorously benchmarking the Multi-Orbital Iterated Perturbation Theory (MO-IPT) impurity solver against the numerically exact Continuous-Time Quantum Monte Carlo (CTQMC) method, specifically through a detailed study of the three-orbital Hubbard model. The two-orbital model serves as a canonical system to test performance under realistic multi-band conditions. Our systematically derived MO-IPT overcomes the critical limitations of CTQMC, including exponential scaling with orbital number, the need for analytic continuation, and the severe fermionic sign problem. By demonstrating high accuracy across varying interaction strengths and temperatures, we validate MO-IPT as an inexpensive, yet reliable, replacement solver essential for future material-specific investigations.
A fast impurity solver for the dynamical mean-field theory (DMFT) named two-mode approximation is proposed based on the Gutzwiller variational approach, which captures the main features of both the coherent and incoherent motion of the electrons. The solver works with real frequency at zero temperature and it provides directly the spectral function of the electrons. It can be easily generalized to multiorbital impurity problems with general on-site interactions, which makes it very useful in local-density approximation plus DMFT. Benchmarks on one- and two-band Hubbard models are presented, and the results agree well with those of exact diagonalization. Lastly a few cartoonlike five-band results are presented, showing that the present solver is really capable of dealing with multiband systems.