Open problem: converting the demonstrated rooting of the nuclear EDF method in ab initio PGCM-PT with MR-IMSRG-preprocessed Hamiltonians into a practical, error-controlled methodology with mean-field-like $O(n_{\rm dim}^4)$ cost

The empirical nuclear energy density functional (EDF) method and ab initio nuclear structure theory have developed as separate worlds: the former delivers mean-field-cost ($O(n_{\rm dim}^4)$) calculations across essentially the whole nuclear chart but rests on phenomenological functionals with uncontrolled errors, while the latter is systematically improvable but computationally prohibitive for heavy doubly open-shell nuclei. Duguet, Ebran, Frosini, Hergert and Somà (Eur. Phys. J. A 59, 13 (2023)) demonstrated, formally and numerically in 20Ne, that the EDF method can be rooted in the ab initio projected-generator-coordinate-method perturbation theory (PGCM-PT) whenever the Hamiltonian is preprocessed by a multi-reference in-medium similarity renormalization group (MR-IMSRG) transformation, whose evolved Hamiltonian $\bar{H}(s)$ plays the role of the EDF effective Hamiltonian. They left open the conversion of this conceptual link into a usable methodology: quantifying and reducing MR-IMSRG truncation and unitarity-breaking errors (especially for excited states), achieving EDF-like computational cost by making perturbative corrections negligible and cheapening the construction of $\bar{H}(s)$, establishing the universality of a single evolved Hamiltonian, and eventually formulating an effective field theory in terms of the one-body density matrix. The overarching objective is so far unresolved.

Background

Two complementary frameworks dominate low-energy nuclear structure theory. The first is the nuclear energy density functional (EDF) method, a self-consistent mean-field approach conceptually analogous to density functional theory: nuclei are described by symmetry-breaking Hartree-Fock-Bogoliubov (HFB) product states (quasiparticle vacua that may break rotational symmetry, particle number and time reversal), optionally restored by projection and mixed by the generator coordinate method (GCM). The GCM diagonalizes the Hamiltonian in a subspace spanned by HFB vacua labeled by collective 'generator coordinates' such as the quadrupole deformation; with quantum-number projection applied beforehand it is called the projected GCM (PGCM). The EDF method's key virtue is a mean-field-like operational cost scaling as $O(n_{\rm dim}^4)$, where $n_{\rm dim}$ is the dimension of the one-body spherical-harmonic-oscillator basis, which authorizes systematic studies over the entire nuclear chart; its key weakness is that the underlying functional is phenomenological and not connected to controlled nuclear forces. The second framework is ab initio theory, which starts from nuclear interactions derived in chiral effective field theory (EFT) — the low-momentum expansion of quantum chromodynamics organized in powers of the resolution scale — and solves the many-body problem with systematically improvable approximations and quantifiable uncertainties. Ab initio methods include coupled cluster, self-consistent Green's functions, the in-medium similarity renormalization group (IMSRG) and (quasi-exact) shell-model expansions. The IMSRG evolves the Hamiltonian through a continuous unitary flow $H(s)=e^{\eta(s)}He^{-\eta(s)}$ driven by an anti-Hermitian generator $\eta(s)$ chosen to decouple a chosen reference (model) space from its complement; the flow parameter $s$ measures the extent of the transformation. In its multi-reference (MR) variant the reference is a correlated superposition such as a PGCM state, and the Hamiltonian is normal-ordered (Wick's theorem reformulated) with respect to that correlated state, producing an evolved Hamiltonian $\bar{H}(s)$ whose zero-, one- and two-body components effectively resum 'dynamical' correlations (weak, short-range, non-collective nucleonic motion), complementing the 'static' correlations (shape deformation and fluctuations) captured by the PGCM itself. A crucial practical point is that the MR-IMSRG flow is truncated (e.g. at the two-body level, MR-IMSRG(2), with construction cost $O(n_{\rm dim}^6)$) and terminated at finite $s$, so the transformation is not exactly unitary and residual couplings between the PGCM subspace and its complement survive. Building on the PGCM-PT formalism (multi-reference many-body perturbation theory formulated on top of the PGCM), Duguet, Ebran, Frosini, Hergert and Somà showed in 2022-2023 that when the PGCM employs an MR-IMSRG-evolved Hamiltonian, the perturbative corrections on top of the PGCM eventually become negligible: the PGCM states become good approximations to low-lying eigenstates of $\bar{H}(s)$, and $\bar{H}(s)$ can thus be identified with the ab initio counterpart of the effective, nucleus-dependent Hamiltonian implicitly at play in the empirical EDF method. This establishes a formal bridge between the two frameworks. What the demonstration does not provide is a practical method: brute-force PGCM-PT(2) costs $O(n_{\rm dim}^8)$ and a naive MR-IMSRG(2) construction of $\bar{H}(s)$ costs $O(n_{\rm dim}^6)$ (MR-IMSRG(3): $O(n_{\rm dim}^9)$), and both would have to be repeated per nucleus and per state, whereas the whole point of the EDF method is its $O(n_{\rm dim}^4)$ cost and its universality across the chart.

