AA Guided Tour of
Ab Initio
Nuclear Many-BodyTheory
Heiko Hergert , ∗ Facility for Rare Isotope Beams and Department of Physics & Astronomy,Michigan State University, East Lansing, MI 48824-1321
Correspondence*:Heiko [email protected]
ABSTRACT
Over the last decade, new developments in Similarity Renormalization Group techniques and nuclearmany-body methods have dramatically increased the capabilities of ab initio nuclear structure and reactiontheory. Ground and excited-state properties can be computed up to the tin region, and from the protonto the presumptive neutron drip lines, providing unprecedented opportunities to confront two- plusthree-nucleon interactions from chiral Effective Field Theory with experimental data. In this contribution,I will give a broad survey of the current status of nuclear many-body approaches, and I will use selectedresults to discuss both achievements and open issues that need to be addressed in the coming decade.
Keywords: nuclear theory, many-body theory, ab initio nuclear structure, ab initio nuclear reactions, similarity renormalization group
Over the past decade, the reach and capabilities of ab initio nuclear many-body theory have grownexponentially. The widespread adoption of Renormalization Group (RG) techniques, in particularthe Similarity Renormalization Group (SRG) [1], and Effective Field Theory (EFT) [2, 3, 4] in the2000s laid the foundation for these developments. Consistent two-nucleon (NN) and three-nucleon(3N) interactions from chiral EFT were quickly established as a new “standard” inputs for a varietyof approaches, which made true multi-method benchmarks possible. The SRG equipped us with theability to dial the resolution scale of nuclear interactions, accelerating model-space and many-bodyconvergence alike. Suddenly, even (high-order) Many-Body Perturbation Theory (MBPT) became aviable tool for rapid benchmarking [5, 6], and exact diagonalization approaches were able to extendtheir reach into the lower sd -shell [7, 8, 9]. A variety of of computationally efficient techniques withcontrolled truncations were readied, like the Self-Consistent Green’s Function method (SCGF) [10],the In-Medium SRG (IMSRG) [11] and Coupled Cluster (CC) [12], the prodigal son [13, 14] whoreturned home after finding success in foreign lands, i.e., quantum chemistry and solid state physics.At the start of the last decade the race was on, and Fig. 1 documents the progress that ensued.Calculations started at closed-shell nuclei [15, 16, 17, 18, 19] and their vicinity before extending tosemi-magic isotopic chains with the development of the Multi-Reference IMSRG [20, 21] and Gor’kovSCGF [22, 23] techniques, and just a couple of years later, the use of CC [24, 25] and IMSRG [26, 27]techniques to construct valence-space interactions opened all nuclei that were amenable to ShellModel calculations for exploration. Owing to very recent developments that extend these combinedapproaches to multi-shell valence spaces, the open region between the nickel and tin isotopic chain a r X i v : . [ nu c l - t h ] A ug ergert A Guided Tour of
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Figure 1.
Progress in ab initio nuclear structure calculations over the past decade. The blue arrowindicates nuclei that will become accessible with new advances for open-shell nuclei in the very nearterm (see Sec. 2.3).is poised to be filled in rapidly [28]. Development of the no-core versions of these methods hascontinued as well, and made direct calculations for intrinsically deformed nuclei possible [29].The growing reach of ab initio many-body methods made it possible to confront chiral NN+3Nforces with a wealth of experimental data, revealing shortcomings of those interactions and sparkingnew efforts toward their improvement. There were other surprises along the way, some good, somebad. Due to the benchmarking capabilities and further developments in many-body theory, we arenow often able to understand the reasons for the failure of certain calculations (see, e.g., Ref. [27]) —hindsight is 2020, as they say .The present collection of Frontiers in Physics contributions provides us with a timely and welcomeopportunity to attempt a look back at some of the impressive results from the past decade and thedevelopments that brought us here, as well as a look ahead at the challenges to come as we enter anew decade.Let us conclude this section with a brief outline of the main body of this work. In Section 2, Iwill discuss the main ingredients of modern nuclear many-body calculations: The input interactionsfrom chiral EFT, the application of the SRG to process Hamiltonians and operators, and eventuallya variety of many-body methods that are used to solve the Schr¨odinger equation. I will review keyideas but keep technical details to a minimum, touching only upon aspects that will become relevantagain later on. Section 3 presents selected applications from the past decade, and discusses both This exhausts my contractually allowed contingent of 2020 vision puns, I swear.
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Nuclear Many-Body Theory the advances they represent as well as open issues. This will provide a starting point for Section 4,which presents ideas for addressing the aforementioned issues and highlights important directionsfor the next decade.Naturally, the discussion in Sections 3 and 4 is highly subjective. While this work grew from a morerestricted scope into a rambling, albeit not random, walk through the landscape of modern nuclearmany-body theory, it still cannot encompass the field in its entirety. The upside is that this reflectsthe breadth of ideas that are being pursued by the ab initio nuclear theory community, includingthose with cross-disciplinary impact, as well as our community’s ability to attract junior researchers.The downside is that the present work can only scratch the tip of the iceberg of impressive resultsfrom the past decade. I hope that the readers will use it as a jumping-off point for delving into thecited literature, including the contributions to this volume.
Quantum Chromodynamics (QCD) is the fundamental theory of the strong interaction betweenquarks and gluons. One of its characteristic features is that the strong coupling, which governsthe strength of interaction processes, is sufficiently small to allow perturbative expansions at highenergies, but large in the low-energy domain relevant for nuclear structure and dynamics [30, 31].This makes the description of all but the lightest nuclei at the QCD level inefficient at best, andimpossible at worst. However, strongly interacting matter undergoes a phase transition that leads tothe confinement of quarks in composite hadronic particles, like nucleons and pions. These particlescan be used as the degrees of freedom for a hierarchy of EFTs that describe the strong interactionacross multiple scales.Following Weinberg [32, 33], one can construct effective Lagrangians that consist of interactionsthat are consistent with the symmetries of QCD and organized by an expansion in ( Q/ Λ). Here, Q isa typical momentum of the interacting system, and Λ is the breakdown scale of the theory, which isassociated with physics that is not explicitly resolved. In chiral EFT with explicit nucleons and pions,Λ = Λ χ is traditionally considered to be in the range 700 − V NN > V > V > . . . . Moreover, onecan readily extend the chiral Lagrangian with couplings to the electroweak sector by gauging thederivatives. In this way, nuclear interactions and electroweak currents depend on the same LECs,and one can use electroweak observables to constrain their values [42, 43, 44, 45]. Last but not least,the existence of a power counting scheme offers inherent diagnostics for assessing the theoretical Frontiers 3 ergert
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Nuclear Many-Body Theory +... +... +...
2N Force 3N Force 4N Force 5N Force LO ( Q/ Λ χ ) NLO ( Q/ Λ χ ) NNLO ( Q/ Λ χ ) N LO ( Q/ Λ χ ) NN 3N 4N + . . . + . . . + . . .
Figure 2.
Chiral two-, three- and four nucleon forces through next-to-next-to-next-to-leading order(N LO) (see, e.g., [37, 41, 2] ). Dashed lines represent pion exchanges between nucleons. The largesolid circles, boxes and diamonds represent vertices that are proportional to low-energy constants(LECs) of the theory (see text).uncertainties that result from working at a given chiral order [34, 35, 36]. This is especially usefulsince issues relating to the regularization and renormalization of these interactions remain (see, e.g.,Refs. [2, 46, 47, 48, 49, 50, 51] and Sec. 4.4).
Renormalization group methods are a natural companion to the hierarchy of EFTs for the stronginteraction. They provide the means to systematically dial the resolution scales and cutoffs of thesetheories, and this makes it possible, at least in principle, to connect the different levels in ourhierarchy of EFTs. The RGs also expand the diagnostic toolkit for assessing the inherent consistencyof EFT power counting schemes, e.g., by tracing the enhancement or suppression of specific operators,or by identifying important missing operators.In nuclear many-body theory, the SRG has become the method of choice. In contrast to WilsonianRG [52], which is based on decimation , i.e., integrating out high-momentum degrees of freedom,SRGs decouple low- and high-momentum physics using continuous unitary transformations. Notethat this concept is not limited to RG applications: we can construct transformations that adapt a
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Nuclear Many-Body Theory many-body Hamiltonian or other observables of interest to our needs, e.g., to extract eigenvalues[11, 53], or impose specific structures on the operator [1, 26, 54, 27, 55].We define the flowing Hamiltonian H ( s ) = U ( s ) H (0) U † ( s ) , (1)where H ( s = 0) is the starting Hamiltonian, and the flow parameter s parameterizes the unitarytransformation. Instead of making an ansatz for U ( s ), we take the derivative of Eq. (1) and obtainthe operator flow equation dds H ( s ) = [ η ( s ) , H ( s )] , (2)where the anti-Hermitian generator η ( s ) is related to U ( s ) by η ( s ) = dU ( s ) ds U † ( s ) = − η † ( s ) . (3)We can choose η ( s ) to achieve the desired transformation of the Hamiltonian as we integrate theflow equation (2) for s → ∞ . Wegner [56] originally proposed a class of generators of the form η ( s ) ≡ [ H d ( s ) , H od ( s )] , (4)that is widely used in applications, although it gives rise to stiff flow equations, and more efficientalternatives exist for specific applications [1, 11, 53]. Wegner generators are constructed by splittingthe Hamiltonian into suitably chosen diagonal ( H d ( s )) and off-diagonal ( H od ( s )) parts. These labelsare a legacy of applying this generator to drive finite-dimensional matrices towards diagonality. Forour purposes, they reflect the desired structure of the operator in the limit s → ∞ : We want to keepthe diagonal part and drive H od ( s ) to zero by evolving it via Eq. (2) (see Refs. [56, 57, 1, 11, 53]).To implement the operator flow equation (23), we need to express η ( s ) and H ( s ) in a basis ofsuitable operators { O i } i ∈ N , η ( s ) = (cid:88) i η i ( s ) O i , (5) H ( s ) = (cid:88) i H i ( s ) O i ( s ) , (6)where η i ( s ) and H i ( s ) are the running couplings of the operators. If the algebra of the operators O i is closed naturally or with some truncation, we have[ O i , O j ] = (cid:88) k c ijk O k (+ . . . ) (7)and Eq. (2) becomes a system of flow equations for the coupling coefficients: dds H i ( s ) = f i ( c , η ( s ) , H ( s )) , (8) Frontiers 5 ergert
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Nuclear Many-Body Theory where the bold quantities collect the algebra’s structure constants and the running couplings,respectively. From this discussion, it is clear that the choice of the O i can have a significant effecton the size of the system of flow equations, as well as the quality of any introduced truncations.An important application of the SRG in nuclear many-body theory is the dialing of the operators’resolution scales. This is achieved by using the Wegner-type generator η ( λ ) = [ T, H ( λ )] (9)to band-diagonalize the Hamiltonian in momentum space, and thereby decouple low- and high-momentum physics in the operators and eigenstates. As indicated in Eq. (9) the flow is typicallyre-parameterized by λ = s − / , which characterizes the width of the band in momentum spaceand controls the magnitude of the momentum transferred in an interaction process. For example, | k i − k f | (cid:46) λ in a two-nucleon system [1, 58].Nowadays, the momentum space evolution is regularly performed for two- and three-nucleon forces[59, 1, 60, 61, 62]. In light of the previous discussion, it can be understood as choosing the operatorbasis B = { a † p a q , a † p a † q a s a r , a † p a † q a † r a u a t a s , . . . } pqrstu... ∈ N , (10)with creation and annihilation operators referring to (discretized) single-particle momentum modes,and truncating four- and higher-body terms that appear when the commutators of the basis operatorsare evaluated. Since the commutator of an M -body and an N -body operator in the basis (10)acts at least on K = max( M, N ) particles, the SRG evolution is exact for A ≤ H ( s ) intwo- and three-nucleon systems, whose entries correspond to the coupling constants in our chosenoperator basis (cf. Eq. (6)). For efficiency, an additional basis change is made to center-of-mass andrelative coordinates.In principle, the strategy for evolving nuclear interactions towards some form of “diagonality”could be used to determine eigenvalues of many-body Hamiltonians, but the computational cost fordealing either with exponentially growing matrix representations or induced terms of high particlerank is prohibitive. This motivates the implementation of the flow equation with a different choiceof basis operators in the In-Medium SRG (see Section 2.3.3). Let us now discuss commonly used many-body methods for solving the nuclear Schr¨odingerequation. Roughly speaking, they fall into two categories: configuration space methods that expandthe nuclear eigenstates on a basis of known many-body states, or coordinate-space methods that workdirectly with the wave function and optimize them in some fashion. Our goal is to use approaches thatsystematically converge to an exact result, e.g., by adding more and more particle-hole excitationsof a selected reference state to the many-body basis of a configuration space, or by exhausting thedistribution of meaningful wave function parameters.The discussion in the following sections will be light on mathematical details, which can be foundin more specialized articles and reviews, including other contributions to the present volume. Thegoal is to review only certain ideas that will become relevant later on.
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Nuclear Many-Body Theory h i | H | j i h i | H IMSRG | j i h i | H CC | j i p h p h p h p h (a) (cid:104) i | H | j (cid:105) (b) (cid:104) i | H IMSRG | j (cid:105) (c) (cid:104) i | H CC | j (cid:105) Figure 3.
Decoupling of particle-hole excitations from a 0p0h reference state: the schematic matrixrepresentation of the initial Hamiltonian H (a) and the transformed Hamiltonians obtained fromIMSRG (b) and CC (c), respectively. (See text for details.) Let us briefly discuss the general setup of the configuration-space approaches. We choose asingle-particle basis, e.g., the eigenstates of a harmonic oscillator, and use it to construct a basis ofSlater determinants for the many-body Hilbert space. Usually, the many-body basis is organized byselecting a reference state | Φ (cid:105) and constructing its particle-hole excitations in order to account forthe natural energy scales of the system under consideration. For further use, we define | Φ a...i... (cid:105) ≡ { a † a . . . a i . . . } | Φ (cid:105) , (11)where particle ( a, b, . . . ) and hole ( i, j, . . . ) indices run over unoccupied and occupied single-particlestates, respectively . The parentheses indicate that the strings of creation and annihilation operatorsare normal ordered with respect to the reference state. They are related to the original operators by a † p a q = { a † p a q } + C qp , (12) a † p a † q a s a r = { a † p a † q a s a r } + C rp { a † q a s } − C sp { a † q a r } + C sq { a † p a r } − C rq { a † p a s } + C rp C sq − C sp C rq , (13)where the indices p, q, . . . run over all single-particle states, and the contractions are defined as C qp ≡ (cid:104) Φ | a † p a q | Φ (cid:105) = ρ qp (14)(see, e.g., Refs. [11, 53] for more details).Let us now consider a Hamiltonian containing up to two-body interactions, for simplicity. Innormal-ordered form, it is given by H = E + (cid:88) pq f pq { a † p a q } + 14 (cid:88) pqrs Γ pqrs { a † p a † q a s a r } , (15) This labeling scheme is commonly used in chemistry [63], and it is used with increasing frequency in nuclear physics as well.
Frontiers 7 ergert
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Nuclear Many-Body Theory where E is the energy expectation value of the reference state, while f and Γ are the mean-fieldHamiltonian and residual two-body interaction, respectively [11, 53]. Our task is to solve themany-body Schr¨odinger equation for this Hamiltonian to determine its eigenvalues and eigenstates,either in an approximate fashion or by exactly diagonalizing its matrix representation, which isshown in Fig. 3(a). Many-Body Perturbation Theory (MBPT) is the simplest configuration-space approach forcapturing correlations in interacting quantum many-body systems. It has enjoyed widespreadpopularity in treatments of the many-electron system since the early days of quantum mechanics,and it comes in a myriad of flavors (see, e.g., Ref. [64] and references therein). A major factor in itssuccess is that the Coulomb interaction is sufficiently weak to make perturbative treatments feasible.Applications in nuclear physics had long been hindered by the strong short-range repulsion andtensor interactions in realistic nuclear forces, despite the introduction of techniques like Brueckner’s G matrix formalism that were meant to resum the strong correlations from these contributions[65, 66, 67, 68]. These issues were overcome with the introduction of the SRG evolution to lowresolution scales, which makes nuclear interactions genuinely perturbative, albeit at the cost ofinducing three-and higher many-body interactions [1]. As a consequence, MBPT has undergonea renaissance in nuclear physics in the past decade [69], leading to efficient applications for thecomputation of ground-state properties [5, 70, 6] and the construction of effective Shell Modelinteractions and operators (see, e.g., Refs. [71, 72, 73, 74], or the reviews [75, 76] and referencestherein). These successes have also motivated the development of novel types of MBPTs [77, 78, 69].In a nutshell, MBPT assumes that the Hamiltonian can be partitioned into a solvable part H and a perturbation H I , H = H + H I , (16)which then allows an order-by-order expansion of its eigenvalues and eigenstates in powers of H I ,usually starting from a mean-field solution. In the Rayleigh-Schr¨odinger formulation of MBPT,which is widely used for its convenience, | Ψ (cid:105) = | Φ (cid:105) + ∞ (cid:88) n =1 (cid:18) H I H − E (0) (cid:19) n | Φ (cid:105) , (17) E = E (0) + ∞ (cid:88) n =0 (cid:104) Φ | H I (cid:18) H I H − E (0) (cid:19) n | Φ (cid:105) , (18)where E (0) is the unperturbed energy. If we assume that the reference Slater determinant | Φ (cid:105) hasbeen variationally optimized by solving the Hartree-Fock equations, E in Eq. (15) is the Hartree-Fockenergy and f is diagonal. Then we can introduce the so-called Møller-Plesset partitioning, H = E + (cid:88) p f p { a † p a p } , H I = 14 (cid:88) pqrs Γ pqrs { a † p a † q a s a r } , (19)and note that the Slater determinants of the basis introduced in Section 2.3.1 are eigenstates of H : H | Φ a...i... (cid:105) = ( E + f a + . . . − f i − . . . ) | Φ a...i... (cid:105) . (20) This is a provisional file, not the final typeset article ergert A Guided Tour of
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The eigenvalues of H then become the unperturbed energies appearing in Eqs. (17),(18), and theenergy including a finite number of correction terms can be evaluated straightforwardly. For example,the ground-state energy through second order is given by E = E − (cid:88) abij | Γ abij | f a + f b − f i − f j . (21)For a more detailed discussion, we refer to Ref. [69] and references therein.The expression (21) can serve to illustrate both advantages and drawbacks of an MBPT treatmentof nuclei. We see that the second-order energy can be evaluated very efficiently, since it requires anon-iterative calculation whose computational effort scales polynomially in the single-particle basissize N , namely as O ( N ). The reason is that the construction of the Hamiltonian matrix (Fig. 3(a))can be avoided. In fact, the computational scaling is even more favorable, because we can distinguishparticle and hole states and achieve O ( N p N h ), and we typically have N h ∼ A (cid:28) N p . Although thereis a proliferation of terms with increasing order [79, 63, 69], MBPT is still fundamentally polynomialand therefore more efficient than an exact diagonalization, whose cost scales exponentially with N . Itis also clear from Eq. (21) that the expansion of the exact eigenvalue will break down if one (or more)of the energy denominators become small due to (near-)degeneracies of the unperturbed energies.Thus, MBPT works best for ground states in systems with a strong energy gap, i.e., closed-shellnuclei, although extensions for more complex scenarios exist (see Refs. [68, 63, 69] and referencestherein). A noteworthy new development is Bogoliubov MBPT, in which particle number symmetryis broken and eventually restored [77, 80, 81].As mentioned at the beginning of this section, MBPT can be used to derive effective interactionsand operators. The primary tool for such efforts is the ˆ Q -box or folded-diagram resummation of theperturbative series (see Refs. [82, 75, 76] and references therein). As already mentioned in our discussion of the SRG in Section 2.2, we could envision applying SRGtechniques not only to preprocess the nuclear interactions, but also to compute eigenvalues andeigenstates. For all but the lightest nuclei, applying the SRG to the Hamiltonian matrix is hopeless,so we work with the operators instead.Let us again consider the matrix representation shown in Fig. 3 (a). We want to design atransformation that will decouple the one-dimensional 0p0h block in the Hamiltonian matrix,spanned by a reference state Slater determinant | Φ (cid:105) , from all excitations as the flow equation (2)is integrated. The matrix element in this block will then be driven towards an eigenvalue (up totruncation errors), and the unitary transformation becomes a mapping between the reference Slaterdeterminant and the exact eigenstate (see below). In principle, we could use a suitably chosenreference to target different eigenstates, e.g., by taking references which are expected to have alarge overlap with the target state (see Section 10.3 in Ref. [58]). In practice, we usually target theground state by using a Hartree-Fock Slater determinant as our reference.To implement the operator flow, we need to choose an operator basis to express H ( s ) and thegenerator η ( s ). Instead of using the basis (10), we switch to operators that are normal ordered with Frontiers 9 ergert
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Nuclear Many-Body Theory respect to the reference state | Φ (cid:105) : B = (cid:110) { a † p a q } , { a † p a † q a s a r } , { a † p a † q a † r a u a t a s } , . . . (cid:111) pqrstu... ∈ N . (22)Commutators of these operators can feed into terms of lower particle rank: For instance, a commutatorof M -body and N − body operators generates | M − N | -body through ( M + N − M, N ) (cf. Section 2.2). As a result, the complexityof the flow equations for the operators’ coupling coefficients increases due to the appearance ofadditional terms that depend on the contractions introduced in Eqs. (12) and (13). These contractionstranslate into density matrices (or occupation numbers) — hence the name In-Medium SRG. At thesame time, we achieve a reduction of the truncation error because only the residual , contraction-independent parts of the operators (12) and (13) are omitted. In the majority of applications todate, we truncate all operators and their commutators at the two-body level, defining the IMSRG(2)truncation scheme. More details can be found in Refs. [11, 53, 58, 76].In the chosen basis we now identify the parts of the Hamiltonian that are responsible for couplingthe reference state to 1p1h and 2p2h excitations, and define the off-diagonal Hamiltonian (cf. 2.2) as H od ≡ (cid:88) ai f ai { a † a a i } + 14 (cid:88) abij Γ abij { a † a a † b a j a i } + H. c. . (23)We use this H od to construct a generator, either using Wegner’s ansatz (4) or an alternativechoice [11, 53]. Plugging the generator into the operator flow equation (2), we obtain a systemof flow equations for the energy E ( s ) and the coefficients f pq ( s ) , Γ pqrs ( s ) , . . . (cf. Eq. (8) andRefs. [11, 53, 76]). By integrating these flow equations, we evolve the Hamiltonian operator so thatits matrix representation assumes the shape shown in Fig. 3 (b). We note that the suppressionof H od not only leads to the desired ground-state decoupling, but also eliminates the outermostband in the Hamiltonian matrix. This simplification makes the evolved Hamiltonian an attractiveinput for other approaches, e.g., configuration interaction (CI) or equation-of-motion methods (seeRefs. [83, 27, 76, 84, 85, 86, 29] and discussion below). Valence-Space IMSRG.
