Ab initio electromagnetic observables with the in-medium similarity renormalization group

TL;DR

This paper develops a formalism for transforming electromagnetic operators within the in-medium similarity renormalization group framework and applies it to medium-mass nuclei, achieving good agreement for some observables but underpredicting others.

Contribution

It introduces a consistent operator transformation method within IMSRG and compares results across different approaches and with experimental data.

Findings

Good agreement between equations-of-motion and valence space methods.

Magnetic dipole observables align reasonably with experiments.

Electric quadrupole and octupole observables are significantly underpredicted.

Abstract

We present the formalism for consistently transforming transition operators within the in-medium similarity renormalization group framework. We implement the operator transformation in both the equations-of-motion and valence-space variants, and present first results for electromagnetic transitions and moments in medium-mass nuclei using consistently-evolved operators, including the induced two-body parts. These results are compared to experimental values, and--where possible--the results of no-core shell model calculations using the same input chiral interaction. We find good agreement between the equations-of-motion and valence space approaches. Magnetic dipole observables are generally in reasonable agreement with experiment, while the more collective electric quadrupole and octupole observables are significantly underpredicted, often by over an order of magnitude, indicating missing physics at the present level of truncation.

Tables

Table 1. Table 1: E c.m. subscript 𝐸 c.m. E_{\textrm{c.m.}} for intrinsic ground state and first 2 + state of 14 C, computed at e m ​ a ​ x subscript 𝑒 𝑚 𝑎 𝑥 e_{max} =14 and ℏ ​ ω Planck-constant-over-2-pi 𝜔 \hbar\omega = 20 MeV with EOM-IMSRG(2,2). Values are given for calculations using H in subscript 𝐻 in H_{\textrm{in}} ( β 𝛽 \beta =0), and H in + β ​ H c.m. subscript 𝐻 in 𝛽 subscript 𝐻 c.m. H_{\textrm{in}}+\beta H_{\textrm{c.m.}} ( β 𝛽 \beta =1).

$\beta$	$E_{\text{c.m.}}\left(0_{gs}^{+}\right)$ (MeV)	$E_{\text{c.m.}}\left(2_1^{+}\right)$ (MeV)
0	0.099	1.298
1	0.068	0.046

Table 2. Table 2: E ​ 2 𝐸 2 E2 transition strengths from first excited 2 + superscript 2 2^{+} state to 0 + superscript 0 0^{+} ground state for even-even nuclei (in e 2 ​ f ​ m 4 superscript 𝑒 2 𝑓 superscript 𝑚 4 e^{2}fm^{4} ). Experiment Pritychenko  et al. ( 2016 ) and Weisskopf  Blatt and Weisskopf ( 1979 ) single particle estimates are compared with IMSRG calculations.

Nucleus	$B{\left(E2\right)}_{exp}$	$B{\left(E2\right)}_{EOM}$	$B{\left(E2\right)}_{VS}$	$B{\left(E2\right)}_W$
6He	$1.1\left(1\right)$		$0.07$	$0.6$
14C	$3.6\left(6\right)$	$4.1$	$3.9$	$2.0$
22O	$4.2\left(1.6\right)$	$0.5$	$0.4$	$3.7$
32S	$59\left(1\right)$	$7.2$	$11.3$	$6.0$
48Ca	$17\left(2\right)$	$2.6$	$2.0$	$10.4$
56Ni	$91\left(17\right)$	$30.7$		$12.7$
60Ni	$186\left(3\right)$	$16.2$		$14.0$

Table 3. Table 3: Effective charges for the E ​ 2 𝐸 2 E2 operator, obtained by decoupling the s ​ d 𝑠 𝑑 sd shell with a reference of 17 O for neutrons ( δ ​ e ν 𝛿 subscript 𝑒 𝜈 \delta e_{\nu} ) and 17 F for protons ( δ ​ e π 𝛿 subscript 𝑒 𝜋 \delta e_{\pi} ), and taking the ratio with the bare matrix elements for protons.

$a$	$b$	$\delta e_{\nu }$	$\delta e_{\pi }$
$0d_{5/2}$	$0d_{5/2}$	$0.213$	$0.026$
$0d_{5/2}$	$0d_{3/2}$	$0.248$	$0.075$
$0d_{5/2}$	$1s_{1/2}$	$0.184$	$0.039$
$0d_{3/2}$	$0d_{3/2}$	$0.120$	$-0.003$
$0d_{3/2}$	$1s_{1/2}$	$0.111$	$-0.007$

Equations

$$\mathcal{S}=S_{0\text{b}}+\sum _{pq}{S}_{pq}\{a_p^{†}{a}_q\}+\frac{1}{4}\sum _{pqrs}{\breve{S}}_{pqrs}^J\{a_p^{†}{a}_q^{†}{a}_s{a}_r\}.$$

$$S_{pq}\equiv ⟨p|S|q⟩$$

$${\breve{S}}_{pqrs}^J\equiv ⟨(pq)J|\breve{S}|(rs)J⟩.$$

$${\breve{S}}_{pqrs}^J\equiv \sqrt{(1+{\delta }_{pq})(1+{\delta }_{rs})}{S}_{pqrs}^J$$

$${\mathcal{T}}_{\mu }^{\lambda }=T_{0\text{b}}^{\lambda }$$

$$+\frac{1}{4}\sum _{pqrs}\sum _{J_1{J}_2}{\breve{T}}_{pqrs}^{(J_1{J}_2)\lambda }\frac{{\left[A_{pq}^{†J_1}\times {\tilde{A}}_{rs}^{J_2}\right]}_{\mu }^{\lambda }}{\sqrt{2\lambda +1}},$$

