Non-Empirical Interactions for the Nuclear Shell Model: An Update
S. Ragnar Stroberg, Scott K. Bogner, Heiko Hergert, Jason D. Holt
NNon-Empirical Interactions for the Nuclear ShellModel: An Update
S. Ragnar Stroberg , , , Heiko Hergert ,Scott K. Bogner , and Jason D. Holt TRIUMF, Vancouver BC, Canada V6T 2A3 Physics Department, Reed College, Portland, OR 97202, USA Department of Physics, University of Washington, Seattle WA, USA Facility for Rare Isotope Beams and Department of Physics & Astronomy,Michigan State University, East Lansing MI 48824, USAxxxxxx 0000. 00:1–57Copyright c (cid:13)
Abstract
The nuclear shell model has been perhaps the most important concep-tual and computational paradigm for the understanding of the struc-ture of atomic nuclei. While the shell model has been predominantlyused in a phenomenological context, there have been efforts stretch-ing back over a half century to derive shell model parameters basedon a realistic interaction between nucleons. More recently, several abinitio many-body methods—in particular many-body perturbation the-ory, the no-core shell model, the in-medium similarity renormalizationgroup, and coupled cluster theory—have developed the capability toprovide effective shell model Hamiltonians. We provide an update onthe status of these methods and investigate the connections betweenthem and potential strengths and weaknesses, with a particular focuson the in-medium similarity renormalization group approach. Three-body forces are demonstrated to be an important ingredient in under-standing the modifications needed in phenomenological treatments. Wethen review some applications of these methods to comparisons withrecent experimental measurements, and conclude with some remainingchallenges in ab initio shell model theory. a r X i v : . [ nu c l - t h ] O c t ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. The (Long) Road Towards
Ab Initio
Shell Model Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Fresh Perspectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Current Status of the ab initio
Shell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4. Organization of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62. Microscopic effective interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1. Quasidegenerate perturbation theory and the ˆ Q -box resummation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2. Okubo-Lee-Suzuki transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3. In-medium similarity renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4. Shell model coupled cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153. Comparison of various approaches to effective interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1. Formal effective interaction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2. Approximation schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234. Three-body forces and the connection with phenomenological adjustments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1. Ensemble normal ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2. Mass dependence of the effective interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.1. Ground and Excited States of sd − Shell Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2. The Calcium Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3. Heavy Nickel and Light Tin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346. Current challenges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.1. Electromagnetic transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.2. The Intruder-state Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397. Other developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.1. EFT for the shell model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447.2. Uncertainty quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457.3. Coupling to the Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46A. IMSRG flow equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49B. Canonical perturbation theory to second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50C. Integration of Magnus flow equation to second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1. Introduction
Since its introduction by Goeppert-Mayer and Jensen almost 70 years ago (1–3), the nuclearshell model has provided the primary conceptual framework for the understanding of nuclearstructure. Its central idea is that protons and neutrons inside a nucleus move within a self-consistently generated mean field. This mean field can be approximated by Woods-Saxon orharmonic oscillator potentials, provided a strong spin-orbit component is added. With theinclusion of the latter, the spectrum of single-particle energies exhibits a shell structure thatexplains the experimentally observed magic proton and neutron numbers. In this picture,the low-lying structure of most nuclei results from the interactions between configurationsof a few valence particles on top of an inert core via a residual nuclear force.From the beginning, it was hoped that the shell model and its residual force couldbe derived from basic building blocks, in particular the interaction between free protonsand neutrons. Despite early successes, this proved to be an enormously difficult task (4–8). eanwhile, increasingly elaborate empirical interactions were used with spectacular successto describe a vast array of experimental data (9–11). In modern language, this is a clearsignal that the shell model contains the relevant degrees of freedom to describe most (ifperhaps not all) phenomena observed in low-energy nuclear structure.In the last two decades, a combination of computational and theoretical advances haveprovided fresh perspectives and opportunities for the systematic derivation of shell modelinteractions from realistic nuclear forces , without the need for ad hoc phenomenologicaladjustments. The primary aim of this review is to describe these developments. The storyof the shell model, and of microscopically-derived effective interactions in particular, is longand full of false starts, dead ends, accidental successes, circling back, and rediscovery of oldwisdom in new language. We are not equal to the task of writing an authoritative history,and we have not attempted to do so. Indeed, we barely touch on recent developments inempirical shell model interactions or the computational aspects of configuration interactioncalculations. Readers interested in such techniques should consult the existing texts onthe subject (12–15). For the status of phenomenological approaches, we refer to severalthorough reviews (16–22). For more details on effective interaction theory, we recommendseveral texts (23–25), and reviews (17, 26–30). Soon after the publication of the seminal works by Goeppert-Mayer and Jensen, first pa-rameterizations of the residual nuclear force based on fits to experimental spectra began toappear (see, e.g., (9, 31–34)). Dissatisfied with such approaches because they hide the linkbetween the effective valence-space interaction and the underlying nucleon-nucleon inter-action, Brown, Kuo and collaborators pioneered the program to systematically derive andexplore this connection (5–7). In the 1960s, considerable progress was made in the con-struction of realistic nucleon-nucleon interactions that described NN scattering phase shiftswith high quality (35, 36). Following a similar strategy as theoretical studies of nuclearmatter, Kuo and Brown used Brueckner’s G matrix formalism (37–39) to treat the strongshort-range correlations induced by these forces, and included up to second-order diagramsin G to account for important core-polarization effects (5). Their work culminated in thepublication of Hamiltonians for the sd and pf shells (7, 8). While first applications weresuccessful, Barrett, Kirson, and others soon demonstrated a lack of convergence in pow-ers of the G matrix (40, 41), and more sophisticated treatments with RPA phonons andnon-perturbative vertex corrections destroyed the good agreement with experiment (42–44).Moreover, Vary, Sauer and Wong found that the use of larger model spaces for intermediate-state summations in diagram evaluations also degraded the agreement with experimentaldata (45). Around the same time, Schucan and Weidenm¨uller (46, 47) demonstrated thatthe presence of low-lying states of predominantly non-valence character (“intruder” states)render the perturbative expansion for the effective interaction divergent.Because of these developments, enthusiasm for the perturbative approach to derivingthe effective interaction dwindled (48). While Kuo and collaborators pursued the ˆ Q -box (orfolded-diagram) resummation of the perturbative series (24, 26, 28, 49, 50), the majorityof efforts in shell model theory were instead focused on the construction and refinement of By “realistic”, we mean interactions that are rooted to some extent in Quantum Chromody-namics and accurately describe few-body scattering and bound-state data. • Non-Empirical Interactions for the Nuclear Shell Model 3 mpirical interactions (10, 51–55). Large-scale calculations with such interactions yieldedimpressive agreement with available nuclear data, and even provided predictive power. Anexample is the “gold standard” universal sd shell interaction (USD) (10, 56, 57), whichachieves a root-mean-square deviation from experimental levels of merely 130 keV through-out the sd shell.Meanwhile, serious efforts were undertaken to develop approaches that circumvent theproblems plaguing the effective interaction methods by starting from the “bare” nuclear in-teractions and treating all nucleons as active particles. Prominent examples are coordinate-space Quantum Monte Carlo (QMC) techniques (58–60), or the no-core shell model (NCSM)(61–63). The late 1970s also saw a wave of nuclear Coupled Cluster calculations (64–66),which use systematic truncations to solve the Schr¨odinger equation at polynomial cost, asopposed to the exponential scaling of the NCSM. By the mid-1990s, computational advancesmade quasi-exact calculations for nuclei feasible (58, 61, 62, 67, 68). However, such calcula-tions were limited to light nuclei by the sheer numerical cost of coordinate-space QMC, andthe slow convergence of configuration-space methods with realistic nuclear interactions. Since the turn of the millennium, nuclear theory has undergone an important philosoph-ical shift with the the adoption of renormalization group (RG) and effective field theory(EFT) concepts. These tools provide a systematic framework for exploring long-existingquestions pertaining to the phase-shift equivalency of significantly different nucleon-nucleoninteractions or the origin and importance of three-nucleon forces (see, e.g., (69, 70)). Mostnoteworthy for the present work is the clarification of the issues that led to the failure ofthe aforementioned G -matrix based approaches, and the capability to reconcile the shellmodel, which is based on an (almost) independent-particle picture analogous to that ofatomic physics, with the notion that strong correlations are induced by (most) realistic NNinteractions. The essential idea of chiral effective field theory (EFT)for the nuclear force is that processes relevant for nuclear structure do not resolve thedetails of short-range interactions between nucleons . There are in fact infinitely-manydifferent potentials, differing at short distances, which all describe low-energy observablesequally well. This is good news, because we can take advantage of this arbitrariness andparameterize the short-range physics in a convenient way, e.g., through a series of contactinteractions. At long distances, the approximate chiral symmetry that chiral EFT inheritsfrom Quantum Chromodynamics (QCD) dictates that interactions are described by (multi-)pion exchange.In a pioneering work (71, 72), Weinberg developed effective Lagrangians to model theinteraction between nucleons in terms of pion exchange and contact interactions, with in-creasingly complicated contributions suppressed by powers of a nucleon’s typical momenta Q ∼ k F or the pion mass m π over the breakdown scale Λ χ of the EFT, ( Q/ Λ χ ) n . This pro-vided a framework to treat two-body forces consistently with three- and higher-body forces,as well as a natural explanation for the relative importance of these terms (see e.g. (73–75) “Short” in this context refers to distances r where k F r (cid:46)
1, with k F ≈ . − the Fermimomentum at saturation density. or recent reviews). Moreover, nuclear transition operators can be derived in a consistentfashion by coupling the chiral Lagrangian to the electroweak fields (see, e.g., (76–80)). De-spite a number of subtle issues which persist to this day, several families of chiral two- plusthree-nucleon interactions have been developed (81–92) that reproduce low-energy observ-ables with an accuracy comparable with phenomenological potentials. These interactionshave become the standard input for modern nuclear theory. The renormalization group (RG), in particular in theformulation developed by Wilson (93–95), is a natural companion to any effective fieldtheory. As discussed above, an EFT requires a cutoff Λ which delineates between “resolved”and “unresolved” physics. The specific form and location of the cutoff (the “scheme” and“scale”) is arbitrary and observables for momenta Q (cid:28) Λ should not depend on this choice.Consequently, there are an infinite number of equivalent theories which differ only in schemeand scale. The RG smoothly connects such equivalent theories.RG methods debuted in low-energy nuclear physics around the turn of the mille-nium (96–103), finally providing a systematic framework which formalized ideas that hadbeen discussed in the nuclear structure community since the 1950s. For instance, bothhard- and soft-core NN potentials can be devised which reproduce NN scattering data, butnuclear matter calculations found that soft potentials do not produce empirical saturationproperties and so soft potentials were disfavored (70). The missing piece in the saturationpuzzle was the connection between the off-shell NN interaction and the 3N interaction, asdemonstrated formally by Polyzou and Gl¨ockle (104). Of course, hard-core potentials aremuch more difficult to handle in many-body calculations, necessitating the use of Brueck-ner’s G -matrix (38, 105–107) to deal with correlations due to the short-range repulsion.From the RG perspective, the hard- and soft-core potentials are related by an RG trans-formation which leaves NN scattering observables unchanged, but shifts strength into in-duced
3N (and higher) interactions. Neglecting these induced terms means that observablesinvolving more than two particles will no longer be preserved. This mechanism provides anexplanation of the Phillips line (108) and the Tjon line (109), which describe correlationsbetween few-body observables calculated using different phase-equivalent NN interactions.In the context of this work, the RG provides a simple explanation of the observation that— after being processed by the Brueckner G -matrix machinery — various NN potentialsproduce very similar spectroscopy (110, 111), as long as they reproduce NN scattering data.This can be understood as an indication that the G -matrix approximately “integrates out”the short-distance physics of the different potentials, leaving the universal long-distancephysics. However, Bogner et al. have shown that the G -matrix can retain significantcoupling between off-shell low- and high-momentum modes, rendering it non-perturbative(102). This explains why the historical efforts to construct the effective interaction pertur-batively from G -matrices were bound to fail. In contrast, methods like the Similarity RG(SRG) (102, 112, 113) achieve a more complete decoupling of the short-distance physics andrender the resulting transformed NN+3N interaction suitable for perturbative expansions(102, 114–116). The SRG has become the tool of choice in nuclear theory for decoupling lowand high momenta because it also provides straightforward means to track induced many-body forces (117–120), and construct consistently transformed observables (121–123). • Non-Empirical Interactions for the Nuclear Shell Model 5 .3. Current Status of the ab initio Shell Model
Over the past decade, the adaptation of EFT and SRG methods has greatly extendedthe reach of ab initio nuclear many-body theory across the nuclear chart. Simply put,the most convenient scale for formulating a theory of nuclear interactions is often not themost convenient scale for solving that theory. The SRG connects one scale to the otherand greatly improves the convergence behavior of nuclear many-body calculations in theprocess. Large-scale diagonalization methods like the NCSM can be used in the lower sd − shell (63, 120, 124, 125), and systematically truncated methods like Self-ConsistentGreen’s Functions, Coupled Cluster (CC) and the In-Medium SRG (IMSRG) can even beapplied to nuclei as heavy as tin (126–129). While SRG-evolved interactions cannot beused easily in QMC due to their nonlocality, new families of local chiral interactions yieldencouraging results in such applications (60, 87, 88, 90).Soon after their introduction to nuclear physics, EFT and RG methods also revitalizedefforts to systematically derive shell model interactions (130, 131). From a practical per-spective, this offered a convenient way to confront RG-evolved chiral two- plus three-nucleoninteractions with the wealth of available spectroscopic data, using existing shell model codes(132–144). At the conceptual level, these interactions validate the independent-particle pic-ture underlying the shell model. They provide sufficient binding already at the mean-fieldlevel, and allow us to use it as the starting point for the treatment of correlations, eitherthrough rapidly converging non-perturbative many-body methods (127, 145, 146), or possi-bly even through finite-order perturbation theory (116, 147). Furthermore, novel approacheslike the valence-space IMSRG (VS-IMSRG) or shell model Coupled Cluster (SMCC), bothdiscussed below, provide both the conceptual framework and practical tools to relate no-coreand valence-space methods, as shown by the consistent ground- and excited-state resultsobtained thus far (see (129, 143) and Sec. 5). Thus, the end of the long and winding roadto ab initio shell model interactions appears to be in sight, though challenges remain, seeSecs. 6 and 8. This work is organized as follows: In Sec. 2, we introduce common approaches to the con-struction of shell model interactions, from the traditional many-body perturbation theoryand the Okubo-Lee-Suzuki method to the valence-space IMSRG and shell model CC. InSec. 3, these approaches are compared within a common formalism to illuminate the rela-tions between them. Section 4 discusses the role of three-nucleon forces in the shell modelcontext, and relates modifications of (semi-)empirical interactions that are supposed to cap-ture such effects to the more systematic treatment of these forces in modern approaches.In Sec. 5, we highlight selected applications of ab initio shell model interactions. Section 6describes the main challenges we are facing today, and analyzes them primarily from theperspective of the VS-IMSRG. New developments like a direct EFT expansion for shellmodel interactions and a novel uncertainty quantification effort are touched upon in Sec. 7.Section 8 provides concluding remarks alongside a list of take-away messages that sum-marize the key aspects of modern ab initio shell model calculations, and clarify commonmisconceptions. Certain technical details are collected in the appendices. . Microscopic effective interactions
The general problem of effective interaction theory is the following: Given a Hamiltonian H expressed in a large (typically intractable) Hilbert space H , we wish to obtain an effectiveHamiltonian H eff which acts in a smaller (tractable) Hilbert space H model , but reproducesa subset of the eigenstates of the large Hilbert space. H | Ψ n (cid:105) = E n | Ψ n (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) Full space Schr¨odinger eq. ⇒ H eff | ψ n (cid:105) = E n | ψ n (cid:105) (cid:124) (cid:123)(cid:122) (cid:125) Model space Schr¨odinger eq. (1)In the context of the nuclear shell model, the large Hilbert space will consist of Slaterdeterminants of single-particle states, typically harmonic oscillator eigenstates. The numberof single-particle states included should be sufficient to obtain convergence. The smallerHilbert space H model is defined by splitting the single-particle states into three categories—core, valence, and excluded — taking the subset of Slater determinants for which all coreorbits are occupied, and all excluded orbits are unoccupied.We note in passing that different partitionings of the Hilbert space can be used forother purposes. For example, choosing H model to be a one-dimensional space correspondsto the single-reference many-body methods for treating the ground state of closed-shellnuclei. Alternatively, defining H model in terms of low-momentum states leads to methodsfor “softening” an interaction, such as V low k (99) or the similarity renormalization group(SRG) (100).An important practical requirement on H eff is that it should obey a rapidly convergingcluster expansion, schematically | V N | (cid:29) | V N | (cid:29) | V N | . . . (2)where V N is the two-body potential, V N is the three-body potential, etc. and the verticalbars indicate some measure of size or importance. This property is essential to the feasibilityof large-scale shell model diagonalizations. Nowadays, such calculations can handle basisdimensions upwards of 10 Slater determinants (18, 20, 148), which would be impossible ifthe full matrix needed to be stored. The limitation to two-body (or possibly three-body)interactions yields a sparse matrix which can be treated efficiently by, e.g., the Lanczos orDavidson methods (149–151).It is worth taking a moment here to motivate why we would expect the cluster expansionto be valid in nuclei, and to consider where it might run into trouble. As mentioned insection 1.2.1, chiral effective field theory naturally generates a hierarchy of the type shownin Eq. (2), with many-body interactions suppressed by increasing powers of the ratio oflow to high scales. On the other hand, for a system of A nucleons, the importance of an n -body term grows combinatorially, accounting for all the different combinations of n particleswhich can interact, suggesting that for heavy nuclei, many-body forces will dominate. Forlarge A , this grows as A ! /n !( A − n )! ∼ A n .Fortunately, we are saved by the short range of the nuclear force and the relatively lowsaturation density of nuclear matter (152). Roughly, each nucleon does not interact withall other nucleons, but instead only with the other nucleons within some interaction volume V ∼ π r with r int the range of the interaction. At density ρ , the expectation value of In chemistry, valence orbits are usually called active states, while excluded orbits are referredto as virtual states. • Non-Empirical Interactions for the Nuclear Shell Model 7 n n -body force will scale as (cid:104) V n (cid:105) ∼ ( ρ V ) n − . With ρ (cid:46) .