Soon after introducing the IMSRG in nuclear physics [87], Tsukiyama,Bogner and Schwenk proposed the use of the IMSRG flow to derive Hamiltonians (and other effectiveoperators) for use in nuclear Shell Model calculations [88]. This is achieved by partitioning thesingle-particle basis into core, valence, and beyond-valence states, normal ordering all operators withrespect to a Slater determinant describing the closed-shell core, and extending the definition of theoff-diagonal Hamiltonian (23) to include all terms that couple valence and non-valence states. Theeigenvalue problem for the evolved Hamiltonian can then be solved in the valence space with widelyavailable Shell model codes [89, 90, 91, 92, 93]. After a study of the oxygen isotopic chain revealedan increasing overbinding away from the chosen core [26], we adopted a normal-ordering schemethat uses an ensemble of Slater determinants to account for partially filled shells in open-shell nuclei[54, 27]. This improved operator basis, along with the valence decoupling procedure and subsequentShell Model diagonalization defines what is nowadays called the valence-space IMSRG (VS-IMSRG)— see Ref. [76] for a recent review.
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Correlated Reference States
H. Hergert - “Progress in
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Techniques in Nuclear Physics”, TRIUMF, Vancouver, March 1, 2018 ! MR-IMSRG(2) . . . MR - IMSRG: build correlations on top of already correlated state (e.g., from a method that describes static correlation well) use generalized normal ordering with 2B,… densities
Figure 4.
Schematic view of correlations in nuclei. Solid circles indicate nucleons, transparentcircles hole states, and dashed ellipses indicate correlations between nucleons. Certain 2p2h, 3p3hand higher correlations (indicated in blue) are built into a correlated wave function that thenserves as the reference state for an MR-IMSRG(2) calculation (capturing correlations indicated inred), while up to an IMSRG(A) calculation would be needed for an equivalent description in theconventional framework.
Correlated Reference States and Multi-Reference IMSRG.
Another important developmentwas the extension of the IMSRG formalism to correlated reference states, in the so-called Multi-Reference IMSRG (MR-IMSRG) [20, 53, 58]. The unitarity of the IMSRG transformation allows usto control to what extent correlations are described by either the Hamiltonian or the reference state.We can see this by considering the stationary Schr¨odinger equation and applying U ( s ): (cid:104) U ( s ) HU † ( s ) (cid:105) U ( s ) | Ψ k (cid:105) = E k U ( s ) | Ψ k (cid:105) . (24)The transformation shifts correlations from the wave function into the evolved, RG-improvedHamiltonian H ( s ) = U ( s ) HU † ( s ), and any many-body method that uses this Hamiltonian as inputnow needs to describe U ( s ) | Ψ k (cid:105) , which should be less correlated than the exact eigenstate | Ψ k (cid:105) . Inthe extreme cases, U ( s ) = 1 and the wave function carries all correlations, or U ( s ) has shifted allcorrelations into the Hamiltonian and | Φ (cid:105) = U ( s ) | Ψ (cid:105) is a simple Slater determinant.Correlated reference states can be particularly useful for the description of systems with strong static or collective correlations, like open-shell nuclei with strong intrinsic deformation or shapecoexistence. Reference states that describe these types of correlations efficiently, e.g., throughsymmetry breaking and restoration (also see Section 2.3.4), are an ideal complement to the IMSRGtransformation, which excels at capturing dynamic correlations, involving the excitation of a fewparticles up to high energies. This complementarity is schematically illustrated in Fig. 4: Collectivecorrelations that would require as much as an IMSRG(A) calculation in the conventional approachare built into the reference state, and an MR-IMSRG(2) calculation is sufficient to treat the bulk ofthe dynamical correlations in the system.Reference state correlations are built into the MR-IMSRG framework by using a generalizednormal ordering [94, 95, 53] that is extended with contractions of higher rank, namely the irreducible k -body density matrices λ ( k ) : λ pq ≡ ρ pq , (25) λ pqrs ≡ ρ pqrs − ρ pr ρ qs + ρ qr ρ ps , (26) Frontiers 11 ergert
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Nuclear Many-Body Theory etc. The irreducible densities matrices encode the correlation content of an arbitrary referencestate | Φ (cid:105) , hence they vanish for Slater determinants. While the basis of normal-ordered operatorssuperficially is the same as in the conventional IMSRG, shown in Eq. (22), the inclusion of theirreducible densities (cf. Eqs. (12) and (13)) equips the basis with the capability to describe thecorrelations that are present in the reference state, which in turn should help to reduce MR-IMSRGtruncation errors. To understand this, let us assume that we know the ground state of our system,and we normal order the Hamiltonian with respect to this correlated state. Then the zero-body partof the normal ordered Hamiltonian already is the exact ground-state energy, and the normal-orderedone-, two- and higher-body parts do not matter at all for our result, and neither does their evolutionunder an exact or truncated MR-IMSRG flow. Thus, the better the reference state matches theground state, the less work the MR-IMSRG evolution and any subsequent many-body method haveto do to obtain the correct ground-state energy. Computational Scaling and Magnus Expansion.
The computational scaling of all threeIMSRG flavors discussed here — traditional, VS-IMSRG, and MR-IMSRG — is governed by thetruncation scheme. If we truncate operators and commutators at the two-body level, as brieflymentioned above, the number of flow equations scales as O ( N ) with the single-particle basis size N ,and the computational effort for evaluating the right-hand sides as O ( N ). This holds despite thegreater complexity of the MR-IMSRG flow equations, which contain terms containing irreducibletwo- and higher-body density matrices.Any observables of interest must, in principle, be evolved alongside the Hamiltonian for consistency,which would create a significant overhead. In practice, we can address this issue by using the so-calledMagnus formulation of the IMSRG [96, 83, 58, 76]: Assuming that the IMSRG transformationcan be written as an explicit exponential, U ( s ) = exp Ω( s ), we can solve a single set of flowequations for the anti-Hermitian operator Ω( s ) instead of evolving observables separately. Alloperators of interest can then be computed by applying the Baker-Campbell-Hausdorff expansion to O ( s ) = exp[Ω( s )] O exp[ − Ω( s )]. IMSRG Hybrid Methods.
As noted earlier in this section, the conventional IMSRG evolutionmakes the matrix representation of the Hamiltonian more diagonal by suppressing couplings betweenthe npnh excitations of the reference state. This implies a decoupling of energy scales of the many-body system, analogous to the decoupling of momentum scales by the free-space SRG, althoughthere are differences in detail that are associated with the operator bases in which the flow isexpressed (cf. Eqs. (10), and (22)).From this realization, it is not a big step to consider using the IMSRG to construct RG-improvedHamiltonians for applications in other methods, defining novel hybrid approaches. In fact, eventhe original IMSRG formulation can be understood from this perspective: The evolution generatesa Hamiltonian that yields the exact ground-state energy (up to truncations) in a Hartree-Fockcalculation, except the HF equations are automatically satisfied for the evolved H , and we can readoff the ground-state energy directly. The same Hamiltonian can then be used as input for EOMmethods to compute excitation spectra [83]. Likewise, the VS-IMSRG produces an RG-improvedHamiltonian that serves as input for a Shell Model diagonalization.Applying the same logic as in the VS-IMSRG case, the IMSRG has been merged with the No-CoreShell Model (NCSM, see Section 2.3.6) into the In-Medium NCSM [84, 97]. In this approach,the IMSRG improves the Hamiltonian with dynamical correlations from high-energy few-nucleon This is a provisional file, not the final typeset article ergert A Guided Tour of
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Nuclear Many-Body Theory excitations that would require enormously large model spaces in the conventional NCSM, and theexact diagonalization in a small model space describes the dynamics of many-nucleon excitations.The NCSM as the “host” method is rooted in the same particle-hole expansion picture as the IMSRGitself, but this is not a requirement. Another new hybrid method is the In-Medium GeneratorCoordinate Method (IM-GCM), which relies on the GCM as a host method to capture collectivecorrelations [85, 86, 29]. In this approach, a many-body basis is generated by restoring the symmetriesof mean field solutions with various types of shape and gauge configuration constraints, which isvery different from the particle-hole excitation basis discussed so far.
The Coupled Cluster (CC) method [63, 12] is an older cousin of the IMSRG approach. It can alsobe understood as a decoupling transformation of the Hamiltonian, but in contrast to the IMSRG, itrelies on a non-unitary similarity transformation (see Fig. 3). Traditionally, CC is motivated by anexponential ansatz for the exact wave function of a system, | Ψ CC (cid:105) = e T | Φ (cid:105) , (27)where | Φ (cid:105) is a reference Slater determinant, and T is the so-called cluster operator . This operator isexpanded on particle-hole excitations, T = (cid:88) ph t ai { a † a a i } + 14 (cid:88) abij t abij { a † a a † b a j a i } + . . . , (28)with the cluster amplitudes t ai , t abij , . . . . In practical applications, the T is truncated to include up to2p2h (CC with Singles and Doubles, or CCSD) or 3p3h terms (CCSDT, including Triples). Variousschemes exist for iteratively or non-iteratively including subsets of Triples [98, 99, 63, 100, 12]. Whenit acts on the reference state | Φ (cid:105) , e T admixes arbitrary powers of few-particle, few-hole excitations.Note, however, that the cluster operator T is not anti-Hermitian because it lacks de-excitationoperators, and therefore e T is not unitary.The cluster amplitudes are determined by demanding that the transformed Hamiltonian, H CC ≡ e − T He T , (29)does not couple the reference to 1p1h and 2p2h states (see Fig. 3). Using notation introduced inSection 2.3.1, the decoupling conditions lead to the following system of non-linear equations: (cid:104) Φ | e − T He T | Φ (cid:105) = E CC , (30) (cid:104) Φ ai | e − T He T | Φ (cid:105) = 0 , (31) (cid:104) Φ abij | e − T He T | Φ (cid:105) = 0 . (32)Here, E CC is the CC ground-state energy, which corresponds to the one-dimensional block in theupper left of Fig. 3 (c) and is analogous to the zero-body part of the IMSRG-evolved Hamiltonian, asdiscussed in the previous section. The other blocks in the first column of the matrix vanish becauseof the CC equations (30)–(32). Frontiers 13 ergert
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Since the CC transformation is non-unitary, one needs to be careful when one evaluates observablesusing the CC wave function, or uses H CC as input for equation-of-motion calculations or otherapplications [63, 12]. For instance, the non-Hermiticity of H CC forces us to consider left and righteigenstates separately. This is a drawback compared to unitary transformation methods like theIMSRG. Coupled Cluster also has advantages, though: For instance, the Baker-Campbell-Hausdorffexpansion appearing in Eqs. (30)–(32) automatically terminates at finite order because the clusteroperator only contains excitation operators. For the same reason, Eq. (31) will automatically solvethe Hartree-Fock equations, so any Slater determinant is equally well suited as a reference state,while MBPT, IMSRG and even exact diagonalization approaches exhibit (some) reference-statedependence. Symmetry Breaking and Collective Correlations.
While most applications of CC theory innuclear physics have enforced and exploited spherical symmetry, the capabilities for performing M -scheme calculations that allow nuclei to develop intrinsic deformation have existed for more thana decade. This is a more natural approach for capturing collective correlations than the constructionof Triples, Quadruples (4p4h) and ever higher particle-hole excitations of a spherical reference(cf. Section 2.3.3). Converging such calculations is challenging because the single-particle basistypically grows by an order of magnitude or more, and the broken symmetries must eventually berestored. The formalism for symmetry restoration in CC has been developed in Refs. [101, 102, 103,104]. In fact, the work of Duguet et al. forms the basis of recent works on symmetry breaking andrestoration in MBPT [77, 80, 81]. Applications are currently underway. Shell-Model CC.
Like the IMSRG, the CC framework can be used to construct effective interactionsand operators for Shell model calculations. Initial work in that direction applied Hilbert spaceprojection techniques (cf. Section 2.3.6) to construct a so-called CC effective interaction (CCEI)[24, 105], but the construction of the model spaces via Equation-of-Motion CC methods provedto be computationally expensive. The CCEI approach is now superseded by the Shell Model CCmethod [25], which applies a second similarity transformation to H CC in Fock space, similar toVS-IMSRG decoupling (cf. Section 2.3.3). Unitary CC.
While almost all applications of CC in nuclear physics use the traditional ansatz(27), unitary CC (UCC) approaches that parameterize the wave function as | Ψ UCC (cid:105) = e T − T † | Φ (cid:105) have been used in numerous studies in quantum chemistry (see, e.g., [106, 107]). Unitary CCwave functions have also become a popular ansatz for the Variational Quantum Eigensolver (VQE)algorithm on current and near-term quantum devices [108, 109]. It is also worth noting that therecently revived Unitary Model Operator Approach (UMOA) is closely related to UCC [110, 111]. Self-Consistent Green’s Function (SCGF) theory is another prominent approach for solving thenuclear many-body problem with systematic approximations [112, 113, 114, 115]. The Green’sFunctions in question are correlation functions of the form g pq...rs ≡ (cid:104) Ψ A | T [ a p ( t p ) a q ( t q ) . . . a † s ( t s ) a † r ( t r )] | Ψ A (cid:105) , (33)which describe the propagation of nucleons in the exact ground state | Ψ A (cid:105) of the system. UsingWick’s theorem, the exact A -body propagator (33) can be factorized into products of irreducibleone-, two-, etc. propagators, similar to the decomposition of density matrices briefly touched upon This is a provisional file, not the final typeset article ergert A Guided Tour of
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Nuclear Many-Body Theory in Section 2.3.3. One can then formulate coupled equations of motion for propagators, and introducetruncations to obtain polynomially scaling methods, again somewhat analogous to IMSRG and CC.We must remain aware that the propagators of SCGF, the induced operators of IMSRG, and the CCamplitudes are all different objects, and while their definitions may make the seem complementaryto each other, there are subtle distinctions. One of these is that the g ( k ) are formally defined withrespect to the exact wave function, while IMSRG and CC use definitions with respect to a referencestate.Practical implementations of the SCGF technique usually work with the Fourier transforms of thepropagators to the energy domain. One needs to solve integral equations of motion of the form g = g + g Σ g , (34)where g is the propagator of the non-interacting system and Σ a kernel that encodes the particles’interactions, which is constructed using diagrammatic techniques. For example, the one-bodypropagator is obtained by solving g pq ( ω ) = g (0) pq ( ω ) + (cid:88) rs g (0) pr Σ rs ( ω ) g sq , (35)the so-called Dyson equation. From this propagator, one can compute the one-body density matrix ρ pq = (cid:104) Ψ A | a † q a p | Ψ A (cid:105) = (cid:90) C + dω πi g pq ( ω ) , (36)where C + indicates an integration contour in the complex upper half plane. Higher-body densitymatrices are connected to the corresponding higher-body propagators in analogous fashion. Usingthe density matrices, one can then evaluate any operator expectation values of interest. For moredetails, we refer to the contributions [10, 115] to the present volume, and the works cited therein.Current applications of SCGF techniques in nuclear physics make use of the so-called AlgebraicDiagrammatic Construction (ADC) scheme, with increasing orders, denoted by ADC(n), convergingto an exact solution. For closed-shell nuclei, calculations up to ADC(3) are be performed regularly,which contain correlations that are roughly comparable to IMSRG(2) with a perturbative 3p3hcorrection (see Section 2.3.3 and Refs. [116, 83, 86]) and CCSD(T) (cf. Section 2.3.4). Som`a andcollaborators have extended the ADC scheme to open-shell nuclei by using Gor’kov Green’s Functionswith explicitly broken particle number symmetry [117, 118]. Applications of this framework haveused a self-consistent second-order scheme, denoted Gor’kov-ADC(2), and the extension to Gor’kov-ADC(3) as well the integration of particle-number projection to restore the broken number symmetryare in progress [80, 114].While the computation of the Green’s Functions tends to be a more involved task than solving theIMSRG flow equations or CC amplitude equations, the propagator contains more information froma single computation than these other methods. For instance, one can immediately extract spectralinformation about the neighboring nuclei and the response of the system [119, 120], which requires theapplication of additional techniques in the IMSRG [83] and CC approaches [121, 122, 12], or, indeed,the computation of the Green’s Function using similarity-transformed operators. Furthermore, thekernels of the equations of motion (34) are energy-dependent effective interactions that govern thedynamics of (few-)nucleon-nucleus interactions. For example, the one-nucleon self-energy in Eq. (35) Frontiers 15 ergert
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Nuclear Many-Body Theory is an ab initio version of an optical potential, as used in reaction theory [123, 124, 125]. We willreturn to this discussion in Section 4.5.
No-Core Configuration Interaction Methods.
