Ab Initio Nuclear Structure Theory

Ab Initio Nuclear Structure Theory Lecture 3: Light Nuclei Robert Roth

Overview Lecture 1: Hamiltonian Prelude Many-ody Quantum Mechanics Nuclear Hamiltonian Matrix Elements Lecture 2: orrelations Two-ody Problem orrelations & Unitary Transformations Similarity Renormalization Group Lecture 3: Light Nuclei Many-ody Problem onfiguration Interaction No-ore Shell Model Applications Lecture 4: eyond Light Nuclei Normal Ordering oupled-luster Theory In-Medium Similarity Renormalization Group Robert Roth - TU Darmstadt - August 217 2

Many-ody Problem

Definition: Ab Initio solve nuclear many-body problem based on realistic interactions using controlled and improvable truncations with quantified theoretical uncertainties numerical treatment with some truncations or approximations is inevitable for any nontrivial nuclear structure application challenges for ab initio calculations are to control the truncation effects quantify the resulting uncertainties reduce them to an acceptable level convergence with respect to truncations is important: demonstrate that observables become independent of truncations continuous transition from approximation to ab initio calculation Robert Roth - TU Darmstadt - August 217 4

onfiguration Interaction Approaches @ h H i 1 A @ (n) 1 A = E n @ (n) 1 A

onfiguration Interaction (I) select a convenient single-particle basis i = n jm tm t i construct A-body basis of Slater determinants from all possible combinations of A different single-particle states i = { 1 2 A} i convert eigenvalue problem of the Hamiltonian into a matrix eigenvalue problem in the Slater determinant representation H int ni = E n ni ni = X (n) i 1 h H int i @ A @ (n) 1 A = E n 1 (n) @ A Robert Roth - TU Darmstadt - August 217 6

Model Space Truncations have to introduce truncations of the single/many-body basis to make the Hamilton matrix finite and numerically tractable full I: truncate the single-particle basis, eg, at a maximum single-particle energy particle-hole truncated I: truncate single-particle basis and truncate the many-body basis at a maximum n-particle-n-hole excitation level interacting shell model: truncate single-particle basis and freeze low-lying single-particle states (core) in order to qualify as ab initio one has to demonstrate convergence with respect to all those truncations there is freedom to optimize the single-particle basis, instead of HO states one can use single-particle states from a Hartree-Fock calculation Robert Roth - TU Darmstadt - August 217 7

Variational Perspective solving the eigenvalue problem in a finite model space is equivalent to a variational calculation with a trial state DX n(d)i = (n) i formally, the stationarity condition for the energy expectation value directly leads to the matrix eigenvalue problem in the truncated model space =1 Ritz variational principle: the ground-state energy in a D-dimensional model space is an upper bound for the exact ground-state energy E (D) E (exact) Hylleraas-Undheim theorem: all states of the spectrum have a monotonously decreasing energy with increasing model space dimension E n (D) E n (D + 1) Robert Roth - TU Darmstadt - August 217 8

Theory Uncertainties model-space truncation is the sole source of uncertainties in the solution of the many-body problem absolute energies are protected by the variational principle, ie, smooth and monotonic dependence on model-space size (not so for other observables) convergence with respect to model-space size is the only thing we have to worry about efficient truncations: closer to convergence with smaller model-space dimension, ie, physics-informed truncation scheme extrapolations: extrapolate observables to infinite model-space from sequence of finite-space calculations uncertainty quantification: extract many-body uncertainty from residual model-space dependence or extrapolation Robert Roth - TU Darmstadt - August 217 9

No-ore Shell Model

No-ore Shell Model (NSM) NSM is a special case of a I approach: single-particle basis is a spherical HO basis truncation in terms of the total number of HO excitation quanta Nmax in the many-body states Φ specific advantages of the NSM: many-body energy truncation (Nmax) truncation is much more efficient than single-particle energy truncation (emax) ˆT Φ equivalent NSM formulation in relative Jacobi coordinates for each Nmax Jacobi-NSM explicit separation of center of mass and intrinsic states possible for each Nmax ˆ Φ Robert Roth - TU Darmstadt - August 217 11