$${\tilde{a}}_p\equiv {\tilde{a}}_{(j_p,m_p)}=(-1{)}^{j_p+m_p}{a}_{(j_p,-m_p)}.$$

$$\begin{array}{rl}{A}_{pq}^{†JM}|0⟩& \equiv {\left[a_p^{†}\times a_q^{†}\right]}_M^J|0⟩\\ & =|(pq)JM⟩\end{array},$$

$${\tilde{A}}_{rs}^{JM}={\left[{\tilde{a}}_s\times {\tilde{a}}_r\right]}_M^J=(-1{)}^{J-M}{A}_{rs}^{J-M}.$$

$$T_{pq}^{\lambda }\equiv ⟨p\Vert T^{\lambda }\Vert q⟩$$

$${\breve{T}}_{pqrs}^{(J_1{J}_2)\lambda }\equiv ⟨(pq)J_1\Vert {\breve{T}}^{\lambda }\Vert (rs)J_2⟩.$$

$${\mathcal{C}}_{\mu }^{\lambda }\equiv [\mathcal{S},{\mathcal{T}}_{\mu }^{\lambda }]=\mathcal{S}{\mathcal{T}}_{\mu }^{\lambda }-{\mathcal{T}}_{\mu }^{\lambda }\mathcal{S}$$

$$U(\infty )|{\Psi }_{\nu }⟩={\bar{X}}_{\nu }^{†}(J^{\Pi })|{\Phi }_0⟩.$$

$$[\bar{H},{\bar{X}}_{\nu }^{†}(J^{\Pi })]|{\Phi }_0⟩=(E_{\nu }-E_0){\bar{X}}_{\nu }^{†}(J^{\Pi })|{\Phi }_0⟩.$$

$${\bar{X}}_{\nu }^{†}(J^{\Pi }M)$$

$$+\frac{1}{4}$$

$$n_{\nu }(1p1h)=\sqrt{\sum _{ia}|{\bar{X}}_{ai}^J(\nu ){|}^2}.$$

$$M_{0\nu }$$

$$\displaystyle=\delta_{\lambda J_{\nu}}(-1)^{J_{\nu}}\Biggr{[}\sum_{ai}\frac{X_{ai}(\nu,J_{\nu}^{\Pi})}{\sqrt{2J+1}}O_{ai}(\lambda,\Pi)$$

$$\displaystyle+\frac{1}{4}\sum_{abij}\sum_{J_{1}J_{2}}\frac{\breve{X}^{J_{1}J_{2}}_{abij}(\nu,J_{\nu}^{\Pi})}{\sqrt{2J+1}}\breve{O}^{J_{1}J_{2}}_{abij}(\lambda,\Pi)\Biggr{]}\,,$$

$$M_{\mu \nu }=⟨{\Phi }_0||[{\bar{X}}_{\mu }(J_{\mu }^{\Pi })\times [{\bar{O}}^{\lambda }\times {\bar{X}}_{\nu }^{†}(J_{\nu }^{\Pi }){]}^{J_{\mu }}{]}^0||{\Phi }_0⟩.$$

$${\mathcal{Y}}_M^J\equiv [{\bar{\mathcal{O}}}^{\lambda }\times {\bar{X}}_{\nu }^{†}(J_{\nu }){]}_M^J=\sum _{M_{\nu }\mu }{C}_{\mu M_{\nu }M}^{\lambda J_{\nu }J}{\bar{\mathcal{O}}}_{\mu }^{\lambda }{\bar{X}}_{\nu }^{†}(J_{\nu }{M}_{\nu }).$$

$${\bar{\mathcal{O}}}^{\lambda }$$

$$={\mathcal{O}}^{\lambda }+[\Omega ,{\mathcal{O}}^{\lambda }]+\frac{1}{2}[\Omega ,[\Omega ,{\mathcal{O}}^{\lambda }]]+\dots$$

$${\mathcal{N}}_{ij}^{\lambda }={\mathcal{O}}_{ij}^{\lambda }+\sum _{aJ{J}^{\prime }}{n}_a(-1)$$

$${\mathcal{N}}_{ijkl}^{J{J}^{\prime }\lambda }$$

$$\bar{H}=UH{U}^{†}={\bar{H}}_{\text{VS}}+{\bar{H}}_{\text{excl}}.$$

$$⟨{\psi }_f\Vert {\mathcal{O}}^{\lambda }\Vert {\psi }_i⟩$$

$$+\frac{1}{4}\sum _{pqrs}\sum _{J_1{J}_2}{\breve{O}}_{pqrs}^{J_1{J}_2\lambda }{\text{TBTD}}_{pqrs}^{J_1{J}_2\lambda }({\psi }_f,{\psi }_i).$$

$${\text{OBTD}}_{pq}^{\lambda }({\psi }_f,{\psi }_i)\equiv \frac{⟨{\psi }_f\Vert [a_p^{†}\times {\tilde{a}}_q{]}^{\lambda }\Vert {\psi }_i⟩}{\sqrt{2\lambda +1}}$$

$${\text{TBTD}}_{pqrs}^{J_1{J}_2\lambda }({\psi }_f,{\psi }_i)\equiv \frac{⟨{\psi }_f\Vert [A_{pq}^{†J_1}\times {\tilde{A}}_{rs}^{J_2}{]}^{\lambda }\Vert {\psi }_i⟩}{\sqrt{2\lambda +1}}.$$