The most straightforward but also mostcomputationally expensive approach to solving the many-body Schr¨odinger equation is to exactlydiagonalize the Hamiltonian in a basis of many-body states. In general, we refer to such approachesas No-Core Configuration Interaction (NCCI). “No core” makes it explicitly clear that all nucleonsare treated as active degrees of freedom, in contrast to the nuclear Shell model discussed below.In light nuclei, the exact diagonalization can be directly formulated in Jacobi coordinates, usingtranslationally invariant harmonic oscillator [126] or hyperspherical harmonic wave functions [127,128]. Since the construction of the basis states themselves and the matrix representation of theHamiltonian becomes increasingly complicated and computationally expensive as the particlenumber grows, one eventually has to switch to Slater determinants in the laboratory system, usinga construction along the lines discussed in Section 2.3.1.A common choice for the single-particle basis in the laboratory system are spherical harmonicoscillator (SHO) states, because they allow an exact separation of center-of-mass and intrinsicdegrees of freedom provided one uses an energy-based truncation for the model space [129, 130].These choices define what we specifically call the No-Core Shell Model (NCSM). A disadvantageof using SHO orbitals is that they are not optimized to the energy scales of specific nuclei, andthey are poorly suited for describing physical features like extended exponential wave functiontails. Other popular choices are Hartree-Fock single-particle states, and perturbatively [131] ornonperturbatively enhanced natural orbitals [132, 133, 134]. Model spaces built on these basesno longer guarantee the separation of center-of-mass and intrinsic coordinates, but fortunately,center-of-mass contaminations either remain small automatically [135], or they can be suppressedusing techniques like the Lawson method [136].
Importance Truncation and Symmetry Adaptation.
As indicated above, the main issuewith exact diagonalization approaches is the exponential (or greater) growth of the Hilbert spacedimension, which is proportional to (cid:0) NA (cid:1) with single particle basis size N and particle number A .A variety of strategies can be used to address this often-quoted “explosion” of the basis size. Onedirection is to avoid the construction of the full model space basis by applying importance-basedtruncation or sampling methods, leading to the Importance-Truncated NCSM [9] or Monte-Carlo(No-Core) CI approaches [137, 138]).Another important research program is the exploration of many-body states that are constructedfrom the irreducible representations (irreps) of the symplectic group Sp(3, R ), which describes anapproximate emergent symmetry of finite nuclei [139, 140]. An exact diagonalization in such asymmetry-adapted basis will offer a much more efficient description of nuclear states with intrinsicdeformation than the conventional NCSM, which would need to use massive model spaces withmany-particle-many-hole excitations. This reduction of the model space dimensions also allows suchsymmetry-adapted NCSM [139, 140] and NCCI approaches [141] to reach heavier nuclei than theconventional versions. Interacting Nuclear Shell Model with a Core (Valence CI).
Instead of treating all of thenucleons as active, one can also factorize the nuclear wave function by introducing an inert core and
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Nuclear Many-Body Theory only treat the interactions of a smaller number of valence nucleons via appropriately transformedinteractions: | Ψ (cid:105) = | Ψ (cid:105) core ⊗ | Ψ (cid:105) valence . (37)This, of course, is the traditional nuclear Shell model approach. Even with the substantial reductionof the single-particle basis to a relatively small number of valence orbitals, the numerical cost for anexact diagonalization quickly becomes unfeasible for many medium-mass and heavy nuclei, especiallyif one needs multi-shell valence-spaces to capture complex nuclear structure features like coexistingintrinsic shapes.In previous sections, we have discussed how a variety of many-body methods can be used to derivevalence-space interactions, hence it is not a surprise that this is possible in NCCI approaches aswell. One strategy is to project solutions of no-core calculations for the core and its neighboringnuclei onto a valence-configuration space to extract the effective Hamiltonian. The viability of thisapproach has been demonstrated in several publications [142, 143, 144, 145], although there areambiguities in the extraction of the valence-space Hamiltonian, and the initial NCCI calculationsthat serve as input for the projection rapidly become expensive. Description of Continuum Effects and Nuclear Dynamics.
An important breakthrough in ab initio calculations for light nuclei has been the merging of the NCSM with resonating groupmethod (RGM) techniques [130, 146]. This makes it possible to describe clustered states as well asreactions between light projectile(s) and targets. In the original NCSM/RGM approach, compactclusters of nucleons are described by NCSM states, which are then used to construct a basis ofconfigurations | χ i (cid:105) that place such clusters at different relative distances. In this basis, one can thensolve the generalized eigenvalue problem, known as the Griffin-Hill-Wheeler equation [147] in theRGM context: H | Ψ (cid:105) = E N | Ψ (cid:105) , (38)where H and N are the so-called Hamiltonian and norm kernels. The latter appears because thechosen basis configurations are not orthogonal in general. The dimension of Eq. (38) is typically small,certainly compared to the NCSM model space, but the computation of the kernels is computationallyexpensive since it relies on the construction of up to three-body transition density matrices. Inrecent years, the NCSM/RGM has been extended to the NCSM with Continuum (NCSMC), whichaccounts for the coupling between the NCSM and RGM sectors of the many-body basis [130]. Itrequires solving the generalized eigenvalue problem (cid:18) h ¯ h ¯ h H (cid:19) (cid:18) Φ χ (cid:19) = E (cid:18) ¯ n ¯ n N (cid:19) (cid:18) Φ χ (cid:19) , (39)where h and are the Hamiltonian and norm kernel in the NCSM sector (the latter being diagonal), H and N the corresponding kernels in the RGM sector (cf. Eq. (38)), and ¯ h and ¯ n encode thecoupling between the sectors of the basis.Alternative approaches to the description of continuum effects in the NCSM are the Single-StateHORSE (Harmonic Oscillator Representation of Scattering Equations) method [148, 149, 150], forwhich the nomen is omen, as well as the No-Core Gamow Shell Model (GSM), a no-core CI approachthat constructs Slater determinants from a single-particle Berggren basis [151] consisting of bound,resonant and scattering states [152, 153, 154, 155]. Frontiers 17 ergert
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The most commonly used Quantum Monte Carlo (QMC) techniques in nuclear physics make useof many-body wave functions in coordinate space representation [156, 157, 158, 159]. As such, theyare well suited for the description of nuclear states with complex intrinsic structures, and theycan readily use interactions with a high momentum cutoff, as opposed to the configuration spacemethods which would exhibit poor convergence in such cases. This allows QMC calculations toexplore physics across the interfaces of the hierarchy of EFTs for the strong interaction (cf. Sections2.1 and 4.4), e.g., for processes that explore energies approaching the breakdown scale of chiral EFT[160, 161, 162, 163].A typical ansatz for a QMC trial state is | Φ T (cid:105) ≡ F ( a ) | Φ( b ) (cid:105) , (40)where F ( a ) is an operator that explicitly imprints correlations on the mean-field like state | Φ( b ) (cid:105) ,and a, b are vectors of tunable parameters. The first step of most QMC calculations is a variationalminimization of the energy in the trial state ,min a , b (cid:104) Φ T | H | Φ T (cid:105)(cid:104) Φ T | Φ T (cid:105) ≥ E , (41)followed by an imaginary-time evolution to project out the true ground state in a quasi-exact fashion: | Ψ (cid:105) ∝ lim τ →∞ e − ( H − E T ) τ | Φ T (cid:105) . (42)This projection can be implemented using Monte Carlo techniques in a variety of ways, which givesrise to different approaches like Green’s Function Monte Carlo (GFMC) or Auxiliary-Field DiffusionMonte Carlo (AFDMC) [156, 158].A major challenge in QMC calculations is that most commonly used algorithms suffer from someform of sign problem [156, 158]. Many quantities of interest like the wave functions or local operatorexpectation values in these wave functions are not positive definite across their entire domain, whichmeans that they cannot be immediately interpreted as probability distributions that the algorithmssample. This is one of the main reasons why QMC methods can only be used with Hamiltoniansthat are either completely local, or have a nonlocality that is at most quadratic in the momenta,e.g., p or l .While QMC applications in ab initio nuclear structure have been focused on coordinate space, there are a wide variety of approaches that merge QMC techniques with the configuration space approachesdiscussed in previous sections. Examples include sampling the intermediate-state summations inMBPT [164], diagrammatic expansions [165, 166, 167], or the coefficients of correlated CC [168] or(No-Core) CI wave functions [137, 138, 169, 170, 171]. Lattice methods are nowadays widely used to simulate the dynamics of nonperturbative field theorieson finite space-time lattices. The most prominent example is Lattice QCD, but implementations ofvarious Effective Field Theories on the Lattice have been developed and applied with impressive
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Nuclear Many-Body Theory outcomes in the past two decades — see, for example, Refs. [172, 173, 174, 175, 173] and referencestherein, which also provide pedagogical introductions to Lattice EFT for nuclear systems.Lattice EFT simulations are built around the partition function, which is defined for a pure state | Ψ (cid:105) as Z ( τ ) = (cid:104) Ψ( τ = 0) | exp ( − Hτ ) | Ψ( τ = 0) (cid:105) . (43)Here, H is an EFT Hamiltonian, typically truncated at a given order of the EFT’s power countingscheme. In practice, the partition function is evaluated as a path integral in which field configurationsare sampled using Monte Carlo techniques. At large τ , one can extract information about the groundstate and low-lying excited states of the system directly from Z (cf. Section 2.3.7), and generalexpectation values can be evaluated using (cid:104) O (cid:105) τ = 1 Z ( τ ) (cid:104) Ψ | exp( − Hτ / O exp( − Hτ / | Ψ (cid:105) . (44)The use of discretized spatial lattices makes Lattice EFT particularly suited for the descriptionof nuclear states with complex geometries like cluster structures [176, 177, 178]. Depending on thesize of the lattices, it will also typically require less computational effort than the imaginary-timeevolution of states that are formulated in continuum coordinates, as in AFDMC or GFMC (seeSection 2.3.7). Moreover, the development of the so-called adiabatic projection method (APM)[179, 180] in recent years has made it possible to compute scattering cross sections for reactions of(light) clusters on the lattice. Conceptually, the APM is reminiscent of the resonating-group methodused to describe reactions in the NCSMC framework discussed in Section 2.3.6.Of course, Lattice EFT is not free of disadvantages, which are usually caused by the discretizationof space(time). The finite size and lattice spacing are related to infrared (long-range, low-momentum)and ultraviolet (short-range, high-momentum) cutoffs of a calculation, which need to be carefullyconsidered. Since the recognition of cutoff scales is an inherent aspect of EFTs, one can systematicallycorrect for these effects [181, 182]. The discrete lattice also breaks continuous spatial symmetriesthat may need to be restored approximately or exactly before comparisons with experimental dataare made [172, 182]. In this section, I will discuss selected achievements of the ab initio nuclear many-body community inthe past decade, and the issues that were encountered in the process. As stated in the introduction,this selection is subjective, and giving full justice to the breadth of research accomplishments isbeyond the scope of this work. I hope that the present discussion will serve as an invitation forfurther exploration, for which the cited literature may serve as a useful starting point.
One of the biggest issues in nuclear theory was the lack of comparability between differentapproaches for describing the structure of medium-mass or heavy nuclei. These nuclei were wellin reach of the Shell Model and nuclear Density Functional Theory (DFT), but whenever issuesemerged, it was unclear whether they resulted from approximations in the many-body method,or deficiencies in the effective interactions, i.e., the valence-space Hamiltonians or energy density
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Nuclear Many-Body Theory �� �� �� �� �� �� �� �� �� - ��� - ��� - ��� - ��� - ��� - �� - �� - �� � � [ � � � ] IT - NCSMMR - IMSRG ( ) VS - IMSRG ( ) CCSD Λ - CCSD ( T ) ADC ( ) Lattice EFT
Figure 5.
Ground-state energies of the oxygen isotopes for various many-body approaches, usingthe chiral NN+3N(400) interaction at λ = 1 .
88 fm − [183]. Details on the Lattice EFT calculationcan be found in Ref. [177]. Gray bars indicate experimental data [184].functionals (EDF). Moreover, one cannot simply perform a valence CI calculation with an EDF,or a DFT calculation with a Shell Model interaction, because the interactions are tailored to theirspecific many-body method.The development of the RG/EFT and many-body methods discussed in Section 2 has opened upa new era for benchmarking the same nuclear interactions across multiple approaches, and on top ofthat, these methods provide a systematic framework for analyzing, and eventually quantifying, thereasons for differences between the obtained results.One of the earliest testing grounds for ab initio calculations of medium-mass nuclei was the oxygenisotopic chain, which was accessible to all of the approaches that emerged at the beginning of thepast decade. Figure 5 shows the ground-state energies of even oxygen isotopes for the same chiralNN+3N interaction, obtained with several of the configuration space approaches introduced inSection 2.3. In addition, results for applying various types of MBPT to the same interaction andnuclei are presented in Ref. [69] — I only refrained from including them here to avoid overloading thefigure. As we can see, the ground-state energies obtained from the different approaches are in goodagreement with each other and with experiment. Since our results include quasi-exact IT-NCSMvalues, the deviation of the other methods’ energies from these values provide us with an estimateof the theoretical uncertainties due to any employed truncations, which is on the order of 1-2%. Aswe can see from Fig. 5, essentially all of the used many-body methods place the drip line in theoxygen isotopic chain at O, although the signal is exaggerated. Continuum effects that have beenomitted in these calculations would lower the energy of the O resonance, which is experimentallyconstrained to be a mere 18(7) keV above the two-neutron threshold [185], and produce a very flattrend in the energies towards O. Similar features were found in calculations for other isotopicchains and other chiral interactions [21, 118, 186, 114]. The O ground state energies obtained forthe employed chiral NN+3N Hamiltonian are also compatible with a Lattice EFT result that wasobtained at a similar resolution scale [177].This last comparison shows that some obstacles to the ideal cross-validation scenario stillremain. Since coordinate-space approaches like Lattice EFT or QMC are truly complementary
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Nuclear Many-Body Theory to configuration-space methods, it would be highly desirable to test the same chiral NN+3NHamiltonians in both types of calculations. However, the Hamiltonians used in configuration spaceare typically given in terms of harmonic oscillator matrix elements (especially if SRG evolved) insteadof the coordinate-space operators required by Lattice EFT or QMC calculations. Furthermore, LatticeEFT and QMC cannot handle all possible types of nonlocality in the Hamiltonian (cf. Section 2.3.7),including the forms generated by the nonlocal regulators that are favored for configuration-spaceHamiltonians. Conversely, local chiral interactions that have been constructed explicitly for QMCapplications [187, 188, 189, 190, 4, 158] exhibit slow model-space convergence in configuration-spacecalculations because they still tend to require a significant repulsive core at short distance to describenucleon-nucleon scattering data, albeit a far weaker one than interactions like Argonne V18 [191].
Ab Initio
Theory
The reach of ab initio many-body theory has increased dramatically over the past decade. Figure1 illustrates this growing coverage of the nuclear chart, but it tells only part of the story. Theexpansion has happened in many “dimensions” besides the mass number A , namely by pushingtowards exotic nuclei via improved treatments of the continuum degrees of freedom, filling in gapsin the coverage that are occupied by doubly open-shell nuclei with strong intrinsic deformation, andexpanding the types of observables that can be computed from first principles. Recalling Section3.1, the ongoing push against the limitations of our many-body approaches will continue to growthe opportunities for benchmarking current- and next-generation chiral Hamiltonians. First calculations for selected nuclei and semi-magic isotopic chains up to tin were already publishedin the first half of the last decade [19, 21, 23]. For the most part, they were using a family of chiralNN+3N interactions that gave a good description of the oxygen ground-state energies (cf. Fig. 5)as well as the spectroscopy of the lower sd -shell region [26, 24]. However, the same interactionsunderpredict nuclear charge radii [192], and start to overbind as we approached the calcium chain(cf. Fig. 7), eventually leading to an overbinding of 1 MeV per nucleon in tin. While model-spaceconvergence in CC, IMSRG and SCGF calculations suggested that calculations for heavier nucleiwould have been technically possible, it made little sense to pursue them.The growing number of results for medium-mass nuclei and the problems they revealed motivateda new wave of efforts to refine chiral interactions. One direction of research aimed to achieve asimultaneous description of nuclear energies and radii up to Ca by including selected many-bodydata in the optimization protocol of the chiral LECs. This work resulted in the so-called NNLO sat interaction [193]. While NNLO sat definitely improved radii [194], its model-space convergence wasfound to become problematically slow already in lower pf -shell nuclei [195, 196, 114].Simultaneously with the efforts to develop new interactions, attention also turned towards anolder, less consistently constructed family of chiral NN+3N interactions that exhibited reasonablesaturation properties in nuclear matter calculations [199, 200]. These forces are referred to as EM λ/ Λ,where λ indicates the resolution scale of the NN interaction, the SRG-evolved N LO potentialof Entem and Machleidt [201], and Λ is the cutoff of an NNLO three-nucleon interaction whoselow-energy constants have been adjusted to fit the triton binding energy and He charge radius[199, 200]. In CC calculations for the nickel isotopes, Hagen et al. demonstrated that the EM1.8/2.0interaction, in particular, allowed a good description of the energies of nuclei in the vicinity of Ni Frontiers 21 ergert
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70 72 74 76 78 80
Mass Number A + E n e r gy ( M e V ) Ni Figure 6.
Energies of the first excited 2 + states from VS-IMSRG [197] and Equation-of-MotionCC [195] calculations for several chiral two- plus three-nucleon interactions. Experimental values[198, 197] are indicated as black bars. Data courtesy of J. D. Holt, J. Men´endez, and G. Hagen.[195]. As shown in Fig. 6, these findings have been reinforced by subsequent VS-IMSRG calculations,as well as the experimental observation of the first excited 2 + state in this nucleus [197].Since this initial application in medium-mass nuclei, the EM λ/ Λ family has seen widespreaduse in ab initio calculations due to its empirical quality, although the Hamiltonian’s theoreticaluncertainties are less well defined than for interactions that obey the chiral power counting morerigorously. Indeed the EM1.8/2.0 interaction was used in VS-IMSRG calculations to produce whatis to my knowledge the first attempt at producing an ab initio mass table for nuclei up to theiron isotopes [186]. For selected nuclei up to the tin region, it also yields converged energies forground and low-lying states that are in good agreement with experimental data [202, 203]. It alsoyields slightly larger radii than previous interactions, although the underprediction is not eliminatedentirely (see Refs. [194, 202] and Section 3.2.3).Multiple applications of the EM λ/ Λ Hamiltonians in support of spectroscopy experiments have beenpublished in recent years (see, e.g., [204, 196, 205, 206, 207]), and additional studies are underway,including an effort to better understand what makes the EM1.8/2.0 Hamiltonian so successful.Furthermore, a new generation of chiral NN+3N interactions is now available for applications inmedium-mass and heavy nuclei [46, 208, 209, 210, 114].
Neutron-rich nuclei are excellent laboratories for disentangling the interplay of nuclear interactions,many-body correlations and the continuum. Thus, data from the experimental push towards thedrip line can offer important constraints for the refinement of chiral interactions if the many-bodytruncations and continuum effects are under control.In practice, ab initio results for observables like the absolute energies of states still exhibitsignificant scale and scheme dependence due to truncations that are made in the EFT, the potentialimplementation of SRG evolutions, and the many-body methods. Since such variations tend to besystematic within families of interactions (and sometimes even across multiple families), differential
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Nuclear Many-Body Theory �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� - ��� - ��� - ��� - ��� - ��� - ��� � � [ � � � ] � �� ���� ��� ����� / ����� + �� ( ��� ) ���� �� �� �� �� �� �� �� �� �� �� �� �� �� �� ������������� � � �� [ � � � ] � �� ���� ��� ����� / ����� + �� ( ��� ) ���� Figure 7.