Problem Statement

The demonstrated rooting of the nuclear EDF method in the ab initio PGCM-PT formalism based on MR-IMSRG-preprocessed Hamiltonians must be converted into a practical, broadly usable ab initio-rooted EDF methodology. The common objective is an EDF-like scheme that computes low-lying ground and excited states (and associated observables) of realistic nuclei, including heavy and doubly open-shell ones, with mean-field-like $O(n_{\rm dim}^4)$ operational cost while retaining ab initio grounding and controlled, quantified errors. Achieving it involves the following interlinked challenges: (i) quantifying and systematically reducing the errors induced by the MR-IMSRG(2) truncation and the finite flow parameter $s$ — truncation and unitarity-breaking errors that leave residual couplings between the PGCM subspace and its complement non-zero, in particular for excited states — for instance through optimized truncation strategies or flow diagnostics (simply going to higher truncation schemes such as MR-IMSRG(3) is ruled out, since it contradicts the $O(n_{\rm dim}^4)$ cost goal); (ii) devising concrete, scalable algorithms that (a) make the perturbative corrections on top of the PGCM negligible for both ground and excited states (e.g. by enriching the PGCM generator coordinates with octupole, triaxial, pairing/cranking degrees of freedom (higher-multipole deformations and rotating-frame constraints) or selected elementary excitations, or by evolving with respect to an ensemble (weighted set) of PGCM states to decouple a whole spectrum at once) and (b) drastically lower the cost of constructing $\bar{H}(s)$ (e.g. via operator bases relying only on one-body density contractions or auxiliary density matrices (approximate densities used for normal ordering), tensor factorization or hypercontraction (compressed tensor decompositions), importance truncation (configuration selection via perturbative importance estimates), natural-orbital bases (one-body bases diagonalizing the correlated one-body density matrix), or emulators such as eigenvector continuation (reconstruction of an eigenstate's dependence on a parameter from a few exactly computed anchor points) for parameter dependencies); (iii) establishing how far a single MR-IMSRG-evolved Hamiltonian, or a small set thereof, can be made universal across nuclei and states via ensemble evolution or smooth parameterizations of $\bar{H}(s)$, and what fidelity trade-offs this entails; (iv) developing explicit, low-cost parameterizations of $\bar{H}(s)$ — and eventually an effective field theory formulated in terms of the one-body density matrix — that retain ab initio grounding while bypassing repeated MR-IMSRG runs; and (v) demonstrating in realistic nuclei the practical design of cost functions, optimization strategies and diagnostics for choosing auxiliary density matrices or truncated operator bases that minimize unitarity violation and residual couplings while preserving $O(n_{\rm dim}^4)$ scaling. These challenges form one program with a single scientific objective, so an answer is a methodology addressing the jointly required core rather than an isolated sub-item. The jointly required core comprises challenge (i) — quantifying and systematically reducing the MR-IMSRG(2)-truncation and finite-flow errors — and challenge (ii): to make the perturbative corrections on top of the PGCM essentially negligible for ground and excited states and to reduce significantly the cost of constructing $\bar{H}(s)$ so that the mean-field-like $O(n_{\rm dim}^4)$ scaling is maintained. The universality of challenge (iii) and the parameterizations and cost functions of challenges (iv)-(v) are complementary research tracks toward this core, and the effective field theory formulated in terms of the one-body density matrix is a longer-term direction building on the know-how collected via these objectives — a theory that remains to be invented — rather than a deliverable required of a single answer.

References

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