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arXiv:2610.38780

An Eigenvector-Continued Green's Function Method

Stefano Brolli, Carlo Barbieri

nucl-thcond-mat.str-elquant-phSubmitted Oct 28, 2026• Announced Oct 30, 2026• 0 citations
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Abstract

Emulating the Green's function using reduced-order equations is challenging because of the amount of information it contains. This information includes the one-particle spectroscopy over the entire energy spectrum, the one-body density matrix, and the ground-state energy. Nevertheless, the computational speedup offered by emulators is essential for applications requiring repeated model evaluations, which are ubiquitous in many-body physics. These include parameter calibration, uncertainty quantification, phase-diagram mapping, and geometry optimization. In this Letter, we introduce a systematically improvable emulator based on eigenvector continuation and moment-matching. The resulting approximation, which we refer to as an ``eigenvector-continued Green's function,'' can be obtained by solving linear equations, enabling an efficient separation of the computational workload into offline and online stages.

AI Summary

The paper addresses reduced-order emulation of many-body Green’s functions for applications requiring repeated model evaluations, including parameter calibration, uncertainty quantification, phase-diagram mapping, and geometry optimization. The challenge is to reduce computational cost while representing the information encoded in the Green’s function: one-particle spectroscopy across the energy spectrum, the one-body density matrix, and the ground-state energy. It introduces an approximation termed an “eigenvector-continued Green’s function,” combining eigenvector continuation with moment matching. The stated result is a systematically improvable emulator whose evaluation can be formulated through linear equations. This formulation permits separation into an offline stage and an online stage, with the intended benefit of making repeated evaluations less expensive. The abstract does not specify the reduced-space construction, which moments are matched, or how the approximation is enlarged to improve accuracy. Only the title and abstract are available, so the evidence establishing accuracy and efficiency cannot be assessed here. No benchmark systems, numerical errors, speedup factors, computational scaling, or convergence bounds are reported in the abstract. It also gives no explicit limitations or conditions under which the method remains accurate. In particular, the available material does not establish how accurately the emulator reproduces each of the spectral and ground-state quantities motivating its construction.

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Written by gpt-6.1-sol from the abstract only. Automated: it can be wrong.