$$|\Psi ⟩=|\psi {⟩}_{\text{in}}\otimes |\psi {⟩}_{\text{c.m.}}.$$

$$⟨{\psi }_{\text{c.m.}}({\Lambda }_{\text{c.m.}}=0)|{\mathcal{O}}_{\text{c.m.}}^{\lambda }|{\psi }_{\text{c.m.}}({\Lambda }_{\text{c.m.}}=0)⟩=0,$$

$$H_{\text{c.m.}}(\tilde{\omega })=\frac{{\mathbf{P}}^2}{2mA}+\frac{1}{2}mA{\tilde{\omega }}^2{\mathbf{R}}^2-\frac{3}{2}\hslash \tilde{\omega }.$$

$$\begin{array}{ccc}⟨R_{\text{c.m.}}^2⟩=\frac{3}{2}{b}^2,& \text{and}& ⟨P_{\text{c.m.}}^2⟩=\frac{3}{2}\frac{{\hslash }^2}{b^2}\end{array},$$

Full text

††preprint: APS/123-QED

I Introduction

Understanding the observed properties of atomic nuclei based upon the underlying hadronic degrees of freedom has long been a major goal of nuclear structure theory. Achieving this goal has become especially important as nuclei become laboratories in the search for physics beyond the Standard Model Engel et al. (1992); Avignone et al. (2008); Menéndez et al. (2012); Gando et al. (2016); Engel and Menéndez (2017). In the treatment of the nuclear physics relevant for these searches, the more traditional phenomenological approaches to nuclear physics—despite their tremendous success in predicting and interpreting existing nuclear data Brown and Richter (2006); Caurier et al. (2005)—suffer from a lack of guidance as to how to incorporate new physics and make meaningful predictions. This is largely due to the fact that, by definition, there is no data for these processes upon which to fix phenomenological parameters. One promising path forward is to construct nuclei ab initio, starting from the underlying degrees of freedom rooted in the Standard Model. The two main tasks in this approach are the formulation of appropriate interactions between nucleons, and the solution of the resulting many-body problem with sufficient accuracy. Substantial progress has been made on the former difficulty by the application of chiral effective field theory (EFT) Epelbaum et al. (2009); Machleidt and Entem (2011); Epelbaum et al. (2015); Entem et al. (2015a, b), though much work certainly remains.

On the many-body front, methods such as the no-core shell model (NCSM) Navrátil et al. (2007, 2009); Barrett et al. (2013) and quantum Monte Carlo (QMC) Carlson et al. (2015) provide exact solutions for $p$-shell nuclei up to finite basis effects and sampling errors. While the application of renormalization group ideas Bogner et al. (2007); Jurgenson et al. (2009); Bogner et al. (2010); Roth et al. (2014) has helped extend the reach of the NCSM, both of these methods encounter prohibitive computational scaling for medium-mass nuclei.

Another class of approximate but systematically improvable many-body methods, namely coupled-cluster (CC) Hagen et al. (2014); Jansen et al. (2011); Binder et al. (2013), self-consistent Green’s functions (SCGF) Cipollone et al. (2013); Somà et al. (2014a); Cipollone et al. (2015); Somà et al. (2014b), many-body perturbation theory (MBPT) Hjorth-Jensen et al. (1995); Tsunoda et al. (2014); Simonis et al. (2016); Tichai et al. (2016), and the in-medium similarity renormalization group (IMSRG) Tsukiyama et al. (2011); Hergert et al. (2016); Bogner et al. (2014); Hergert (2016), have enabled applications to nuclei beyond the $fp$-shell Somà et al. (2013); Binder et al. (2014a); Hergert et al. (2014); Hagen et al. (2016); Tichai et al. (2016); Simonis et al. (2017). Each of these methods may be formulated in terms of summed Goldstone diagrams (including some classes of diagrams to all orders), and each employs normal ordering with respect to a reference state in order to approximately treat three- and higher-body terms. With these methods, immense progress has been made in the calculation of nuclear binding energies, radii, and excited state spectra, where it is now possible to calculate these observable quantities consistently using two- and three-nucleon forces throughout the expanses of the medium-mass nuclear landscape. At the present time, the deficiencies in the nuclear interactions have become the main source of error for many calculations, as opposed to truncation errors in the solution of the many-body problem.

As alluded to above, a major advantage of ab initio methods which start from chiral EFT is the possibility to obtain transition operators consistent with a given interaction. A consistent treatment of operators is essential to address open questions in nuclear physics such as the source of axial-vector quenching in-medium Wildenthal et al. (1983); Martínez-Pinedo et al. (1996) , and to do away with phenomenological concepts such as effective charges for $E2$ transitions. It will also be indispensable for reliably calculating quantities relevant for searches for physics beyond the Standard Model, such as neutrinoless double beta decay Engel and Menéndez (2017). Finally, it remains to be demonstrated that the success of diagrammatic-expansion methods in calculating energies and radii carries over to other observables.