16 fm − , the cluster hierarchywill be maintained so long as r int (cid:46) r π ∼ . ρ , but the quasiparticle density which will typically be significantly smaller (152).When we derive an effective interaction, we are eliminating degrees of freedom, namelyorbits outside of the valence space. For high-lying orbits, the relevant interaction matrix ele-ment will be dominated by high-momentum (short-distance) components, in which case theabove argument holds and the induced terms should still exhibit a cluster hierarchy (153).However, for excitations near the fermi surface we have no short-distance argument. Indeed,as we discuss in section 6.1, low-lying collective excitations can be a source of trouble.It is important to keep in mind that the above argument only holds if the observable inquestion can be expressed in terms of connected diagrams (i.e. one can trace a continuouspath through the diagram between any two points on it). If a four-body term consists oftwo disconnected two-body terms, then there is no reason why all four particles would needto be within some interaction volume. We return to this point in Sec. 3.2.2.In the remainder of this section, we describe the most popular approaches to derivingeffective interactions for the nuclear shell model, using the notation that appears in thenuclear physics literature. In section 3, we treat these methods within a more generalframework to illuminate the relationships between them. ˆ Q -box resummation Let us introduce the projection operator P and its complement Q such that P H P = H model and P + Q = 1. If we assume | ψ n (cid:105) = P | Ψ n (cid:105) , that is, that the eigenstates of the effectiveHamiltonian are just the projection of the full eigenstates onto the model space, then theeffective Hamiltonian should satisfy P H eff P | Ψ n (cid:105) = E n P | Ψ n (cid:105) , QH eff P = 0 . (3)Straightforward manipulation then yields the Bloch-Horowitz energy-dependent effectiveHamiltonian (154, 155) H BH ( E n ) = P HP + P HQ E n − QHQ QHP. (4)One important aspect of Eq. (4) is that the effective interaction depends on the eigenvalue E n , and so it must be solved self-consistently. A second point is that different valence-space eigenstates will in general not be orthogonal, because they are eigenstates of differentoperators. The energy dependence can be removed by expanding the denominator aboutsome starting energy E , yielding (138, 156) H eff = H BH ( E ) + ∞ (cid:88) k =1 k ! (cid:20) d k dE k H BH ( E ) (cid:21) ( H eff − E ) k . (5) he expression (5) may also be obtained in the context of time-dependent perturbationtheory (24), or by a similarity transformation combined with an iterative solution (157) forthe decoupling condition (3).A simplification may be obtained by partitioning the Hamiltonian into a zero-orderpiece and a perturbation H = H + V , and assuming that the eigenvalues of H in thevalence space are degenerate, with energy (cid:15) . Then one uses Eq. (5) with E = (cid:15) , and( H eff − (cid:15) ) = V eff . In this context, a popular approach is to define the “ ˆ Q -box” (24, 49, 50),indicated ˆ Q ( (cid:15) ). ˆ Q ( (cid:15) ) = P V P + P V Q (cid:15) − QHQ QV P. (6)The operator ˆ Q ( (cid:15) ) is not to be confused with the projection operator Q . The effectivevalence space interaction is, in analogy to Eq. (5), V eff = ˆ Q ( (cid:15) ) + ∞ (cid:88) k =1 k ! d k ˆ Q ( (cid:15) ) d(cid:15) k ( V eff ) k . (7)The commonly adopted strategy for evaluating Eq. (7) is to expand the inverse operatorin Eq. (6) perturbatively (typically to second or third order in V ) and to solve Eq. (7)self-consistently by iteration, evaluating the derivatives numerically by calculating ˆ Q ( (cid:15) ) forseveral starting energies (cid:15) .A technical point arises because the ˆ Q -box contains one-body pieces, arising from e.g.core polarization diagrams (5–7). For computational convenience, the one-body part isembedded in the two-body part, with an accompanying spectator nucleon. This leads todisconnected two-body terms in Eq. (7) which contain arbitrary numbers of interactions in-volving one-particle, but no interactions between the two. These disconnected contributionscan be understood as the dressed one-body part of the effective interaction embedded intoa two-body interaction. They can be removed by solving Eq. (7) using only the one-bodypiece of the ˆ Q -box, which is called the ˆ S -box (158, 159). The resulting effective one-bodyinteraction is subtracted off from V eff , leaving only connected terms. In principle these samediagrams should then be added self-consistently to the degenerate single-particle energies ε i = (cid:104) i | H | i (cid:105) + ˆ S i ( ε i ). In practice, it is often discarded and the single particle energies aretaken from experiment . As an exception to this, when three-body forces are included inthe normal ordering approximation (see Secs. 2.3.1 and 4), and the single-particle energiesare computed so that the starting energy (cid:15) corresponds to the centroid of the valence-spacesingle-particle energies, no additional adjustments are required (136, 137). In general, one would like to be able to usenon-degenerate valence orbits, for example when using a Hartree-Fock basis, or when gen-erating an interaction for two (or more) major harmonic oscillator shells. Then one shoulduse Eqs. (4) and (5) rather than Eqs. (6) and (7). This approach has only recently beenexplored (138, 144, 160, 161). In the case of a valence space spanned by two major os-cillator shells, taking harmonic oscillator single-particle energies can lead to zero energydenominators in Eq. (7), while the parameter E in Eq. (5) may be chosen to avoid zero Determining the experimental single-particle energies is ambiguous, as one must choose whichexperimental state is “the” shell model one, or consider an average of multiple states (see, e.g.,(10, 56)) • Non-Empirical Interactions for the Nuclear Shell Model 9 enominators. Of course, for a P space with a very large spread in single-particle energies,one would expect that the size of certain contributions to ( H eff − E ) k in Eq. (5) wouldbe comparable to the corresponding energy denominators, and the Taylor series (5) mightconverge slowly, if at all. However, for a modest spread of energies ( ∼
10 MeV), this doesnot appear to be a problem (138, 144).
Another approach to the effective interaction is called the Okubo-Lee-Suzuki approach,often employed in conjunction with the NCSM (62, 63, 162), the paradoxical-sounding“NCSM with a core” or Double OLS approach (163–165), or more recently, the coupledcluster method (140, 166) (“coupled cluster effective interaction” (CCEI)). The idea is toobtain a unitary transformation U which diagonalizes H in the large Hilbert space H , sothat U H U † = E (8)where E is a diagonal matrix. The effective Hamiltonian H eff acting in the smaller Hilbertspace H model = P H P is given by (163) H eff = U † P (cid:113) U † P U P E U P (cid:113) U † P U P (9)where U P ≡ P U P is the projection of the transformation U onto the model space. One caneasily confirm that the transformation in Eq. (9) is unitary, and so the eigenvalues of H eff in the model space will be a subset of the eigenvalues of H in the full space.So far, this does not appear to be a helpful procedure, because the first step was tosolve the eigenvalue problem in the large Hilbert space, and the goal of effective interactiontheory is to allow applications for which the full solution is not tractable. The benefit comeswhen one assumes that the effective interaction H eff will also provide a good approximationfor other systems described in the same model space. Assuming a cluster expansion, oneobtains H eff for a few active particles in the model space and then applies H eff to systemswith more active particles.As a concrete example, consider the system of two particles in the no-core shell model.This can readily be diagonalized in a large space of many harmonic oscillator states, say N ≤ N max = 500, where N = 2 n + (cid:96) is the number of oscillator quanta. On the otherhand, even a light nucleus like Li with only six particles cannot possibly be diagonalizedin such a space. Using the OLS transformation (9) one can obtain an effective interactionfor a manageable model space (say N max = 10) which exactly reproduces the low-lyingeigenvalues of the two-body calculation in the large space. The application of this effectiveinteraction to Li then provides a reasonable approximation to the eigenvalues one wouldobtain in the large space.We note that there are essentially two main assumptions here: (i) the effective interac-tion one would obtain if one could apply the OLS procedure directly to the six-body systemhas a rapidly convergent cluster expansion | V N | > | V N | > | V N | . . . , and (ii) the two-body-cluster component V N of the full effective interaction H eff for the six-body system is wellapproximated by the effective interaction obtained for the two-body system. While both Perhaps this would more appropriately be called the Okubo-Lee-Suzuki-Okamoto approach.
10 Stroberg, Hergert, Bogner, and Holt
QPQ P QPQ P QPQ
U U † P (a) (b) (c) Figure 1: A schematic of how the OLS approach obtains the effective interaction. (a)The original Hamiltonian. (b) The Hamiltonian is diagonalized by transformation U . (c)The transformation U † P approximately inverts U in the P space and yields the effectiveinteraction H eff .of these assumptions are plausible and encouraging results have been obtained using them,we know of no rigorous proof. Indeed, as we discuss briefly in Sec. 3.2.2, there is potentialcause for concern related to disconnected diagrams. In the in-medium similarity renormalization group (IMSRG)(139, 143, 146, 167, 168), theeffective Hamiltonian is also expressed in terms of a unitary transformation U acting on theinitial Hamiltonian H eff = UHU † . (10)In contrast to the OLS approach, the IMSRG transformation is obtained without solvingthe eigenvalue problem for a particular many-body system. Instead, it is parameterized bya continuous flow parameter s , and applied to the Hamiltonian through the flow equation dH ( s ) ds = [ η ( s ) , H ( s )] , (11)where the generator η ( s ) is formally defined as η ( s ) ≡ dU ( s ) ds U † ( s ) = − η † ( s ) . (12)We split the flowing Hamiltonian H ( s ) into two pieces, a “diagonal” and an “off-diagonal”piece H ( s ) = H d ( s ) + H od ( s ) (13)such that H od ( s ) = P H ( s ) Q + QH ( s ) P. (14)where the projection operators P and Q have the same meaning as in the previous sections.Our goal is then to devise a generator η ( s ) such thatlim s →∞ H od ( s ) = 0 (15)and therefore lim s →∞ H d ( s ) = H eff . (16) • Non-Empirical Interactions for the Nuclear Shell Model 11 n the language of the renormalization group, H eff is a fixed point of the RG flow.One choice for η ( s ), which is used in the calculations we will describe here is the Whitegenerator (145, 169) η Wh ( s ) ≡ H od ( s )∆( s ) . (17)For present and future use, we have introduced a convenient superoperator notation(cf. (170)), in which we indicate division of the operator O by a suitably defined energydenominator ∆ is defined as (cid:104) φ i | O ∆ | φ j (cid:105) ≡ (cid:104) φ i | O | φ j (cid:105) (cid:15) i − (cid:15) j (18)which can be thought of as element-wise division. Here (cid:15) i , (cid:15) j are energies associated withthe basis states φ i , φ j . The quantity O ∆ itself is an operator whose Fock-space expression is O ∆ = (cid:88) ij O ij (cid:15) i − (cid:15) j a † i a j + 14 (cid:88) ijkl O ijkl (cid:15) i + (cid:15) j − (cid:15) k − (cid:15) l a † i a † j a l a k + . . . (19)Returning to the flow equation, it is clear that if H od →
0, then η → dH ( s ) ds →
0, so H eff is indeed a fixed point of the flow. One potential issuewith the generator (17) is that a vanishing energy denominator will cause η to diverge. Analternative, also suggested by White (169) (see also (171)), is η atan ( s ) ≡
12 atan (cid:18) H od ( s )∆( s ) (cid:19) . (20)The arctangent—motivated by the solution of a 2 × P QPQ P QPQ P QPQ dHds dHds (a) (b) (c)
Figure 2: A schematic representing of how the IMSRG approach obtains the effective in-teraction H eff by progressively suppressing the off-diagonal terms of H . (a) s = 0, (b) s = 5,(c) s = 30The IMSRG is formulated in terms of Fock-space operators, and so its computationalcost scales polynomially with the basis size N , but not explicitly with the number of particlesbeing treated. In practical applications, we truncate all operators at a consistent particlerank to close the system of flow equations arising from Eq. (11) (see Appendix A). Wealso set up the decoupling conditions to be minimally invasive to avoid an uncontrolled
12 Stroberg, Hergert, Bogner, and Holt ccumulation of truncation errors, as discussed in detail in Ref. (145). In VS-IMSRG weperform the decoupling in two stages for this reason, decoupling the reference state fromexcitations as in a direct ground-state calculation before decoupling the valence space in asecond evolution (cf. Secs. 5 and 6).