Ground-state and two-neutron separation energies for several chiral NN+3N interactionsfrom MR-IMSRG(2) calculations. Experimental data are indicated by black bars [184, 211].quantities like separation and excitation energies or transition matrix elements often exhibit a weakerscale and scheme dependence — note, for example, the small systematic variation of the first excited2 + states of the neutron-rich nickel isotopes for EM λ/ Λ interactions. This makes energy differencesan ideal observable for confronting ab initio results with experimental data.Let us consider two-neutron separation energies as a concrete example. Sudden drops in theseobservables are a signal of (sub)shell closures (albeit not universally [194]) and in the neutron-richdomain, they are important indicators for the proximity of the drip line. Figure 7 shows MR-IMSRG ground-state and two-neutron separation energies of the calcium isotopes, obtained with theNN+3N(400) interaction used in Fig. 5, as well as the NNLO sat and EM1.8/2.0 interactions brieflydiscussed in the previous section. We note the overbinding produced by NN+3N(400) and the bafflingaccuracy of the EM1.8/2.0 results, given the approximations that went into the construction of thisforce, as well as the MR-IMSRG truncation. Common to all three interactions is the emergenceof a very flat trend in the ground-state and separation energies in neutron-rich calcium isotopes,which will likely be further enhanced by the inclusion of continuum effects, and extended beyond theshown mass range. Similar flat trends emerge in many isotopic chains, as shown both in ab initio surveys based on chiral interactions [114, 10, 186] as well as a sophisticated Bayesian analysis ofempirical EDF models [212]. Naturally, this will make the precise determination of the neutron dripline in the medium-mass region a challenging task, but also suggests that interesting features likealternating patterns of unbound odd nuclei and weakly-bound even nuclei with multi-neutron haloscould emerge. This is an exciting prospect for the experimental programs at rare-isotope facilities.With the exception of the NCSMC and HORSE methods discussed in Section 2.3.6, the inclusionof continuum degrees in configuration-space techniques has been focused on the use of the Berggrenbasis [151]. While such calculations are challenging due to the significantly increased single-particlebasis size and the difficulties of handling the resulting complex symmetric Hamiltonians, applicationsin CC (see Refs. [12, 213] and references therein), both valence and No-Core Gamow Shell Model[153, 154, 155, 214, 215] and IMSRG [216] calculations have been published. Common to all theseapproaches is that a configuration space interaction that is given in terms of SHO matrix elementsis expanded on a basis containing SHO and Berggren states, hence it is still an open question how adirect implementation of the interactions in a basis with continuum degrees of freedom might modify
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Nuclear Many-Body Theory
Figure 8.
NCSMC spectrum of Be with respect to the n + Be threshold. Dashed black linesindicate the energies of the Be states. Light boxes indicate resonance widths. See Ref. [223] fordetails. Figure reprinted with permission from the American Physical Society.existing results. It is worth noting that such a construction has been achieved for phenomenologicalGSM interactions that have been tuned for light nuclei [217, 218, 219, 220, 221, 222].In light nuclei, the NCSMC has been applied with impressive success to describe a variety ofexotic nuclei with up to three-cluster structures. For example, Calci et al. [223] carried out NCSMCcalculations for Be with several chiral NN+3N interactions to investigate the parity inversion ofthe ground and first-excited states in this nucleus from first principles. The authors found thatthe coupling between the NCSM and RGM sectors of the generalized eigenvalue has strong effects,but that among the tested interactions, only NNLO sat can produce the experimentally observedordering of the states (see Fig. 8). However, it still underpredicts the splitting of these levels andas a result, overestimates the cross section for the photodisintegration Be( γ, n ) Be. Additionalapplications of the NCSMC for exotic nuclei can be found in the review [130] and references therein,as well as the more recent works [224, 225, 226].
The capabilities of ab initio approaches have also significantly expanded when it comes to theevaluation of observables other than the energies.
Nuclear Radii.
Figure 9 shows MR-IMSRG results for the charge radii of calcium isotopes. Theleft panel illustrates the reasonable reproduction of the Ca and Ca charge radii that can beobtained for NNLO sat . The MR-IMSRG(2) results are slightly smaller than the experimental datadue to differences in the truncations from the CCSD charge radius calculations that were used inthe NNLO sat optimization protocol [193]. Note the steep increase in the experimental charge radiibeyond Ca: At the time of the measurement, NNLO sat was the only chiral NN+3N interactionexhibiting this feature, although other more recent interactions can replicate this trend as well[114, 10]. Also note that none of the calculations are able to reproduce the inverted arc of thecharge radii between Ca and Ca. In a CI picture, it is caused by strong mixing with 4p4hexcitations into the pf -shell [227]. Since the MR-IMSRG(2) calculations shown here included only upto (generalized) 2p2h excitations and used particle-number projected Hartree-Fock Bogoliubov vacua This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● �� �� �� �� �� �� �� �� �� �������������������������� � � � � [ � � ] � �� ������ - ���� �� ���� ���� ����� ��� ������� ��� ∆ R c h ( f m ) L (MeV) Ca − Ar PREXthis work GW1708170.140.150.160.170.180.19 ∆ R c h ( f m ) Ca − S ∆ R c h ( f m ) L (MeV) Ca − Ar PREXthis work GW1708170.140.150.160.170.180.19 ∆ R c h ( f m ) Ca − S (cid:4)(cid:3) (cid:78)(cid:77)(cid:78)(cid:77) (cid:4)(cid:3) EM2.0/2.0 (PWA)EM2.2/2.0EM2.0/2.0EM1.8/2.0 (cid:4)(cid:3) (cid:78)(cid:77)(cid:78)(cid:77) (cid:4)(cid:3) (a) (b)
Figure 9.
Panel ( a ) Calcium charge radii from MR-IMSRG(2) calculations with NNLO sat . Theshaded area indicates uncertainties from basis convergence. Black bars and orange circles indicateexperimental data [228, 194]. Panel ( b ): Mirror charge radius difference of Ca and S versus theslope of the symmetry energy, L , at nuclear saturation, for the EM λ/ Λ interactions (symbols asindicated in the legend), compared to Skyrme functionals (solid circles) and Relativistic Mean Fieldmodels (crosses). The band indicates the experimental result from the BECOLA facility at NSCL.See Ref. [229] for details.as reference states that do not contain collective correlations (cf. Section 2.3.3), it is not surprisingthat the inverted arc cannot be reproduced. We will return to this issue of missing collectivity later.While the EM λ/ Λ interactions underpredict the absolute charge radii, they fare quite well in thedescription of radius differences, as suggested in the previous section. Figure 9(b) is adapted from arecent study that suggests a correlation between the charge radius difference of mirror nuclei, ∆ R ch ,and the slope of the symmetry energy in the nuclear matter equation of state [229]. We see thatthe MR-IMSRG results for ∆ R ch are actually compatible with results from a multitude of SkyrmeEDFs, and the value for the magic EM1.8/2.0 interaction falls into the uncertainty band of theexperimental result. Electromagnetic Transitions.
Since the second half of the past decade, ab initio calculations fortransitions in medium-mass nuclei have become more frequent, owing to the appropriate extensions ofthe IMSRG, CC and SCGF methods [230, 204, 231]. While results for transitions that are dominatedby a few nucleons, e.g., M β decays (see Ref. [232] and the discussion below)can be quite good, the description of collective transitions is hampered by inherent truncations ofthese many-body methods, which are better suited for dynamical, few-particle correlations (seeSections 2.3.3 and 2.3.4). Results from the SA-NCSM [139, 140] and the IM-GCM discussed inSection 2.3.3 show that the modern chiral interactions themselves adequately support the emergenceof nuclear collectivity.Consider for example Fig. 10, which shows VS-IMSRG(2) results for the quadrupole transitionfrom the first excited 2 + state to the ground state in C, O and S [230]. The picture is fairlyconsistent for all four chiral NN+3N interactions that were used in the study: The 2 + energies aredescribed quite well, but energies are not very sensitive to the details of the nuclear wave functions.In C, the E E Frontiers 25 ergert
A Guided Tour of
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Nuclear Many-Body Theory C E (2 + ) (MeV) R pp (fm ) h +1 k E k +1 i ( e fm )024 O N LO sat EM1 . / . LO lnl NN + ( ) exp theory S exp theory exp theory
Figure 10.
Energies of the first excited 2 + state, proton mean square radius and quadrupoletransition matrix elements for selected nuclei, based on VS-IMSRG(2) calculations with multiplechiral NN+3N interactions. See Refs. [230] and [76] for more details. Experimental values (withuncertainties indicated by bands) are taken from [228, 233]. Figure courtesy of R. Stroberg.while the matrix element for the collective transition in S is underpredicted by 25-50%. TheNN+3N(400) interaction gives a particularly poor result, but this is also related to the significantunderestimation of the point-proton radius we obtain for this Hamiltonian, as discussed earlier.The result for O deserves special attention. The E O onlyhas neutrons in an sd valence space, so the E E psd valence space,so that the proton dynamics is treated explicitly instead of implicitly by valence-space decoupling.Until recently, we were unable to perform such a multi-shell decoupling because of the IMSRGversion of the intruder-state problem, but a promising workaround was introduced in Ref. [28]. Gamow-Teller Transitions.
In recent years, there have been concerted efforts to understandthe mechanisms behind the empirically observed quenching of Gamow-Teller (GT) transitions inmedium-mass nuclei, in part due to its relevance to neutrinoless double-beta decay searches (seebelow). In Ref. [232], the authors show that this issue is largely resolved by properly accountingfor the scale and scheme dependence of configuration-space calculations. By dialing the resolutionscale to typical values favored by approaches like NCSM, CC and VS-IMSRG, correlations areshifted from the wave functions into induced two- and higher-body contributions to the renormalizedtransition operator, just as in the quadrupole case discussed above.The transition operator, including two-body currents, is consistently evolved to lower resolutionscale alongside the nuclear interactions, keeping the induced contributions. The transition matrix
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Ab Initio
Nuclear Many-Body Theory E x [ M e V ] + + + + + + + + + + + + + + + ( β ) ( β , ω ) EXP ( β , ω ) ( β )IMGCM(EM1.8/2.0) IMGCM(EM2.0/2.0) Ti
75 100 125 150 B ( E + → + ) [e fm ]0 . . . . . M ν Extrap. RamanEM1.8/2.0(12)EM1.8/2.0(16)EM2.0/2.0(16) Pritychenko e max = e max = e max = Figure 11.
IM-GCM description of the neutrinoless double beta decay Ca → Ti, using EM λ/ Λinteractions: Low-lying spectrum of Ti and its compression through the admixing of crankedconfigurations ( a ) and the nuclear matrix element vs. B(E2) transition probability ( b ). See text andRef. [29] for details. Panel ( a ) courtesy of J. M. Yao, panel (b) reprinted with permission from theAmerican Physical Society.elements of the evolved operator are then computed with the NCSM in light nuclei, and VS-IMSRGin sd - and pf -shell nuclei, leading to agreement with experimental GT strengths within a few %. Incontrast, the bare GT operator must be quenched by 20-25% via the introduction of an effectiveaxial coupling, g eff A < g A , to yield agreement with experimental beta decay rates.The GT transitions in light nuclei have also been evaluated in the GFMC, most recently withconsistently constructed local chiral interactions and currents [234, 235]. Interestingly, the inclusionof two-body currents seems to consistently enhance the GT matrix elements, while it tends toquench the matrix element in NCSM calculations. Since this is almost certainly related to thedifferences in the resolution scale and calculation scheme, the disentanglement of these observablesmight yield further insights into the interplay of wave function correlations and the renormalizationof the transition operators. Neutrinoless Double Beta Decay.
Due to the high impact the observation of neutrinoless doublebeta decay (or lack thereof) would have on particle physics and cosmology, the computation ofnuclear matrix elements (NMEs) for neutrinoless double beta decay is a high priority for nuclearstructure theory. Precise knowledge of the NMEs for various candidate nuclei is required to extractkey observables like the absolute neutrino mass scale from the measured lifetimes (or at least, anynew bounds that would be provided by experiment). Most calculations of the NME to date weresubject to the lack of comparability between phenomenological nuclear structure results that wasdiscussed in Section 3.1, hence a new generation of ab initio calculations with quantified uncertaintiesis required.A major step in that direction was the first calculation of the NME for the decay Ca → Tibased on chiral interactions [29]. The IM-GCM approach discussed in Section 2.3.3 was used todescribe the structure of the intrinsically deformed daughter nucleus Ti, achieving a satisfactoryreproduction of the low-lying states and their quadrupole transitions (see Fig. 11). Since the initialpublication (blue spectra in Fig. 11(a)), the description of the excited states has been improvedfurther through the admixing of cranked configurations (red spectra), without affecting the NME
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Nuclear Many-Body Theory (Fig. 11(b)). Work on quantifying the uncertainties due to the many-body method, the Hamiltonian,and the transition operator is underway, in preparation for the computation of the NMEs of morerealistic candidate nuclei like Ge and
Xe.
From the computation of transitions between low-lying levels, it is only a small step to thecomputation of nuclear response functions and cross sections, although the implementation can bechallenging and the applications are often computationally expensive.
Nuclear Response Functions.
In light nuclei, GFMC is a powerful yet numerically heavy toolfor computing exact nuclear response functions (see, e.g., Refs. [236, 237]). In medium-mass nuclei,applications of SCGF and CC techniques to the computation of the nuclear response have beenpublished in recent years. As mentioned in Section 2.3.5, the Green’s functions computed in thestandard or Gor’kov ADC Green’s function schemes inherently contain information about the nuclearresponse that has been used to study both electromagnetic and weak processes of medium-massnuclei [120, 238, 239, 119, 240].In the Coupled Cluster framework, response functions have been computed by merging CC with upto Triples excitations with the Lorentz Integral Transformation (LIT) technique [241, 242, 243, 244,245]. Immediately after its inception, this approach was used to for the first ab initio calculations ofdipole response and the related photodisassociation cross section of medium-mass closed-shell nuclei[241, 242]. More recently, it was used to compute the electric dipole polarizability α D of nuclei like Ca [243, 246, 244] and Ni [247]. Together with measurements of the charge radius, this quantitycan be used to constrain ab initio calculations that will in turn allow the theoretical extraction ofthe neutron point radius as well as the thickness of the neutron skin.An important application for nuclear response calculations is to map out the neutrino response of Ar, the primary target material in detectors for the short-baseline [248] and long-baseline neutrinoexperiments, like the Deep Underground Neutrino Experiment (DUNE) Far Detector [249, 250].At low energies, the cross section for coherent neutrino elastic scattering is essentially determinedby the weak form factor of Ar, which has recently been computed using CC techniques [251].This work is complementary to SCGF calculations of the neutrino response in the region of thequasi-elastic peak by Barbieri et al. [238].
Nuclear Reactions.
As discussed in Section 2.3, there has been enormous progress in thedevelopment of unified treatments of ab initio nuclear structure and reactions. Here, I wantto highlight two among a bevy of impressive results. Figure 12(a) shows S − and D − wave phaseshifts for α − α scattering, computed order by order in Lattice EFT [179, 180]. These calculations aremade possible by the lattice’s capability to describe clustered states (also see Refs. [176, 177, 178]),as well the development of the APM and associated algorithms. The results for the phase shiftsshow the desired order-by-order improvement, and the inclusion of higher-order terms of the chiralexpansion is expected to improve agreement with experimental data. The near identical NLO andNNLO phase shifts in the S − wave appear to be the result of an accidental cancellation that is notoccurring in the D − wave phase shifts.In Ref. [252], the authors studied deuterium-tritium (D-T) fusion using the NCSMC. One of themain results of this work is shown in Fig. 12(b), which compares the NCSMC D–T reaction ratesfor polarized and unpolarized fuels to each other, as well as rates obtained with several widely used This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory δ ( d e g r ee s ) E Lab (MeV) Afzal et al.
LO (no Coulomb)NLONNLO 0 40 80 120 160 200 0 2 4 6 8 10 12 0 40 80 120 160 200 0 2 4 6 8 10 12 δ ( d e g r ee s ) E Lab (MeV) 0 40 80 120 160 200 0 2 4 6 8 10 12 0 40 80 120 160 200 0 2 4 6 8 10 12 δ ( d e g r ee s ) E Lab (MeV) 0 40 80 120 160 200 0 2 4 6 8 10 12 0 40 80 120 160 200 0 2 4 6 8 10 12 δ ( d e g r ee s ) E Lab (MeV)Afzal et al.
