Emergence of rotational properties from an ab initio

Emergence of rotational properties from an ab initio perspective 2016 Thomas DUGUET CEA/SPh. N, Saclay, France IKS, KU Leuven, Belgium NSCL, Michigan State University, USA Ab initio methods Shapes and symmetries in nuclei: from experiment to theory workshop CSNSM, Orsay, France, November 7 -11 2016

Outline ○ Rotational properties: from experiment to theory ○ First test cases for ab initio many-body calculations ○ p-shell nuclei from no core shell model ○ sd-shell nuclei from valence-space shell model ○ sd-shell nuclei from symmetry-broken/restored coupled cluster theory

Rotational properties: from experiment to theory I. The phenomenology II. The symmetry Symmetry IRREP A. Bohr, Nobel Lecture (1975) Observable patterns dictated by SU(2) symmetry Observable patterns o Energies labeled by J and independent of M o Set of states organized according to Selection rules for EM(l) transitions initio viewpoint o Strong E 2 transitions in the set Ab scaling with nuclear A-bodyo problem But no more than that! o Happens for nuclei bunched in between doubly magic ones Do rotational properties emerge from basic interactions between the nucleons? ►Non-trivial as ��energy ≪ scale for individual excitations IV. The interpretation = emergent symmetry breaking III. The model = adiabatic rotational model ► 2 N (+3 N) are adjusted on 2 -body (+3 -body) systems Do perturbations due to coupling to vibrations and individual dof emerge? True eigenstate Intrinsic state Cannot be anticipated from dof + H Is not fully realized Similar for other symmetries of H Rotational motion Features 1) Lessons o Links a specific subset of states together o Excellent account of idealized patterns o Built in separation of rotational degrees of freedom o Disturbed by coupling of rot. to vib. and ind. dynamics 2) 1) Spontaneous breaking of SU(2) o GS has lower symmetry than H o GS = wave packet mixing IRREPs o Goldstone boson = rotations o Higgs modes = vibrations 2) Finite system = breaking only emergent o Symmetry is actually enforced o Lower symmetry imprints excitations o Rotational bands and transitions

Outline ○ Rotational properties: from experiment to theory ○ First test cases for ab initio many-body calculations ○ p-shell nuclei from no core shell model ○ sd-shell nuclei from valence-space shell model ○ sd-shell nuclei from symmetry-broken/restored coupled cluster theory

Ab initio many-body methods: theoretical scheme 2 N, 3 N… interactions from c. EFT Ø Account for symmetries of QCD Ø Systematic expansion Ø A priori uncertainty at given order [E. Epelbaum, PPNP 67, 343 (2012)] Ø Soften through RG Comp data [S. K. Bogner et al. , PPNP 65, 94 (2010)] Input Reach as of 2016 Ab-initio many-body theories Ø Effective structure-less nucleons Ø 2 N + 3 N + … inter-nucleon interactions Ø Solve A-body Schrödinger equation Ø Thorough assessment of errors needed

Two strategies to deal with symmetries, e. g. SU(2), of H A! cost A. Enforced throughout = symmetry-conserving methods ⦿ Diagonalization methods No core shell model ⦿ Imaginary time propagation Green’s function monte carlo Ap cost Symmetry-conserving basis expansion ⦿ Expansion methods Symmetry-conserving wave operator Symmetry-conserving reference state Coupled cluster, self consistent Green’s function, In-medium similarity renormalization group Ap cost B. Allowed to break at low order before being restored = symmetry-broken and -restored methods ⦿ Expansion methods Symmetry-broken wave operator Symmetry-breaking reference state Coupled cluster, (self consistent Green’s function), In-medium similarity renormalization group Explicit symmetry restoration Efficient in doubly (singly) open-shell nuclei to capture quadrupole (pairing) via breaking/restoration of SU(2) (U(1))