4He: NSM onvergence worst case scenario for NSM convergence: Argonne V18 potential = fm 4 = 3 fm 4 4 2 4 6-1 NNonly 2-15 E [MeV] 8 1-2 -2 E AV18 Eexp 2 4 6 8 hω [MeV] 12 14 16-25 2 4 6 8 hω [MeV] Robert Roth - TU Darmstadt - August 217 12

NSM asis Dimension Vary et al; J Phys: onf Series 18, 1283 (29) 1x1 12 1x1 11 1x1 1 2N Matrix Dimension 1x1 9 1x1 8 1x1 7 1x1 6 1x1 5 1x1 4 3N 4N 6 Li 1 24 Mg 28 Si 1x1 3 12 32 S 1x1 2 16 O 4 a 1x1 1 1x1 2 4 6 8 1 12 14 N max Robert Roth - TU Darmstadt - August 217 13

Robert Roth - TU Darmstadt - August 217 omputational Strategy key properties of the computational problem: only interested in a few low-lying eigenstates Hamilton matrix is very sparse (typically <1% non-zeros) Laczos-type algorithms for an iterative solution of the eigenvalue problem amount of fast storage for non-zero matrix elements & a few eigenvectors sets the limits and drives parallelization strategies 14 @ h H i 1 A @ (n) 1 A = E n @ (n) 1 A

Lanczos Algorithm Lanczos Algorithm: convert the eigenvalue problem of a huge matrix H in an iterative process to an eigenvalue problem of small matrices Tm which converge to the same extremal eigenvalues H = @ 1 1 1 1 1 A ~ := ~ ~ 1 := any norm vector 1 := for = 1,m do ~ := H ~ ~ 1 := ~ ~ ~ := ~ ~ +1 := ~ ~ +1 := ~ / +1 T m = @ 1 2 2 2 3 3 3 A m m m 1 end for Robert Roth - TU Darmstadt - August 217 15

Lanczos Algorithm Lanczos Algorithm: convert the eigenvalue problem of a huge matrix H in an iterative process to an eigenvalue problem of small matrices Tm which converge to the same extremal eigenvalues Robert Roth - TU Darmstadt - August 217 16

Importance Truncation

Importance Truncation converged NSM calculations limited to lower & mid p-shell nuclei example: full Nmax=1 calculation for 16 O would be very difficult, basis dimension D > 1 1 E [MeV] -11-12 -13-14 -15 16 O NN only α = 4 fm 4 hω = 2 MeV 2 4 6 8 112 14 16 18 2 Robert Roth - TU Darmstadt - August 217 18

Importance Truncation converged NSM calculations limited to lower & mid p-shell nuclei example: full Nmax=1 calculation for 16 O would be very difficult, basis dimension D > 1 1 Importance Truncation reduce model space to the relevant basis states using an a priori importance measure derived from MPT E [MeV] E [MeV] -11-12 -13-14 -15-11 -12-13 -14-15 16 O NN only α = 4 fm 4 hω = 2 MeV 2 4 6 8 112 14 16 18 2 IT-NSM + full NSM 2 4 6 8 112 14 16 18 2 Robert Roth - TU Darmstadt - August 217 19

Importance Truncation starting point: approximation Ψ ref for the target state within a limited reference space M ref Ψ ref = (ref) ν ν ν M ref measure the importance of individual basis state ν M ref via first-order multiconfigurational perturbation theory κ ν = ν H Ψ ref Δε ν construct importance-truncated space M(κ min ) from all basis states with κ ν κ min solve eigenvalue problem in importance truncated space M IT (κ min ) and obtain improved approximation of target state Robert Roth - TU Darmstadt - August 217 2