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5 pages, 4 figures

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Emulating the Green's function using reduced-order equations is challenging because of the amount of information it contains. This information includes the one-particle spectroscopy over the entire energy spectrum, the one-body density matrix, and the ground-state energy. Nevertheless, the computational speedup offered by emulators is essential for applications requiring repeated model evaluations, which are ubiquitous in many-body physics. These include parameter calibration, uncertainty quantification, phase-diagram mapping, and geometry optimization. In this Letter, we introduce a systematically improvable emulator based on eigenvector continuation and moment-matching. The resulting approximation, which we refer to as an ``eigenvector-continued Green's function,'' can be obtained by solving linear equations, enabling an efficient separation of the computational workload into offline and online stages.
The paper addresses reduced-order emulation of many-body Green’s functions for applications requiring repeated model evaluations, including parameter calibration, uncertainty quantification, phase-diagram mapping, and geometry optimization. The challenge is to reduce computational cost while representing the information encoded in the Green’s function: one-particle spectroscopy across the energy spectrum, the one-body density matrix, and the ground-state energy.
It introduces an approximation termed an “eigenvector-continued Green’s function,” combining eigenvector continuation with moment matching. The stated result is a systematically improvable emulator whose evaluation can be formulated through linear equations. This formulation permits separation into an offline stage and an online stage, with the intended benefit of making repeated evaluations less expensive. The abstract does not specify the reduced-space construction, which moments are matched, or how the approximation is enlarged to improve accuracy.
Only the title and abstract are available, so the evidence establishing accuracy and efficiency cannot be assessed here. No benchmark systems, numerical errors, speedup factors, computational scaling, or convergence bounds are reported in the abstract. It also gives no explicit limitations or conditions under which the method remains accurate. In particular, the available material does not establish how accurately the emulator reproduces each of the spectral and ground-state quantities motivating its construction.
Repeated evaluation of the single-particle Green's function (GF) across parameter space is a recurring bottleneck in many-body methods, limiting the resolution at which phase boundaries may be probed and the frequency resolution of computed spectra. We introduce MOR+EC, a framework that constructs reusable reduced-order models for computing single-particle Green's functions by combining eigenvector continuation (EC) for parameter space exploration and model order reduction (MOR) for extrapolation in frequency space. Contrary to conventional parameterized MOR, we construct these reduced-order models with a parameter-independent resolvent, further reducing the number of required full-space evaluations. Benchmarking against exact diagonalization as the impurity solver for our example case of DMFT calculations for single- and two-band models, MOR+EC reproduces the DMFT-converged impurity GF to an average relative error below 10−310^{-3}10−3, with a median wall-time speedup of 9.59.59.5-92×92\times92×. This accuracy and efficiency together resolve a fine-grained scan of the impurity occupation as doping is tuned and produce a high-resolution phase diagram of an orbital-selective Mott transition. The reduced-order model also reproduces real- and imaginary-frequency spectra at no additional cost in full-space evaluations. This framework applies broadly to GF-based methods requiring repeated parametric evaluation and high resolution of the frequency axis.
First principles calculations of atomic nuclei based on microscopic nuclear forces derived from chiral effective field theory (EFT) have blossomed in the past years. A key element of such ab initio studies is the understanding and quantification of systematic and statistical errors arising from the omission of higher-order terms in the chiral expansion as well as the model calibration. While there has been significant progress in analyzing theoretical uncertainties for nucleon-nucleon scattering observables, the generalization to multi-nucleon systems has not been feasible yet due to the high computational cost of evaluating observables for a large set of low-energy couplings. In this Letter we show that a new method called eigenvector continuation (EC) can be used for constructing an efficient and accurate emulator for nuclear many-body observables, thereby enabling uncertainty quantification in multi-nucleon systems. We demonstrate the power of EC emulation with a proof-of-principle calculation that lays out all correlations between bulk ground-state observables in the few-nucleon sector. On the basis of ab initio calculations for the ground-state energy and radius in 4He, we demonstrate that EC is more accurate and efficient compared to established methods like Gaussian processes.
The numerical treatment of quantum systems often requires large amounts of computing power and time. As a result, performing calculations repeatedly for different values of the input parameters is often not feasible. One remedy is using eigenvectors describing the system that are analytic functions that vary smoothly for real values of the input parameters. This allows one to replace computationally expensive calculations with emulators that project onto a reduced-basis set. This Colloquium explores a particular class of reduced-basis methods known as eigenvector continuation and its applications, with emphasis on nuclear physics.
We present the reduced basis method as a tool for developing emulators for equations with tunable parameters within the context of the nuclear many-body problem. The method uses a basis expansion informed by a set of solutions for a few values of the model parameters and then projects the equations over a well-chosen low-dimensional subspace. We connect some of the results in the eigenvector continuation literature to the formalism of reduced basis methods and show how these methods can be applied to a broad set of problems. As we illustrate, the possible success of the formalism on such problems can be diagnosed beforehand by a principal component analysis. We apply the reduced basis method to the one-dimensional Gross-Pitaevskii equation with a harmonic trapping potential and to nuclear density functional theory for 48Ca^{48}\mathrm{Ca}48Ca, achieving speed-ups of more than $\ifmmode\times\else\texttimes\fi{}150$ and $\ifmmode\times\else\texttimes\fi{}250$, respectively, when compared to traditional solvers. The outstanding performance of the approach, together with its straightforward implementation, show promise for its application to the emulation of computationally demanding calculations, including uncertainty quantification.
Reduced-basis methods provide a powerful framework for building efficient and accurate emulators. Although widely applied in many fields to simplify complex models, reduced-basis methods have only been recently introduced into nuclear physics. In this Letter we build an emulator to study the single-particle structure of atomic nuclei. By scaling a suitable mean-field Hamiltonian, a ``universal'' reduced basis is constructed capable of accurately and efficiently reproduce the entire single-particle spectrum of a variety of nuclei. Indeed, the reduced-basis model reproduces both ground- and excited-state energies as well as the associated wave functions with remarkable accuracy. Our results bode well for more demanding applications that use Bayesian optimization to calibrate nuclear energy density functionals.
We develop a class of emulators for solving quantum three-body scattering problems. They are based on combining the variational method for scattering observables and the recently proposed eigenvector continuation concept. The emulators are first trained by the exact scattering solutions of the governing Hamiltonian at a small number of points in its parameter space, and then employed to make interpolations and extrapolations in that space. Through a schematic nuclear-physics model with finite-range two and three-body interactions, we demonstrate the emulators to be extremely accurate and efficient. The computing time for emulation is on the scale of milliseconds (on a laptop), with relative errors ranging from ${10}^{\ensuremath{-}13}$ to ${10}^{\ensuremath{-}4}$ depending on the case. The emulators also require little memory. We argue that these emulators can be generalized to even more challenging scattering problems. Furthermore, this general strategy may be applicable for building the same type of emulators in other fields, wherever variational methods can be developed for evaluating physical models.