Ab initio many-body methods: rotational properties 1) 8 Be Ideal test cases ○ No-core shell model ○ Symmetry conserving 2) 20 Ne and 24 Mg ○ Ab initio valence-space shell model ○ Symmetry conserving 3) 24 Mg 20 N e 24 Mg ○ Coupled cluster theory ○ Symmetry broken/restored 8 Be

No-core shell model calculation of 8 Be isotopes [M. A. Caprio et al. , JMPE 24 (2015) 1541002] Yrast positive parity band Jmax=4 in pure p shell No core shell model calculation of 8 Be Two-nucleon interactions Ø Chiral 2 N (N 2 LO ; L 2 NF = 500 Me. V/c) E(4+)/E(2+)=3. 46 Me. V Pure rotational model Axis scales as J(J+1) [Ekstrom et al. , PRL 110 (2013) 192502] Results Rotational behavior emerge very convincingly in 8 Be o Energies, E 2/M 1 moments and transitions o Converged quadrupole strength ratios (absolute moment unsettled) o Null spin contribution to m(J) consistent with a-clustering o Consistent analysis of (un)natural parity/excited bands in 7, 9 Be Good account of experimental energies in 7 -9 Be (not shown) Robust against (modest) variation of 2 N interaction (not shown) o Add 3 N force and test at various EFT orders (LO, NLO…) Must confront data systematically and for less ideal cases Similar for GFMC calculations [R. B. Wiringa et al. , PRC 62 (2000) 014001] Orbital =g. R J =0. 49 J Spin

Valence-space shell model calculation of 20 Ne and 24 Mg [S. R. Stroberg et al. , PRC 93 (2016) 051301] Yrast spectroscopy of ideal rotor nuclei sd shell model calculation of 20 Ne and 24 Mg 1) 2) sd-shell effective interaction from IMSRG calculation of 16 O Inter-nucleon interactions Ø Chiral 2 N (N 3 LO ; L 2 NF = 500 Me. V/c) [D. R. Entem, R. Machleidt, PRC 68, 041001 (2003)] Ø Chiral 3 N (N 2 LO ; L 3 NF = 400 Me. V/c) [P. Navratil, FBS 41, 117 (2007)] Ø SRG evolved down to l = 2. 0 fm-1 Perpectives Ø Check in non-ideal rotor nuclei Ø Various EFT orders and uncertainty propagation Ø Investigate E 2/M 1 moments and transitions Results Ø Rotational bands emerge convincingly Ø Quantitatively as good as empirical model Ø Insensitive to 3 N interaction at low spins Unlike overall spectroscopy in sd shell Similarly for CC-based valence-space shell model [G. Hagen et al. , Phys. Scr. 91 (2016) 063006]

SU(2) broken &restored MBPT (CC) calculation of 24 Mg [T. D. JPG 42 (2015) 025107] [S. Binder, T. D. , G. Hagen, T. Papenbrock, unpublished] Yrast states in 24 Mg sd shell emax=7 y r a in 1)+2) m i l re Deformed MBPT (CC) calculation & J restoration P Two-nucleon interactions Ø Chiral 2 N (N 2 LO ; L 2 NF = 500 Me. V/c) [Ekstrom et al. , PRL 110 (2013) 192502] Results Perpectives Ø Ø Inclusion of 3 N interactions Full CC beyond perturbation theory SU(2) restored PES Extension to non-yrast states Ø Ø Rotational bands emerge convincingly J=4, 6 not converged yet with respect to emax No 3 N interaction yet Moment of inertia reduced by correlations

Long-term perspectives of ab initio methods 2004 Emergence from nucleons and their interactions? ○ Binding, size, limit of existence, collectivity, superfluidity… Limits of such a description with A/in accuracy? 20? ? Rotational properties v Dominance of prolate over oblate? v Superdeformation? ○ Modified ab initio effective theory when A increases? v Physics of transitional nuclei? ○ More effective but explicitly connected approaches? v Shape coexistence phenomenon? Detailed and systematic description of nuclei v New features? What specific features of AN interactions probed?