The effort to obtain consistent effective operators for use in nuclear structure calculations is certainly not new  (see, e.g., Da Providencia and Shakin (1964); Brandow (1967); Lo Iudice et al. (1971); Barrett (1975); Ellis and Osnes (1977)), and has long been a difficult problem for nuclear theory, though some progress has been made in recent years  Anderson et al. (2010a); Paar et al. (2006); Schuster et al. (2014); Stetcu et al. (2005); Navrátil et al. (1997); More et al. (2015). The IMSRG presents a straightforward framework for deriving consistent effective operators, because it is formulated in terms of a series of unitary transformations. In order to reduce the storage needed for calculations, the IMSRG, like the other diagrammatic expansion methods, is generally formulated in an angular momentum coupled basis. As a result, additional formal developments are required for the treatment of spherical tensor operators—i.e., operators that carry angular momentum—which are necessary for the calculation of transition strengths, electromagnetic moments and response functions. In this work, we present a streamlined effective operator formalism for spherical tensors, using the recently developed equations-of-motion IMSRG Parzuchowski et al. (2017) (EOM-IMSRG) and valence-space IMSRG Tsukiyama et al. (2012); Bogner et al. (2014); Stroberg et al. (2017) (VS-IMSRG). The two methods offer complementary approaches to the problems of nuclear spectroscopy and decay, each with different benefits and drawbacks: The EOM-IMSRG works with large single-particle spaces, but limits the type of particle-hole excitations, while VS-IMSRG treats all particle-hole excitations in a small single-particle valence space exactly, but relies on a truncated IMSRG decoupling to account for excitations outside of that valence space. As we will discuss in the following, operators that appear in the IMSRG flow equations are truncated at the two-body level, and higher induced operators are neglected. We will demonstrate that both methods are capable of consistently describing excited states and transitions for a certain class of states. In some cases we find results consistent with experiment, while in others we make note of discrepancies.

This work is organized as follows. In section II, we give the relevant commutator expressions for the calculation of effective tensor operators, and lay out the formalism for the EOM-IMSRG and VS-IMSRG. In section  III, we present results of calculations of transitions and moments for several nuclei ranging in mass from the deuteron to 60Ni, and we present conclusions in section IV.

II Formalism

Here, we lay out the framework for evaluating matrix elements of spherical tensor operators in the IMSRG. For a review of the theory and formalism of IMSRG, we refer the reader to Ref. Hergert et al. (2016).

II.1 Commutator expressions

The main new development required for the transformation of tensor operators is the expression for the commutator between an operator of spherical tensor rank $\lambda$ with a scalar operator ($\lambda=0$). We truncate all operators at the two-body level in the following discussion. We write a scalar operator $\mathcal{S}$ in normal-ordered form as

[TABLE]

The braces $\{\}$ indicate normal ordering with respect to the reference state $|\Phi_{0}\rangle$. The zero-body term is given by $S_{0\textrm{b}}=\langle\Phi_{0}|\mathcal{S}|\Phi_{0}\rangle$. The coefficients $S_{pq}$ and $\breve{S}^{J}_{pqrs}$ are defined by

[TABLE]

Our two-body states are antisymmetrized but unnormalized, so that expressions may be written in terms of unrestricted sums. The unnormalized two-body matrix elements, indicated by a breve111In previous works, we have indicated unnormalized two-body matrix elements with a tilde ($\sim$). However, to avoid confusion in the present work we reserve the tilde to indicate spherical tensor annihilation operators. ˘, are related to conventional normalized matrix elements via

[TABLE]

We write a spherical tensor operator $\mathcal{T}^{\lambda}_{\mu}$ of rank $\lambda$ and projection $\mu$ as

[TABLE]

where $[\times]$ indicates a tensor product. Note that a tensor operator ($\lambda\neq 0$) that is normal-ordered with respect to a spherical reference state (as used in all calculations here) will have a zero-body piece $T^{\lambda}_{0\textrm{b}}=0$. The tilde in eq. (5) indicates the usual transformation of the annihilation operator $a_{p}$ to a spherical tensor operator Bohr and Mottleson (1969); Suhonen (2007):

[TABLE]

$A^{\dagger JM}_{pq}$ is a creation operator for a two-particle state with total angular momentum $J$ and projection $M$:

[TABLE]

with a corresponding definition for $\tilde{A}^{JM}_{rs}$

[TABLE]

The coefficients $T^{\lambda}_{pq}$ and $\breve{T}^{(J_{1}J_{2})\lambda}_{pqrs}$ are defined by the following reduced matrix elements, using the convention of Edmonds Edmonds (1960); Suhonen (2007),

[TABLE]

The commutator $\mathcal{C}^{\lambda}_{\mu}$ of the operators $\mathcal{S}$ and $\mathcal{T}^{\lambda}_{\mu}$ will be a spherical-tensor operator of rank $\lambda$:

[TABLE]

The coefficients $C^{\lambda}_{pq}$ and $\breve{C}^{(J_{1}J_{2})\lambda}_{pqrs}$ are given by equations (44) and (45) in appendix B.

II.2 Equations-of-motion IMSRG

In the equations-of-motion (EOM) formulation of the IMSRG, we first perform a single reference ground state calculation, which maps the reference $|\Phi_{0}\rangle$ to the ground state $|\Psi_{0}\rangle$ via a continuous sequence of unitary transformations $U(s)$ that are labeled by the flow parameter $s$. We then describe the excited states in the IMSRG-transformed frame using a ladder operator $\bar{X}^{\dagger}_{\nu}$ acting on the reference state

[TABLE]

Here the bar indicates that the ladder operator is expressed in the transformed frame. The Schrödinger equation for the IMSRG rotated Hamiltonian $\bar{H}$ may then be written as

[TABLE]

As a result of the ground-state decoupling, there is no correlation between the ground state and excited states in the rotated frame, so $\bar{X}^{\dagger}_{\nu}(J^{\Pi})$ will consist only of excitation operators of the form $a^{\dagger}_{a}a^{\dagger}_{b}\cdots a_{i}a_{j}\cdots$, where $a,b,c,\ldots$ and $i,j,k,\ldots$ denote orbitals that are unoccupied and occupied, respectively, in the reference state. Note that evaluating the l.h.s. of (13) requires a scalar-tensor commutator as defined in (11).