An important feature of the IMSRG method is the use of operatorsin normal-ordered form (see, e.g., (145, 146)). Starting with the free-space Hamiltonianwritten as a Fock-space operator with two- and three-body interactions H = (cid:88) ij t ij a † i a j + (cid:88) ijkl V ijkl a † i a † j a l a k + (cid:88) ijklmn V ijklmn a † i a † j a † k a n a m a l , (21)we may use Wick’s theorem to express the strings of creation and annihilation operators innormal order with respect to some reference state | Φ (cid:105) (172). We denote the normal orderingwith braces, and the normal order of a pair of operators is defined so that their expectationin the reference is zero, i.e. (cid:104) Φ |{ a † i a j }| Φ (cid:105) = 0 . (22)Whether the normal order is a † i a j or a j a † i depends on whether or not the states created andannihilated are present in the reference | Φ (cid:105) . If we choose | Φ (cid:105) to be a single Slater determi-nant such as the Hartree-Fock ground state of the system of interest, then application ofWick’s theorem allows us to write H as H = E + (cid:88) ij f ij { a † i a j } + (cid:88) ijkl Γ ijkl { a † i a † j a l a k } + (cid:88) ijklmn W ijklmn { a † i a † j a † k a n a m a l } (23)where the new coefficients can be obtained from the old coefficents by E = (cid:88) a n a t aa + (cid:88) ab n a n b V abab + (cid:88) abc n a n b n c V abcabc f ij = t ij + (cid:88) a n a V iaja + (cid:88) ab n a n b V iabjab Γ ijkl = V ijkl + (cid:88) a n a V ijakla W ijklmn = V ijklmn . (24)Operators other than the Hamiltonian can be rewritten in the same way.In Eq. (24), n a is the occupation of orbit a in the reference, i.e. n a = (cid:104) Φ | a † a a a | Φ (cid:105) , andfor a Slater determinant reference, n a is either zero or one. In section 4.1, we discuss adifferent choice of reference for which a can have fractional occupation. One may also usea correlated reference, constructed out of a linear combination of Slater determinants, inwhich case one must use the generalized normal ordering of Kutzelnigg and Mukherjee (173).This is the basis of the multireference IMSRG (MR-IMSRG) method that will be used inground-state energy comparisons in later sections (125, 146).The advantage of expressing operators in normal ordered form is that it puts as muchinformation as possible from the high particle-rank (i.e. many-body) operators into the In actual calcualtions, one subtracts off the center-of-mass kinetic energy, and so the kineticterm has a two-body piece (146). We neglect that here for simplicity. • Non-Empirical Interactions for the Nuclear Shell Model 13 ower rank operators. This is evident in Eq. (24), where the normal-ordered zero-body term E contains contributions from the free one-, two- and three-body terms. If the reference | Φ (cid:105) is a good approximation of the exact wave function | Ψ (cid:105) , then the expectation value (cid:104) Ψ |{ a † a † a † aaa }| Ψ (cid:105) ≈
0, and even formally non-vanishing 3N interactions W ijklmn can beneglected to a good approximation. Consequently, normal ordering may be thought of as away to improve the convergence of the cluster expansion described in the beginning of thissection. A particularly convenient formulation of the IMSRG approachrelies on the Magnus expansion (174, 175). The idea is to express the more general unitaryIMSRG transformation as the true exponential of the anti-Hermitian Magnus operatorΩ( s ) = − Ω † ( s ). The evolved Hamiltonian can then be expressed in terms of an infiniteseries of nested commutators H ( s ) = e Ω( s ) H (0) e − Ω( s ) = H (0) + [Ω( s ) , H (0)] + 12 [Ω( s ) , [Ω( s ) , H (0)]] + . . . (25)This formulation of the IMSRG allows for a more transparent comparison with canonicaltransformation theory (169, 176) and unitary coupled cluster method (177) used in quantumchemistry, as well as canonical perturbation theory (178), where the expansion in Eq. (25)is evaluated perturbatively.Considering the flow equation (11), we see that under an infinitesimal step ds we maywrite H ( s + ds ) = H ( s ) + [ η ( s ) , H ( s )] ds = e η ( s ) ds H ( s ) e − η ( s ) ds = e η ( s ) ds e Ω( s ) H (0) e − Ω( s ) e − η ( s ) ds . (26)Expressing H ( s + ds ) in Magnus form as well, we obtain an expression for Ω( s + ds ): e Ω( s + ds ) = e η ( s ) ds e Ω( s ) . (27)Because Ω( s ) and η ( s ) do not in general commute, we use the Baker-Campbell-Hausdorffformula to take the logarithm on both sides and obtainΩ( s + ds ) = Ω( s ) + η ( s ) ds + [ η ( s ) , Ω( s )] ds + [Ω( s ) , [Ω( s ) , η ( s )]] ds + . . . (28)This may be expressed in compact form as (suppressing explicit s dependence) d Ω ds = ∞ (cid:88) k =0 B k k ! ad ( k )Ω ( η ) (29)where B k are the Bernoulli numbers, and the adjoint ad ( k )Ω ( η ) signifies a recursively definednested commutator: ad ( k )Ω ( η ) = [Ω , ad ( k − ( η )] , ad (0)Ω ( η ) = η. (30)Fortunately, in most practical applications, only the first few terms of the infinite seriesin Eqs. (25) and (28) are important, and so the commutator may be evaluated iterativelyuntil the size of a given term is below some numerical threshold (for an example of anexception to this, see Sec. 6.2). A major practical advantage of the Magnus method is thatby solving for Ω( s ), we can compute arbitrary effective operators besides the Hamiltonianin a consistent and efficient way (see Sec. 6.1 and Refs. (123, 174)).
14 Stroberg, Hergert, Bogner, and Holt .4. Shell model coupled cluster
P QPQ P QPQ P QPQ ¯¯ H ( t ) ¯¯ H ( t ) (a) (b) (c) Figure 3: A schematic depicting how the SMCC approach obtains the effective interaction¯¯ H ( t ). Note that, in contrast to the H eff of Fig. 2, ¯¯ H is non-Hermitian. (a) t = 0, (b) t = 5,(c) t = 30First attempts to derive an effective shell model interaction with coupled cluster methodswere similar in spirit to the NCSM-based Double OLS approach (see Sec. 2.2 and Refs. (162–165)). Equation-of-motion CC (EOM-CC) states defined in a space of up to 4p2h excitationsare subsequently projected into the shell model space via the OLS method, yielding the so-called coupled cluster effective interaction (CCEI) (140, 166). The cost of the EOM-CCcalculations, however, presented a significant obstacle to widespread applications of thismethod. A much more efficient alternative is the recently introduced shell model coupledcluster (SMCC) method (179), which is formulated in Fock space and can be viewed as anon-unitary cousin of the IMSRG.In the standard coupled cluster method (25, 127), a similarity transformation is per-formed to decouple a single closed-shell reference state | Φ (cid:105) from all particle-hole excitations e − T He T | Φ (cid:105) = E corr e T | Φ (cid:105) (31)where E corr is the correlation energy and T is the cluster operator which is written as T = (cid:88) ia t ai { a † a a i } + (cid:88) abij t abij { a † a a † b a j a i } + . . . (32)Here the indices a, b, c . . . denote unoccupied orbits and i, j, k . . . denote occupied orbits.The similarity-transformed Hamiltonian is written as¯ H = e − T He T . (33)Shell model coupled cluster extends the idea by performing a similarity transformationwhich decouples a valence space rather than a single configuration. Denoting this transfor-mation with an S , we have ¯¯ H = e − S ¯ H e S , Q ¯¯ H P = 0 . (34)The operator S is obtained by a flow equation closely mirroring the one used in IMSRG dSdt = − η ( ¯¯ H ( t )) (35) • Non-Empirical Interactions for the Nuclear Shell Model 15 here η is the generator of the flow. As in the IMSRG, there is considerable freedom forchoosing η as long as the decoupling condition (34) is realized in the limit t → ∞ . InRef. (179), adapted variants of the White (Eq. (17)) and arctangent generators (Eq. (20))are used. The essential difference from the IMSRG formulation is that T and S are notanti-Hermitian operators, and so the transformation is not unitary, and the resulting effec-tive Hamiltonian is not Hermitian. The inconvenience of a non-Hermitian Hamiltonian iscompensated by the greater simplicity of the equations which need to be solved.
3. Comparison of various approaches to effective interactions
To investigate how the methods described in section 2 (and a few others) are related toone another, we consider the general structure of effective interactions, and show howthe above methods sum the perturbation series. For more details, we refer readers toRefs. (170, 178, 180–183).
We begin by expressing the effective Hamiltonian in termsof a similarity transformation of the original Hamiltonian, parametrized as the exponentialof a generator G H eff = e G He −G = H + [ G , H ] + [ G , [ G , H ]] + . . . (36)and the decoupling condition QH eff P = 0 . (37)We partition the original Hamiltonian into an exactly-solvable zero-order part H and aperturbation V H = H + V (38)and consider an expansion of the generator G and the interaction H eff in powers of V G = G [1] + G [2] + G [3] + . . . , H eff = H [0]eff + H [1]eff + H [2]eff + . . . (39)For convenience, we define the partial sum of the series up to order n as G [ n ] ≡ n (cid:88) m =1 G [ m ] . (40)The n th order contribution to the effective Hamiltonian is then H [ n ]eff = (cid:16) e G [ n ] He − G [ n ] (cid:17) [ n ] . (41)One can easily verify that G [ n ] only contributes to a single term in Eq. (41). Peeling thisterm off we have H [ n ]eff = [ G [ n ] , H ] + (cid:16) e G [ n − He − G [ n − (cid:17) [ n ] . (42)Enforcing the decoupling condition (37) yields an equation for G [ n ] in terms of lower-ordercontributions Q [ H , G [ n ] ] P = Q (cid:16) e G [ n − He − G [ n − (cid:17) [ n ] P. (43)
16 Stroberg, Hergert, Bogner, and Holt quation (43) is of the general form [ H , X ] = Y . (44)If we work in the eigenbasis of H such that H | φ i (cid:105) = (cid:15) i | φ i (cid:105) , the commutator can be easilyevaluated in terms of the unperturbed energies: (cid:104) φ i | [ H , X ] | φ j (cid:105) = ( (cid:15) i − (cid:15) j ) (cid:104) φ j | X | φ j (cid:105) ≡ ∆ ij (cid:104) φ j | X | φ j (cid:105) . (45)This suggests that a solution to Eq. (44) can be written as X = Y ∆ + Z , (46)where Z is some arbitrary function which commutes with H , and the superoperator nota-tion introduced in Eq. (19) is used for brevity.Following this line of reasoning we solve Eq. (43) as Q G [ n ] P = Q (cid:16) e G [ n − He − G [ n − (cid:17) [ n ] ∆ P. (47)As we can see, the decoupling condition applies to the Q G P block of the generator G , andwe have some freedom to choose the rest of G , namely P G P , P G Q , Q G Q . The variouschoices, which we will outline below, result in different effective Hamiltonians. There are afew important consequences of these choices.First, in order for the transformation to be unitary, we need the generator to be anti-Hermitian: G = −G † . Consequently, the popular choice P G Q = 0 will result in a non-unitary transformation, and a non-Hermitian effective Hamiltonian. All else being equal, aHermitian effective Hamiltonian is preferable, but the significant simplifications that comewith taking P G Q = 0 can make this choice attractive. c o r e v a l e n cee x c l ud e d c o r e v a l e n cee x c l ud e d Figure 4: A schematic illustration showing that the same operator (indicated by the arrow)can connect
Left: a P space configuration to a Q space configuration as well as Right: a Q space configuration to a Q space configuration.Second, the choice Q G Q = 0 cannot be enforced in a Fock-space formulation, and so thischoice can only be made when working directly in the A -body Hilbert space. To understand The additional term like Z in Eq. (46), which commutes with H , will vanish when sandwichedbetween Q and P , so we need not include that term here. • Non-Empirical Interactions for the Nuclear Shell Model 17 his point consider a one-body Fock-space operator which excites a particle from thevalence space to the excluded space. As illustrated in Fig. 4, if this operator acts on aconfiguration which belongs to the P space, it will generate a configuration which belongs tothe Q space. However, if that same operator acts on a Q -space configuration which alreadyhas some other particle-hole excitation, it will generate a distinct Q -space configuration.So the operator also connects Q -space configurations to Q -space configurations. In orderto enforce Q G Q = 0 while allowing Q G P (cid:54) = 0, the operator G needs to ensure that it actsonly on P -space states, which means G must be an A -body operator.Third, if we choose both P G P = 0 and P G Q = 0, then we have P e G = P (1 + G + G + . . . ) = P. (48)If we denote the eigenstate of the full Hamiltonian by | Ψ i (cid:105) and the corresponding eigenstateof the effective Hamiltonian | ψ i (cid:105) , these are related by the similarity transformation | Ψ i (cid:105) = e −G | ψ i (cid:105) , (49)and since | ψ i (cid:105) exists entirely in the P space, Eq. (48) implies that | ψ i (cid:105) = P | Ψ i (cid:105) . (50)Physically, the eigenstate of the effective Hamiltonian is just given by the projection ofthe full-space wave function to the P space. This means that | Ψ i (cid:105) and | ψ i (cid:105) cannot besimultaneously normalized to one, and we must employ an “intermediate normalization” (cid:104) Ψ i | ψ i (cid:105) = 1 . On the other hand, if we do not require P G Q = 0, then the above argument nolonger holds and the eigenstates of the effective Hamiltoninan will in general not be simplyprojections of the full-space eigenstates.Finally, before investigating various choices of G , we consider an iterative method forsumming the perturbative series to all orders. To do this, we notice that selecting outthe n th order contribution on the right-hand side of Eq. (47) quickly leads to complicatedformulas (see, e.g., section III of Ref. (170)). Overall, the right-hand side of Eq. (47) isof order n and higher, since the lower order terms in G have been selected to eliminatethe undesired components of H eff to their respective orders. This means that if we do notspecifically select the n th order terms on the right hand side, but instead take everything,we obtain a contribution G [ n ] which suppresses the n th order term in H eff , as well as somecontribution from higher order terms. These higher order terms will be suppressed duringlater iterations. Because this is no longer a strict order-by-order perturbative expansion, weuse a subscript to denote the n th iteration, to distinguish it from the superscript indicatingthe n th-order contribution: Q G n P = Q e G n − He − G n − ∆ P. (51)Here, we have defined G n ≡ (cid:80) nm =1 G m in analogy with Eq. (40). Defining the transformedHamiltonian after n iterations as H n ≡ e G n He − G n , this may be written as Q G n P = Q H n − ∆ P. (52) To our knowledge, this point was first made in the chemistry literature by Kutzelnigg (180, 181),but it has not been explicitly stated in the nuclear physics literature.