LO (no Coulomb)NLONNLO 0 40 80 120 160 200 0 2 4 6 8 10 12 0 40 80 120 160 200 0 2 4 6 8 10 12 δ ( d e g r ee s ) E Lab (MeV) 0 40 80 120 160 200 0 2 4 6 8 10 12 T hermonuclear reaction rates of light nuclei are critical tonuclear science applications ranging from the modeling ofbig-bang nucleosynthesis and the early phases of stellarburning to the exploration of nuclear fusion as a terrestrial sourceof energy. The low-energy regime (tens to hundreds of keV)typical of nucleosynthesis and fusion plasmas is challenging toprobe due to low counting rates and the screening effect ofelectrons, which in a laboratory are bound to the reacting nuclei.A predictive understanding of thermonuclear reactions is there-fore needed alongside experiments to achieve the accuracy and/orprovide part of the nuclear data required by these applications. Asalient example is the fusion of deuterium (D) with tritium ( H orT) to generate a He nucleus ( α -particle), a neutron, and 17.6MeV of energy released in the form of kinetic energy of theproducts. This reaction, used at facilities such as ITER and NIF in the pursuit of sustained fusion energy production, is char-acterized by a pronounced resonance at the center-of-mass (c.m.)energy of 65 keV above the free D and T nuclei due to the for-mation of the J π = + resonance of the unbound He nucleus.Fifty years ago, it was estimated that, in the ideal scenario inwhich the spins of the reactants are perfectly aligned in a total-spin 3/2 con fi guration and assuming that the reaction is isotropic,one could achieve an enhancement of the cross section by a factorof δ = . However, while the unpolarized cross section and someanalyzing-power data exist, no correlation coef fi cients have beenmeasured yet to con fi rm this prediction . More generally, whatlittle is known about the properties of the polarized DT fusionwas inferred from measurements of the D He reaction .The DT fusion is a primary example of a thermonuclearreaction in which the conversion of two lighter elements to aheavier one occurs through the transfer of a nucleon from theprojectile (D) to the target (T). Despite the fairly small number ofnucleons involved in this process, arriving at a comprehensiveunderstanding — in terms of the laws of quantum mechanics andthe underlying theory of the strong force (quantum chromody-namics) — of the interweaving of nuclear shell structure andreaction dynamics giving rise to the DT fusion already representsa formidable challenge for nuclear theory.Towards this goal, a pioneering ab initio study of the DT fusionwas carried out in ref. , using a nucleon-nucleon (NN) interac-tion that accurately describes two-nucleon data and representingthe wave function on a basis of continuous “ microscopic-cluster ” states made of D + T and n + He pairs in relative motion withrespect to each other. However, this approach was unable to yieldresults of adequate fi delity, due to the omission of the three-nucleon (3N) force — disregarded for technical reasons. Numerousstudies have shown that this component of the nuclear interac-tion is essential for the reproduction of single-particle proper-ties – , masses – , and spin properties , all impactful in thepresent case. Besides the 3N force, the approach of ref. alsolacked a complete treatment of short-range fi ve-nucleon corre-lations, which are crucial to arrive at the accurate description ofthe 3/2 + resonance. The formation of this rather long-livedresonance as a correlated, localized system of fi ve nucleons builtup during the fusion process is integral to the reaction mechan-ism. Finally, no polarization observables were calculated in thestudy of ref. .In the following, we report on ab initio predictions for thepolarized DT fusion using validated NN and 3N forces derived inthe framework of chiral effective fi eld theory (EFT) , a pow-erful tool that enables the organization of the interactions amongprotons and neutrons in a systematically improvable expansionlinked to the fundamental theory of quantum chromodynamics.The quantum-mechanical fi ve-nucleon problem is solved usingthe no-core shell model with continuum (NCSMC) , where the model space includes D + T and n + He microscopic-clusterstates, plus conventional static solutions for the aggregate Hesystem . This enables a fully integrated description of the reac-tion in the incoming (outgoing) channel, where the reactants(products) are far apart, as well as when all fi ve nucleons are closetogether. We show that this approach yields an accurate repro-duction of the DT cross section for unpolarized reactants, dis-criminating among reaction rates from phenomenologicalevaluations and demonstrating in detail the small contribution ofanisotropies in the vicinity of the 3/2 + resonance. The maximumenhancement of the polarized cross section varies as a function ofthe deuterium incident energy, dropping signi fi cantly above 0.8MeV. However, such variation is slow in the narrow range ofoptimal energies for the reaction, resulting in a rather constantenhancement of the rate of fusion, compatible with the historicapproximate estimate. Results
Validation of model for unpolarized reaction observables . Webegin our study with a validation of our ab initio reactionmethod on existing experimental data for the unpolarized DT × ab × S – f a c t o r ( b × M e V ) × –1 × –2 × –3 × –4 –3 –2 –1 ( b × s r –1 ) ! Ω × × × E c.m. (keV) E D (MeV) × × –2 × –1 × Experimental dataNCSMC–phenoNCSMC NCSMC–phenoEvaluated dataNCSMC –2 –1 Fig. 1
Unpolarized DT cross sections. a Astrophysical S-factor as a functionof the energy in the center-of-mass (c.m.) frame, E c.m. , compared toavailable experimental data – (with error bars indicating the associatedstatistical uncertainties). b Angular differential cross section ∂ σ ∂ Ω ! " as afunction of the deuterium incident energy, E D , at the c.m. scattering angle of θ c.m. =
0° compared to the evaluated data of ref. . In the fi gures “ NCSMC ” and “ NCSMC-pheno ” stand for the results of the presentcalculations before and after a phenomenological correction of − + resonance ARTICLE
NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08052-6 compilation , which is intended for applications in astrophysicssimulations. Overall, we fi nd that they agree well even at energiesabove the resonance. In more detail, our calculation agrees bestwith the phenomenological R -matrix evaluation, particularly athigher energies where data are typically scarcer. In our case, theuncertainties due to the fi niteness of the model space are indistinguishable from the line width. The convergence of our abinitio model is discussed in Supplementary Note 1 (see alsoSupplementary Figs. 1-5 and Supplementary Table 2). A furtheranalysis of the systematic and statistical uncertainties associatedwith the adopted nuclear interaction model, such as thosestemming from the order of the chiral expansion or the uncer-tainty in constraining its parameters, is presently computationallyprohibitive (see also Supplementary Discussion). The phenom-enological correction induces a global shift towards the reactionthreshold, commensurate with that of the resonance centroid. Inpractice, this fi ne tuning is tightly constrained by the requirementto match S-factor data in the energy range below the resonantpeak. The polarized reaction rate shows the same shape,albeit globally enhanced by a factor of ~1.32, in agreementwith the approximate estimate for the chosen polarization.This result follows from the rather slow variation of theenhancement factor of the reaction cross section as a function ofthe energy in the narrow Gamow window (deuteron incidentenergies below a few hundred keV) where the product ofthe Maxwell – Boltzmann distribution with the tunnelingprobability of the nuclei through their Coulomb barrier is sig-ni fi cantly different from zero. It is interesting to note that withpolarization a reaction rate of equivalent magnitude as the apex ofthe unpolarized reaction rate is reached at lower temperatures,that is less than 30 keV compared to 65 keV (where bothrates peak), as highlighted in Fig. 5 by the arrows. As a naiveillustration, this means that by using polarized DT fuel the outputof a standard fusion reactor could either be enhanced by 32% orits operational temperature decreased by as much as 45%. A morecomprehensive discussion of the economics of using polarizedfuel in the case of inertial con fi nement fusion can be foundin ref. . Angular distribution of the polarized reaction products . Whilethe deviations from a pure J π ¼ = þ ; ‘ ¼ a m a x ! p z q z , p zz p zz p z q z –1.0 –0.5 0.0 0.5 1.0 –2.0–1.00.01.00.50.8 ! NCSMC–pheno; fullNCSMC–pheno; s –wave J " = , E D (MeV) 1.2 1.6 b Fig. 4
Enhancement factor of the polarized DT reaction cross section. a Present results for the enhancement factor ( δ ) of the polarized DT reaction crosssection at the deuterium incident energy of E D =
100 keV as a function of the vector ( p z q z ) and tensor ( p zz ) polarization of the deuterium and tritium. b Computed maximum enhancement factor (over all possible values of p z q z and p zz ) of the polarized DT cross section as a function of the deuteron incidentenergy. The maximum enhancement is always found for p z q z ; p zz ¼
1. The “ NCSMC-pheno; full ” label stands for the results of the present calculationsincluding ‘ ≠ − + resonance. Due to the energy scale of the fi gure, theenhancement factor obtained without such phenomenological correction (that is, the NCSMC result) is indistinguishable from the NCSMC-pheno curve.Also shown is the maximum enhancement factor obtained by retaining only the ‘ ¼ J π ¼ = þ and 1/2 + partial waves, labeled as “ NCSMC-pheno; s -wave J π ¼ ð = þ ; = þ Þ ” × × × × N A 〈 $ 〉 ( c m × m o l –1 × s –1 ) × × × T (keV) × × × PolarizedUnpolarizedBosch and HaleDescouvemontNACRE
Fig. 5
DT reaction rate with and without polarization. Comparison betweenthe computed DT reaction rate ( N A σ ν h i , with N A the Avogadro number) forunpolarized and polarized fuel with aligned spins as a function of thetemperature, T . The “ Polarized ” and “ Unpolarized ” labels stand for thepresent results obtained with the phenomenological correction of − + resonance (dubbed NCSMC-pheno). We usereactants ’ polarization parameters achievable in the laboratory, that is p z ; p zz ¼ : q z ¼ :
8. Also shown for comparison are the unpolarizedreaction rates obtained from the widely adopted parametrization of the DTfusion cross section of Bosch and Hale (labeled as “ Bosh and Hale ” ), fromthe R -matrix fi t of Descouvemont (labeled as “ Descouvemont ” ) and fromthe NACRE compilation (labeled as “ NACRE ” ). The arrows in the fi gureshow that, with polarization, a reaction rate of equivalent magnitude as theapex of the unpolarized reaction rate is reached at lower temperaturesNATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-018-08052-6 ARTICLE (a) (b) Figure 12.
Ab initio calculations of nuclear reactions. Panel (a) : S ( δ ) and D -wave phase shifts( δ ) for α - α scattering at various orders in Lattice EFT. For details, see [179, 180]. Figure courtesy ofS. Elhatisari. Panel (b) : NCSMC results for the deuterium-tritium (D–T) fusion cross section (top)and reaction rate (bottom). The figure compares the rates for unpolarized and polarized fuel, as wellas rates obtained from widely adopted parametrization of the fusion cross section (see Ref. [252] fordetails). The arrows are included to the guide the reader’s eye (see text). Figure reprinted from[252] under a CC BY 4.0 license.parameterizations of the D–T fusion cross section. The NCSMC calculations indicate that for anexperimentally realizable polarized fuel with aligned spins, a reaction rate of the same magnitude asfor unpolarized fuel can be achieved at about half the temperature. Naturally, this suggests thatpolarized D-T fuels will allow a more efficient power generation in thermonuclear reactors. Ab Initio
Calculations
The progress in ab initio calculations over the past decade has not only led to impressive resultsfor nuclear observables, but also revealed the long-surmised underpinnings of empirical models ofnuclear structure. In many cases, the ideas that led to the formulation of such models were shown tobe correct, but they could not be verified at the time because RG and EFT techniques or sufficientcomputing power for a more thorough exploration were not available.
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OFNe Na 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 OFNe Na Mg Al Si P S Cl ArKCa rms=647 keVEM 1.8/2.0+ENO+VS-IMSRG
OFNe Na Mg Al Si P S Cl ArKCa rms=220 keV
391 excitationenergies in sd shell USDB ∆ E [ M e V ] EM1.8/2.0 bare, O ref. ∆ E rms = 1696 keV EM1.8/2.0, VS-IMSRG ∆ E rms = 647 keV USDB ∆ E rms = 220 keV Figure 13.
Deviations between theoretical and experimental excitation energies of 391 sd -shellstates, for (a) the EM1.8/2.0 interaction without valence decoupling, (b) the same interactiontransformed with VS-IMSRG , and (c) the USDB interaction [258]. The points correspond to therespective root-mean-square deviations for each interaction. Figure courtesy of R. Stroberg. The Nuclear Shell Model.
The first prominent example I want to discuss is the nuclear ShellModel and some of the “folklore” surrounding it. We can immediately make the observation that theShell model picture is inherently a low-momentum description of nuclear structure. It is based onthe assumption that nucleons are able to move (almost) independently in a mean field potential, andthat nuclear spectra can be explained by the mixing of a few valence configurations above an inertcore via the residual interaction. As we know now, the existence of a bound mean-field solution anda weak, possibly perturbative residual interaction relies on the decoupling of low and high momentain the nuclear Hamiltonian [1, 6, 253], e.g., by an SRG transformation. Historical approaches toexploit this connection to construct the Shell model from realistic nuclear forces [254, 255, 256]failed in part because the decoupling of the momentum scales via Brueckner’s G − matrix formalism[65, 66, 67] was not as good as believed [1].In addition to the momentum-space decoupling, one must also decouple the valence spaceconfigurations from the excluded space. This can be achieved using a variety of techniques (cf. Sections2.3.3–2.3.6), and either by performing transformations in sequence, or designing a single procedurethat achieves both types of decoupling simultaneously. In practice, the former strategy tends to bemore efficient and less prone to truncation errors — an example is the VS-IMSRG decoupling ofHamiltonians that have been evolved to a low resolution scale by means of a prior SRG evolution (seeSections 2.2 and 2.3.3, as well as Ref. [76]). An added benefit of using low-momentum interactions isthat the Shell Model wave functions will qualitatively resemble those obtained by a no-core methodusing the same Hamiltonian without valence decoupling. This facilitates qualitative comparisonsand allow us to apply the same intuitive picture. For quantitative comparisons, the effects of allunitary transformations must be carefully taken into account [257].Figure 13 illustrates the effect of the discussed transformations via the deviations between thecomputed and experimental energies of close to 400 levels in the sd -shell. Since the EM1.8/2.0interaction used in these calculations has a low resolution scale, simply using the valence-spacematrix elements of the input Hamiltonian without any further valence-space decoupling yields aroot-mean-square (rms) deviation of “only” about 1 . This is a provisional file, not the final typeset article ergert A Guided Tour of
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Nuclear Many-Body Theory improved rms deviation of approximately 650 keV, which is better than for some of the older sd -shellinteractions, albeit not as good as the USDB Hamiltonian, which is shown for comparison [258, 259].This is not really surprising: USDB essentially represents the best possible fit to experimental dataunder the model assumptions, i.e., the choice of a pure sd -shell valence space, the restriction toa two-body Hamiltonian, the omission of isospin-breaking effects from the Coulomb interactionand the nuclear interactions, and the empirical A -dependence multiplying the two-body matrixelements (TBMEs). The accuracy of the VS-IMSRG results, on the other hand, is affected bypossible deficiencies in the input Hamiltonian and the use of the VS-IMSRG(2) truncation. Naturally,both of these aspects will be improved systematically in future calculations.Phenomenological adjustments of effective Shell Model interactions like the A -dependent scalingfactors in the USD Hamiltonians or Zuker’s monopole shift [260] are typically attributed to thechanges in the nuclear mean field away from the core, as well as missing three-body interactions. InRef. [76], the VS-IMSRG is used to demonstrate that this is indeed the case. As described in Section2.3.3, upon normal ordering, the three-body force gives contributions to operators of equal andlower particle rank, which in the Shell Model case amounts to the core energy, single-particle energy,and two-body matrix elements. All of these contributions become A -dependent in the VS-IMSRG,but one can shift the A -dependent parts completely into the TBMEs, like in phenomenologicalinteractions, without changing the Hamiltonian matrix in the many-body Hilbert space or itseigenvalues.Procedures like the VS-IMSRG decoupling also let us track in detail how operators besides thenuclear interactions evolve when they are subject to the valence-decoupling transformation. Recallfrom the discussion in Section 3.2.3 that this can even quantitatively explain the quenching of theGamow-Teller strength in phenomenological Shell Model calculations, provided two-body currentcontributions to the initial transition operator are taken into account as well. For electromagnetictransitions, the renormalization of the one-body transition operator and the appearance of inducedterms generate at least some part of the usual phenomenological effective charges, but a more completetreatment of nuclear collectivity (cf. Section 2.3.3) as well the inclusion of current contributions tothese operators are developments that need to be undertaken in the coming years. Emergence of Collectivity.
Both NCCI and VS-IMSRG calculations with chiral NN+3Ninteractions have demonstrated that these interactions do indeed produce the telltale featuresof collective behavior in nuclear spectra [26, 213, 262, 263, 141]. Upon a bit of reflection, it isnot surprising that reasonable results on rotational bands, for instance, should be found in theseapproaches: While they rely on particle-hole type expansions, the exact diagonalization is done in acomplete model space of up to A v h A v p excitations, where A v is the number of valence nucleons. Incontrast, euation-of-motion methods that typically employ 1p1h or 2p2h truncations struggle withthe description of collectivity in low-lying states [122, 83, 203], but they do work reasonably well forgiant resonances [241, 242].As argued in Sections 2.3.3 and 2.3.6, bases built on particle-hole type expansions are not ideallysuited to the description of collective correlations. The SA-NCSM [139] instead uses irreduciblerepresentations of SU(3) or Sp(3, R ), the dynamical symmetry groups of collective models [264],to achieve a much more efficient description of collective behavior in nuclei. This is illustrated forthe case of Ne in Fig. 14. The SA-NCSM calculations [140] based on the two-nucleon NNLO opt potential [261] describe the ground-state rotational band extremely well, all the way to the J = 8 + Frontiers 31 ergert
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Nuclear Many-Body Theory -10 10 30 50 70 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 P r edo m i nan t s hape c on t r i bu t i on ( % ) Energy (MeV)
Equilibrium deformation + vibrations Rotation Giant resonance Expt. (MeV)
0 1.63 4.25 8.78 11.95 + + + + + + J π % % % %
0 0.1 0.2 0.3 0.4 β γ Predominant shape contribution to states of Ne β γ “equilibrium shape” “dynamical shapes” Energy (MeV)
0 0.1 0.2 0.3 0.4 β γ + gs
0 0.1 0.2 0.3 0.4 β γ + (a) (b) Figure 14.
SA-NCSM results for Ne in an SU(3)-adapted basis, using the two-nucleon interaction[261]. Panel (a) : Excitation energies (horizontal axis) of the ground-state rotational band ( J π = 0 + through 8 + ) and 0 + states, and the dominant shape in each state (vertical axis), indicated usingthe ab initio one-body density profiles in the intrinsic (body-fixed) frame. Panel (b) : Distributionof the equilibrium shapes that contribute to the ground state and first excited 2 + state, indicatedby the average deformation parameters ( β, γ ). See Ref. [140] for additional details. Figure reprintedwith permission from the American Physical Society.state. It is dominated by a single SU(3) irrep, associated with the axially elongated shape of thecomputed intrinsic density profile that is also shown in the figure. A substantial part of the appeal of methods like CC, IMSRG and SCGF is their polynomial scaling.For the purposes of uncertainty quantification (UQ), we need to be able to evaluate at least twoconsecutive truncation levels to assess the convergence of the many-body expansion in nuclei forwhich exact calculations are not feasible. Efforts in that direction have been in progress for sometime, and while some methods are at a more advanced stage than others, the improved truncationsshould be available for regular use within the next couple of years [12, 100, 244, 116, 86, 265, 10].In part, this is owing to the development of computer tools that automate tasks like diagrammaticevaluation or angular momentum coupling [266, 267]. The computational scaling of these approacheswill be of order O ( N ) or O ( N ), which makes applications a task for leadership-class computingresources for the foreseeable future. It is clear that it will not be feasible to just push the calculationsfurther, since we would then face a (naive) O ( N ) scaling.Applications where we would expect to need high-order truncations involve nuclear states withstrong collective correlations, provided we work from a spherical reference state. As explained This is a provisional file, not the final typeset article ergert A Guided Tour of
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Nuclear Many-Body Theory in Section 2.3, this issue can likely be addressed either by using mean-field reference states withspontaneously broken symmetries (cf. Section 2.3.4) or using correlated reference states in thefirst place (cf. Section 2.3.3), and the first applications of the IM-GCM give credence to that idea.Moreover, there is first evidence that the CC and IMSRG truncations converge much more rapidlyfor observables that are sensitive to collectivity [268], i.e., the current state-of-the-art truncationsmay be sufficiently precise.The IMSRG framework also offers perspectives for the construction of further IMSRG hybridmethods (cf. Section 2.3.3). Based on the successes of both the IM-NCSM and IM-GCM it would beworthwhile to use IMSRG-evolved Hamiltonians in the SA-NCSM or techniques like the DensityMatrix Renormalization Group, which is also capable of efficiently describing strong collectivecorrelations under certain conditions [269, 270].
The progress in ab initio many-body calculations is not simply due to the availability of increasinglypowerful computational resources, but also due to dedicated collaborations with computer scientiststo ensure that the available high-performance computers are used efficiently. Such collaborationswill only grow more important as hardware architectures change rapidly and a growing demandfor computing time requires users to demonstrate sufficient efficiency to be granted access tosupercomputers.Measures to boost the numerical efficiency can also be taken at the many-body theory level.Efficient calculations rely on finding optimal representations of the relevant physical informationthat is encoded in the Hamiltonian. Algorithmic gains are possible whenever there is a mismatch,either because we made convenient choices, e.g., by expanding many-body states in terms of simpleSlater determinants, or because we were not able to recognize simplifications beforehand, e.g., dueto hidden or dynamical symmetries.The SRG has played a key role in addressing the first points at the level of the nuclear interactionover the past two decades, and SRG and IMSRG can be applied in novel ways to explore dynamicalsymmetries [55]. In the construction of a configuration space, the selection of the single-particle basisleaves room for optimization. Indeed, the natural orbitals introduced in Ref. [131] lead to fastermodel-space convergence in NCSM and CC calculations, implying a more compact Hamiltonianmatrix in natural orbital representation. The efficiency of this representation can be leveragedfurther by making robust importance truncations based on analytical measures, e.g., in MBPT, CC,or IT-NCSM [271, 9].The aforementioned steps make use of prior theoretical knowledge, e.g., to identify desireddecoupling patterns in interactions, or define analytical measures for the importance of basis states.If such knowledge is not available, or we want to avoid bias, we can leverage a myriad of PrincipalComponent Analysis (PCA) methods to factorize interactions or intermediate quantities in many-body calculations [272, 271]. This can potentially even give us control over the computational scalingof nuclear many-body methods (see, e.g., [273, 274, 275, 276, 277]).A very noteworthy development with origins in nuclear physics is Eigenvector Continuation (EVC)[279, 280], a method for learning manifolds of eigenvector trajectories of parameter-dependentHamiltonians. The method has been employed in several contexts, e.g., to stabilize high-orderMBPT expansions [81] and to construct emulators for nuclear few- and many-body calculations
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Nuclear Many-Body Theory − − − − G r o und - s t a t ee n e r g y ( M e V ) SP-CC(5) (points 1-5)SP-CC(3) (points 1-3)CCSD2 . . . . . C S (10 GeV − )2 . . . . . C h a r g e r a d i u s ( f m ) O NN L O s a t − − − . . . . × . . . . . . . . × ˜ C ( n p ) S ˜ C ( pp ) S ˜ C ( nn ) S C S ˜ C S C S C E C P C P C P C P c D c E c c c S e n s i t i v i t y ( % ) Energy Total effect S Ti Energy Main effect S i Radius Total effect S Ti Radius Main effect S i (a) (b) Figure 15.