Calculations of this type are subject to two sources of systematically improvable error, namely truncations of the IMSRG equations and truncations of the EOM ladder operator. In this work, both truncations will be made at the two-body level (EOM(2)-IMSRG(2) $\equiv$ EOM-IMSRG(2,2)). The normal ordering with respect to the reference state is crucially important to control the quality of these truncations, because it allows us to retain in-medium contributions from 3N forces in the normal ordered zero-, one-, and two- body pieces of our operators. Beyond the IMSRG framework, the truncation of input interactions and operators at the normal-ordered two-body level is known as the normal-ordered two-body (NO2B) approximation Hagen et al. (2007); Roth et al. (2012); Binder et al. (2014a); Ekström et al. (2014).

Our ladder operators are linear combinations of one- and two-body excitation operators coupled to desired spin $J^{\Pi}$

[TABLE]

The amplitudes $X^{J}_{ai}(\nu)$ and $\breve{X}^{(J_{1}J_{2})J}_{abij}(\nu)$, as well as excitation energies, are obtained by solving the eigenvalue problem (13). Note that this formulation is equivalent to configuration interaction with singles and doubles (CISD), i.e. diagonalizing the transformed Hamiltonian in the space of 1p1h and 2p2h excitations out of $|\Phi_{0}\rangle$.

To quantify the importance of the EOM ladder operator truncation, we compute the 1p1h partial norms,

[TABLE]

For states with $n_{\nu}(1p1h)$ near one, we expect small error in the EOM portion of the calculation. A small 1p1h partial norm indicates that the rotated wave-function for the state in question contains higher-order many-body excitations which are not captured by the ladder operator in (14).

Operator matrix elements for transitions to the ground state may be written

[TABLE]

and for transitions between excited states, or expectation values of excited states,

[TABLE]

Equation (17) requires the calculation of the full tensor product

[TABLE]

The matrix elements of $\mathcal{Y}$ are given by equations 49 and 50 in appendix B. In equations (16)–(18), we use a transition operator which is transformed consistently with the Hamiltonian. To achieve this, we express the unitary transformation as the exponential of an anti-Hermitian generator: $U=e^{\Omega}$, with $\Omega^{\dagger}=-\Omega$ Morris et al. (2015). Any operator $\mathcal{O}^{\lambda}$ can then be consistently transformed by

[TABLE]

where we again use the scalar-tensor commutators of (11).

In the formulas presented in Appendix B, transition operators are assumed to be normal-ordered with respect to the reference $|\Phi_{0}\rangle$. If $\mathcal{O}^{\lambda}$ is initially a one-body operator with $\lambda\neq 0$, then this requires no additional work. If $\mathcal{O}^{\lambda}$ has a two-body component—as is the case if we include meson-exchange currents, or if the bare operator has been SRG evolved in free space—then we need the formula for obtaining the normal-ordered form (indicated $\mathcal{N}^{\lambda}$) of $\mathcal{O}^{\lambda}$:

[TABLE]

This may be obtained by beginning with the usual $m$-scheme formula Hergert et al. (2016) and applying (51). Here $n_{a}$ is the occupation fraction of orbit $a$, defined so that $0\leq n_{a}\leq 1$.

II.3 Valence space IMSRG

In the valence-space (VS) formulation of the IMSRG, the unitary transformation $U$ decouples a valence space Hamiltonian $H_{\textrm{VS}}$ from the remainder of the Hilbert space (the excluded space) $H_{\textrm{excl}}$,

[TABLE]

The eigenstates are obtained by a subsequent diagonalization of $\bar{H}_{\textrm{VS}}$ within the valence space.

The expectation value of $\mathcal{O}^{\lambda}$ between initial state $|\psi_{i}\rangle$ and final state $|\psi_{f}\rangle$ may be obtained by combining the matrix elements of $\mathcal{O}^{\lambda}$ with the one- and two-body transition densities (working with the consistently-transformed valence-space operators and wave functions)

[TABLE]

The one-body transition densities are defined by

[TABLE]

and the two-body transition densities are

[TABLE]

There is a clear parallel between (22) and (16), due to the fact that the amplitudes $X_{ai}^{J}(\nu)$ and $\breve{X}_{abij}^{(J_{1}J_{2})J}(\nu)$ correspond to the one- and two-body transition densities, respectively, between $|\Psi_{\nu}\rangle$ and the ground state. For all the valence space results presented here, the diagonalizations were performed with the shell model code NuShellX@MSU Brown and Rae (2014). As NuShellX does not provide functionality to calculate the two-body transition densities for spherical tensor operators, an additional code has been developed Stroberg (2017a).

For open-shell nuclei, we use the ensemble normal ordering (ENO) approach presented in Ref. Stroberg et al. (2017). After the valence space is decoupled, we change the normal ordering reference to be the core of the valence space, which requires the use of (20).

We note that the only approximation made in this procedure is the truncation to normal-ordered two-body operators. Of course, the quality of this approximation depends on the choice of reference and valence space.