18 Stroberg, Hergert, Bogner, and Holt r Q G n P = Q G n − P + Q H n − ∆ P. (53)Beginning with G = 0 and iterating Eq. (53) successively eliminates contributions to QH eff P of increasing powers of the perturbation V , and for n → ∞ yields the exact gener-ator: G ∞ = G .In the following subsections, we consider choices found in the literature for fully speci-fying G , and the consequences of these choices. We begin with the most restrictive combination Q G Q = P G P = P G Q =0, which allows for the greatest simplification. Following the notation of Suzuki andLee (157), for this choice we write the generator as G = − ω . The effective Hamiltonian is H eff = e − ω He ω (54)and we have the great simplification that ω = 0 so that e ω = 1+ ω . As we remarked above,this choice will yield a non-Hermitian effective Hamiltonian which is necessarily formulatedin the A -body Hilbert space. The decoupling condition is QH eff P = (1 − ω ) H (1 + ω )= QV P + QHωP − QωH P − QωV P − QωV ωP = 0 , (55)where in the second line we have used H = H + V . If we take a degenerate P space withenergy (cid:15) so that QωP H P = (cid:15)QωP , we may rearrage to solve for ωω = 1 (cid:15) − QHQ ( QV P − ω ( P V P + P V Qω )) . (56)As shown by Suzuki and Lee (157), defining H eff = P H P + P V P + P V Qω (57)and iteratively inserting Eq. (56) into itself yields the Q -box folded diagram expansion (7)or, with a different iteration scheme, R n = 11 − ˆ Q − n − (cid:80) m =2 ˆ Q m n − (cid:81) k = n − m +1 R k ˆ Q (58)where V eff = R ∞ , ˆ Q is the same ˆ Q -box defined in Eq. (6), and ˆ Q m ≡ d m d(cid:15) m ( ˆ Q ( (cid:15) ).As we have pointed out in section 3.1.1, the requirement QωQ = 0 implies that thisapproach must be formulated in the A -body space. However, as we see next, a Fock-spaceformulation yields the same effective interaction. If we drop the requirement Q G Q = 0, we have P G Q = P G P = 0. Following convention, we express the transformation in terms of the Møller waveoperator and its inverse Ω = e −G , Ω − = e G . (59) We use Ω for the Møller wave operator for consistency with existing literature in the presentsubsection, but caution that it should not be confused with the Magnus operator used in the IMSRG. • Non-Empirical Interactions for the Nuclear Shell Model 19 oting that Q Ω − (1 − Ω P ) = Q Ω − , we can satisfy the decoupling condition (37) if(1 − Ω P ) H Ω P = 0 . (60)Again, splitting up H = H + V and rearranging, this may be written as Q [Ω , H ] P = QV Ω P − Q Ω P V Ω P (61)which is the generalized Bloch equation (184).The effective interaction in the P space is given by P H eff P = P H Ω P = P H P + P V Ω P. (62)Expanding Eqs. (61) and (62) in powers of V yields a linked expansion for the Rayleigh-Schr¨odinger perturbation series (23, 156). In the nuclear case, the order-by-order conver-gence of this series is questionable (cf. Sec. 1).The second term on the right hand side of Eq. (61) can be represented by “folded”diagrams (23, 156). The ˆ Q -box method described in section 2.1 amounts to a perturbativeexpansion of the first term in Eq. (61), followed by a summation of certain higher-order termsin the folded diagram series. We note that if we express the wave operator as Ω = P + χ ,where χ is called the “correlation operator”, then Eq. (61) is equivalent to Eq. (55) with χ = ω . Interestingly, while the Lee-Suzuki approach required QωQ = 0, and therefore couldnot be expressed in a Fock-space formulation, the Bloch equation approach does not usethat constraint and so may be formulated in Fock-space. Evidently, the Bloch equationapproach does not make any reference to Q G Q , and so setting it to zero does not alter theresulting effective Hamiltonian. For a discussion of the differences between the Lee-Suzukischeme and the ˆ Q -box approach, see e.g. Ref. (28). Instead of working with the wave operator, we maydirectly work with the generator G . Following the notation of Ref. (179), we write G as − S .We employ the iterative procedure laid out in section 3.1.1 to obtain an iterative expressionfor S : S n = S n − − Q H n − ∆ P (63)where we have denoted the similarity-transformed Hamiltonian at the n th step H n = e − S n He S n . The effective Hamiltonian is then H eff = H ∞ . In arriving at Eq. (63), we haveimplicitly assumed that the terms in QSQ are only those which also contribute to
QSP .Consequently, if a term in S only connects Q configurations, then it is taken to be zero.Often, the iterations are better behaved with the help of a convergence factor, whichwe denote dt to connect with the formulation of Ref. (179). Multiplying the second term ofEq. (63) and taking the limit dt →
0, we can reinterpret it as a flow equation: dSdt = − η ( t ) ≡ − Q H ( t )∆ P. (64)Taking the P space to be a single Slater determinant, Eq. (63) yields an iteration scheme tosolve the coupled cluster equations, while defining the P space in terms of a valence space In practical applications, the procedure is applied to the CC Hamiltonian (33), (c.f. Sec. 2.4)
20 Stroberg, Hergert, Bogner, and Holt ields the shell model coupled cluster approach (179) described in section 3.1.4. As discussedabove, the requirement
P SQ = 0 means that H eff is not Hermitian in this approach.Additionally, we see that the SMCC effective interaction is equivalent to the other twonon-Hermitian effective interactions discussed in sections 3.1.2 and 3.1.3, as long as noapproximations are made. We next drop the condition Q G P = 0, enabling usto enforce G † = −G so that the transformation is unitary and H eff is Hermitian. Weretain the condition Q G Q = 0, with the consequence that we cannot express the theory interms of Fock-space operators. However, this restriction greatly simplifies the analysis. Forconsistency with the literature, we write the generator as G = − G . It can be shown (178,185) that the operator G is related to the operator ω from section 3.1.2 by G = arctanh (cid:16) ω − ω † (cid:17) (65)and the transformation is (185) e G = (1 + ω − ω † )(1 + ωω † + ω † ω ) − / . (66)The effective Hamiltonian obtained in this approach is the Hermitized version of the effec-tive Hamiltonian resulting from Eq. (55). To connect this result with the OLS approachpresented in section 2.2, we write that transformation out explicitly: H eff = U † P (cid:113) U † P U P U H U † U P (cid:113) U † P U P . (67)By inserting a sum over the eigenstates of H , and using | Ψ i (cid:105) = (1 + ω ) P | Ψ i (cid:105) , one canshow (162, 163) that U † U P = (1 + ω ) P , and the OLS effective interaction is equivalent to P e − G He G P .An iterative scheme very similar to the one described in section 3.1.1 was proposedby Suzuki (171), but not directly pursued further. The unitary model operator approach(UMOA) (185) follows this formalism, with the valence cluster expansion (see section 3.2.2)carried out on the generator, rather than on the effective Hamiltonian. So far, studies withthe UMOA have focused on ground state energies of closed-shell nuclei (186), so we will notdiscuss it further here. If we desire a Hermitian effective operatorwith a Fock-space decomposition, then we should drop the restriction Q G Q = 0, leavingonly P G P = 0. Writing out Eq. (47) order-by-order with the requirement G † = −G yieldsthe canonical perturbation theory of Primas (187) and Klein (170). Interestingly (181), theresulting expansion is different from the expansion obtained in section 3.1.5, i.e. the Fock-space and A -body space formulations are not equivalent, in contrast to what was found forthe non-Hermitian formulation. This approach has not been pursued in the nuclear physicsliterature. Here, the hyperbolic arctangent of an operator is defined in terms of its Taylor series expan-sion (178). • Non-Empirical Interactions for the Nuclear Shell Model 21 .1.7. Unitary coupled cluster.
Alternatively, we may take P G P = 0 and G † = −G andfollow the iterative procedure of section 3.1.1 to obtain G n = G n − + Q H n − ∆ P + P H n − ∆ Q. (68)This yields a unitary coupled cluster expansion for the effective interaction. As with theSMCC solution, this may be recast as a differential equation (here we use s instead of t ) d G ( s ) ds = Q H ( s )∆ P + P H ( s )∆ Q. (69)This approach has also not been persued in nuclear physics, although it is very closelyrelated to the Magnus formulation of the IMSRG, as we will show below. Finally, we may drop the constraint P G P = 0 and instead specify G by the requirement that it should reproduce the flowing Hamiltonian H ( s ) along its entiretrajectory. Following Ref. (174), we write G = Ω( s ), which we call the Magnus operator,and require e Ω( s ) He − Ω( s ) = H ( s ) (70)where H ( s ) is the solution of the flow equation (11). This may be reorganized as a flowequation for the Magnus operator Ω( s ), as described in section 2.3.2. Considering the firstfew terms in the series, we have d Ω( s ) ds = η ( s ) −
12 [Ω( s ) , η ( s )] + . . . (71)If we choose the White generator (17), which may be written as η Wh ( s ) = Q H ( s )∆ P + P H ( s )∆ Q, (72)and neglect all terms on the right hand side of Eq. (71) aside from the first one, we recoverthe unitary coupled cluster equation (69). The difference between Magnus IMSRG andunitary coupled cluster then lies in the commutator terms of Eq. (71). A perturbativeanalysis reveals that the leading-order effect of the first commutator term is to inducecontributions to P Ω P and Q Ω Q at third order. If the transformation is evaluated exactly,these terms of course have no effect on the resulting observables. However, if approximationsare made—as they inevitably must be—then these terms may produce a different result.This has not yet been investigated in detail.Integrating Eq. (71) numerically with a step size ds = 1 —again neglecting all but thefirst term— we find, following the discussion leading to Eq. (53), that the first integrationstep yields a generator which satisfies the decoupling condition to first order in perturbationtheory (cf. Appendix C). Likewise, the second step in ds satisfies the decoupling condition tosecond order, and the n th step satisfies decoupling to n th order. Thus, numerical integrationof the flow equation with step size ds = 1 corresponds to an order-by-order summation ofthe perturbation series. If instead we take a smaller step size, ds = 0 .
5, then after the firstintegration step we will have only suppressed half of the first-order term in the decouplingcondition. After the second integration step, taking us to s = 1, we have suppressed halfof the remaining first order term, as well as half of the second order term. Taking the
22 Stroberg, Hergert, Bogner, and Holt ontinuous limit ds →
0, we find the first-order off-diagonal piece suppressed as e − s , withthe higher-order terms also suppressed at the same rate.In light of this discussion, we can view the numerical integration of the flow equation(71) with some finite step-size ds as a summation of the perturbative expansion (145, 146),with the step size specifying anything from an order-by-order summation ( ds = 1) to allorders at once ( ds → An exact evaluation of the formulas for H eff presented in the preceding subsections willinevitably be at least as expensive as a direct diagonalization of the Hamiltonian in the fullHilbert space—precisely the task we set out to avoid. The utility of the effective interactionframework is that it facilitates approximations which greatly reduce the required effortwhile minimally impacting the accuracy of the computed quantities of interest, namelyobservables related to low-lying eigenstates. Within the shell-model context, this impliesthe need for some sort of cluster truncation.Perhaps the most straightforward approximation scheme is a truncation in in powers ofthe residual interaction V , i.e. perturbation theory. Unfortunately, in nuclear physics theeffective interaction often converges slowly in powers of V , and as discussed in section 6.2,the intruder-state problem suggests that in most cases the perturbation series is divergent.Nonperturbative truncation schemes have been made essentially along two lines: eithera cluster truncation is imposed within a Fock-space formulation, as for IMSRG and CC(cf. Secs. 2.3.2 and 3.1.4), or the problem is solved directly in the A -body system for a fewvalence particles, followed by a cluster expansion. When working in a Fock-space formulation, it is nat-ural to perform a cluster truncation on the generator G or on all operators, typically limitingthem to consist of zero-, one- and two-body pieces. From a practical point of view, sucha truncation is a necessity; keeping three-body terms is unpleasant but feasible, while theneed for e.g. six-body terms would be sufficiently onerous to render the method useless.For the special case of a one-dimensional P space (i.e. a single-reference calculation),using the non-hermitian formulation of section 3.1.4, truncating G to one- and two-bodyoperators is equivalent to coupled cluster with singles and doubles (CCSD) (25). Here,we find the desirable feature that the Baker-Campbell-Hausdorff expansion (36) formallytruncates after a finite number of nested commutators (four in the CCSD approximation,if H has at most two-body terms).For the case of interest in the context of the shell model, with the dimension of the P space greater than one, the Baker-Campbell-Hausdorff expansion does not formally trun-cate (179). One approach to this issue is to truncate the series at a finite order of perturba-tion theory, or else at a finite power of G (see, e.g. (177)). Alternatively, one can specify aform for the Fock-space operators, e.g. retaining one- and two-body terms while discardingthe rest, allowing the series to be evaluated iteratively (169, 175). While the series remainsinfinite with this truncation, in most cases of interest it is found that the series convergesso that for a given precision only a finite number of nested commutators must be evaluated.Importantly, this truncation scheme retains only connected diagrams, and so maintains sizeextensivity. This approach is used in the VS-IMSRG and SMCC described above, and in • Non-Empirical Interactions for the Nuclear Shell Model 23 he canonical transformation theory of Yanai et al (176, 188).When operators are normal ordered with respect to a finite-density reference, many-body operators can feed back into fewer-body operators through the commutators in theBaker-Campbell-Hausdorff expansion. However, the reduction in particle-rank of an op-erator always comes with an occupation number (see the flow equation in Appendix A),corresponding to a factor of the density, and so the discussion in Sec. 2 about the clusterhierarchy justifies this truncation.
The other approximation scheme is to work within the A -body Hilbert space formulation and build up the effective interaction in order of increasingcluster rank (163). One diagonalizes the A core , A core + 1 , A core + 2 systems successively, andextracts the consistent core energy, single-particle energies, and two-body matrix elementsby subtracting the contributions of lower particle rank. One could continue this procedureto obtain higher-body effective interaction, with rapidly increasing effort. Instead, assumingthat the effective interaction has a sufficiently convergent cluster expansion, the effectiveinteraction obtained in the zero-, one-, and two-valence-particle systems can then be appliedto systems with more valence particles. For self-bound systems like nuclei, one must takecare to properly treat the mass-dependence of the intrinsic kinetic energy in the constructionprocedure for the effective interaction (163, 165, 166), although this effect becomes lessimportant for heavier systems.This scheme is used in the Okubo-Lee-Suzuki approaches based on coupled cluster(CCEI) (140, 166) and the no-core shell model (162, 163, 165), as well as the ˆ Q -box ap-proach of Kuo and collaborators (26, 30, 158). One potential drawback of the valence clusterexpansion is that the optimal cluster decomposition for two valence particles might be sig-nificantly different from the optimal decomposition for many valence particles. Consideringthe sd shell as a specific example, the appropriate mean field for an empty valence space,corresponding to O, will be quite different from the appropriate mean field for a filledvalence space, corresponding to Ca, and so one would expect that different single-particleenergies would be optimal.Another potentially more serious issue arises from a perturbative analysis. As wasdiscussed in Sec. 3.1.1, the requirement Q G Q = 0 cannot be enforced in a Fock-spaceformulation. If it is enforced in the A -body formulation, an analysis reveals that discon-nected diagrams arise (181, 189). For example, in a system with four valence particlesand a purely two-body interaction, a disconnected four-body contribution to the effectiveinteraction arises at fourth order. Such a contribution — essentially two-body interactionsbetween two independent pairs — is not subject to the arguments at the beginning of Sec. 2about short-range interactions at low density, because it does not depend on the separationbetween the clusters. We may expect such terms to be combinatorially enhanced, whichwould be a serious problem — this calls for further investigation.The above point may provide some explanation for the surprising finding in Ref. (166),where a single interaction for the sd shell obtained with the CCEI method produced im-pressive agreement with experimental binding energies throughout the shell. While at firstglance such a result is cause for celebration, a closer look suggests trouble. As a specific ex-ample, other ab initio calculations (126, 143, 190, 191) (including coupled cluster) using thesame input interaction find Ca to be over-bound by nearly 40 MeV, with relatively smallvariation among the calculations, while the CCEI result is under-bound by just 2 MeV. Thesupplemental material of Ref. (166) makes note of this, since it is unreasonable to expect
24 Stroberg, Hergert, Bogner, and Holt hat the CCEI method should be more accurate than the coupled cluster method uponwhich it is based. In the present context, we may speculate that the combination of (i) the over-binding inherent in the input force, (ii) missing valence many-body effects (seeSec. 4.2) enhanced by the effect of disconnected diagrams, and (iii) truncation errors inhigh-lying eigenvalues from the equation-of-motion coupled cluster method (192) inciden-tally conspire to cancel out in the sd shell. However, one can and should not rely on sucha cancellation in general.
4. Three-body forces and the connection with phenomenological adjustments
Shortly after Yukawa’s formulation of the nuclear interaction in terms of pion exchange, itwas pointed out (193) that a description of a quantum field theory in terms of an instan-taneous (or, equivalently, energy-independent) potential inevitably leads to three-body andhigher-body forces. The connection between these many-body forces and nuclear satura-tion was also suspected early on (70, 194, 195), although the calculations were necessarilyschematic. Likewise, mean-field calculations using a Skyrme (196, 197) or Gogny (198)parameterization of the force require a three-body, or density-dependent two-body, term.Figure 5: Three-body forces generated from
Left: elimination of the ∆ isobar degree offreedom, and
Right: elmination of an excitation to a Q -space configuration.Of course, even if the initial interaction were solely of a two-body nature, the effectiveinteraction in the valence space will still in general contain three-body and higher-bodyforces. In fact, these “effective” or “induced” 3N forces and the “genuine” 3N forces areessentially of the same origin: the elimination of degrees of freedom. This is illustrated inFig. 5, where the elmination of the ∆ isobar degree of freedom and the elimination of anexcitation toa Q -space configuration both lead to effective 3N interactions. Some previousstudies have found the induced and “genuine” 3N forces are of comparable magnitude (199,200), although this is depends on the renormalization scheme and scale of the interaction.As a practical matter—due to the ambiguity in producing a three-body term consistentwith the two-body interaction, as well as the difficulty in handling a three-body term in amany-body calculation—explicit three-body forces have historically been neglected in shellmodel treatments, although there were some exploratory calculations (see e.g. (199–203)).There have also been more recent calculations evaluating 3N forces in the valence spaceeither perturbatively (204) or explicitly (179). Zuker and collaborators (18, 205) argued thatthe main effect of the three-body forces should be to modify the monopole (i.e. diagonal, J -averaged) component of the effective interaction. This argument simultaneously justifiedthe omission of explicit three-body terms as well as the phenomenological adjustment ofmonopole terms in the effective interaction, which resulted in excellent reproduction ofthe experimental data. Further supporting this point of view were (i) the fact that the • Non-Empirical Interactions for the Nuclear Shell Model 25 arious realistic NN interactions produced similar shell model matrix elements , allowinglittle room for improvement, and (ii) the observed improvement in spectroscopy of lightnuclei obained of quantum Monte Carlo and no-core shell model calculations when explicitthree-body forces were included (207, 208).An important demonstration of the effect of three-body forces in the shell model was acalculation showing that three-body forces could help explain the location of the neutrondripline in oxygen (132), followed by an explanation of the N = 28 magic number inthe calcium isotopes (134). These calculations used a normal-ordering approximation (seeSec. 2.3.1) for the three-body force and obtained essentially the monopole effect describedby Zuker, although they used empirical single-particle energies and scaled the two-bodymatrix elements by A / as in phenomonological calculations. The same effect was soonconfirmed in ab initio calculations without phenomenological adjustments (125, 209, 210).The first VS-IMSRG calculations of the oxygen isotopes did not obtain the correctdripline (139), even though three-body forces were included in the normal-ordered approx-imation. More troubling, the heavier oxygen isotopes were systematically overbound byapproximately 10 MeV. The issue was that the normal ordering of the Hamiltonian usedthe core wave function as a reference state in these initial VS-IMSRG calculations, hencethe effects of three-body interactions between valence nucleons were not properly captured.This deficiency was remedied by the use of ensemble normal ordering (ENO) (143), whichenables an approximate treatment of the effect of three-body forces that does not degradeas valence particles are added. This echoes the results of previous investigations of theeffects of three-body forces in the shell model (29, 199, 200, 211). Since only a brief accountof ENO has been given in the literature (143), we provide a more detailed description inthe following section. When performing a VS-IMSRG calculation, a natural choice for the normal ordering ref-erence | Φ (cid:105) is the core of the valence space. This allows an approximate treatment of 3Nforces in which a sum over particles in the core yields effective one-body and two-bodyforces in the valence space. However, VS-IMSRG calculations of the oxygen isotopic chainusing chiral NN+3N forces overpredicted the binding energy of neutron-rich oxygen nucleicompared to an earlier MR-IMSRG study with the same interactions (125, 139). Calcula-tions involving both protons and neutrons in the valence space yielded even more significantoverbinding (142).This discrepancy was essentially due to the fact that the normal ordering in the MR-IMSRG calculation is performed directly with respect to the system of interest, not withrespect to the core of the valence space. This meant that, in O for example, the MR-IMSRG was better capturing the 3N interactions between the 8 valence neutrons. Indeed,taking the normal ordering reference to be the nearest closed-shell nucleus brought theVS-IMSRG binding energies back in line with the MR-IMSRG results (142).This approach was then generalized to treat systems that are not close to any closedsub-shell, by allowing fractional occupation numbers. As an example, consider O, whichin a naive shell model picture has three neutrons in the 0 d / orbit on top of a closed O This observation is easily understood from the RG/EFT point of view—the various potentialsdiffer in their high-momentum content but reproduce the same low-momentum physics (206).