Sensitivity analysis using subspace-projected CC (SPCC) method [278]. Panel (a) illustrates the capability of the SPCC Hamiltonian constructed from 3–5 sample points to predictfull CCSD ground-state energies and charge radii for O over a wide range of values of the chiralLEC C S . Panel (b) shows the global sensitivity of the O ground-state energy and charge radiusto chiral LECs, determined by evaluating over 1,000,000 quasi-MC samples from a 64-dimensionalSPCC Hamiltonian. Vertical bars indicate 95% confidence intervals. For details, see Ref. [278].Figure reprinted with permission from the American Physical Society.[281, 278]. As an example, Fig. 15 shows a global sensitivity analysis of CC results for O undervariations of the chiral LECs [278]. Eigenvector continuation was used to learn representations ofthe CCSD Hamiltonian and charge radius operators in a 64-dimensional subspace of the space ofCCSD ground-state wave functions for interactions with 16 varying LECs. The subspace-projectedHamiltonian was then sampled more than a million times on laptop, while full CCSD calculationsof the same ensemble would be completely unfeasible. The successful applications of EVC suggestthat the method should be further explored as a tool for improvement, emulation and UQ in othermany-body methods in the (near) future.
In typical nuclear many-body calculations as discussed in Secs. 2 and 3 the main sources oftheoretical uncertainties are the EFT truncation of the observables and the many-body wavefunction, either due to many-body expansion and/or model space truncations in configuration spaceapproaches, lattice discretization effects in Lattice EFT, or the specific form of the wave functionansatz in QMC. If an SRG evolution is applied, there is an additional uncertainty associated withthe truncation of induced operators (see Section 2.2).The application of Bayesian methods has led to enormous progress in the quantification of theEFT uncertainties [282, 283, 284, 34, 36, 35, 285], and it would be highly desirable to apply thesame approach to the many-body uncertainties as well. The most challenging amongst these are thetruncation of the many-body expansion in methods like CC, IMSRG or SCGF, and the truncation ofthe free-space SRG flow of observables. In contrast, the infrared effects of finite-basis size truncations
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Nuclear Many-Body Theory in HO bases — or general orbitals that are at some point expanded in an HO basis — are wellunderstood for the energy and other observables, and they can be systematically corrected for[286, 287, 288, 289, 290]. The situation is less clear for ultraviolet basis-size errors [291], but thiserror can be suppressed by working at appropriate values of the HO frequency.An ideal uncertainty analysis would combine the exploration of EFT and many-body uncertaintiesfor nuclear observables of interest, using EC or Machine Learning (ML) to construct emulators thatallow an efficient sampling of the parameter space. In such an effort, the generation of sufficienttraining data poses a significant challenge, because it would require calculations at several truncationlevels (see Section 4.1). A possible strategy for mitigating this issue is to combine non-perturbativemethods with cheaper high-order MBPT in Bayesian mixed models (see Refs. [292, 212, 293] forapplications in nuclear physics). The successful application of factorization methods to the nuclearmany-body problem could likely resolve the issue once and for all by reducing the computationalscaling of high-order truncations, at the cost of introducing an additional uncertainty from thefactorization procedure.On the road towards the destination represented by such a “complete” UQ framework, theintermediate milestones will already provide valuable insights into open issues in the EFTs of thestrong interactions, and enable the design of better protocols for constraining and refining EFT-basedinteractions and operators (see, e.g., Refs. [294, 295] and references therein.)
Strong interaction physics is a multi-scale problem, and there are good reasons for making betteruse of the hierarchy of Effective (Field) Theories at our disposal. At the top level, we have QCD,followed by EFTs involving hyperons that can be eliminated progressively until we arrive at the“traditional” pionful and pionless chiral EFTs (see Refs. [296, 297] and references therein). Ateven lower scales, one can formulate an EFT for nuclear halos (or clusters) [297] and make theconnection to nuclear DFT and collective models, which can be understood as EFTs as well[298, 299, 300, 301, 302, 303, 304, 305].At least in principle, the different levels of this hierarchy can be connected either by computingobservables with different theories and matching the LECs, or using RG flows to track in detailhow the theories evolve from one into another. While matching procedures have been appliedsuccessfully to modern EFts in nuclear physics [306, 307, 308, 309, 310, 311] as well as effortsto match more traditional models of nuclear structure to ab initio calculations [312, 313, 314],making the connection through RG methods is a more daunting task. While I must admit to biasin this regard, I still consider this an effort worth undertaking. The success of SRG techniques innuclear physics demonstrate how these methods reveal the most effective degrees of freedom evenin situations were the separation of scales is not perfectly clear. Moreover, RGs would also revealunexpected features of the power counting schemes, like the enhancement or inadvertent omission ofcertain operators (see Ref. [51] and references therein).
Tackling Power Counting Issues.
Throughout this work, I have alluded to shortcomings andissues of the current generation of chiral interactions, like the struggle to achieve a good simultaneousdescription of nuclear binding energies and radii (see Section 3.1). Recent efforts to construct new,accurate chiral interactions have revealed that this issue is connected to the use of local or nonlocalregulators, with the latter being favored for better descriptions [208, 114]. In another exploration ofnonlocally regularized chiral forces [209, 210], a tension between the simultaneous description of
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Nuclear Many-Body Theory nuclear matter and finite was observed in the attempts to fit the chiral LECs. In QMC calculations,it was demonstrated that the use of local regulators breaks the equivalence of parameterizations ofthe interaction that are connected by Fierz identities, in certain cases with disastrous consequences[188]. Meanwhile, Epelbaum and co-workers have proposed the use of a more nuanced semilocalregularization scheme that applies local regulators to the long-range pion exchange and nonlocalregulators to the short-range contact terms [46, 3]. While physical arguments can be made in favorof different regularization schemes, perhaps especially the semilocal one, the significant schemedependence is at odds with the principles of EFT, which would predict regulator artifacts to bepushed beyond the order at which one currently works.It has also been suggested that the scales of the chiral EFT interaction and the inherent scales ofthe many-body configuration space (e.g., IR and UV cutoffs inherited from a HO basis, see Section4.3) or coordinate space wave functions should not be treated independently, and that by doing so,current many-body approaches might at least contribute to power counting issues. There have beena few efforts to explore this problem, but more work is clearly required [315, 316, 317, 318, 319, 320].
Application Needs.
Aside from the formal need to make progress on the power counting issue,there are also concrete application needs that call for a tighter coupling between QCD and thenuclear EFTs. For example, the chiral EFT transition operator for neutrinoless double-beta decay(see Section 3.2.3) contains counter terms whose LECs can only be determined from Lattice QCD[321, 322, 323, 324].The dawning of a new age in our understanding of neutron stars, heralded by the detection ofgravitational waves from the neutron-star merger GW170817, has taken the demand for accurateneutron and nuclear matter equations of state to a new level (see, e.g., Ref. [159] and referencestherein). While ab initio calculations of infinite matter up to the saturation region based on chiralinteractions are reasonably well controlled [285, 325, 190, 159], the supranuclear densities probed inmerger events are beyond the range of validity of regular pionful chiral EFT. To increase its validity,hyperons must be taken into account as dynamical degrees of freedom (see [296] and referencestherein), and the entire set of nuclear and hyperon LECs must be readjusted at the increasedbreakdown scale. For the NN sector, this is unproblematic due to the plethora of available scatteringdata. Since no direct experiments on three-neutron or three-proton systems are feasible, the onlyavailable experimental constraints come from finite nuclei, which implies that the correspondingchannels of the 3N interaction are only constrained at sub-saturation densities. The world databaseof hyperon-nucleon scattering data is also quite limited. Thus, a high-precision interaction fordescribing the equation of state at high density can only be constructed with the help of LatticeQCD constraints on the 3N and hypernucleon LECs.
The final important research direction for the coming decade I want to discuss here are effortsto interface the advanced ab initio nuclear structure methods at our disposal with reaction theory[326].As discussed in Sections 2.3.6 and 3.2.4, the NCSMC has been applied with great success to thereactions of light nuclei at low energies, but its computational complexity makes this approachunfeasible for nuclei beyond A ≈ −
20. Work has begun on a similar approach that combinesSA-NCSM with the RGM, leveraging the efficiency of the symmetry-adapted basis to reach medium-mass nuclei [327] (cf. Sections 2.3.6 and 3.3). Since the RGM is just a special case of a Generator
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Coordinate Method, the IM-GCM discussed in Sections 2.3.3 and 3.2.3 is a promising candidate forextending this type of reaction calculations to even heavier nuclei.Methods that are similar in spirit to these combinations of structure approaches witht the RGMare the APM, which can provide an interface between structure and scattering in Lattice EFT(cf. Sections 2.3.8 and 3.2.4), as well as the GSM Coupled Channel (GSM-CC) approach, which wasdeveloped to describe reactions between light projectiles and targets that are treated in the GSM witha core [328, 221, 329]. Thus far, applications of the GSM-CC have been based on phenomenologicalvalence-space interactions, but new efforts are underway to directly construct suitable Hamiltoniansbased on EFT principles [220, 330], or derive the effective interactions from chiral forces with thetechniques discussed in Section 2.3 (see [214, 215]). Of course, the GSM-CC ideas could also beapplied to the No-Core GSM [153, 155, 218], although the computational complexity would limitsuch an approach to light nuclei.A complementary strategy for bridging nuclear structure and reactions for medium-mass nuclei isthe construction of optical potentials for use in traditional reaction calculations. In SCGF theory,the optical potential for elastic nucleon-nucleus scattering is given by the one-body self energy,which is obtained as a byproduct of a nuclear structure calculation, and can be used with little effortin reaction codes [125]. Similarly, Rotureau et al. constructed optical potentials by extracting theself-energy from the Coupled Cluster Green’s Function [123, 124, 331]. One can roughly view thisprocedure as performing a GF calculation with the similarity-transformed CC Hamiltonian, whichdoes not require self-consistent iterations because of the CC decoupling (cf. Section 2.3.4). Opticalpotentials can also be constructed by folding scattering T -matrices with ab initio density matrices.This technique was applied for NCSM density matrices by two collaborations in Refs. [332, 333] and[334, 335], respectively, and more applications are underway.While the published results from the optical-potential based approaches are promising, an importantaspect of these calculations must be checked carefully in the near term: The optical potential dependson the resolution scale of the used chiral interactions, and the calculation scheme, which encompassesthe truncations in the operators and many-body method, as well as the choice of regulator in theinteraction [336, 257]. To produce scale- and scheme-independent observables, these choices must bematched by the reaction theory. Matching the resolution scales is probably the easier of the twochecks, but it will require the analysis of free-space SRG transformations on the reaction theoryside. Once structure and theory are defined at a matching resolution scale, any residual schemedependence of the observables will give rise to the remaining theoretical uncertainty of the combinedcalculation. Thus concludes our little excursion through the landscape of state-of-the-art ab initio nuclearmany-body theory, but of course, the road goes ever on. I hope that this guided tour has contributedto your appreciation of the immense progress the community has made in the last ten years, as wellas the challenges that we are facing on the next stage of the road. None of the obstacles in our pathare unsurmountable, and while we chip away at them, results from ab initio calculations can makemeaningful contributions to the analysis and planning of nuclear physics and fundamental symmetryexperiments. With new facilities launching in the next couple of years, the fun will begin in earnest,so here’s looking forward to the next decade!
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ACKNOWLEDGMENTS
This work has been shaped by an enormous amount of discussions over the past decade, and namingall discussion partners would likely require multiple pages — it is safe to say that if your work is citedhere, we have likely talked in person at some point. I am deeply grateful for these conversations, andthe spirit of the ab initio nuclear structure and reactions community that fosters such exchanges.I would like to thank the current (and former) members of my research group as well as colleaguesat NSCL/FRIB, who had the most immediate impact on this work by virtue of being a short walkaway. Particular thanks go out to S. Bogner, K. Fossez, M. Hjorth-Jensen, R. Wirth, and J. M. Yao.Special thanks are owed to R. Stroberg and S. Elhatisari, who helped out with sudden requests forfigures.I am also grateful to the Institute of Nuclear Theory for its hospitality, which was the venue formany of the aforementioned discussions, most recently during the program INT-19-2a, “NuclearStructure at the Crossroads”.The preparation of this work has been supported by the U.S. Department of Energy, Office ofScience, Office of Nuclear Physics under Awards No. DE-SC0017887, DE-SC0018083 (NUCLEI2SciDAC-4 Collaboration), and DE-SC0015376 (DBD Topical Theory Collaboration).
LIST OF ACRONYMS
ADC Algebraic Diagrammatic Construction (for Self-Consistent Green’s Functions)AFDMC Auxiliary Field Diffusion Monte CarloAPM Adiabatic Projection Method (in Lattice EFT)BMBPT Bogoliubov Many-Body Perturbation TheoryCI Configuration InteractionCC Coupled ClusterCCSD Coupled Cluster with Singles and Doubles excitationsCCSDT Coupled Cluster with Singles, Doubles and Triples excitationsCCSD(T) Coupled Cluster with Singles, Doubles and perturbative Triples excitations χ EFT Chiral Effective Field TheoryDFT Density Functional TheoryEVC Eigenvector ContinuationEDF Energy Density FunctionalEFT Effective Field TheoryGCM Generator Coordinate MethodGFMC Green’s Function Monte CarloGHW Griffin-Hill-Wheeler (equation)HF Hartree-FockHFB Hartree-Fock-BogoliubovIM-GCM In-Medium Generator Coordinate Method (a combination of IMSRG and GCM)IM-NCSM In-Medium No-Core Shell Model (a combination of IMSRG and NCSM)IMSRG In-Medium Similarity Renormalization GroupLEFT Lattice Effective Field TheoryLO Leading Order (Effective Field Theory)MBPT Many-Body Perturbation Theory
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MR-IMSRG Multi-Reference In-Medium Similarity Renormalization GroupNCCI No-Core Configuration InteractionNCSM No-Core Shell ModelNCSMC No-Core Shell Model with ContinuumNLO Next-to-Leading Order (EFT)NNLO Next-to-Next-to-Leading Order (EFT)N LO Next-to-Next-to-Next-to-Leading Order (EFT)N LO Next-to-Next-to-Next-to-Next-to-Leading Order (EFT)QCD Quantum ChromodynamicsQMC Quantum Monte CarloRG Renormalization GroupRGM Resonating Group MethodSCGF Self-Consistent Green’s FunctionsSRG Similarity Renormalization GroupTBME two-body matrix elements (typically in the discussion of Shell Model interactions)UCC Unitary Coupled ClusterUMOA Unitary Model Operator ApproachUQ Uncertainty QuantificationVMC Variational Monte CarloVS-IMSRG Valence-Space In-Medium Similarity Renormalization Group
REFERENCES [
1] Bogner SK, Furnstahl RJ, Schwenk A. From low-momentum interactions to nuclear structure.
Prog. Part. Nucl. Phys. (2010) 94–147. doi:10.1016/j.ppnp.2010.03.001. [
2] Machleidt R, Sammarruca F. Chiral eft based nuclear forces: achievements and challenges.
Physica Scripta (2016) 083007. [
3] Epelbaum E, Krebs H, Reinert P. High-precision nuclear forces from chiral eft: State-of-the-art,challenges, and outlook.
Frontiers in Physics (2020) 98. doi:10.3389/fphy.2020.00098. [
4] Piarulli M, Tews I. Local nucleon-nucleon and three-nucleon interactions within chiral effectivefield theory.
Frontiers in Physics (2020) 245. doi:10.3389/fphy.2019.00245. [
5] Roth R, Langhammer J. Pad´e-resummed high-order perturbation theory for nuclear structurecalculations.
Phys. Lett. B (2010) 272 – 277. doi:10.1016/j.physletb.2009.12.046. [
6] Tichai A, Langhammer J, Binder S, Roth R. Hartree–fock many-body perturbation theory fornuclear ground-states.
Physics Letters B (2016) 283–288. doi:http://dx.doi.org/10.1016/j.physletb.2016.03.029. [
7] Navr´atil P, Gueorguiev VG, Vary JP, Ormand WE, Nogga A. Structure of a = 10˘13 nucleiwith two- plus three-nucleon interactions from chiral effective field theory. Phys. Rev. Lett. (2007) 042501. doi:10.1103/PhysRevLett.99.042501. [
8] Roth R, Navr´atil P.
Ab Initio study of Ca with an importance-truncated no-core shell model.
Phys. Rev. Lett. (2007) 092501. doi:10.1103/PhysRevLett.99.092501. [
9] Roth R. Importance truncation for large-scale configuration interaction approaches.
Phys.Rev. C (2009) 064324. doi:10.1103/PhysRevC.79.064324. [
10] Som`a V. Self-consistent Green’s function theory for atomic nuclei.
ArXiv e-prints (2020)arXiv:2003.11321.
Frontiers 39 ergert
A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [
11] Hergert H, Bogner SK, Morris TD, Schwenk A, Tsukiyama K. The in-medium similarityrenormalization group: A novel ab initio method for nuclei.
Physics Reports (2016)165–222. doi:http://dx.doi.org/10.1016/j.physrep.2015.12.007. [
12] Hagen G, Papenbrock T, Hjorth-Jensen M, Dean DJ. Coupled-cluster computations of atomicnuclei.
Rept. Prog. Phys. (2014) 096302. [
13] Coester F. Bound states of a many-particle system.
Nucl. Phys. (1958) 421 – 424.doi:http://dx.doi.org/10.1016/0029-5582(58)90280-3. [
14] Coester F, K¨ummel H. Short-range correlations in nuclear wave functions.
Nucl. Phys. (1960) 477 – 485. doi:http://dx.doi.org/10.1016/0029-5582(60)90140-1. [
15] Hagen G, Hjorth-Jensen M, Jansen GR, Machleidt R, Papenbrock T. Continuum effects andthree-nucleon forces in neutron-rich oxygen isotopes.
Phys. Rev. Lett. (2012) 242501.doi:10.1103/PhysRevLett.108.242501. [
16] Hagen G, Hjorth-Jensen M, Jansen GR, Machleidt R, Papenbrock T. Evolution of shellstructure in neutron-rich calcium isotopes.
Phys. Rev. Lett. (2012) 032502. doi:10.1103/PhysRevLett.109.032502. [
17] Hergert H, Bogner SK, Binder S, Calci A, Langhammer J, Roth R, et al. In-medium similarityrenormalization group with chiral two- plus three-nucleon interactions.
Phys. Rev. C (2013)034307. doi:10.1103/PhysRevC.87.034307. [
18] Cipollone A, Barbieri C, Navr´atil P. Isotopic chains around oxygen from evolved chiral two-and three-nucleon interactions.
Phys. Rev. Lett. (2013) 062501. doi:10.1103/PhysRevLett.111.062501. [
19] Binder S, Langhammer J, Calci A, Roth R. Ab initio path to heavy nuclei.
Phys. Lett. B (2014) 119 – 123. doi:http://dx.doi.org/10.1016/j.physletb.2014.07.010. [
20] Hergert H, Binder S, Calci A, Langhammer J, Roth R.
Ab Initio calculations of even oxygenisotopes with chiral two-plus-three-nucleon interactions.
Phys. Rev. Lett. (2013) 242501.doi:10.1103/PhysRevLett.110.242501. [
21] Hergert H, Bogner SK, Morris TD, Binder S, Calci A, Langhammer J, et al. Ab initiomulti-reference in-medium similarity renormalization group calculations of even calcium andnickel isotopes.
Phys. Rev. C (2014) 041302. doi:10.1103/PhysRevC.90.041302. [
22] Som`a V, Barbieri C, Duguet T.
Ab initio gorkov-green’s function calculations of open-shellnuclei.
Phys. Rev. C (2013) 011303. doi:10.1103/PhysRevC.87.011303. [
23] Som`a V, Cipollone A, Barbieri C, Navr´atil P, Duguet T. Chiral two- and three-nucleon forcesalong medium-mass isotope chains.
Phys. Rev. C (2014) 061301. doi:10.1103/PhysRevC.89.061301. [
24] Jansen GR, Engel J, Hagen G, Navratil P, Signoracci A. Ab-initio coupled-cluster effectiveinteractions for the shell model: Application to neutron-rich oxygen and carbon isotopes.
Phys.Rev. Lett. (2014) 142502. doi:10.1103/PhysRevLett.113.142502. [
25] Sun ZH, Morris TD, Hagen G, Jansen GR, Papenbrock T. Shell-model coupled-cluster methodfor open-shell nuclei.
Phys. Rev. C (2018) 054320. doi:10.1103/PhysRevC.98.054320. [
26] Bogner SK, Hergert H, Holt JD, Schwenk A, Binder S, Calci A, et al. Nonperturbativeshell-model interactions from the in-medium similarity renormalization group.