III Results

For all of the calculations presented here, with the exception of the results in section III.5, we employ the chiral NN interaction of Entem and Machleidt Entem and Machleidt (2003) at N3LO with a cutoff $\Lambda_{NN}=500$ MeV, and the local 3N interaction of Refs. Navrátil (2007); Gazit et al. (2009); Roth et al. (2012) at N2LO with a cutoff $\Lambda_{3N}=400$ MeV. We use an additional three-body energy truncation $E_{3max}\equiv e_{1}+e_{2}+e_{3}\leq 14$, where $e_{i}=2n_{i}+l_{i}$ corresponds to the $i$th single particle shell in the harmonic oscillator basis. The interactions are consistently SRG evolved Bogner et al. (2007); Roth et al. (2014) to a scale $\lambda_{SRG}=2.0~{}\textrm{fm}^{-1}$. This interaction has been shown to give an excellent reproduction of the binding energies in the vicinity of the oxygen isotopes Hergert et al. (2013); Cipollone et al. (2013, 2015), but it produces radii which are too small by roughly 10% Lapoux et al. (2016). Since we consider $E2$ transitions and moments, and the $E2$ operator goes as $r^{2}Y^{(2)}$, we might expect quadrupole moments and $B(E2)$ strengths to be too small by 20% and 35%, respectively. However, because these observables are dominated by the particles near the Fermi surface, while the radii are a bulk property, it is not obvious that this naive scaling should actually apply.

In most of the figures presented in the following, we present an observable calculated for various values of model space truncation $e_{max}$ and basis frequency $\hbar\omega$. If the result is converged with respect to the model space truncation, it should not change as $e_{max}$ is increased, and it should be independent of $\hbar\omega$, corresponding to a horizontal line in our figures.

III.1 Center-of-mass factorization

Before presenting results for electromagnetic moments and transitions, we investigate the role of center-of-mass motion for our calculations. The structure of self-bound nuclei is governed by a translationally-invariant Hamiltonian, which is why we expect factorization of the intrinsic and center-of-mass (c.m.) components of the wave function:

[TABLE]

This is particularly important for our current investigation because we do not use translationally-invariant transition operators $\mathcal{O}^{\lambda}$ in order to avoid the inclusion of cumbersome recoil corrections Eisenberg and Greiner (1970). If the c.m. wave function has angular momentum $\Lambda_{\textrm{c.m.}}=0$, then by the Wigner-Eckart theorem,

[TABLE]

and there is no error incurred by including the c.m. part of the operator. The IMSRG is formulated in a lab-frame harmonic oscillator basis with a truncation on the single particle energies ($2n+l\leq e_{max}$), and consequently we cannot ensure rigorous factorization of the c.m. and intrinsic wave functions. We seek instead to demonstrate approximate factorization and, if necessary, project out spurious c.m. contamination.

III.1.1 Calculation of $H_{\textrm{c.m.}}$

The form of the c.m. Hamiltonian is taken to be that of a harmonic trap, with the zero-point energy removed:

[TABLE]

We can compute properties of the c.m. wave function in a manner similar to the discussion in Refs. Hergert et al. (2016); Hagen et al. (2009, 2010). If the center-of-mass wave function is a Gaussian with oscillator length $b$, then it will have

[TABLE]

which implies

[TABLE]

The deviation of $\xi_{\textrm{c.m.}}$ in (29) from zero indicates the deviation of the c.m. wave function from a pure Gaussian. Once the Gaussian form is confirmed, the appropriate trapping frequency $\hbar\tilde{\omega}$ may be obtained from (28), with ${b^{2}=\hbar/Am\tilde{\omega}}$ or, equivalently,

[TABLE]

Figure 1 shows $\xi_{\textrm{c.m.}}$ results from IMSRG ground-state calculations for 14C. Also shown are two ways of estimating $\hbar\tilde{\omega}$ from the expectation values of $T_{\textrm{c.m.}}$ and $R_{\textrm{c.m.}}^{2}$. The right column of Figure 1 shows the same quantities, but with a c.m. trap (as described in the next section) with $\beta=3$ and $\hbar\tilde{\omega}=16$ MeV. Clearly, the trap makes the c.m. wave function more Gaussian, though not perfectly Gaussian, and it speeds up the convergence of the c.m. wave function.

III.1.2 Treatment for excited states

Spurious excited states manifest as nearly degenerate intrinsic states in nuclear spectra.

These states can be removed via the Lawson-Gloeckner method Gloeckner and Lawson (1974), where the intrinsic Hamiltonian is augmented with a scaled center-of-mass trap of the form of eq. 27,

[TABLE]

Here, the scale factor $\beta$ can be taken to arbitrarily large values if sufficient factorization is achieved in calculations using $H_{\textrm{in}}$ only. This process effectively shifts spurious states out of the spectrum by adding a large c.m. excitation energy.

Figure 2 demonstrates this procedure for 14C, for the ground state, first 2+ excited state, and $B(E2)$ value. Quantities are calculated with the EOM-IMSRG(2,2) method. The energies are approximately independent of $\beta$, which may be taken naively as evidence of factorization for these states. However, the $B(E2)$ value undergoes a sudden downward shift as the Lawson-Gloeckner term is introduced, but it saturates eventually and displays $\beta$-independence as we go to higher $\beta$. Of course, the quadrupole operator is more sensitive to structural details of the wave function than the energy, and since we do not use it in a translationally-invariant form, it is not surprising that the $B(E2)$ value would be affected by the imperfect factorization of the wave functions. The fact that we eventually obtain a $\beta$-independent result suggests that the Lawson-Gloeckner method is an adequate alternative to explicitly including recoil corrections in the operator Eisenberg and Greiner (1970).

Table 1 gives the computed $E_{\textrm{c.m.}}$ for calculations with and without explicit inclusion of a center-of-mass trap via the Lawson-Gloeckner term.