26 Stroberg, Hergert, Bogner, and Holt o r e v a l e n c ee x c l u d e d decoupledecouple (b) c o r e v a l e n c ee x c l u d e d decoupledecouple (c)
12 16 20 24 28 A − − − − − − − E g s ( M e V ) A O (Z=8)
ExperimentSCGFGGFCCSD(T)CRCCIT-NCSMMR-IMSRGVS-IMSRG no ENOVS-IMSRG +ENO (a)
Figure 6: (a) The binding energy as a function of mass number A for the oxygen isotopes,calculated with a variety of many-body methods(125, 126, 140, 212, 213). The blue curvelabeled IMSRG(SM) corresponds to the scheme in (b) with the core taken as the normal-ordering reference, while the red curve labeled IMSRG(ENO) corresponds to (c) with anensemble reference. Adapted from (143).core. Equivalently, it could be considered as three neutron holes in O. When using an Oreference, the occupation number for the neutron 0 d / orbit would be 0, while it wouldbe 1 for an O reference. The O reference will underestimate the missing three-bodyeffects, while the O reference will overestimate them. The compromise is then to takethe occupation to be 0.5, i.e. filling the orbit half way. This strategy is frequently used inmean-field theory, and it is known as the “equal-filling approximation” or simply the “fillingapproximation”. (For an application in chemistry, see (188)).The question then arises: what reference state (if any) is actually being used whenwe select fractional occupation numbers? As explained in Ref. (214, 215) the equal fillingapproximation can be framed in terms of a mixed state or ensemble, in the sense of quantumstatistical mechanics, specified by a density matrix ρ = (cid:88) α c α | Φ α (cid:105)(cid:104) Φ α | (73)for some set of coefficients c α . (Here α labels different Slater determinants). The expectationvalue of an operator O in the ensemble is obtained by a trace over the density matrix: (cid:104)O(cid:105) = Tr [ O ρ ] = (cid:80) α c α (cid:104) Φ α |O| Φ α (cid:105) .As discussed in Sec. 2.3, for a single reference | Φ (cid:105) , the normal order of a pair of creationand annihilation operators is the one which gives zero expectation value in the reference. Not to be confused with the one-body density matrix ρ pq = (cid:104) Ψ | a † p a q | Ψ (cid:105) . • Non-Empirical Interactions for the Nuclear Shell Model 27 he normal order of a string of more than two creation/annihilation operators can be chosenso that every pair of operators in the string is in normal order.Wick’s theorem (172), which expresses a string of creation/annihilation operators interms of normal-ordered strings and contractions, was extended to more general referencestates by Kutzelnigg and Mukherjee (173). In this case, the normal ordering is still definedso that the reference expectation value of a normal ordered string of creation/annihilationoperators vanishes. However, the concept of a contraction now involves one-body, two-body, and higher-body density matrices, which encode the correlations contained in thereference. This generalized Wick’s theorem is used, for instance, to formulate the MR-IMSRG (125, 146).As shown by Kutzelnigg and Mukherjee, Wick’s theorem also applies to a mixed-state,or ensemble, reference: (cid:104){ a † p a q }(cid:105) = Tr (cid:104) { a † p a q } ρ (cid:105) = (cid:88) α c α (cid:104) Φ α |{ a † p a q }| Φ α (cid:105) = 0 . (74)This is an extension of the original formulation of the finite-temperature Wick’s theo-rem (216–218), which only applied to the expectation value of an operator in the ensemble.The formulation of Kutzelnigg and Mukherjee, on the other hand, is an operator identity,just like the original zero-temperature Wick’s theorem.Our goal is then to find an ensemble such that contractions have the same form as inthe single-reference case, except that the occupation of an orbit may have some non-integervalue between 0 and 1. That is, we want all two-body and higher-body irreducible densitymatrices, as well as the off-diagonal one-body density matrix, to vanish. Such an ensemblemust necessarily contain a variable number of particles. This may be easily understood byconsidering a single particle placed in two levels. If we require that there always be a fixedtotal number of particles (like in canonical or micro-canonical ensembles, for instance),then the occupation of one level implies that the other level must be empty and so theoccupations are correlated, leading to an irreducible two-body density matrix. A grandcanonical ensemble, on the other hand will meet our needs.The ensemble that has been used in VS-IMSRG calculations published thus far corre-sponds to the zero-temperature limit of a finite-temperature Hartree-Fock calculation (218),with the chemical potential chosen to fix the average particle number. To illustrate the ap-plication of Wick’s theorem, we consider a single level with degeneracy d = 2 j + 1, so aconfiguration has N particles with 0 ≤ N ≤ j + 1. The contraction of two operators isgiven by a † p a q = (cid:104) a † p a q (cid:105) = δ pq Z j +1 (cid:88) N =0 (cid:32) j N − (cid:33) e β ( (cid:15) − µ ) N = δ pq Z j +1 (cid:88) N =0 (cid:32) j + 1 N (cid:33) N j + 1 e β ( (cid:15) − µ ) N = δ pq j + 1 (cid:104)N (cid:105) = δ pq n p . (75)Here β is the inverse temperature, µ is the chemical potential, (cid:15) is the energy of the level, Z is the partition function, and the binomial coefficients count how many of the configurationswith N particles will have orbit p occupied. Of course, other ensembles may be selected,and they need not be thermal ensembles, i.e. multiple levels could be fractionally occupied
28 Stroberg, Hergert, Bogner, and Holt nd there need not be a connection between the energy of a level and its occupation (215).Another very reasonable choice of occupations is to use natural orbitals, or a perturbativeapproximation of them (147).In actual calculations, this ensemble need not be explicitly constructed; we only usethe corresponding definition of the contraction when we use Wick’s theorem. In fact, thereis another reference which can be constructed to produce the same fractional occupations.Instead of employing an ensemble state, we may use a single-determinant reference builtfrom a single-particle basis which is slightly different from the one used in the calculation.To fractionally fill an orbit p , we admix in some other “inert” orbit Q which is orthogonalto all of the single-particle states used in our calculation (cid:32) | p (cid:105)|Q(cid:105) (cid:33) → (cid:32) | ¯ p (cid:105)| ¯ Q(cid:105) (cid:33) = (cid:32) √ n √ − n −√ − n √ n (cid:33) (cid:32) | p (cid:105)|Q(cid:105) (cid:33) (76)where 0 ≤ n ≤
1. If we choose a reference in which the orbit ¯ p is filled, i.e. (cid:104) Φ | a † ¯ p a ¯ p | Φ (cid:105) = 1,then the occupation in terms of the original orbit p is (cid:104) Φ | a † p a p | Φ (cid:105) = n . Because the referenceΦ is a single Slater determinant, all higher-body density matrices vanish automatically. Inaddition, there will be a non-zero occupation of the inert orbit Q : (cid:104) Φ | a †Q a Q | Φ (cid:105) = (1 − n ), aswell as off-diagonal one-body densities (cid:104) Φ | a † p a Q | Φ (cid:105) = (cid:112) n (1 − n ), which are not desirable.However, we have asserted that the orbit Q is inert. By this we mean that a †Q and a Q do notappear in any operator we consider, and we may neglect terms involving orbit Q withoutchanging the physics. While the introduction of inert orbits might seem contrived, it is nomore contrived than the ensemble with variable particle number. Indeed, we could say thatthe inert orbits live somewhere in the reservoir that supplies the additional particles.Again, for practical purposes, it is irrelevant whether the reference is an ensemble or isconstructed with an inert orbit mixed in. What matters is that we may use Wick’s theoremwith fractional occupation numbers, and that this procedure constitutes an exact rewritingof our operators—given an operator which is normal ordered with fractional occupations,we can reconstruct the operator normal-ordered with respect to the true vacuum. Theimportance of this point is that by employing fractionally-filled orbitals we have not intro-duced an additional approximation. If we retain all the induced operators up to A -bodyoperators, then the IMSRG calculation is exact. What the fractional filling does is reducethe impact discarding the residual three-body terms has on the low-lying states. Becausestandard shell model codes typically work with valence particles (not valence holes), afterthe IMSRG decoupling, we again use Wick’s theorem to rewrite all operators in normalorder with respect to the core (which is a single Slater determinant) .Certainly, an uncorrelated ensemble reference is a crude approximation of the exactwave function, and one might envision that a correlated reference state as used in the MR-IMSRG could do better (125, 146). However, as argued in Ref. (188), it is not clear thatthis is the best way to proceed in a valence space context. If correlation effects are includedin the reference in order to better approximate a particular state, then this might welldeteriorate the description of other low-lying states (which contain different correlations),leading to a worse overall description of the spectroscopy. This “re-normal-ordering” is easily achieved by using (24), replacing n a → ( n new a − n old a ). • Non-Empirical Interactions for the Nuclear Shell Model 29 .2. Mass dependence of the effective interaction
A significant consequence of the ensemble normal ordering (ENO) procedure is that adifferent valence space interaction is obtained for each nucleus. It is important to emphasizehere that because the procedure does not involve any fitting to data, there is no loss ofpredictive power . The ENO should be considered as a technique for reducing the impact ofthe truncation to two-body operators. In terms of computational effort, the need to generatea new interaction for each nucleus makes a study of the full sd shell more laborious, butstill manageable. For nuclei in middle of the pf shell, the exponential scaling of the valencespace diagonalization catches up with the polynomial scaling of the VS-IMSRG and sogenerating the effective interaction takes about as long as the shell model calculation thatuses it.The need for some mass-dependence of the effective interaction has been known fora long time. The sd shell interactions of Kuo and collaborators (6, 7) yielded a gooddescription of spectroscopy for a few valence particles or valence holes, but agreement de-teriorated for mid-shell systems (219). Investigations by Chung and Wildenthal (10, 52)suggested that a single phenomenological adjustment could not remedy the situation, anda scaling of the two-body matrix elements according to A . was introduced. This pre-scription has been adopted in many later treatments (11, 30, 134). The scaling is typicallyjustified in terms of the increasing nuclear radius changing the optimal harmonic oscilla-tor frequency (10, 11, 56, 144). While such an argument would suggest that the core andsingle-particle energies should also change with mass, these effects could in principle beabsorbed into the scaling of the two-body matrix elements (10). On the other hand, theneed for mass-dependence of two-body matrix elements could be interpreted as a signalof non-negligible three-body terms in the effective interaction, and indeed this has beensuggested a number of times (199, 200, 211).We may expect that ensemble normal ordering should capture both the effects of achanging mean field and of the residual three-body effective interaction . Fig. 7 displaysbinding energies per nucleon obtained for oxygen isotopes and N = Z nuclei in the sd -shell nuclei with the USDB interaction, both with and without the mass scaling of theTBMEs. These are compared to the binding energies of the same nuclei calculated usingthe VS-IMSRG with and without ensemble normal ordering. It is evident that the ensemblenormal ordering has qualitatively the same effect as the scaling of the TBMEs, athough aninvestigation of the VS-IMSRG TBMEs reveals no such smooth scaling (the effect is largelycaptured in the core and single-particle energies).Figure 8 shows the single-particle energies and monopoles of two-body matrix elements(TBMEs) obtained for a Si reference with and without explicit 3N forces. Including the3N force has a significant impact on the single-particle energies; indeed, the neutron SPEsare shifted closer to the USD (57) values (USD does not include the Coulomb interaction).In panel (b) of Fig. 8, we see that the effect of the 3N interaction on the TBME monopoles isrepulsive, as expected from binding energy calculations, and that they shift the monopolestowards the USDB values. In panel (c) of Fig. 8 we show the difference between each ofthe two-body matrix elements obtained with the NN only and NN+3N interactions (blackcrosses) and the difference when the NN only monopoles have been shifted to the NN+3N Indeed, these effects are not entirely distinct; the induced three-body interaction depends onthe choice of reference.
30 Stroberg, Hergert, Bogner, and Holt A − . − . − . − . − . − . E / A ( M e V ) N = Z nuclei in sd shell VS-IMSRGUSDBVS-IMSRG, no ENOUSDB, no A . scaling
16 20 24 28 A − . − . − . − . − . E / A ( M e V ) Oxygen isotopes
Figure 7: Energy per nucleon for
Left the oxygen isotopes 16 ≤ A ≤
28, and
Right the N = Z nuclei in the sd shell, obtained with VS-IMSRG using the EM1.8/2.0 interactioncompared with the results obtained with the USDB interaction. The thinner lines indicatethe effect of turning off the ensemble normal ordering (ENO) in the VS-IMSRG calculation,or turning off the A . scaling of two-body matrix elements in the USDB interaction. ( d d ) ( d d ) ( d s ) ( d d ) ( d s ) ( s s ) V pn (b) ( d d ) ( d d ) ( d s ) ( d d ) ( d s ) ( s s ) − − − − M o n o p o l e ( M e V ) V nn NN onlyNN+3NUSDB protons neutrons − − − − S i n g l e p a r t i c l ee n e r g y ( M e V ) d / d / s / NNonly NN+3N NNonly NN+3N
USDB (a)
TBME (arbitrary order) − − V NN + N − V NN (c) no V mon corr.with V mon corr. Figure 8: (a) Single particle energies in the sd shell obtained with the VS-IMSRG normalordered with respect to a Si reference, using an NN interaction, with or without the 3Npiece. (b)Neutron-neutron and proton-neutron monopoles of the two-body matrix elements,with and without the 3N force. (c) The difference between matrix elements obtained withNN only and NN+3N, with and without a monopole correction. All calculations use theEM 1.8/2.0 interaction of Ref. (220).values. The monopole shift does not yield perfect agreement—there is still some scatter inthe red markers—but the remaining discrepancy is approximately Gaussian and centeredon zero. It is not unreasonable that there would be moderate cancellation between the • Non-Empirical Interactions for the Nuclear Shell Model 31 emaining terms, and that the monopole correction would approximately account for themissing 3N forces, as claimed by Zuker et al. (205). For a related approach using densityfunctionals to inform the monopole correction, see Ref. (221).Based on our discussion, we can conclude that the “standard” phenomenological ad-justments made to shell model interactions can be understood essentially in terms of theeffect of missing (normal-ordered) three-body forces (see e.g. the discussion in section 8 ofRef. (156), as well as Refs. (199, 200, 211, 222)). • Historically, the core energy was taken from experiment (typically no effort was madeto calculate it consistently from the input force), and modern ab initio calculationshave confirmed the importance of three-body forces to binding energies (190, 209,223). • Likewise, single-particle energies were typically taken from experiment, as the onesobtained from the NN interaction did not reproduce the spectra of one-valence-particlesystems. The normal-ordered contribution of three-body forces to the single-particleenergies essentially accounts for this discrepancy. • Even with the core and single-particle energies taken from experiment, realistic NNforces typically did not give good spectroscopy, and needed phenomenological adjust-ment. Zuker (205) argued that the most important adjustment was of the two-bodymonopoles, and that this shift should be understood in terms of missing three-bodyforces. Indeed, when the three-body contribution to the normal-ordered two-bodyinteraction is taken into account, no phenomenological shifts are needed. Moreover,as shown in Figure 8, the bulk of the discrepancy between an interaction derivedfrom only NN forces and one including 3N effects can be corrected by a shift ofthe monopoles. The remaining discrepancy (the “multipole” terms) is approximatelyGaussian and centered on zero, so that the net effect will be in general small. • Finally, the ∼ A / scaling of two-body matrix elements can be understood as a wayto capture the bulk effects of three-body forces among valence particles. This samephysics is captured by employing ensemble normal ordering (see also (204, 224)).