Phys. Rev. Lett. (2014) 142501. doi:10.1103/PhysRevLett.113.142501. [
27] Stroberg SR, Calci A, Hergert H, Holt JD, Bogner SK, Roth R, et al. Nucleus-dependentvalence-space approach to nuclear structure.
Phys. Rev. Lett. (2017) 032502. doi:10.1103/PhysRevLett.118.032502.
This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [
28] Miyagi T, Stroberg S, Holt J, Shimizu N. Ab initio multi-shell valence-space Hamiltoniansand the island of inversion.
ArXiv e-prints (2020) arXiv:2004.12969. [
29] Yao JM, Bally B, Engel J, Wirth R, Rodr´ıguez TR, Hergert H. Ab initio treatment of collectivecorrelations and the neutrinoless double beta decay of Ca.
Phys. Rev. Lett. (2020)232501. doi:10.1103/PhysRevLett.124.232501. [
30] Gross DJ, Wilczek F. Ultraviolet behavior of non-abelian gauge theories.
Phys. Rev. Lett. (1973) 1343–1346. doi:10.1103/PhysRevLett.30.1343. [
31] Politzer HD. Reliable perturbative results for strong interactions?
Phys. Rev. Lett. (1973)1346–1349. doi:10.1103/PhysRevLett.30.1346. [
32] Weinberg S. Effective chiral lagrangians for nucleon-pion interactions and nuclear forces.
Nuclear Physics B (1991) 3–18. doi:https://doi.org/10.1016/0550-3213(91)90231-L. [
33] Weinberg S.
The Quantum Theory of Fields, Vol. II. Modern Applications (UK: CambridgeUniversity Press), 2nd edn. (1996). [
34] Melendez JA, Wesolowski S, Furnstahl RJ. Bayesian truncation errors in chiral effective fieldtheory: Nucleon-nucleon observables.
Phys. Rev. C (2017) 024003. doi:10.1103/PhysRevC.96.024003. [
35] Melendez JA, Furnstahl RJ, Phillips DR, Pratola MT, Wesolowski S. Quantifying correlatedtruncation errors in effective field theory.
Phys. Rev. C (2019) 044001. doi:10.1103/PhysRevC.100.044001. [
36] Wesolowski S, Furnstahl RJ, Melendez JA, Phillips DR. Exploring bayesian parameterestimation for chiral effective field theory using nucleon–nucleon phase shifts (2019) 045102.doi:10.1088/1361-6471/aaf5fc. [
37] Epelbaum E, Hammer HW, Meißner UG. Modern theory of nuclear forces.
Rev. Mod. Phys. (2009) 1773–1825. doi:10.1103/RevModPhys.81.1773. [
38] Rodriguez Entem D, Machleidt R, Nosyk Y. Nucleon-nucleon scattering up to n5lo in chiraleffective field theory.
Frontiers in Physics (2020) 57. doi:10.3389/fphy.2020.00057. [
39] Ekstr¨om A. Analyzing the nuclear interaction: Challenges and new ideas.
Frontiers in Physics (2020) 29. doi:10.3389/fphy.2020.00029. [
40] Ruiz Arriola E, Amaro JE, Navarro P´erez R. Nn scattering and nuclear uncertainties.
Frontiersin Physics (2020) 1. doi:10.3389/fphy.2020.00001. [
41] Machleidt R, Entem D. Chiral effective field theory and nuclear forces.
Phys. Rept. (2011)1 – 75. doi:10.1016/j.physrep.2011.02.001. [
42] Gazit D, Quaglioni S, Navr´atil P. Three-nucleon low-energy constants from the consistency ofinteractions and currents in chiral effective field theory.
Phys. Rev. Lett. (2009) 102502.doi:10.1103/PhysRevLett.103.102502. [
43] Pastore S, Girlanda L, Schiavilla R, Viviani M. Two-nucleon electromagnetic charge operatorin chiral effective field theory ( χ eft) up to one loop. Phys. Rev. C (2011) 024001.doi:10.1103/PhysRevC.84.024001. [
44] K¨olling S, Epelbaum E, Krebs H, Meißner UG. Two-nucleon electromagnetic current in chiraleffective field theory: One-pion exchange and short-range contributions.
Phys. Rev. C (2011) 054008. doi:10.1103/PhysRevC.84.054008. [
45] Piarulli M, Girlanda L, Marcucci LE, Pastore S, Schiavilla R, Viviani M. Electromagneticstructure of a=2 and 3 nuclei in chiral effective field theory.
Phys. Rev. C (2013) 014006.doi:10.1103/PhysRevC.87.014006. Frontiers 41 ergert
A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [
46] Reinert P, Krebs H, Epelbaum E. Semilocal momentum-space regularized chiral two-nucleonpotentials up to fifth order.
The European Physical Journal A (2018) 86. doi:10.1140/epja/i2018-12516-4. [
47] Lynn JE, Tews I, Carlson J, Gandolfi S, Gezerlis A, Schmidt KE, et al. Chiral three-nucleoninteractions in light nuclei, neutron- α scattering, and neutron matter. Phys. Rev. Lett. (2016) 062501. doi:10.1103/PhysRevLett.116.062501. [
48] Lynn JE, Tews I, Carlson J, Gandolfi S, Gezerlis A, Schmidt KE, et al. Quantum monte carlocalculations of light nuclei with local chiral two- and three-nucleon interactions.
Phys. Rev. C (2017) 054007. doi:10.1103/PhysRevC.96.054007. [
49] Valderrama MP, S´anchez MS, Yang CJ, Long B, Carbonell J, van Kolck U. Power counting inperipheral partial waves: The singlet channels.
Phys. Rev. C (2017) 054001. doi:10.1103/PhysRevC.95.054001. [
50] S´anchez MS, Yang CJ, Long B, van Kolck U. Two-nucleon s amplitude zero in chiral effectivefield theory. Phys. Rev. C (2018) 024001. doi:10.1103/PhysRevC.97.024001. [
51] van Kolck U. The problem of renormalization of chiral nuclear forces.
Frontiers in Physics (2020) 79. doi:10.3389/fphy.2020.00079. [
52] Wilson KG. The renormalization group: Critical phenomena and the kondo problem.
Rev.Mod. Phys. (1975) 773–840. doi:10.1103/RevModPhys.47.773. [
53] Hergert H. In-medium similarity renormalization group for closed and open-shell nuclei.
Phys.Scripta (2017) 023002. [
54] Stroberg SR, Hergert H, Holt JD, Bogner SK, Schwenk A. Ground and excited states of doublyopen-shell nuclei from ab initio valence-space hamiltonians.
Phys. Rev. C (2016) 051301.doi:10.1103/PhysRevC.93.051301. [
55] Johnson CW. Unmixing symmetries.
Phys. Rev. Lett. (2020) 172502. doi:10.1103/PhysRevLett.124.172502. [
56] Wegner F. Flow equations for hamiltonians.
Ann. Phys. (Leipzig) (1994) 77. [
57] Kehrein S.
The Flow Equation Approach to Many-Particle Systems , Springer Tracts in ModernPhysics , vol. 237 (Springer Berlin / Heidelberg) (2006). [
58] Hergert H, Bogner SK, Lietz JG, Morris TD, Novario SJ, Parzuchowski NM, et al. In-mediumsimilarity renormalization group approach to the nuclear many-body problem. Hjorth-JensenM, Lombardo MP, van Kolck U, editors,
An Advanced Course in Computational NuclearPhysics: Bridging the Scales from Quarks to Neutron Stars (Cham: Springer InternationalPublishing) (2017), 477–570. doi:10.1007/978-3-319-53336-0 {\ } [
59] Jurgenson ED, Navr´atil P, Furnstahl RJ. Evolution of Nuclear Many-Body Forces withthe Similarity Renormalization Group.
Phys. Rev. Lett. (2009) 082501. doi:10.1103/PhysRevLett.103.082501. [
60] Hebeler K. Momentum-space evolution of chiral three-nucleon forces.
Phys. Rev. C (2012)021002. doi:10.1103/PhysRevC.85.021002. [
61] Wendt KA. Similarity renormalization group evolution of three-nucleon forces in ahyperspherical momentum representation.
Phys. Rev. C (2013) 061001. doi:10.1103/PhysRevC.87.061001. [
62] Calci A.
Evolved Chiral Hamiltonians at the Three-Body Level and Beyond . Ph.D. thesis, TUDarmstadt (2014). [
63] Shavitt I, Bartlett RJ.
Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory (Cambridge University Press) (2009).
This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [
64] Kutzelnigg W. How many-body perturbation theory (mbpt) has changed quantum chemistry.
International Journal of Quantum Chemistry (2009) 3858–3884. doi:10.1002/qua.22384. [
65] Brueckner KA, Levinson CA, Mahmoud HM. Two-body forces and nuclear saturation. i.central forces.
Phys. Rev. (1954) 217–228. doi:10.1103/PhysRev.95.217. [
66] Brueckner KA, Levinson CA. Approximate Reduction of the Many-Body Problem for StronglyInteracting Particles to a Problem of Self-Consistent Fields.
Phys. Rev. (1955) 1344–1352.doi:10.1103/PhysRev.97.1344. [
67] Day BD. Elements of the brueckner-goldstone theory of nuclear matter.
Rev. Mod. Phys. (1967) 719–744. doi:10.1103/RevModPhys.39.719. [
68] Brandow BH. Linked-cluster expansions for the nuclear many-body problem.
Rev. Mod. Phys. (1967) 771–828. doi:10.1103/RevModPhys.39.771. [
69] Tichai A, Roth R, Duguet T. Many-body perturbation theories for finite nuclei.
Frontiers inPhysics (2020) 164. doi:10.3389/fphy.2020.00164. [
70] Langhammer J, Roth R, Stumpf C. Spectra of open-shell nuclei with pad´e-resummed degenerateperturbation theory.
Phys. Rev. C (2012) 054315. doi:10.1103/PhysRevC.86.054315. [
71] Otsuka T, Suzuki T, Holt JD, Schwenk A, Akaishi Y. Three-body forces and the limit ofoxygen isotopes.
Phys. Rev. Lett. (2010) 032501. doi:10.1103/PhysRevLett.105.032501. [
72] Holt JD, Engel J. Effective double- β -decay operator for ge and se. Phys. Rev. C (2013)064315. doi:10.1103/PhysRevC.87.064315. [
73] Tsunoda N, Takayanagi K, Hjorth-Jensen M, Otsuka T. Multi-shell effective interactions.
Phys. Rev. C (2014) 024313. doi:10.1103/PhysRevC.89.024313. [
74] Holt JD, Men´endez J, Simonis J, Schwenk A. Three-nucleon forces and spectroscopy ofneutron-rich calcium isotopes.
Phys. Rev. C (2014) 024312. doi:10.1103/PhysRevC.90.024312. [
75] Coraggio L, Covello A, Gargano A, Itaco N, Kuo TTS. Shell-model calculations and realisticeffective interactions.
Progress in Particle and Nuclear Physics (2009) 135–182. doi:https://doi.org/10.1016/j.ppnp.2008.06.001. [
76] Stroberg SR, Hergert H, Bogner SK, Holt JD. Nonempirical interactions for the nuclearshell model: An update.
Annual Review of Nuclear and Particle Science (2019) 307–362.doi:10.1146/annurev-nucl-101917-021120. [
77] Tichai A, Arthuis P, Duguet T, Hergert H, Som`a V, Roth R. Bogoliubov many-bodyperturbation theory for open-shell nuclei.
Physics Letters B (2018) 195–200. doi:https://doi.org/10.1016/j.physletb.2018.09.044. [
78] Tichai A, Gebrerufael E, Vobig K, Roth R. Open-shell nuclei from no-core shell model withperturbative improvement.
Physics Letters B (2018) 448 – 452. doi:https://doi.org/10.1016/j.physletb.2018.10.029. [
79] Kucharski SA, Bartlett RJ. Fifth-order many-body perturbation theory and its relationshipto various coupled-cluster approaches (Academic Press),
Advances in Quantum Chemistry ,vol. 18 (1986), 281 – 344. doi:http://dx.doi.org/10.1016/S0065-3276(08)60051-9. [
80] Ripoche J, Tichai A, Duguet T. Normal-ordered k-body approximation in particle-number-breaking theories.
The European Physical Journal A (2020) 40. doi:10.1140/epja/s10050-020-00045-8. [
81] Demol P, Duguet T, Ekstr¨om A, Frosini M, Hebeler K, K¨onig S, et al. Improved many-bodyexpansions from eigenvector continuation.
Phys. Rev. C (2020) 041302. doi:10.1103/PhysRevC.101.041302.
Frontiers 43 ergert
A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [
82] Hjorth-Jensen M, Kuo TTS, Osnes E. Realistic effective interactions for nuclear systems.
Phys.Rept. (1995) 125 – 270. doi:http://dx.doi.org/10.1016/0370-1573(95)00012-6. [
83] Parzuchowski NM, Morris TD, Bogner SK. Ab Initio Excited States from the In-MediumSimilarity Renormalization Group.
Phys. Rev. C (2017) 044304. doi:10.1103/PhysRevC.95.044304. [
84] Gebrerufael E, Vobig K, Hergert H, Roth R. Ab initio description of open-shell nuclei: Mergingno-core shell model and in-medium similarity renormalization group.
Phys. Rev. Lett. (2017) 152503. doi:10.1103/PhysRevLett.118.152503. [
85] Yao JM, Engel J, Wang LJ, Jiao CF, Hergert H. Generator-coordinate reference states forspectra and 0 νββ decay in the in-medium similarity renormalization group.
Phys. Rev. C (2018) 054311. doi:10.1103/PhysRevC.98.054311. [
86] Hergert H, Yao JM, Morris TD, Parzuchowski NM, Bogner SK, Engel J. Nuclear structurefrom the in-medium similarity renormalization group.
Journal of Physics: Conference Series (2018) 012007. [
87] Tsukiyama K, Bogner SK, Schwenk A. In-medium similarity renormalization group for nuclei.
Phys. Rev. Lett. (2011) 222502. doi:10.1103/PhysRevLett.106.222502. [
88] Tsukiyama K, Bogner SK, Schwenk A. In-medium similarity renormalization group foropen-shell nuclei.
Phys. Rev. C (2012) 061304. doi:10.1103/PhysRevC.85.061304. [
89] Caurier E, Mart´ınez-Pinedo G, Nowacki F, Poves A, Zuker AP. The shell model as a unified viewof nuclear structure.
Rev. Mod. Phys. (2005) 427–488. doi:10.1103/RevModPhys.77.427. [
90] Brown B, Rae W. The shell-model code nushellx@msu.
Nuclear Data Sheets (2014) 115 –118. doi:http://dx.doi.org/10.1016/j.nds.2014.07.022. [
91] Engeland T, Hjorth-Jensen M. The oslo fci code (2017). [
92] Johnson CW, Ormand WE, McElvain KS, Shan H. BIGSTICK: A flexible configuration-interaction shell-model code. arXiv:1801.08432 (2018). [
93] Shimizu N, Mizusaki T, Utsuno Y, Tsunoda Y. Thick-restart block lanczos method forlarge-scale shell-model calculations.
Computer Physics Communications (2019) 372–384.doi:https://doi.org/10.1016/j.cpc.2019.06.011. [
94] Kutzelnigg W, Mukherjee D. Normal order and extended wick theorem for a multiconfigurationreference wave function.
J. Chem. Phys. (1997) 432–449. doi:10.1063/1.474405. [
95] Kong L, Nooijen M, Mukherjee D. An algebraic proof of generalized wick theorem.
J. Chem.Phys. (2010) 234107. doi:10.1063/1.3439395. [
96] Morris TD, Parzuchowski NM, Bogner SK. Magnus expansion and in-medium similarityrenormalization group.
Phys. Rev. C (2015) 034331. doi:10.1103/PhysRevC.92.034331. [
97] D’Alessio A, et al. Precision measurement of the E +1 state of C. ArXiv e-prints (2020) arXiv:2005.04072. [
98] Taube AG, Bartlett RJ. Improving upon ccsd(t): Lambda ccsd(t). i. potential energy surfaces.
J. Chem. Phys. (2008) 044110. doi:10.1063/1.2830236. [
99] Taube AG, Bartlett RJ. Improving upon ccsd(t): Lambda ccsd(t). ii. stationary formulationand derivatives.
J. Chem. Phys. (2008) 044111. doi:10.1063/1.2830237. [ Phys. Rev. C (2013) 054319. doi:10.1103/PhysRevC.88.054319. [ J. Phys. G (2015) 025107. This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ Ab initio bogoliubov coupled cluster theoryfor open-shell nuclei.
Phys. Rev. C (2015) 064320. doi:10.1103/PhysRevC.91.064320. [ Journal of Physics G: Nuclear and Particle Physics (2017) 015103. [ Phys. Rev. C (2019) 044301.doi:10.1103/PhysRevC.99.044301. [ sd -shell nuclei from firstprinciples. Phys. Rev. C (2016) 011301. doi:10.1103/PhysRevC.94.011301. [ Int. J. QuantumChem. (2006) 3393–3401. doi:10.1002/qua.21198. [ Rev. Mod. Phys. (2007) 291–352. doi:10.1103/RevModPhys.79.291. [ Phys. Rev. Lett. (2018) 210501.doi:10.1103/PhysRevLett.120.210501. [ Phys. Rev. A (2019) 012320.doi:10.1103/PhysRevA.100.012320. [ Phys. Rev. C (2017) 054312. doi:10.1103/PhysRevC.96.054312. [ Phys. Rev. C (2019) 034310. doi:10.1103/PhysRevC.100.034310. [ Prog. Part. Nucl. Phys. (2004) 377 – 496. doi:http://dx.doi.org/10.1016/j.ppnp.2004.02.038. [ An Advanced Course in Computational Nuclear Physics (Springer),no. 936 in Lecture Notes in Physics, chap. 11 (2017). [ Phys. Rev. C (2020) 014318. doi:10.1103/PhysRevC.101.014318. [ ArXiv e-prints (2020)arXiv:2006.10610. [ Systematic Improvements of Ab Initio In-Medium Similarity RenormalizationGroup Calculations . Ph.D. thesis, Michigan State University (2016). [ Ab initio self-consistent gorkov-green’s function calculationsof semimagic nuclei: Formalism at second order with a two-nucleon interaction.
Phys. Rev. C (2011) 064317. doi:10.1103/PhysRevC.84.064317. [ Phys. Rev. C (2014) 024323. doi:10.1103/PhysRevC.89.024323. Frontiers 45 ergert
A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ Phys. Rev. C (2019) 054327. doi:10.1103/PhysRevC.99.054327. [ Phys. Rev. C (2018) 025501. doi:10.1103/PhysRevC.98.025501. [ Phys. Rev. C (2011) 054306. doi:10.1103/PhysRevC.83.054306. [ Phys. Rev. C (2013)024305. doi:10.1103/PhysRevC.88.024305. [ Phys. Rev. C (2017) 024315. doi:10.1103/PhysRevC.95.024315. [ Phys. Rev. C (2018) 044625. doi:10.1103/PhysRevC.98.044625. [ Phys. Rev. Lett. (2019) 092501. doi:10.1103/PhysRevLett.123.092501. [ Phys. Rev. C (2000) 044001. [ Nuclear Physics A (2001) 565 – 578.doi:http://dx.doi.org/10.1016/S0375-9474(01)00794-1. [ (2004) 155–167. doi:10.1007/s00601-004-0066-y. [ Prog. Part. Nucl. Phys. (2013) 131 – 181. doi:10.1016/j.ppnp.2012.10.003. [ (2016) 053002. doi:10.1088/0031-8949/91/5/053002. [ Phys. Rev. C (2019) 034321. doi:10.1103/PhysRevC.99.034321. [ C testnucleus.