We expect a perfectly factorized wave-function to have $E_{\textrm{c.m.}}$=0 MeV, since our choice of $H_{\textrm{c.m.}}$ ensures that the c.m. ground state has zero energy. For either case, the ground state wave function demonstrates limited contamination from spurious c.m. excitations, with $E_{\textrm{c.m.}}<$ 100 keV. The 2+ state of $H_{in}$ does not exhibit this level of factorization, with $E_{\textrm{c.m.}}$=1.298 MeV, indicating a small admixture of spurious states. This level of contamination is ostensibly negligible for excitation energies, but evidently has important effects when the state is probed by the quadrupole operator. When the c.m. trap is explicitly added, $E_{\textrm{c.m.}}(2^{+}_{1})$ is diminished to below 100 keV and accordingly, we see a shift in the $B(E2)$ value which corresponds to a recoil correction. For the results presented below, we have checked and found that 14C is the only system where the c.m. trap has a noticeable effect.

III.2 The deuteron

As a first illustration, we consider ground-state properties of the deuteron. This is useful for a few reasons. First, the system consists of only two particles and so induced three-body forces are irrelevant. Further, the reference is taken to be the true vacuum, so the neglected three body forces do not feed back into the two body terms. We should therefore expect the IMSRG(2) to be exact. Second, full configuration interaction (FCI) calculations are easily performed for modest model spaces, allowing a direct evaluation of the precision of the IMSRG transformation. Finally, we may treat the deuteron in the $0s$ valence space where the bare quadrupole moment is identically zero. In this case, any non-zero quadrupole moment we obtain is entirely due to effects of the IMSRG evolution.

Figure 3 shows the ground-state energy, root-mean-square charge radius, quadrupole moment and magnetic moment of the deuteron, computed both with FCI and using the IMSRG to decouple the $0s$ valence space, followed by a trivial diagonalization. We can see that the IMSRG calculation indeed reproduces the FCI.

Here again we see the effect of c.m. spuriosities in the deuteron wave function. While the energy and dipole moment converge to the exact values with little alteration from c.m. contamination, the charge radius overshoots it drastically. Although we have not reached convergence for the charge radius, it is evident that Lawson-Gloeckner scaling significantly reduces its value. To get a sense of the rate of convergence for these observables in an oscillator basis, we have performed calculations in a relative Jacobi basis, where it is possible to go much higher in $e_{max}$. We observe that the charge radius converges slowly in the Jacobi basis as well.

III.3 $p$-shell nuclei: comparison with NCSM

The deuteron is, of course, an exceptionally simple case, due to the fact that there is not really a “medium”, and so the IMSRG is really a free-space SRG evolution. Once additional particles are considered, the NO2B approximation is used, and the IMSRG is no longer exact. To test this approximation, we consider $p$-shell nuclei which may also be treated in the no-core shell model (NCSM). For these calculations, we use the same input Hamiltonian and include the 3N force completely, without using the NO2B approximation222Errors from the NO2B approximation in NCSM calculations will be different from those in IMSRG(2) calculations, as additional NO2B errors accumulate during the IMSRG(2) flow due to induced many-body forces. The NCSM calculations are presented as a function of the truncation parameter $N_{max}$ which limits the total number of oscillator quanta allowed above the minimum value. For the $A=14$ systems, the $N_{max}=8$ results have been obtained using an importance truncation Roth and Navrátil (2007). We note that in the NCSM, the c.m. factorization is exact for any $N_{max}$ truncation.

We begin by considering 6Li, which was previously studied in Ref. Navrátil et al. (1997) in the context of consistently-transformed electromagnetic transition operators using the Okubo-Lee-Suzuki method. Figure 4 presents several observables for 6Li, calculated with the valence-space IMSRG, compared to NCSM and experiment. We first observe that there is overall good agreement between the VS-IMSRG and NCSM, as well as with experiment, for the energy and quadrupole moment of the ground state. In Ref. Lisetskiy et al. (2009), where an effective $p$-shell $E2$ operator was obtained via an Okubo-Lee-Suzuki transformation, the small ground-state quadrupole moment was found to be the result of cancellations between the one and two-body pieces of the effective $E2$ operator. We find a similar effect in this work333Since the IMSRG and Okubo-Lee-Suzuki transformations are not identical, there is no requirement that the breakdown into one- and two-body operators be the same in both approaches., though even greater in magnitude – for example, for the $e_{max}=12$, $\hbar\omega$=20 calculation we find $Q_{1\textrm{b}}=-0.454$ $e$b and $Q_{2\textrm{b}}=0.301$ $e$b. The results for observables involving the unbound $3^{+}$ excited state converge much more slowly in the NCSM, indicating missing continuum effects. Such effects could be included using the NCSM with continuum Hupin et al. (2015); Romero-Redondo et al. (2016), but for our present concerns, this is unnecessary. Despite the importance of continuum effects, the VS-IMSRG(2) converges rapidly for observables involving the $3^{+}$ state. This indicates that errors incurred through the NO2B truncation hide the effects of the continuum. This produces excellent convergence properties by mistake; the VS-IMSRG(2) converges to an incorrect result without continuum degrees of freedom.

A striking disagreement is found between experiment and the calculations of the $B(E2;3^{+}_{1}\rightarrow 1^{+}_{1})$ strength. As we will see, this will be a recurring observation. Finally, we note that the $M1$ transition strength displays reasonable convergence and good agreement with experiment.

Another interesting case in the $p$ shell is 6He, which in the naive shell model consists of two neutrons outside a 4He core. In this picture, any electric multipole observables are identically zero because all valence particles are electrically neutral. This problem has historically been addressed by the introduction of an effective charge for the neutrons Bohr and Mottleson (1969). As in the deuteron case, 6He therefore allows us to test how the IMSRG evolution incorporates physics from outside of the valence space into the evolved operator, building up an effective charge in the process.