5. Applications
As discussed previously, shell model calculations with ab initio interactions allow us to con-front our starting point, the underlying chiral two- plus three-nucleon force of our choice,with a wealth of available experimental data. Until recently, applications have been pri-marily focused on ground- and excited state energies, with very encouraging results. In thefollowing, we will highlight selected examples. sd − Shell Nuclei
In Sec. 4, we discussed the importance of chiral three-nucleon forces for the correct de-scription of nuclear shell structure. Their effect on the location of the oxygen dripline wasone of the first high-profile applications of ab initio interactions in the nuclear shell model(132), which has subsequently been confirmed in more consistent calculations with bothvalence-space and no-core methods (cf. Fig. 6).Multiple studies for sd -shell nuclei with a progressively more consistent perturbativeconstruction of the valence-space interaction followed (134–136, 141, 204), including firstworks for the derivation of multi-shell interactions (138, 144), until the non-perturbative VS-
32 Stroberg, Hergert, Bogner, and Holt
FNeNa Mg Al Si P S Cl ArKCa − − D e v i a t i o n f r o m e x p t . ( M e V ) O ref, ¯ hω =16 MeVrms=1696 keVEM 1.8/2.0 Bare OFNeNa Mg Al Si P S Cl ArKCa rms=647 keVEM 1.8/2.0+ENO+VS-IMSRG
OFNeNa Mg Al Si P S Cl ArKCa rms=220 keV
391 excitationenergies in sd shell USDB
Figure 9: Deviation from experiment for excited states throughout the sd shell, obtainedwith Left: the EM 1.8/2.0 interaction without transformation,
Middle:
EM 1.8/2.0 trans-formed with the VS-IMSRG using ensemble normal ordering,
Right: the USDB interaction.IMSRG and CCEI/SMCC emerged (139, 140, 142, 143, 179). The VS-IMSRG, in particular,has been widely used to compute ground and excited-state energies (129, 142, 143, 225–238), although theoretical uncertainties stemming from the method still prove challenging(see Sec. 6).In Fig. 9, we show results from a VS-IMSRG survey of 391 levels in the sd -shell, startingfrom the EM1.8/2.0 chiral two- plus three-nucleon interaction (220). The points indicatethe deviation between the computed and experimental energies for all of these levels, whichcontribute to the specified cumulative root-mean-squared deviation between theory andexperiment. In the left panel of Fig. 9, we have simply used the “bare” matrix elements ofthe EM1.8/2.0 interaction in the sd -shell valence space, while the center panel shows theresults obtained by applying the VS-IMSRG with ensemble normal ordering, as describedin Secs. 2 and 4.1. Since our starting interaction has been evolved to a low resolution scale,correlations due to the strong short-range repulsion and the tensor force have largely beenaccounted for. Thus, the shell model picture is reasonable: Low-lying nuclear states arebound and excitation energies are at least of the correct order of magnitude, with a sizablerms deviation of 1696 keV.The deviations from experiment are reduced significantly when we use the VS-IMSRG todecouple the sd -shell valence space from other excitations, accounting for core polarizationand other types of long-range, many-body correlations (see Sec. 2). With an rms deviationof 647 keV, we are not doing as well as the gold-standard USDB interaction for whichthe deviation is merely 220 keV for the selected levels (and only ∼
130 keV for all 600+measured sd -shell levels). This is not unexpected: USDB is essentially the best possiblefit to experimental data under the chosen model assumptions, including the choice of avalence space containing only the 1 s / , d / and 0 d / orbitals, the mass-dependenceof the two-body matrix elements, and the omission of residual three- and higher-bodyeffective interactions. The accuracy of the VS-IMSRG results is subject to the uncertaintiesof the input interaction and the truncation used in the method. Both can and will besystematically improved in future applications. • Non-Empirical Interactions for the Nuclear Shell Model 33 hereMðNÞshows theatomic mass of thenneutrons. This quantity is known as the oparameter of second difference [20]. It is rthe Δ atoddN canbeassociatedwiththepaWenoteherethedifferencebetween δ eantwo-neutron shell gap Δ ≡ S ðNÞ − S which is frequently used to demonstratevolution in nuclei. The Δ shell gap clthe δ e through the relation Δ ¼2½ δ e −Δ ðN þ 1Þþ Δ ðNwhere N is an even number Hence the Δ )() (a)(b)
24 26 28 30 32 34-8-6-4-202468 V (+3.5 MeV) Ti Sc (-3.5 MeV) Δ n [ M e V ] Neutron Number Ca (-7.0 MeV) TITAN
Figure 10:
Left:
Two-neutron separation energies of neutron-rich calcium isotopes from re-cent measurements at RIKEN, compared to VS-IMSRG and MBPT results obtained withthe EM1.8/2.0 interaction, as well as results for phenomenological interactions. Adaptedfrom (235).
Right:
VS-IMSRG (solid lines) results for three-point energy differences inthe calcium isotopes and neighboring chains, compared to both AME data and new tita-nium measurements at TITAN. VS-IMSRG results used the EM1.8/2.0 interaction, whileGor’kov Green’s Function results for the scandium chain (dashed line) use a different chiralinteraction. Reprinted from (233). See original references for additional details.
Soon after the successful application of perturbatively constructed shell model interactionsin the sd -shell, first results for the calcium isotopes followed, including a successful predic-tion of the two-neutron separation energies at the sub-shell closure in Ca (133, 137, 239).More recently, the masses of − Ca were measured at RIKEN, showing the onset of a flattrend in the separation energies beyond Ca that would be consistent with the filling ofthe neutron 0 f / shell (see Fig. 10). Such a trend had also been found in MR-IMSRG andGor’kov Greens Function (GGF) ground-state calculations using chiral interactions (212),although absolute two-neutron separation values ( S n ) could not be determined preciselybecause of theoretical uncertainties in the interactions and the many-body methods.Recent high-precision mass measurements of the titanium isotopes at TITAN aimedto shed new light on the evolution of the N = 32 shell closure (233). The right panel ofFig. 10 shows three-point energy differences ∆ n ≡ S n ( N, Z ) − S n ( N + 2 , Z ) extractedfrom the new data alongside AME data, in comparison with results from the VS-IMSRGfor isotopic chains in the lower pf -shell and GGF calculations for the scandium chain.While the theoretical ∆ n compare favorably with experimental data overall, the strengthof the N = 32 closure is overestimated with increasing Z . This artificial enhancement ofshell closures is frequently observed in calculations with current chiral interactions (see,e.g., (191, 212, 240)), and might provide important clues toward the refinement of next-generation forces. For sufficiently soft interactions, IMSRG and CC calculations for nuclei in the upper pf andlower sdg shells can be converged (126, 128, 129). The limiting factor is a truncation in thethree-body matrix elements e + e + e ≤ E where e = 2 n + (cid:96) . Memory constraints have
34 Stroberg, Hergert, Bogner, and Holt
Mass Number A + E n e r gy ( M e V ) Ni Figure 11: Energies of the first excited 2 + states in Ni isotopes from VS-IMSRG andCoupled Cluster calculations including triples corrections (128), using EM1.8/2.0 and otherchiral two- plus three-nucleon interactions as input (see (220) for details).restricted calculations to E ≤
18. The dimension of the valence space also becomes anissue during the diagonalization of the effective interaction, but approaches like the MonteCarlo shell model (241) or importance-truncated configuration interaction (IT-CI) (242)can be used to tackle this problem.In Figure 11, we show the evolution of the first excited 2 + state in neutron-rich nickelisotopes, which serves as a strong indicator for (sub-)shell closures. The jump in the 2 + energy at Ni suggests that this nucleus is indeed doubly magic. The VS-IMSRG repro-duces the available experimental data (243) well, and the energies are insensitive under(admittedly small) variations of the interaction’s resolution scales or low-energy constants— see (220) for more details on these Hamiltonians.Recently, Hagen et al. also computed the 2 + (128) energies of , Ni using the Equation-of-Motion Coupled Cluster approach. In Fig. 11, we include their excitation energies fromthe so-called EOM-CCSD(T) method, which are about 1 MeV lower than the VS-IMSRGresults with the corresponding interactions. This difference can be traced back to the effectsof triples (i.e., 3p3h) correlations and continuum effects that are currently not included inthe VS-IMSRG.Moving to even heavier nuclei, the structure of the lightest tin isotopes was the subject ofa recent joint EOM-CC and VS-IMSRG study (129). Figure 12 shows results for the energygap between the two lowest-lying states in light odd-mass tin isotopes and
Te. Theno-core EOM-CC and the VS-IMSRG results for
Sn are consistent, and the VS-IMSRGproduces a near-degeneracy of the J π = 5 / + and J π = 7 / + states that is compatiblewith experiment, the systematic uncertainties of the method must be properly quantified— and, most likely, reduced — before one can make spin assignments with confidence. • Non-Empirical Interactions for the Nuclear Shell Model 35 Sn Sn Sn Sn Sn Sn0.60.40.20.00.2 E / + E / + [ M e V ] CC 1.8/2.0(EM)CC 2.0/2.0(EM)VS-IMSRG 1.8/2.0(EM)VS-IMSRG 2.0/2.0(EM)ExpExp − − − E [ M e V ] / + Sn / + Sn / − − − − / + Sn / + Sn / T max − − − − E [ M e V ] / + Te / + Te T max − − − − / + Te / + Te Figure 12: VS-IMSRG and EOM-CC results for the ground and first excited states of odd-mass tin isotopes and
Te, using chiral NN+3N interactions (220). The right panel showsthe convergence of the states as a function of the model space truncation in the IT-CIdiagonalization. Figure adapted from Ref. (129).
6. Current challenges
While great strides have been made in deriving effective interactions for the shell model,challenges remain. Here we focus on two in particular, and analyze them from the perspec-tive of the VS-IMSRG.First, electric quadrupole (E2) observables which are sensitive to low-lying collectiveexcitations and which historically have been treated phenomenologically by introducingeffective charges, are not captured well with present techniques. This can qualitatively beunderstood in the context of the cluster expansion discussed at the beginning of Sec. 2. Itis precisely the low-lying collective modes that are expected to violate the cluster hierarchyupon which the IMSRG relies.Second, several regions of the nuclear chart—e.g. the “islands of inversion” (244, 245),or the charge radii of the calcium isotopes (246, 247)—display features which suggest thata naive valence space of a single major harmonic oscillator shell is not an appropriate first-order description. However, the derivation of effective interactions for non-standard valencespaces leads to difficulties related to the well-known intruder-state problem.
The first attempt at a microscopic treatment of electric quadrupole ( E
2) observables was thework of Horie and Arima (248), investigating the role of configuration mixing on quadrupolemoments. A series of investigations by Seigel and Zamick (249–251) demonstrated theimportance of terms beyond first order in perturbation theory. Specifically, they investigatedthe impact of Tamm-Dancoff (TDA) and random-phase approximation (RPA) graphs tothe effective charge, with the physical interpretation that the effective charge comes largelyfrom a coupling to the giant quadrupole resonance. A subsequent calculation by Kirson (44)indicated that a self-consistent treatment including screening effects essentially canceled theeffect obtained with RPA. For a discussion, see Ref. (27).An important development came with the application of the OLS approach to an effec-tive interaction for Li in the p -shell (162), where the resulting effective E
36 Stroberg, Hergert, Bogner, and Holt roduce the collective effects of E p shell carry over to heavier masses.As discussed in Sec. 2.3.2, the Magnus formulation of the IMSRG provides a straight-forward way to construct effective valence space operators for general observables. Alloperators are consistently transformed according to O eff = e Ω O e − Ω = O + [Ω , O ] + [Ω , [Ω , O ]] + . . . (77)A first application of this approach was to electromagnetic transitions in light and mediummass nuclei (123), where it was found that the observables were well-converged with respectto the model space truncation (i.e. frequency and number of major shells included in theinitial harmonic oscillator basis). However, the computed values for collective observableslike magnetic moments or electric quadrupole and octupole transitions were substantiallysmaller than experimental data.The possible explanations for this discrepancy are that either that the truncation ofEq. (77) to two-body operators is insufficient to capture this type of collectivity, or elsethe input chiral interactions are deficient in some way. Most likely, both are in effect tosome degree. The interaction used in Ref. (123) is known to underpredict charge radii inthese same nuclei (240). Given that the electric quadrupole operator is proportional to r , where r is the point proton radius, and that the transition strengths B ( E
2) go as r ,one would naturally expect some underestimation of the quadrupole strength. However, asdemonstrated in Fig. 13, this cannot be the whole story. C E (2 + ) (MeV) R pp (fm ) h +1 k E k +1 i ( e fm )024 O N LO SAT
EM 1.8/2.0 N LO L/NLEM 500/400 exp th S exp th exp th Figure 13: Electric quadrupole transition matrix element (cid:104) + (cid:107) E (cid:107) + (cid:105) in C, O, and Scomputed using the VS-IMSRG with several choices of input chiral interaction. Also shownare the energy of the 2 +1 state and the point proton radius squared. Experimental radii arefrom (252), energies and transition matrix elements are from (243).The point proton radius squared, indicated R pp in Fig. 13, is underpredicted at approx-imately the same level in C and S. On the other hand, while the E Cis reasonably reproduced, in S it is underpredicted by ∼ Ois underpredicted by ∼ E S cannot beexplained solely by the radius deficiencies. • Non-Empirical Interactions for the Nuclear Shell Model 37 (a) Core pol. x (b) TDA x (c) RPA x (d) IMSRGFigure 14: Exaples of diagrams contributing to the one-body part of the effective E E sd shell nuclei yielded significantly larger E E sd shell obtained with first-order core polarization,including TDA and RPA graphs to all orders, and from IMSRG. These results are obtainedwith the EM1.8/2.0 interaction (220), in a Hartree-Fock basis constructed from an oscillatorbasis with e max = 10, (cid:126) ω = 16 MeV. d / d / d / d / d / d / d / s / d / s / Core. pol. 0.110 0.035 0.064 0.034 0.026TDA 0.121 0.037 0.062 0.040 0.031RPA 0.119 0.037 0.061 0.038 0.030IMSRG 0.202 0.098 0.222 0.163 0.093
Table 1 presents the effective charge for a neutron in the sd shell in these various levelsof approximation. The orbit-dependent effective charge is obtained as (251) e ab = (cid:104) a (cid:107)O E (cid:107) b (cid:105)(cid:104) πa (cid:107)O E (cid:107) πb (cid:105) (78)where in the denominator, we take the matrix element for the corresponding proton or-bit. We work in a Hartree-Fock basis constructed from an oscillator basis with frequency (cid:126) ω = 16 MeV and e max = 10. In this basis, we obtain a bare proton matrix element (cid:104) πd (cid:107)O E (cid:107) πd (cid:105) = − . e fm , and we see that in order to reproduce the experimentalquadrupole moment of O ( Q = − . e fm (253)), we require an effective neutron charge
38 Stroberg, Hergert, Bogner, and Holt f e n ≈ .
37. Likewise, the bare proton matrix element (cid:104) πd (cid:107)O E (cid:107) πs (cid:105) = − . e fm ,and so to reproduce the experimental transition strength B ( E
12 + →
52 + ) = 6 . e fm we require an effective neutron charge e n ≈ .