Threshold Extrapolation E [MeV] E [MeV] -146-1465 -147-1475 -148-1485 -15-151 -152-153 16 O NN-only α = 4 fm 4 hω = 2 MeV = 8 repeat calculations for a sequence of importance thresholds κ min observables show smooth threshold dependence and systematically approach the full NSM limit use aposterioriextrapolation κ min of observables to account for effect of excluded configurations -154-155 = 12 2 4 6 8 1 κ min 1 5 uncertainty quantification via set of extrapolations Robert Roth - TU Darmstadt - August 217 21

4He: Ground-State Energy Roth, et al; PRL 17, 7251 (211); PRL 19, 5251 (212) NN only NN+3N ind NN+3N full -23-24 hω = 2 MeV E [MeV] -25-26 -27-28 -29 2 4 6 8 1 12 14 16 Exp 2 4 6 8 1 12 14 2 4 6 8 1 12 14 α = 4 fm 4 α = 5 fm 4 α = 625 fm 4 α = 8 fm 4 α = 16 fm 4 Λ = 224 fm 1 Λ = 211 fm 1 Λ = 2 fm 1 Λ = 188 fm 1 Λ = 158 fm 1 Robert Roth - TU Darmstadt - August 217 22 5

7Li: Ground-State Energy Roth, et al; PRL 17, 7251 (211); PRL 19, 5251 (212) NN only NN+3N ind NN+3N full -22 hω = 2 MeV E [MeV] -24-26 -28-3 -32 Exp -34 2 4 6 8 1 12 14 2 4 6 8 1 12 2 4 6 8 1 12 α = 4 fm 4 α = 5 fm 4 α = 625 fm 4 α = 8 fm 4 α = 16 fm 4 Λ = 224 fm 1 Λ = 211 fm 1 Λ = 2 fm 1 Λ = 188 fm 1 Λ = 158 fm 1 Robert Roth - TU Darmstadt - August 217 23 6

12: Ground-State Energy Roth, et al; PRL 17, 7251 (211); PRL 19, 5251 (212) NN only NN+3N ind NN+3N full -6 hω = 2 MeV -7 E [MeV] -8-9 Exp -1-11 2 4 6 8 1 12 14 2 4 6 8 1 12 2 4 6 8 1 12 α = 4 fm 4 α = 5 fm 4 α = 625 fm 4 α = 8 fm 4 α = 16 fm 4 Λ = 224 fm 1 Λ = 211 fm 1 Λ = 2 fm 1 Λ = 188 fm 1 Λ = 158 fm 1 Robert Roth - TU Darmstadt - August 217 24 7

16O: Ground-State Energy Roth, et al; PRL 17, 7251 (211); PRL 19, 5251 (212) -8 NN only NN+3N ind NN+3N full hω = 2 MeV -1 E [MeV] -12-14 Exp -16-18 2 4 6 8 1 12 14 2 4 6 8 1 12 signature of induced 4N interactions Nbeyond m x mid p-shell 2 4 6 8 1 12 α = 4 fm 4 α = 5 fm 4 α = 625 fm 4 α = 8 fm 4 α = 16 fm 4 Λ = 224 fm 1 Λ = 211 fm 1 Λ = 2 fm 1 Λ = 188 fm 1 Λ = 158 fm 1 Robert Roth - TU Darmstadt - August 217 25 8

16O: Ground-State Energy Roth, et al; PRL 17, 7251 (211); PRL 19, 5251 (212) -8 NN only NN+3N ind NN+3N full hω = 2 MeV -1 E [MeV] -12-14 Exp Λ3N=4 MeV -16 Λ3N=5 MeV -18 2 4 6 8 1 12 14 2 4 6 8 1 12 2 4 6 8 1 12 α = 4 fm 4 α = 5 fm 4 α = 625 fm 4 α = 8 fm 4 α = 16 fm 4 Λ = 224 fm 1 Λ = 211 fm 1 Λ = 2 fm 1 Λ = 188 fm 1 Λ = 158 fm 1 Robert Roth - TU Darmstadt - August 217 26 9