Figure 5 shows the results of VS-IMSRG and NCSM calculations for the ground-state energy, $2^{+}$ excitation energy, and $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1}$) for 6He. Like 6Li, the excited states of this nucleus are unbound, and in addition, the 6He ground state can be characterized as a two-neutron halo Zhukov et al. (1993), which is difficult to describe in a truncated oscillator basis. Nevertheless, we see that the ground state energy displays excellent agreement between the VS-IMSRG, NCSM, and experiment. There is reasonable agreement as well for the energy of the $2^{+}$ state, although the NCSM result is not converged with respect to $N_{max}$ (again likely reflecting missing continuum effects). However, for the B(E2), there is serious disagreement between all three. The NCSM result is much lower than the experimental value, and shows no sign of convergence with respect to $N_{max}$. This is perhaps not surprising, as the $E2$ operator is of long range, and therefore more sensitive to the halo effects. The VS-IMSRG result appears converged with respect to $e_{max}$, but is smaller than the NCSM result as well as experiment—the latter by a factor of approximately 15—indicating that the NO2B approximation is insufficient in this case.

As a third test in the $p$ shell, we consider 14C. Because this is a closed-shell nucleus, we may employ the EOM-IMSRG as well as the VS-IMSRG, and a system of 14 particles is still feasible with the NCSM. Figure 6 displays results for the $2^{+}_{1}$ excitation energy and $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ for 14C. Here, we find excellent agreement between NCSM and both variants of the IMSRG. We remind the reader that the IMSRG calculations are performed with an explicit center-of-mass trap, as in eq. 31, using $\beta=1.0$ for 14C. This treatment only serves to remove spurious c.m. contamination of the $2^{+}_{1}$ state.

Of note are the excellent convergence properties of the IMSRG calculations. For the EOM-IMSRG, observables are nearly independent of the specified $\hbar\omega$ for the single-particle basis. VS-IMSRG calculations have not used the exhaustive model spaces of the EOM-IMSRG, but they too demonstrate desirable convergence features. The NCSM has begun to show convergence at $N_{max}$=8, but extrapolation methods must be used to reveal fully converged values. Hence the utility of the IMSRG: For light nuclei such as 14C, convergence is obtainable without extrapolation, and for heavier nuclei, we expect to be able to identify convergence trends clearly enough to make extrapolation procedures relatively painless compared to the prohibitively large uncertainties one would incur when exact methods such as NCSM are used. Of course, the effect of the additional NO2B approximation must be fully investigated.

As a final test in the $p$ shell, we analyze the isobaric neighbor nucleus 14N. Here the EOM-IMSRG requires the use of a charge-exchange formalism, i.e., ladder operators which exchange one neutron for a proton. Figure 7 displays the $0^{+}_{1}$ excitation energy for 14N, the ground state magnetic dipole moment, and the $M1$ transition strengths $B(M1;0^{+}_{1}\rightarrow 1^{+}_{1})$ and $B(M1;1^{+}_{2}\rightarrow 0^{+}_{1})$. The agreement among methods is moderate, with the exception of the transition $B(M1;0^{+}_{1}\rightarrow 1^{+}_{1})$ to the ground state. We note that this relatively weak transition, which is an analog of the Gamow-Teller beta decay of 14C, was found to result from a subtle cancellation between various contributions Maris et al. (2011); Ekström et al. (2014), so that small errors on an absolute scale appear large on a relative scale. Regardless, the disagreement between VS-IMSRG and EOM-IMSRG will be investigated in the future.

III.4 $sd$ and $fp$ shell systems

Ultimately, the power of IMSRG approaches to excited states and effective operators will be the ability to describe these properties in medium- to heavy-mass regions where exact methods are not computationally tractable. In this section we investigate the quality of these calculations for several medium-mass nuclei, again using the electric quadrupole and magnetic dipole operators as case studies.

III.4.1 Electric quadrupole observables

Figure 8 displays the first 2+ excitation energies and $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$ strengths for several nuclei in the $sd$ and $pf$ shells. We find excellent convergence properties, as we did in the $p$ shell, and we see reasonable agreement with experiment for the excitation energies. However, transition strengths are generally underpredicted by an order of magnitude. These results are strikingly consistent between the two methods. A tentative explanation for the diminished strength in 22O and 48Ca is provided by the lack of valence protons. In order to describe the transition in these nuclei, valence neutrons must be dressed consistently as quasi-neutrons possessing an effective charge.

The absence of any appreciable strength in the two IMSRG calculations appears to be convincing evidence that IMSRG evolutions, when restricted to the two-body operator level (i.e., VS-IMSRG(2) and EOM-IMSRG(2,2)), do not sufficiently renormalize the neutron charges. However, this discrepancy is evident in many nuclei, regardless of shell structure; we see the same underpredictions in 32S, and 56,60Ni, which lie in middle of their respective major shells, with plenty of valence protons to model an electromagnetic transition.

Table III.4.1 compiles the results from several of the calculations presented here, where $B(E2)$ corresponds to $B(E2;2^{+}_{1}\rightarrow 0^{+}_{1})$. In the far right column, we include the the Weisskopf estimate for the transition Blatt and Weisskopf (1979). The Weisskopf estimate, given by

[TABLE]

models the transition as a single proton excitation from a core with the empirical nuclear radius $r_{0}A^{1/3}$, where $r_{0}=1.2$ fm. Excitations that are dominated by a single 1p1h transition will yield experimental $B(E2)$ values near the Weisskopf estimate. This picture certainly falls short of describing those nuclei with magic proton numbers, such as 22O, but it is nonetheless instructive to consider what the single particle estimates are for even these nuclei, as they describe neutrons with an effective charge in this case.