38. We find that while the IMSRG generatesa larger neutron effective charge than the other methods, the result is still well below theexperimental value.These IMSRG effective charges are essentially the same as those found in a previousstudy (123) with a different chiral interaction. That study also found proton effectivecharges close to 1, i.e. with almost no renormalization. As discussed in Ref. (27), this canpotentially be understood by considering that in order to “dress” a valence nucleon, thatnucleon must excite a proton out of the core. A valence neutron can do this through the T = 0 channel, while a valence proton must act in the weaker T = 1 channel. Over the last few decades, experimental investigations of nuclei far from stability haverevealed the existence of several “islands of inversion”, where nuclei near traditional shellclosures have ground states that indicate significant deformation or correlated particle-holeexcitations out of the closed shell (244, 245, 254). The classic examples are Na and Mg, both with N = 20. Na has a ground state spin-parity of
32 + , while shell modelcalculations predicted
52 + , and Mg has a 2 + excitation energy of 885 keV, far lower thanexpected for a closed neutron shell. Both have greater binding energies than predicted inthe shell model. If these correlations are sufficiently important, then it is possible thatthe ground state will not be among the subset of eigenstates reproduced in the valencespace diagonalization. Indeed, Watt et al. (255) found that by explicitly allowing neutronexcitations out of the standard sd shell and into the f / shell, the discrepancies for Naand Mg could be understood. Of course, even if the correlated ground state is formallyamong the valence states, it is likely that incorporating the correlated excitations into aneffective Hamiltonian would require large many-body forces.It is therefore desirable to be able to produce an effective interaction for a valencespace that spans more than one major shell, such as the sd − fp space, and indeed phe-nomenological interactions for such a space have been successful at describing the island ofinversion effects (245), as well as the charge radii of the calcium isotopes (246). Unfortu-nately, in deriving such an interaction from first principles, one runs into the well-knownintruder-state problem which we discuss below. In fact, attempts to include effects of thecontinuum—essential for studies near the driplines—suffer from the same problem (256).Understanding and solving this problem, particularly in the context of a nonperturbativeapproach, is clearly of great interest. As demonstrated by Schucan andWeidenm¨uller (46, 47), there are serious reasons to doubt the convergence of the perturba-tive expansion for the effective interaction. To illustrate this, we split up the Hamiltonianas before into a zero-order piece and a perturbation, with a dimensionless power-countingparameter x H ( x ) = H + xV (79) With our definition of the E Q = (cid:112) π/ (cid:104) J, M = J |O E | J, M = J (cid:105) • Non-Empirical Interactions for the Nuclear Shell Model 39 uch that H (0) is the zero-order Hamiltonian and H (1) is the full Hamiltonian. The pertur-bative expansion of H eff can therefore be seen as a Taylor expansion about x = 0 evaluatedat x = 1. The trouble arises if one of the states belonging to the Q space has an energylower than one of the P space states. Such a state is called an “intruder state”. Assum-ing the P states are all at lower energy than the Q states at x = 0, this implies a levelcrossing for some x ∈ [0 , x at which such a crossing occurs (even if it is anavoided crossing) corresponds to a branch point which places an upper limit on the radiusof convergence of the effective Hamiltonian (47).Unfortunately such level crossings are the rule, not the exception. Moreover, if thezeroth-order levels in the valence space are non-degenerate—e.g. if a Hartree-Fock basis isused—then as more particles are added to the valence space, the energy of the highest P space can quickly become higher than the energy of the lowest Q space, even without theresidual interaction. There are a few reasons why one might expectthe IMSRG to avoid the intruder-state problem. First, it is formally a non-perturbativemethod, so the above argument does not directly apply. Second, because it is formulated inFock space, the energies of the A -body system do not enter into any energy denominators,and so one would not naively expect divergences due to crossings in the A -body system.Unfortunately, the IMSRG suffers from a related, but distinct intruder-state problem.An illustrative example of the type of behavior encountered is shown in Fig. 15. Here weaim to decouple a valence space consisting of the p and sd major shells from a large spaceconstructed from 7 major harmonic oscillator shells ( e max = 6). We use an O Hartree-Fock reference state, which is indicated schematically in Fig. 15. Also shown in Fig. 15 arethe zero-body term E ( s ), the norm of the generator (cid:107) η ( s ) (cid:107) and the norm of the Magnusoperator (cid:107) Ω( s ) (cid:107) as a function of the flow parameter s . As usual, we do this in two steps,first decoupling excitations out of the core ( He in this case), followed by a decoupling ofthe valence space (cf. Sec. 2.3). The core decoupling is achieved at s ≈
12. At this point,we observe a jump in (cid:107) η ( s ) (cid:107) because our new definition of ‘off-diagonal” now includes manymore matrix elements. In a well-behaved calculation, these terms would then be suppressedby the IMSRG flow. Indeed, the size of η initially decreases, but it soon begin to growagain, and the caclulation fails to converge. We also observe that the flow of the zero-bodyterm E turns around and diverges, and the Magnus operator Ω grows indefinitely. At somepoint, Ω grows beyond the radius of convergence of the BCH expansion. As a result, noeffective interaction is obtained.We mention in passing that there has been some success using the IMSRG to decouplevalence spaces other than those defined by a single major harmonic oscillator shell, solong as they are reasonably well separated by a shell gap. These spaces correspond tothe “extruded-intruded” spaces described by the Strasbourg group (18), where due to thespin-orbit potential the orbit with the largest- j orbit drops out (is “extruded”) and thelargest- j orbit from the next shell up comes down (it “intrudes”). An example is the spaceconsisting of the orbits 1 p / , 1 p / , 0 f / , 0 g / . This space (for neutrons), was used totreat heavy chromium isotopes (236). However, the results obtained there suggested thatthis space was not sufficient to describe the ground states of those isotopes.The connection between intruders and failed convergence of the IMSRG flow can be un-derstood schematically by considering the flow equation formulation dds H = [ η, H ]. Imaginewe have two levels labeled p and q , both with degeneracy greater than 2, and with single
40 Stroberg, Hergert, Bogner, and Holt rotons neutrons − − S i n g l e - p a r t i c l ee n e r g y ( M e V ) O, psd valence space − − E ( M e V ) k η k s k Ω k Begin valencedecouplingBegin coredecoupling
Figure 15:
Left:
Decoupling of the psd valence space using an O reference, shown withthe Hartree-Fock single-particle spectrum.
Right:
The zero-body part of the flowing Hamil-tonian, the norm of the generator η and the norm of the Magnus operator Ω as a functionof the flow parameter s . At s ∼
12, the core is decoupled and the decoupling of the valencespace begins.particle energies (cid:15) q > (cid:15) p . We intend to decouple the q orbit from the Hilbert space, whichmeans suppressing terms like V qqpp a † q a † q a p a p which excite particles from the p level to the q level. The flow equation for V qqpp is, schematically, dds V qqpp ∼ V qqpp (cid:15) p − (cid:15) q + V pppp − V qqqq (cid:15) q − (cid:15) p + . . . (80)If the one-body terms dominate the right hand side of (80), then dds V qqpp ∼ − V qqpp andthe off-diagonal term is suppressed exponentially. On the other hand, if the interactionterms V are larger than the one-body terms, and of opposite sign, then V qqpp will beexponentially enhanced. This can be achieved if V qqqq is negative (attractive) and V pppp is positive (repulsive), and a positive numerator in (80) corresponds to an inversion of thestates | pp (cid:105) and | qq (cid:105) .In fact, terms like V pppp and V qqqq can be included in the denominator by repartitioningthe Hamiltonian so that the diagonal (i.e. bra=ket) parts of V are included in H , avoidingthe exponential growth. However, intruders can also be driven by collective effects whichcannot be tamed by a straightforward repartitioning. Consider the case where we havemultiple included and excluded levels p, p (cid:48) . . . q, q (cid:48) . . . In this case, we should also considercontributions like dds V qqpp ∼ V qqp (cid:48) p (cid:48) (cid:15) q − (cid:15) p (cid:48) V p (cid:48) p (cid:48) pp − V qqq (cid:48) q (cid:48) V q (cid:48) q (cid:48) pp (cid:15) q (cid:48) − (cid:15) p + . . . (81)If there are many such terms involving V p (cid:48)(cid:48) p (cid:48)(cid:48) pp , etc. and these terms add coherently, withthe V qqq (cid:48)(cid:48) q (cid:48)(cid:48) type terms having opposite sign, they can compete with the contributions in(80) and potentially lead to growth of the off-diagonal terms. Such a situation will also • Non-Empirical Interactions for the Nuclear Shell Model 41 ead to a crossing of collective levels. Clearly, this situation and the previous one will beexacerbated by the small energy denominators which occur in multi-shell valence spaces.There is another way in which intruders can cause trouble, and this is by spoiling thecluster hierarchy. We illustrate this with a toy system in the next section.
To illustrate how intruders and levelcrossings can lead to large induced many-body terms, we consider the problem of three kindsof fermion—which could be, say, spin-up neutron, spin-down neutron, spin-up proton—living in a three-level Hilbert space. We require three particles because we wish to monitorinduced three-body forces.The initial Hamiltonian is H ( x ) = H + xV where H ≡ (cid:88) i (cid:15) i a † i a i , V ≡ (cid:88) ijkl V ijkl a † i a † j a l a k . (82)Additionally, three-body terms will be induced by the transformation. All three specieshave the same single-particle energies: ( (cid:15) , (cid:15) , (cid:15) ) = (0 , , V are: v QQ = V = V = V = V = V = V v PP = V v PQ = V = V = V = V . (83)We take v PP =8, v QQ = − v PQ =1. Essentially v PP and v QQ mix configurations within the P and Q spaces, respectively, leading to a collective Q state coming down in energy as theinteraction is turned on, while a collective P state is pushed up, and eventually the statescross. The remaining term v PQ , is initially the term we want to suppress. It couples the P and Q states and makes the level crossing an avoided crossing. To reduce somewhat thesize of the problem, we exclude the highest level for the third particle (call it the proton),reducing the three-body Hilbert space to 3 × × v QQ and v PQ to act only between neutrons, while v PP acts on all species.The eigenstates of this problem may easily be found by forming the 18 ×
18 Hamiltonianmatrix and diagonalizing. However, our aim here is to first decouple the P and Q spaces,and then diagonalize within the decoupled spaces.We perform a non-perturbative decoupling in the 3-body Hilbert space using the it-erative method outlined in section 3.1.6 (this is essentially the approach proposed bySuzuki (171) to deal with the intruder-state problem). The first step is to construct the ma-trix H , where the subscript denotes iterations. Next, we form an anti-hermitian generator G n which is defined as (cid:104) q | G n | p (cid:105) = (cid:104) q | H n | p (cid:105)(cid:104) q | H n | q (cid:105) − (cid:104) p | H n | p (cid:105) (84)(here p and q label A -body configurations belonging to the P and Q spaces, respectively)and obtain the next iteration of H by the Baker-Campbell-Hausdorff expansion H n +1 = H n + [ G n , H n ] + [ G n , [ G n , H n ]] + . . . (85)The nested commutators are evaluated until the norm of the last nested commutator fallsbelow 10 − . The iteration in n is performed until the norm of G n falls below 10 − , at whichpoint the P and Q spaces are decoupled.
42 Stroberg, Hergert, Bogner, and Holt I M S R G ( ) E i g e n v a l u e QP − I M S R G ( ) E i g e n v a l u e . . . . . . x C l u s t e r n o r m k H b kk H b kk H b k Figure 16:
Top:
The eigenvalues after decoupling the P and Q spaces in the IMSRG(2) ap-proximation (the gray lines indicate the exact values), Middle:
Eigenvalues after decouplingwith the full IMSRG(3),
Bottom:
The cluster decomposition of the P -space component ofthe transformed Hamiltonian, all as a function of the interaction strength parameter x .We also perform an IMSRG decoupling , using the flow equation formulation, directlyon the Fock space representation of the Hamiltonian (82). We use the flow equation for-mulation because in the Magnus formulation for x (cid:38) .
5, the Magnus operator Ω growssufficiently large that the Baker-Campbell-Hausdorff expansion does not converge. We per-form an IMSRG(2) calculation, discarding 3-body terms, and we also perform an IMSRG(3)calculation, including the full three-body commutators, so the calculation is exact for thethree-body problem. The results are presented in Fig. 16 as a function of the perturbationstrength parameter x . The gray lines in panels (a) and (b) are the results of the decouplingin the A -body space following the iteration procedure in Eqs. (84) and (85). The purple andred lines and symbols correspond to the IMSRG(2) and IMSRG(3) solution in Fock-space. Strictly speaking, because there is no core, and no normal ordering is performed, there is no“medium” and so this is really just an SRG calculation. • Non-Empirical Interactions for the Nuclear Shell Model 43 e immediately make two observations. First, as shown in the top panel of Fig. 16,before the level crossings the IMSRG(2) eigenvalues are in agreement with the exact ones,while after the level crossings they go astay. Second, as shown in the bottom panel ofFig. 16, at the first level crossing near x ≈ . P -space three-body contribution: dds V ∼ η V − V η + . . . (86)The level crossing prevents the off-diagonal two-body matrix elements like V from beingrapidly suppressed, and keeps the door open for strength to leak into the three-body sectorvia terms like (86).This investigation of a toy problem illustrates how the effects which lead to intruderconfigurations also cause problems with decoupling within the IMSRG framework, evenwithout small denominators, and even if the flow equation converges. Presumably, sucheffects will also arise in the SMCC framework because of the strong similarity between thetwo approaches. This remains an open and important problem, and we hope that a betterunderstanding of these effects will lead to a solution in the near future.
7. Other developments7.1. EFT for the shell model
As we mentioned in the introduction, the success of shell model phenomenology stronglyindicates that the shell model provides the relevant degrees of freedom for nuclear structure.It is therefore tempting to formulate the shell model as an effective theory with some schemefor systematic improvement. The main difficulty is in identifying a separation of scales whichone can use to form an expansion.20 years ago, Haxton et al. put forward the idea of formulating the shell model as aneffective theory (258–261). The method presented in that work amounts to an effectivetheory for the NN interaction with the harmonic oscillator basis serving as the regulator,and the Bloch-Horowitz effective interaction (4) cast as an RG flow equation. Similar ideashave been pursued in Refs. (262–264), and by the Oak Ridge group (265, 266). This typeof approach is appealing because it is formulated in the harmonic oscialltor basis and soyields an interaction well-suited to a number of popular many-body methods.Using the harmonic oscillator basis as a regulator is conceptually distinct from formulat-ing the standard shell model directly as an EFT. In the latter case, one should use the shellmodel to define the degrees of freedom, and write down all possible terms in the Hamilto-nian consistent with the relevant symmetries (parity, rotational invariance, charge, baryonnumber, etc.). Then one should assign an importance to those terms based on some power
44 Stroberg, Hergert, Bogner, and Holt ounting. A recent attempt more along these lines (267) employs a Weinberg chiral powercounting in a shell-model basis, modified by the Galilean invariance breaking terms due tothe presence of the core. While a rigorous basis for the use of Weinberg’s power counting isstill lacking– core excitations introduce a new scale, for instance, that might make it morenatural to treat the Fermi momentum as a hard scale– very encouraging order-by-orderconvergence was obtained.Another possibility might be, a-la Landau-Migdal theory (268), to exploit the simi-larities between the valence shell model philosophy and Landau’s Fermi liquid theory forinfinite systems. In modern parlance, the latter can be viewed as an effective field theory forlow-lying excitations (“quasi-particles”) in the vicinity of the Fermi surface. As with anyEFT, the effective Hamiltonian of Landau’s theory incorporates the underlying symmetriesof the system, and the low-energy couplings– the Landau parameters– can either be fixed byexperiment or calculated microscopically based on the underlying theory. Lending credenceto the analogy with the valence shell model, Shankar, Polchinski, and others have shownthat Landau’s theory can be understood as an IR fixed point of the RG as one integratesout modes away from the Fermi surface (269–271). Intriguingly, their analysis shows that (i) three- and higher-body quasi-particle interactions are irrelevant in the sense of the RG,which might shed light as to why phenomenological shell model interactions with 1- and2-body terms are so effective, and (ii) the natural small parameter is the ratio of excitationenergy to the Fermi energy, which could provide guidance for formulating an appropriatepowercounting for an EFT tailored to the shell model.
