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Real-Frequency Correlation Functions from Neural Quantum States via Operator Lanczos Approach

Jonas B. Rigo, Markus Schmitt

Published Mar 20, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.21203/rs.3.rs-8976411/v1

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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.
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.
Understanding dynamical properties of quantum many-body systems remains a central challenge because excitations generally require information beyond a ground-state wave function. Here we introduce a neural polaron ansatz that represents quasiparticle excitations by neural many-body operators acting on a correlated ground state. Instead of learning an independent excited-state wave function, the method parameterizes a local dressing operator through a compact neural head defined on the feature map of a pretrained ground-state network. This operator-based construction builds in translation symmetry, momentum resolution, and quasiparticle locality, while separating ground-state correlations from excitation-specific dressing. We benchmark the method on the square-lattice J1J_1J1​-J2J_2J2​ Heisenberg model, where it accurately reproduces magnon dispersions and spectral weights over a broad range of frustration. In particular, it captures nontrivial many-body features including the (π,0)(π,0)(π,0) anomaly and its progressive softening with increasing J2/J1J_2/J_1J2​/J1​. These results establish neural operators as a physically transparent route for extending neural quantum states from ground-state properties to dynamical response.
In this work, we use neural quantum states (NQS) to describe the high-dimensional wave functions of electron-phonon coupled systems. We demonstrate that NQS can accurately and systematically learn the underlying physics of such problems through a variational Monte Carlo optimization of the energy with minimal incorporation of physical information, even in highly challenging cases. We assess the ability of our approach across various lattice model examples featuring different types of couplings. The flexibility of our NQS formulation is demonstrated via application to ab initio models parametrized by density functional perturbation theory consisting of electron or hole bands coupled linearly to dispersive phonons. We compute accurate real-frequency spectral properties of electron-phonon via a response formalism based on NQS. Our work establishes a general framework for exploring diverse ground-state and dynamical phenomena arising in electron-phonon systems, including the nonperturbative interplay of correlated electronic and electron-phonon effects in systems ranging from simple lattice models to realistic models of materials parametrized by ab initio calculations.
Spectral functions are central to link experimental probes to theoretical models in condensed matter physics. However, performing exact numerical calculations for interacting quantum matter has remained a key challenge especially beyond one spatial dimension. In this work, we develop a versatile approach using neural quantum states to obtain spectral properties based on simulations of the dynamics of excitations initially localized in real or momentum space. We apply this approach to compute the dynamical structure factor in the vicinity of quantum critical points (QCPs) of different two-dimensional quantum Ising models, including one that describes the complex density wave orders of Rydberg atom arrays. When combined with deep network architectures we find that our method reliably describes dynamical structure factors of arrays with up to 24×24 spins, including the diverging timescales at critical points. Our approach is broadly applicable to interacting quantum lattice models in two dimensions and consequently opens up a route to compute spectral properties of correlated quantum matter in yet inaccessible regimes.
Density matrix embedding theory (DMET) [Phys. Rev. Lett. 109, 186404 (2012)], introduced an approach to quantum cluster embedding methods whereby the mapping of strongly correlated bulk problems to an impurity with finite set of bath states was rigorously formulated to exactly reproduce the entanglement of the ground state. The formalism provided similar physics to dynamical mean-field theory at a tiny fraction of the cost but was inherently limited by the construction of a bath designed to reproduce ground-state, static properties. Here, we generalize the concept of quantum embedding to dynamic properties and demonstrate accurate bulk spectral functions at similarly small computational cost. The proposed spectral DMET utilizes the Schmidt decomposition of a response vector, mapping the bulk dynamic correlation functions to that of a quantum impurity cluster coupled to a set of frequency-dependent bath states. The resultant spectral functions are obtained on the real-frequency axis, without bath discretization error, and allows for the construction of arbitrary dynamic correlation functions. We demonstrate the method on the one- (1D) and two-dimensional (2D) Hubbard model, where we obtain zero temperature and thermodynamic limit spectral functions, and show the trivial extension to two-particle Green's functions. This advance therefore extends the scope and applicability of DMET in condensed-matter problems as a computationally tractable route to correlated spectral functions of extended systems and provides a competitive alternative to dynamical mean-field theory for dynamic quantities.