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Physical Review C

Comment on “Improved many-body expansions from eigenvector continuation”

Péter R. Śurján, Ágnes Szabados

Published Jun 28, 2021• 0 citations
Publisher

Abstract

In a recent paper [Phys. Rev. C 101, 041302(R) (2020)] Demol et al. published an interesting method to improve results from finite-order perturbation theory. Their applications on nuclear many-body problems show that the numerical results are very accurate, capable of converting divergent series to fast-converging sequences. We argue that one reason for this impressive accuracy is connected to those of related methods which proved to be useful in many-electron theory.

DOI
10.1103/physrevc.103.069801

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In a recent paper [Phys. Rev. C 101, 041302(R) (2020)] Demol et al. published an interesting method to improve results from finite-order perturbation theory. Their applications on nuclear many-body problems show that the numerical results are very accurate, capable of converting divergent series to fast-converging sequences. We argue that one reason for this impressive accuracy is connected to those of related methods which proved to be useful in many-electron theory.
For pt.II see ibid., vol.26, no.13, p.1885-96 (1993). The authors employ the method developed in a previous paper for small systems-Be, LiH, H 2 -where full CI calculations are available for monitoring convergence of many-body perturbation theory. It is shown that divergent series, in particular for excited states, can be transformed into fast converging ones. In essence their method consists in performing infinite subsummations of perturbation series in order to improve convergence: coupling constants are redefined such that singularities are incorporated in a non-perturbative manner and remaining correlations can be expanded in a larger domain of the complex coupling constant plane. It is in this way that the notion of 'improved convergence' has a well defined meaning.
Eigenvector continuation is a computational method that finds the extremal eigenvalues and eigenvectors of a Hamiltonian matrix with one or more control parameters. It does this by projection onto a subspace of eigenvectors corresponding to selected training values of the control parameters. The method has proven to be very efficient and accurate for interpolating and extrapolating eigenvectors. However, almost nothing is known about how the method converges, and its rapid convergence properties have remained mysterious. In this Letter, we present the first study of the convergence of eigenvector continuation. In order to perform the mathematical analysis, we introduce a new variant of eigenvector continuation that we call vector continuation. We first prove that eigenvector continuation and vector continuation have identical convergence properties and then analyze the convergence of vector continuation. Our analysis shows that, in general, eigenvector continuation converges more rapidly than perturbation theory. The faster convergence is achieved by eliminating a phenomenon that we call differential folding, the interference between nonorthogonal vectors appearing at different orders in perturbation theory. From our analysis we can predict how eigenvector continuation converges both inside and outside the radius of convergence of perturbation theory. While eigenvector continuation is a nonperturbative method, we show that its rate of convergence can be deduced from power series expansions of the eigenvectors. Our results also yield new insights into the nature of divergences in perturbation theory.
Abstract In reply to Kaldor's [Int. J. Quantum Chem. XX, XXX (1985)] criticism of our study of simple four‐electron models, in which the degree of quasi‐degeneracy can be continuously varied, by the finite‐order nondegenerate many‐body perturbation theory, we examine in more detail a simple two‐state model that the used to substantiate his claim that “the low order sum of the perturbation series is not very meaningful” in view of its divergence. It is shown that in contrast to Kaldor's claim, the partitioning used increases the radius of convergence of the considered perturbation series and is in principle capable to make it convergent. It is also shown that the convergence of the series is not very essential and that even divergent series can provide useful estimate of the exact result, particularly when the resummation techniques, such as Padè approximants or continued fractions, are employed. Finally, the shortcomings of the existing multi‐reference perturbation approaches, which Kaldor advocates, are pointed out.