12: Excitation Spectrum Roth, et al; PRL 17, 7251 (211); PRL 19, 5251 (212) 18 NN only NN+3N ind NN+3N full 16 14 2 1 2 1 1 4 2 1 2 1 1 4 2 1 2 1 1 4 12 1 1 1 E [MeV] 1 8 6 12 hω = 16 MeV 4 2 2 2 2 2 4 6 8 Exp 2 4 6 8 Exp 2 4 6 8 Exp α = 4 fm 4 α = 8 fm 4 Λ = 224 fm 1 Λ = 188 fm 1 Robert Roth - TU Darmstadt - August 217 27 22

From Dripline to Dripline

Ground States of Helium Isotopes E [MeV] -6-65 -7-75 -22 E [MeV] -24-26 -28 3 He hω = 2 MeV 6 He -2-22 -24-26 -28-3 -22-24 -26-28 4 He hω = 2 MeV 7 He -2-22 -24-26 -28-3 -22-24 -26-28 5 He hω = 16 MeV 8 He hω = 16 MeV α = 8 fm 4, E 3m x = 12-3 -32 hω = 16 MeV 4 6 8 1 12 14 16 18 2-3 -32 hω = 16 MeV 4 6 8 1 12 14 16 18 2-3 -32 NN+3N-induced NN+3N-full 4 6 8 1 12 14 16 18 2 Robert Roth - TU Darmstadt - August 217 29

Ground States of Helium Isotopes -6-8 exp NN+3N-induced chiral NN interaction cannot reproduce ground-state systematics E [MeV] -1-22 -24-26 -28-3 -32 3 He 4 He 5 He 6 He 7 He 8 He α = 8 fm 4, E 3m x = 12 Robert Roth - TU Darmstadt - August 217 3

Ground States of Helium Isotopes E [MeV] -6-8 -1-22 -24-26 exp NN+3N-induced NN+3N-full(5) chiral NN interaction cannot reproduce ground-state systematics inclusion of chiral 3N interaction improves trend significantly -28-3 -32 3 He 4 He 5 He 6 He 7 He 8 He α = 8 fm 4, E 3m x = 12 Robert Roth - TU Darmstadt - August 217 31

Ground States of Helium Isotopes E [MeV] -6-8 -1-22 -24-26 -28-3 -32 exp NN+3N-induced NN+3N-full(5) NN+3N-full(4) 3 He 4 He 5 He 6 He 7 He 8 He chiral NN interaction cannot reproduce ground-state systematics inclusion of chiral 3N interaction improves trend significantly systematics is sensitive to details of the 3N interaction, test for new chiral Hamiltonians continuum needs to be included: NSM with ontinuum α = 8 fm 4, E 3m x = 12 Robert Roth - TU Darmstadt - August 217 32

Oxygen Isotopes oxygen isotopic chain has received significant attention and documents the rapid progress over the past years Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL 15, 3251 (21) 21: shell-model calculations with 3N effects highlighting the role of 3N interaction for drip line physics Hagen, Hjorth-Jensen, Jansen, Machleidt, Papenbrock, PRL 18, 24251 (212) 212: coupled-cluster calculations with phenomenological two-body correction simulating chiral 3N forces Hergert, inder, alci, Langhammer, Roth, PRL 11, 24251 (213) 213: ab initio IT-NSM with explicit chiral 3N interactions and first multi-reference in-medium SRG calculations ipollone, arbieri, Navrátil, PRL 111, 6251 (213) ogner, Hergert, Holt, Schwenk, inder, alci, Langhammer, Roth, PRL113, 14251(214) Jansen, Engel, Hagen, Navratil, Signoracci, PRL 113, 14252 (214) since: self-consistent Green s function, shell model with valencespace interactions from in-medium SRG or Lee-Suzuki, Robert Roth - TU Darmstadt - August 217 33