Phys. Rev. C (2016) 024302. doi:10.1103/PhysRevC.93.024302. [ sd -shell nuclei. Phys. Rev. C (2017) 044315. doi:10.1103/PhysRevC.95.044315. [ The European Physical Journal A (2017) 49. doi:10.1140/epja/i2017-12232-7. [ Phys. Lett. B (2009) 334–339. doi:10.1016/j.physletb.2009.07.071. [ Phys. Lett. B (1974) 313 – 318.doi:http://dx.doi.org/10.1016/0370-2693(74)90390-6. This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ Progress in Particle and Nuclear Physics (2001) 319–400. doi:https://doi.org/10.1016/S0146-6410(01)00157-0. [ (2017) 063001. doi:10.1088/1402-4896/aa65e4. [ Progressin Particle and Nuclear Physics (2016) 101–136. doi:https://doi.org/10.1016/j.ppnp.2016.02.001. [ Phys. Rev. Lett. (2020) 042501. doi:10.1103/PhysRevLett.124.042501. [ The European Physical Journal A (2020) 120. doi:10.1140/epja/s10050-020-00112-0. [ Ab-initio shell modelwith a core.
Phys. Rev. C (2008) 044302. doi:10.1103/PhysRevC.78.044302. [ Phys. Rev. C (2009) 024315. doi:10.1103/PhysRevC.80.024315. [ Ab initio effectiveinteractions for sd -shell valence nucleons. Phys. Rev. C (2015) 064301. doi:10.1103/PhysRevC.91.064301. [ sd shell. Phys. Rev. C (2019) 054329. doi:10.1103/PhysRevC.100.054329. [ Phys. Rev.Lett. (2017) 062501. doi:10.1103/PhysRevLett.119.062501. [ The Nuclear Many-Body Problem (Springer), 1st edn. (1980). [ Phys. Rev.C (2016) 064320. doi:10.1103/PhysRevC.94.064320. [ α scatteringand resonances in He and Li with jisp16 and daejeon16 NN interactions. Phys. Rev. C (2018) 044624. doi:10.1103/PhysRevC.98.044624. [ Annals ofPhysics (2000) 299–335. doi:https://doi.org/10.1006/aphy.1999.5992. [ Nucl. Phys. A (1968) 265–287. doi:http://dx.doi.org/10.1016/0375-9474(68)90593-9. [ J. Phys. G (2009) 013101. [ Phys. Rev. C (2013) 044318.doi:10.1103/PhysRevC.88.044318. Frontiers 47 ergert
A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ Few-Body Systems (2013) 725–735. doi:10.1007/s00601-012-0455-6. [ (2017) 075103. doi:10.1088/1361-6471/aa6cb7. [ Rev. Mod. Phys. (2015) 1067–1118. doi:10.1103/RevModPhys.87.1067. [ Annual Review of Nuclear and Particle Science (2019) 279–305. doi:10.1146/annurev-nucl-101918-023600. [ Frontiers in Physics (2020) 117. doi:10.3389/fphy.2020.00117. [ Frontiers in Physics (2020) 153. doi:10.3389/fphy.2020.00153. [ Phys. Rev. Lett. (2015) 092301. doi:10.1103/PhysRevLett.114.092301. [ Phys. Rev. C (2018) 034005. doi:10.1103/PhysRevC.98.034005. [ ArXiv e-prints (2019) arXiv:1907.03658. [ (2020) 045109.doi:10.1088/1361-6471/ab6af7. [ The Journal of Physical Chemistry A (2014) 655–672. doi:10.1021/jp410587b. [ Phys. Rev. Lett. (2007) 250201. doi:10.1103/PhysRevLett.99.250201. [ Nature Physics (2012) 366–370. doi:10.1038/nphys2273. [ The Journal of Physical Chemistry Letters (2019) 925–935. doi:10.1021/acs.jpclett.9b00067. [ Phys. Rev. Lett. (2014) 221103. doi:10.1103/PhysRevLett.112.221103. [ The Journal of Chemical Physics (2009) 054106. doi:10.1063/1.3193710. [ Journal of Chemical Theory and Computation (2019). doi:10.1021/acs.jctc.9b00049.
This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ The Journal of Chemical Physics (2013)164126. doi:10.1063/1.4802766. [ An Advanced Course in Computational Nuclear Physics (Springer), no. 936in Lecture Notes in Physics, chap. 5 (2017). [ Frontiers in Physics (2020) 174.doi:10.3389/fphy.2020.00174. [ Nuclear Lattice Effective Field Theory . No. 957 in Lecture Notes inPhysics (Springer) (2019). [ An Advanced Course in Computational Nuclear Physics (Springer), no. 936 in Lecture Notes in Physics, chap. 6 (2017). [ Phys. Rev. Lett. (2012) 252501. doi:10.1103/PhysRevLett.109.252501. [ Phys. Rev. Lett. (2014) 102501. doi:10.1103/PhysRevLett.112.102501. [ Phys. Rev. Lett. (2017) 222505. doi:10.1103/PhysRevLett.119.222505. [ Nature (2015) 111–114. [ TheEuropean Physical Journal A (2019) 144. doi:10.1140/epja/i2019-12844-9. [ TheEuropean Physical Journal A (2018) 233. doi:10.1140/epja/i2018-12676-1. [ The European Physical Journal A (2018) 121. doi:10.1140/epja/i2018-12553-y. [ nn + 3 n interactions for the Ab Initio description of C and O . Phys. Rev. Lett. (2011) 072501.doi:10.1103/PhysRevLett.107.072501. [ Chinese Physics C (2017) 030002. [ O: Abarely unbound system beyond the drip line.
Phys. Rev. Lett. (2016) 102503. doi:10.1103/PhysRevLett.116.102503. [ ArXiv e-prints (2019) arXiv:1905.10475. [ Phys. Rev. C (2014)054323. doi:10.1103/PhysRevC.90.054323. Frontiers 49 ergert
A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ Phys. Rev. C (2018) 044318. doi:10.1103/PhysRevC.97.044318. [ Phys. Rev. Lett. (2018) 052503. doi:10.1103/PhysRevLett.120.052503. [ Phys. Rev. Research (2020) 022033. doi:10.1103/PhysRevResearch.2.022033. [ Phys. Rev. C (1995) 38–51. doi:10.1103/PhysRevC.51.38. [ Phys. Rev. Lett. (2016) 052501.doi:10.1103/PhysRevLett.117.052501. [ Phys. Rev. C (2015) 051301.doi:10.1103/PhysRevC.91.051301. [ Nat. Phys. (2016) 594–598. [ Ni from first-principles computations.
Phys. Rev. Lett. (2016) 172501. doi:10.1103/PhysRevLett.117.172501. [ n = 32 shell closure seen through precision mass measurements of neutron-richtitanium isotopes. Phys. Rev. Lett. (2018) 062503. doi:10.1103/PhysRevLett.120.062503. [ Nature (2019) 53–58.doi:10.1038/s41586-019-1155-x. [ . [ Phys.Rev. C (2004) 061002. doi:10.1103/PhysRevC.70.061002. [ Phys. Rev. C (2011) 031301. doi:10.1103/PhysRevC.83.031301. [ Phys. Rev. C (2003) 041001. [ Phys. Rev. C (2017) 014303. doi:10.1103/PhysRevC.96.014303. [ Phys. Rev. Lett. (2018) 152503. doi:10.1103/PhysRevLett.120.152503. [ Physics Letters B (2018) 468–473. doi:https://doi.org/10.1016/j.physletb.2018.05.064. [ e Phys. Rev. C (2019) 024306. doi:10.1103/PhysRevC.99.024306. This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ n = 34 subshell closure? first spectroscopy of Ar.
Phys. Rev. Lett. (2019) 072502.doi:10.1103/PhysRevLett.122.072502. [ Sc and , Ti nuclides: The n = 32 subshell closure in scandium. Phys. Rev. C (2019)064303. doi:10.1103/PhysRevC.99.064303. [ ArXiv e-prints (2019) arXiv:1911.04955. [ Phys. Rev. Lett. (2019) 042501. doi:10.1103/PhysRevLett.122.042501. [ Phys. Rev. C (2019) 024318.doi:10.1103/PhysRevC.100.024318. [ Ca: First mass measurements of −− Ca.
Phys. Rev. Lett. (2018) 022506.doi:10.1103/PhysRevLett.121.022506. [ Phys. Rev. Lett. (2019) 062502. doi:10.1103/PhysRevLett.122.062502. [ Phys. Scripta (2016) 063006. [ Physics Letters B (2017) 227–232. doi:https://doi.org/10.1016/j.physletb.2017.03.054. [ Physics Letters B (2020) 135206. doi:https://doi.org/10.1016/j.physletb.2020.135206. [ Phys. Rev. C (2019) 061302. doi:10.1103/PhysRevC.99.061302. [ psd -shell nuclei. Phys. Rev. C (2017) 054316. doi:10.1103/PhysRevC.96.054316. [ Phys. Rev. Lett. (2017) 032501. doi:10.1103/PhysRevLett.119.032501. [ Phys. Rev. C (2017) 024308. doi:10.1103/PhysRevC.96.024308. [ Phys. Rev. C (2018) 061302. doi:10.1103/PhysRevC.98.061302. [ , O. Phys. Rev. C (2019) 054302. doi:10.1103/PhysRevC.99.054302. [ ArXiv e-prints (2020) arXiv:2004.02981.
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Nuclear Many-Body Theory [ Ab Initio theoryexplain the phenomenon of parity inversion in Be?
Phys. Rev. Lett. (2016) 242501.doi:10.1103/PhysRevLett.117.242501. [ C. Phys. Rev. Lett. (2017)262502. doi:10.1103/PhysRevLett.118.262502. [ He nucleus from the no-core shell model with continuum.
Phys. Rev. C (2018) 034314.doi:10.1103/PhysRevC.97.034314. [ Be and Li nuclei within the no-core shellmodel with continuum.
Phys. Rev. C (2019) 024304. doi:10.1103/PhysRevC.100.024304. [ Physics Letters B (2001) 240–244. doi:http://dx.doi.org/10.1016/S0370-2693(01)01246-1. [ Atomic Data and Nuclear Data Tables (2013) 69–95. doi:http://dx.doi.org/10.1016/j.adt.2011.12.006. [ Ca − S and Ca − Ar difference in mirror charge radii on the neutron matter equationof state.
Phys. Rev. Research (2020) 022035. doi:10.1103/PhysRevResearch.2.022035. [ Phys. Rev. C (2017)034324. doi:10.1103/PhysRevC.96.034324. [ Phys. Rev. C (2019) 024317. doi:10.1103/PhysRevC.100.024317. [ Nature Physics (2019) 428. doi:10.1038/s41567-019-0450-7. [ Atomic Data and Nuclear Data Tables (2016) 1–139.doi:https://doi.org/10.1016/j.adt.2015.10.001. [ a = 6 − −
10 nuclei.
Phys. Rev. C (2018) 022501.doi:10.1103/PhysRevC.97.022501. [ ArXiv e-prints (2020)arXiv:2004.05263. [ C. Phys. Rev. Lett. (2014) 182502. doi:10.1103/PhysRevLett.112.182502. [ ν − C inclusive quasielastic scattering.
Phys. Rev. C (2018)022502. doi:10.1103/PhysRevC.97.022502. [ Ar and Ti in the quasielastic peakregion.
Phys. Rev. C (2019) 062501. doi:10.1103/PhysRevC.100.062501.
This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ Phys. Rev. C (2019) 025502. doi:10.1103/PhysRevC.99.025502. [ Frontiers in Physics (2020)116. doi:10.3389/fphy.2020.00116. [ O . Phys. Rev. Lett. (2013) 122502. doi:10.1103/PhysRevLett.111.122502. [ He, , O, and Ca from chiral nucleon-nucleon interactions.
Phys. Rev. C (2014) 064619. doi:10.1103/PhysRevC.90.064619. [ Phys. Rev. C (2016) 034317. doi:10.1103/PhysRevC.94.034317. [ Cawith increased precision.
Phys. Rev. C (2018) 014324. doi:10.1103/PhysRevC.98.014324. [ The European Physical Journal A (2019) 241. doi:10.1140/epja/i2019-12825-0. [ Ca and implications for the neutron skin.
Phys. Rev. Lett. (2017)252501. doi:10.1103/PhysRevLett.118.252501. [ Ni and correlation with the dipole polarizability.
Phys. Rev. Lett. (2020)132502. doi:10.1103/PhysRevLett.124.132502. [ ArXiv e-prints (2015) arXiv:1503.01520. [ [ [ Ar from first principles.
Phys. Rev. C (2019) 061304. doi:10.1103/PhysRevC.100.061304. [ Nature Communications (2019) 351. doi:10.1038/s41467-018-08052-6. [ Phys. Rev. C (2017) 054002. doi:10.1103/PhysRevC.96.054002. [ Nuclear Physics (1965)234–240. doi:http://dx.doi.org/10.1016/0029-5582(65)90262-2. [ Nucl. Phys. (1966) 40 – 86. doi:10.1016/0029-5582(66)90131-3. [ Nucl. Phys. A (1967) 199 – 208. doi:http://dx.doi.org/10.1016/0375-9474(67)90749-X. Frontiers 53 ergert
A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ Phys. Rev. C (2015) 034313. doi:10.1103/PhysRevC.92.034313. [ Phys. Rev. C (2006)034315. doi:10.1103/PhysRevC.74.034315. [ Phys. Rev.C (2020) 064312. doi:10.1103/PhysRevC.101.064312. [ Phys. Rev. Lett. (2003) 042502. doi:10.1103/PhysRevLett.90.042502. [ Phys. Rev. Lett. (2013)192502. doi:10.1103/PhysRevLett.110.192502. [ Physics Letters B (2013) 179–184. doi:https://doi.org/10.1016/j.physletb.2012.12.064. [ InternationalJournal of Modern Physics E (2015) 1541002. doi:10.1142/S0218301315410025. [ Physica Scripta (2016) 033003. [ Phys. Rev. C (2018) 054308. doi:10.1103/PhysRevC.97.054308. [ ComputerPhysics Communications (2019) 202–227. doi:https://doi.org/10.1016/j.cpc.2018.11.023. [ SU (2) algebra. ArXiv e-prints (2020)arXiv:2002.05011. [ private communication (2020). [ Phys. Rev. C (2009) 014304. doi:10.1103/PhysRevC.79.014304. [ Phys. Rev. C (2015) 051303. doi:10.1103/PhysRevC.92.051303. [ The EuropeanPhysical Journal A (2019) 90. doi:10.1140/epja/i2019-12758-6. [ Phys.Rev. C (2019) 034320. doi:10.1103/PhysRevC.99.034320. [ J. Chem. Phys. (2012)044103. doi:http://dx.doi.org/10.1063/1.4732310. [ J. Chem. Phys. (2012) 224106. doi:http://dx.doi.org/10.1063/1.4768233.
This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ J. Chem. Phys. (2012) 221101. doi:http://dx.doi.org/10.1063/1.4768241. [ J. Chem. Phys. (2014) 181102. doi:http://dx.doi.org/10.1063/1.4876016. [ n -body potentials in many-body quantum problems. Phys. Rev. Lett. (2013) 132505.doi:10.1103/PhysRevLett.111.132505. [ Phys.Rev. Lett. (2019) 252501. doi:10.1103/PhysRevLett.123.252501. [ Phys. Rev. Lett. (2018) 032501. doi:10.1103/PhysRevLett.121.032501. [ ArXiv e-prints (2020)arXiv:2004.07651. [ ArXiv e-prints (2019) arXiv:1909.08446. [ [ Phys. Rev. C (2015) 024005. doi:10.1103/PhysRevC.92.024005. [ Journal of Physics G: Nuclear and Particle Physics (2016)074001. [ ArXiv e-prints (2020) arXiv:2004.07805. [ Phys. Rev. C (2013) 044326. doi:10.1103/PhysRevC.87.044326. [ Phys. Rev. C (2014) 044301. doi:10.1103/PhysRevC.89.044301. [ Phys. Rev. C (2015) 061301. doi:10.1103/PhysRevC.91.061301. [ Phys. Rev. C (2016) 044331. doi:10.1103/PhysRevC.93.044331. [ Phys. Rev. C (2018) 034328.doi:10.1103/PhysRevC.97.034328. [ Phys. Rev. C (2014) 064007. doi:10.1103/PhysRevC.90.064007. [ [ Phys. Rev. C (2020) 044307. doi:10.1103/PhysRevC.101.044307.
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Nuclear Many-Body Theory [ (2019) 095101. doi:10.1088/1361-6471/ab2b14. [ ArXiv e-prints (2020)arXiv:2004.11307. [ The European Physical Journal A (2020) 91. doi:10.1140/epja/s10050-020-00100-4. [ Rev. Mod. Phys. (2020) 025004. doi:10.1103/RevModPhys.92.025004. [ EFT for DFT (Berlin, Heidelberg: Springer Berlin Heidelberg) (2012), 133–191.doi:10.1007/978-3-642-27320-9 {\ } [ The European Physical Journal A (2020) 85. doi:10.1140/epja/s10050-020-00095-y. [ Phys. Rev. C (2014) 014334. doi:10.1103/PhysRevC.89.014334. [ Journal of Physics G: Nuclear and Particle Physics (2015) 105103. [ Phys. Rev. C (2015) 064309. doi:10.1103/PhysRevC.92.064309. [ e Phys. Rev.C (2015) 014323. doi:10.1103/PhysRevC.92.014323. [ Phys.Rev. C (2016) 054316. doi:10.1103/PhysRevC.94.054316. [ ArXive-prints (2020) arXiv:2005.11865. [ Physics Letters B (2017)839–848. doi:https://doi.org/10.1016/j.physletb.2017.07.048. [ Phys. Rev. Lett. (2013) 132501. doi:10.1103/PhysRevLett.111.132501. [ ab initio calculations of trapped neutron drops. Phys. Rev.C (2011) 044306. doi:10.1103/PhysRevC.84.044306. [ Phys. Rev. C (2017) 054314. doi:10.1103/PhysRevC.95.054314. [ Phys. Rev. C (2018) 054304. doi:10.1103/PhysRevC.97.054304. [ Phys. Rev. C (2018) 064306. doi:10.1103/PhysRevC.98.064306. This is a provisional file, not the final typeset article ergert A Guided Tour of
Ab Initio
Nuclear Many-Body Theory [ Phys. Rev. Lett. (1996) 2416–2419. doi:10.1103/PhysRevLett.76.2416. [ Eur. Phys. J. ST (2008) 207–215.doi:10.1140/epjst/e2008-00618-x. [ Phys. Rev. C (2019) 034322. doi:10.1103/PhysRevC.99.034322. [ Phys. Rev. C (2016)064004. doi:10.1103/PhysRevC.94.064004. [ TheEuropean Physical Journal A (2020) 96. doi:10.1140/epja/s10050-020-00104-0. [ Phys. Rev. C (2016) 044332. doi:10.1103/PhysRevC.93.044332. [ Phys. Rev. C (2018) 054301. doi:10.1103/PhysRevC.98.054301. [ Physics Letters B (2019) 134880. doi:https://doi.org/10.1016/j.physletb.2019.134880. [ The European Physical Journal A (2020) 119. doi:10.1140/epja/s10050-020-00097-w. [ Journal of High Energy Physics (2018)97. doi:10.1007/JHEP12(2018)097. [ β decay. Phys. Rev. Lett. (2018) 202001.doi:10.1103/PhysRevLett.120.202001. [ β decay ineffective field theory: The light-majorana neutrino-exchange mechanism. Phys. Rev. C (2018) 065501. doi:10.1103/PhysRevC.97.065501. [ β decay. Phys. Rev. C (2019) 055504. doi:10.1103/PhysRevC.100.055504. [ ArXiv e-prints (2020) arXiv:2004.07232. [ ArXiv e-prints (2019)arXiv:1912.00451. [ ArXiv e-prints (2019)arXiv:1910.00638. [ Ne.
Phys. Rev. C (2014) 034624. doi:10.1103/PhysRevC.89.034624. [ He( d, d ) elasticscattering reactions.
Phys. Rev. C (2019) 044606. doi:10.1103/PhysRevC.99.044606. Frontiers 57 ergert
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