A major deficiency in essentially all shell model applications to date is the absence ofquantified theoretical uncertainties. This is no small thing; at a fundamental level, a theo-retical prediction without some confidence interval cannot be falsified. In particular, sinceexperimental binding energies and excitation energies can routinely be measured at parts-per-million precision, whether or not theory and experiment are compatible is entirelydependent on the theoretical uncertainty.In fact, there are various meanings one can assign to error bars. Perhaps the moststraightforward form of shell model uncertainty is the root-mean-squared deviation fromexperiment (see Fig. 9). As mentioned above, the phenomenological USDB interaction(57) has a very small rms deviation of 130 keV throughout the sd shell, and this maybe interpreted in the following way. There exist a large number of states which can beinterpreted as shell model configurations mixed by the same universal residual interaction.In the context of effective interaction theory these are the states which get mapped to the P space. We may then interpret levels where the deviation is much larger than 130 keVas not belonging to the P space. A clear illustration of this can be found in Ref. (57),where the ground state energies of , Ne, , Na, and , Mg have conspicuously largedeviations—a clear signal that these “island of inversion” nuclei have a different characterthan the others.This notion of uncertainty has been recently made more quantitatively rigorous byYoshida et al. (272), who explored the various possible effective interactions in the p shell andobtained marginal distributions for each parameter based on the deviation from experiment.These distributions were then propagated to the calculated spectra, yielding an error barindicating the range of values that could be obtained by a universal p -shell interaction. • Non-Empirical Interactions for the Nuclear Shell Model 45 omparison to experiment then yields a well-defined quantification of how appropriate thenaive shell model picture of a given state is.A different type of uncertainty is sought by ab initio approaches employing chiral in-teractions. There, one should quantify and propagate three sources of uncertainty: (i) contributions from truncated higher orders in the EFT expansion, (ii) the experimentaluncertainty in data used to fit the parameters of the EFT, and (iii) uncertainty due to ap-proximations made in solving the many-body problem. In this case, because one has begunwith the most general Lagrangian compatible with the symmetries of the Standard Model,the resulting theoretical error bar would then indicate compatibility with the StandardModel. While we certainly expect such compatibility from the known nuclear energy lev-els, this becomes very important for testing extensions to the Standard Model through e.g.searches for neutinoless double beta decay (273) or dark matter (274). While uncertaintyquantification is feasible for quantum Monte Carlo or no-core shell model calculations, thereis as yet no rigorous means for uncertainty quantification of ab initio shell model effectiveinteractions. This is an important avenue of future work.
Throughout the previous sections we have briefly touched upon the need to account forcontinuum effects in theoretical calculations. Of course, this will be especially relevant aswe seek to understand the structure for increasingly neutron-rich nuclei. While a variety ofapproaches for coupling the shell model and other many-body methods to the continuumexist (see, e.g., (275–277), as well as the reviews (278, 279)), methods based on the Berggrenbasis (280) appear to be most suitable in the context of VS-IMSRG and SMCC.The Berggren basis adds resonant and scattering states to the single-particle basis fromwhich many-body states are constructed. In valence-space configuration interaction calcu-lations, one obtains what is colloquially known as the Gamow Shell Model (GSM), whichentails the large-scale diagonalization of a complex symmetric Hamiltonian (278). Applica-tions of this method to weakly bound nuclei have been quite successful (281–289), and thereis a push to move from the commonly used phenomenological interactions to fundamentalones (289). The Berggren basis has been used successfully in ground- and excited-stateCC calculations (see, e.g., (128, 210)), hence the inclusion in VS-IMSRG and SMCC istechnically straightforward. However, the proliferation of states due to the inclusion of thecontinuum aggravates the intruder-state problem discussed in Sec. 6. If this issue can besolved, it would allow us to properly account for the continuum coupling in the derivationof effective interactions. For the time being, work is underway to at least treat the impactof the continuum on the dynamics of the valence particles via the GSM.
8. Concluding remarks
In this work, we have reviewed the current state of efforts to derive effective interactionsfor the shell model from modern nuclear forces, with an emphasis on the impact of RGand EFT ideas on our understanding of the shell model itself. We have summarized thepopular approaches, and discussed their relations at a formal level. We emphasized theimportance of three-body forces in eliminating the need for phenomenological adjustmentsand presented the ensemble normal ordering (ENO) approach to efficiently including three-body effects. We presented highlights from recent applications of ab initio shell model
46 Stroberg, Hergert, Bogner, and Holt alculations, and discussed the current challenges of low-lying collective excitations andintruder states.Before concluding, we would like to make some remarks and clarify some commonmisconceptions about ab initio valence space methods.
SUMMARY POINTS • The shell model picture is inherently a low-momentum description ofnuclear structure.
The basic assumption of the shell model is that nucleons are(almost) independent particles moving in a mean field potential, and that nuclearspectra can be explained by the mixing of a few valence configurations above aninert core via a residual interaction. The bound mean-field solution and weak(possibly even perturbative) residual interaction that are the foundation of thisintuitive picture can only be obtained if low and high momenta are decoupled inthe Hamiltonian (102, 115, 116).Of course, nuclear observables — energies, radii, transition rates — must be inde-pendent of the resolution scale at which a theory operates. In principle, there isnothing that prevents one from microscopically constructing a valence shell modelHamiltonian starting from a high-resolution description, e.g., using an input in-teraction with a highly repulsive core. However, not only does such a choice makecomputations more difficult, but it complicates interpretations as the resulting shellmodel wave functions bear little resemblance to the exact ones, which contain sizablecontributions from a vast number of configurations and defy a simple interpretation.In contrast, the exact wave functions of a low-resolution Hamiltonian at least qual-itatively resemble those that come out of the shell model diagonalization, providinga simple and intuitive picture. • Approaches such as the VS-IMSRG, SMCC, or the ˆ Q -box resumma-tion, are methods for solving the nuclear many-body problem, not newshell model interactions. The approaches described in this article combine thederivation of effective interactions with a shell model diagonalization. This shouldbe understood as an efficient alternative to a large-scale, full no-core configura-tion interaction (e.g., NCSM) calculation that would yield exact results for nuclearspectra, but is infeasible in most cases. • A careful comparison with experimental data or theoretical results re-quires that the method and underlying nuclear interaction be specified.
When comparing two phenomenological shell model calculations to experimentaldata, the interpretation is generally straightforward: the interaction that betterreproduces the data is the better interaction. With ab initio approaches, such aninterpretation is no longer appropriate.Disagreements between theory and experiment must be caused either by deficienciesin the underlying nuclear interactions, or the approximations employed in derivingthe effective interaction (provided the shell model calculation is done without furtherapproximations of its own). It is therefore crucially important to specify both themethod and the input interaction when comparisons with experimental data orother theoretical results are presented.It should also be kept in mind that the major advantage of ab initio approaches isthe ability to systematically improve the precision of the theoretical result by lifting • Non-Empirical Interactions for the Nuclear Shell Model 47 pproximations, or improving the input nuclear interactions. • Three-body forces are inevitable and non-negligible in nuclear struc-ture.
So long as we choose to use protons and neutrons as our active degreesof freedom—excluding explicit Deltas, anti-nucleons, etc.—there will be “genuine”(in the traditional language) many-body forces accounting for these integrated outdegrees of freedom. So long as we wish to work in a valence space, there will be“effective” many-body forces accounting for excitations outside the valence space.The relative importance of these many-body forces will depend on the details ofthe implementation (scheme and scale). Indeed, an excellent description of a localregion of the chart can be obtained with a purely two-body interaction, like USDBfor the sd -shell (57). But such an interaction will only work locally, and will needmodification (e.g., scaling of matrix elements with mass — again think of USDB)in order to be used over a wider range of nuclei. The theoretical evidence supportsthe expectation that three-body forces are the underlying source of such ad hocmodifications (see Sec. 4). • The mass-dependence of modern effective interactions does not imply aloss of predictive power.
Effective interactions for different target nuclei are de-rived from the same two- plus three-nucleon force, and there are no parameter refitsor phenomenological modifications . Again, the derivation of the effective interactionand subsequent shell model diagonalization are merely an efficient alternative to afull no-core configuration interaction calculation. • All observables, not just the Hamiltonian, must be treated consistentlyto produce a true ab initio result.
As discussed in section 3, the effective inter-action corresponds to a similarity transformation of the original Hamiltonian, andin order to perform a consistent calculation, all operators must also be transformed.As a simple example, consider computing the deuteron ground state by generatingan effective interaction for the 0 s shell. By construction, the energy obtained bya (trivial) diagonalization in the 0 s shell would be identical to the result from adiagonalization in the full space with the bare Hamiltonian. Now, if one were tocalculate the deuteron quadrupole moment using the bare E s -wave), the result would bezero. Using a consistently transformed E E sd -shell oxygen isotopes,which only have neutrons in the valence space. Along these same lines, the useof phenomenological effective charges in conjunction with an ab initio shell modelinteraction should be considered inappropriate. In general, it is difficult to makemeaningful conclusions based on inconsistent calculations. • The use of an inert core does not constitute an ad hoc approximation.
Calculations based on effective interction theory do not formally rely on an assump-tion that excitations out of the core are “negligible”. Such excitations are accountedfor by the effective interaction. Certainly, there will be states in the experimental
48 Stroberg, Hergert, Bogner, and Holt pectrum that are not generated in the valence space calculation even with a perfecteffective interaction—these belong to the excluded Q -space. However, those statesthat are generated will not be improved by, e.g., allowing core excitations describedby a schematic interaction, as this would amount to double-counting. Instead, toinclude core excitations explicitly, one should re-define the P and Q spaces andderive a new effective interaction.The shell model has been the primary intellectual and computational framework for lowenergy nuclear structure for the past 70 years. While the computational work has beenlargely phenomenological over that time, an enormous amount of knowledge and intuitionhas been developed. At the same time, perhaps no problem in nuclear structure has sostubbornly resisted a satisfactory solution as the microscopic derivation of shell modelinteractions. The general path has been more or less known for over half a century, but itis only recently that the available computational power, combined with a more systematicway of thinking about nuclear forces and the many-body problem, has allowed a directconnection between the shell model, the forces applicable to few-body scattering, and theunderlying physics of the Standard Model.We are not quite yet in the promised land. While there are certainly many detailsremaining to be worked out (including those mentioned here), and several clear extensionsto be made (continuum effects, reactions), there are still two major hills to climb: a fullyconsistent and satisfactory power-counting for the interaction, and a rigorous uncertaintyquantification for our many-body methods. We hope that progress can be made on thesefronts in the near future, enabling a broadly applicable, quantitatively predictive theory ofnuclear structure. ACKNOWLEDGMENTS
We would like to thank B. Alex Brown, Takayuki Miyagi, Titus Morris, Petr Navr´atiland Zhonghao Sun for helpful discussions, and Gaute Hagen for providing coupled clusterresults. S.R.S is supported by the U.S. DOE under contract DE-FG02-97ER41014. H.H.acknowledges support by the National Science Foundation under Grant No. PHY-1614130,as well as the U.S. Department of Energy, Office of Science, Office of Nuclear Physics underGrants No. de-sc0017887 and de-sc0018083 (NUCLEI SciDAC Collaboration). S.K.B.acknowledges support by the National Science Foundation under Grant No. PHY-1713901,as well as as the U.S. Department of Energy, Office of Science, Office of Nuclear Physicsunder Grant No. de-sc0018083 (NUCLEI SciDAC Collaboration).
APPENDIXA. IMSRG flow equations
For reference, we now present the IMSRG(2)/VS-IMSRG(2) flow equations (145, 146, 167).Ground-state and valence-space decoupling only differ by the choice of the generator η (seeRefs. (146, 168)).The system of flow equations for the zero-, one-, and two-body parts of H ( s ) result from • Non-Empirical Interactions for the Nuclear Shell Model 49 valuating dHds = [ η ( s ) , H ( s )] (87)with normal-ordered Fock-space operators that are truncated at the two-body level : H = E + (cid:88) ij f ij { a † i a j } + 14 (cid:88) ijkl Γ ijkl { a † i a † j a l a k } (88) η = (cid:88) ij η ij { a † i a j } + 14 (cid:88) ijkl η ijkl { a † i a † j a l a k } . (89)The flow equations are then dE ds = (cid:88) ab n a ¯ n b ( η ab f ba − f ab η ba ) + 14 (cid:88) abcd n a n b ¯ n c ¯ n d ( η abcd Γ cdab − Γ abcd η cdab ) (90) df ij ds = (cid:88) a ( η ia f aj − f ia η aj ) + (cid:88) ab ( n a − n b )( η ab Γ biaj − f ab η biaj )+ 12 (cid:88) abc ( n a n b ¯ n c + ¯ n a ¯ n b n c ) ( η ciab Γ abcj − Γ ciab η abcj ) (91) d Γ ijkl ds = (cid:88) a (1 − P ij )( η ia Γ ajkl − f ia η ajkl ) − (1 − P kl )( η ak Γ ijal − f ak η ijal )+ 12 (cid:88) ab ( n a n b − ¯ n a ¯ n b )( η ijab Γ abkl − Γ ijab η abkl ) − (cid:88) ab ( n a − n b )(1 − P ij )(1 − P kl ) η bjal Γ aibk (92)where P ij exchanges indices i and j , n a is the occupation of orbit a and ¯ n a ≡ − n a . B. Canonical perturbation theory to second order
We partition the Hamiltonian into a zeroth-order piece and a perturbation, H = H + xV , (93)and we consider a perturbative expansion in powers of the dimensionless order parameter x , where in the end we will take x = 1. We further distinguish between “diagonal” and “off-diagonal” components, V = V d + V od , where “off-diagonal” generically means the termswe wish to suppress by the transformation H eff = e G He −G . (94)As in the discussion leading to Eq. (39), we use the superoperator notation to express acommutator with H in terms of an energy denominator ∆. Through second order in x , weobtain for G G []1] = V od ∆ G [2] = (cid:104) G [1] , V d (cid:105) od / ∆ + (cid:104) G [1] , V od (cid:105) od / ∆ (95) These expressions can be easily adapted to evaluate the nested commutators appearing in theMagnus formulation of the IMSRG.
50 Stroberg, Hergert, Bogner, and Holt he second term in G [2] will vanish for A -body Hilbert space formulations, but not in generalfor a Fock space formulation. This is related to the different meanings of “off-diagonal” inthe two formulations. The transformed Hamiltonian through second order is H [0]eff = H H [1]eff = V d H [2]eff = (cid:104) G [1] , V d (cid:105) d + (cid:104) G [1] , V od (cid:105) d (96) C. Integration of Magnus flow equation to second order
Here, we integrate the flow equation (28) explicitly to second order. As in appendix B, wepartition the Hamiltonian into a zero order piece H , and a perturbation V , and we split upthe perturbation into “diagonal” and “off-diagonal” pieces. We use the White generator,which we write as η ( s ) ≡ H od ( s )∆ (97)using the super-operator notation introduced in Sec. 2.3. To first order in x , the flowequation for Ω is d Ω [1] ds = η [1] ( s ) = H [1] od ( s )∆ = V od ∆ + [Ω [1] ( s ) , H ] od / ∆ = V od ∆ − Ω [1] ( s ) . (98)A differential equation for Ω [2] ( s ) may be obtained in a similar manner. The solutions giventhe initial condition Ω(0) = 0 areΩ [1] ( s ) = (1 − e − s ) V od ∆Ω [2] ( s ) = (1 − e − s − se − s ) (cid:20) V od ∆ , V d (cid:21) od / ∆ + (1 − e − s ) (cid:20) V od ∆ , V od (cid:21) od / ∆ . (99)The flowing Hamiltonian through second order is H [0] = H H [1] = V d + e − s V od H [2] = (1 − e − s ) (cid:20) V od ∆ , V d (cid:21) d + (1 − e − s ) (cid:20) V od ∆ , V od (cid:21) d + se − s (cid:20) V od ∆ , V d (cid:21) od + e − s (1 − e − s ) (cid:20) V od ∆ , V od (cid:21) od . (100)We see that at first order, the off-diagonal part of the perturbation is exponentially sup-pressed. At s = 0, the second order piece is by definition zero. As s increases, we initiallyinduce both diagonal and off-diagonal second-order terms, and eventually, the inducedsecond-order terms are suppressed exponentially, leaving a purely diagonal the second-ordercorrection.Taking the limit s → ∞ , we obtain the same generator and effective Hamiltonian as thecanonical perturbation theory in appendix B. Note that this equivalence requires the samedefinition of “off-diagonal”, and that the results will differ at higher orders if we include thecommutator terms on the right hand side of the Magnus flow equation in (28). • Non-Empirical Interactions for the Nuclear Shell Model 51
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