Ground States of Oxygen Isotopes Hergert et al, PRL 11, 24251 (213) -4 NN+3N ind (chiral NN) 12 O NN+3N full (chiral NN+3N) 12 O -6 E [MeV] -8-1 -12-14 -16 2 4 6 8 1 12 14 16 18 14 O 16 O 18 O 2 O 22 O 24 O 26 O 2 4 6 8 1 12 14 16 18 14 O 16 O 18 O 2 O 22 O 26 O 24 O Λ 3N = 4 MeV, α = 8 fm 4, E 3m x = 14, optimal hω 23 Robert Roth - TU Darmstadt - August 217 34

Ground States of Oxygen Isotopes Hergert et al, PRL 11, 24251 (213) E [MeV] -4-6 -8-1 -12-14 NN+3N ind (chiral NN) A O NN+3N full (chiral NN+3N) experiment IT-NSM -16 parameter-free ab initio calculations -18 with explicit chiral 123N 14 interactions 16 18 2 22 24 highlights 26 12 predictive 14 16 18 2 A power of chiral NN+3N A 22 24 26 Λ 3N = 4 MeV, interactions α = 8 fm 4, E 3m x = 14, optimal hω Robert Roth - TU Darmstadt - August 217 35

Spectra of Oxygen Isotopes Hergert et al, PRL 11, 24251 (213) & in prep E [MeV] E [MeV] 7 6 5 4 3 2 1 6 5 4 3 2 1 18 O 2 4 6 Exp 2 2 4 2 21 O 9 2 7 2 3 2 1 2 5 4 3 2 1 9 8 7 6 5 4 3 2 1 19 O 2 4 6 Exp 3 2 7 2 9 2 1 2 3 5 2 2 22 O 4 2 3 2 6 2 O 5 4 3 2 1 7 6 5 4 3 2 1 2 4 6 Exp 4 2 4 2 23 O 3 2 5 2 NN+3N full (chiral NN+3N) Λ 3N = 4 MeV, α = 8 fm 4, hω = 16 MeV 2 4 6 Exp 5 2 2 4 6 Exp 2 4 6 Exp 1 2 Robert Roth - TU Darmstadt - August 217 36 25

12: Testing hiral Hamiltonians standard N3LO+N2LO 5 MeV optimized Epelbaum N2LO+N2LO N2LO+N2LO 5/7 MeV 45/5 MeV 6/5 MeV 55/6 MeV 45/7 MeV 15 2 1 1 2 1 4 1 E [MeV] 1 5 2 NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N Exp = 8, α = 8 fm 4, hω = 16 MeV 26 Robert Roth - TU Darmstadt - August 217 37

12: Testing hiral Hamiltonians standard N3LO+N2LO 5 MeV optimized Epelbaum N2LO+N2LO N2LO+N2LO 5/7 MeV 45/5 MeV 6/5 MeV 55/6 MeV 45/7 MeV 15 2 1 1 2 1 4 1 E [MeV] 1 5 2 NN +3N NN +3N NN +3N NN +3N NN +3N NN +3N Exp = 8, α = 8 fm 4, hω = 16 MeV Robert Roth - TU Darmstadt - August 217 38

The NSM Family NSM HO Slater determinant basis with Nmax truncation Jacobi NSM relative-coordinate Jacobi HO basis with Nmax truncation Importance Truncated NSM HO Slater determinant basis with Nmax and importance truncation Monte arlo NSM Monte arlo sampling to select important basis states Symmetry Adapted NSM group-theoretical basis with SU(3) deformation quantum numbers & truncations Gamow NSM/I Slater determinant basis including Gamow single-particle resonance states NSM with ontinuum NSM for sub-clusters with explicit RGM treatment of relative motion Robert Roth - TU Darmstadt - August 217 39