ab initio nuclear structure from chiral nn+3n hamiltoniansrobert roth – tu darmstadt – 04/2013...
TRANSCRIPT
Ab Initio Nuclear Structure
from Chiral NN+3N Hamiltonians
Robert Roth
1
From QCD to Nuclear Structure
Robert Roth – TU Darmstadt – 04/2013
Nuclear Structure
Low-Energy QCD
2
From QCD to Nuclear Structure
Robert Roth – TU Darmstadt – 04/2013
Nuclear Structure
Low-Energy QCD
NN+3N Interactionfrom Chiral EFT
chiral EFT based on the rel-evant degrees of freedom &symmetries of QCD
provides consistent NN, 3N,...interaction plus currents
3
From QCD to Nuclear Structure
Robert Roth – TU Darmstadt – 04/2013
Nuclear Structure
Low-Energy QCD
NN+3N Interactionfrom Chiral EFT
Unitary / SimilarityTransformation
adapt Hamiltonian to trun-cated low-energy model space
• tame short-range correlations• improve convergence behavior
transform Hamiltonian & ob-servables consistently
4
From QCD to Nuclear Structure
Robert Roth – TU Darmstadt – 04/2013
Nuclear Structure
Low-Energy QCD
NN+3N Interactionfrom Chiral EFT
Unitary / SimilarityTransformation
Exact & Approx.Many-Body Methods
accurate solution of the many-body problem for light & inter-mediate masses (NCSM, CC,...)
controlled approximations forheavier nuclei (MBPT,...)
all rely on truncated modelspaces & benefit from unitarytransformation
5
From QCD to Nuclear Structure
Robert Roth – TU Darmstadt – 04/2013
Nuclear Structure
Low-Energy QCD
NN+3N Interactionfrom Chiral EFT
Unitary / SimilarityTransformation
Exact & Approx.Many-Body Methods
How can we extend current ab-initiomethods to describe open-shell anddeformed nuclei?
How can we include the effects ofthree-nucleon forces in a computa-tionally efficient manner?
How can we describe the onset ofpairing in nuclei within various ab-initio frameworks?
Benchmarking and accuracy: canwe develop reliable theoretical errorestimates?
How can we bridge structure and re-actions in a consistent fashion?
How can we generate reliable predic-tions for the drip-lines?
6
Nuclear Interactions
from Chiral EFT
7
Nuclear Interactions from Chiral EFT
Robert Roth – TU Darmstadt – 04/2013
Weinberg, van Kolck, Machleidt, Entem, Meissner, Epelbaum, Krebs, Bernard,...
low-energy effective field theoryfor relevant degrees of freedom (π,N)based on symmetries of QCD
long-range pion dynamics explicitly
short-range physics absorbed in con-tact terms, low-energy constants fit-ted to experiment (NN, πN,...)
hierarchy of consistent NN, 3N,...interactions (plus currents)
NN 3N 4N
LO
NLO
N2LO
N3LO
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many ongoing developments
• 3N interaction at N3LO, N4LO,...
• explicit inclusion of Δ-resonance
• YN- & YY-interactions
• formal issues: power counting,renormalization, cutoff choice,...
8
Chiral NN+3N Hamiltonians
Robert Roth – TU Darmstadt – 04/2013
standard Hamiltonian:
• NN at N3LO: Entem / Machleidt, 500 MeV cutoff
• 3N at N2LO: Navrátil, local, 500 MeV cutoff, fit to T1/2(3H) and E(3H, 3He)
standard Hamiltonian with modified 3N:
• NN at N3LO: Entem / Machleidt, 500 MeV cutoff
• 3N at N2LO: Navrátil, local, with modified LECs and cutoffs, refit to E(4He)
consistent N2LO Hamiltonian:
• NN at N2LO: Epelbaum et al., 450,...,600 MeV cutoff
• 3N at N2LO: Epelbaum et al., nonlocal, 450,...,600 MeV cutoff
consistent N3LO Hamiltonian:
• coming soon...
9
Similarity
Renormalization Group
Roth, Calci, Langhammer, Binder — in preparation (2013)
Roth, Langhammer, Calci et al. — Phys. Rev. Lett. 107, 072501 (2011)
Roth, Neff, Feldmeier — Prog. Part. Nucl. Phys. 65, 50 (2010)
Roth, Reinhardt, Hergert — Phys. Rev. C 77, 064033 (2008)
Hergert, Roth — Phys. Rev. C 75, 051001(R) (2007)
10
Similarity Renormalization Group
Robert Roth – TU Darmstadt – 04/2013
Wegner, Glazek, Wilson, Perry, Bogner, Furnstahl, Hergert, Roth, Jurgenson, Navratil,...
continuous transformation drivingHamiltonian to band-diagonal form
with respect to a chosen basis
unitary transformation of Hamiltonian (and other observables)
eHα = Uα†HUα
evolution equations for eHα and Uα depending on generator ηαd
dαeHα =ηα, eHα
d
dαUα = −Uαηα
dynamic generator: commutator with the operator in whoseeigenbasis H shall be diagonalized
ηα = (2μ)2Tint, eHα
simplicity and flexibilityare great advantages of
the SRG approach
solve SRG evolutionequations using two-,
three- & four-body matrixrepresentation
11
SRG Evolution in Three-Body Space
Robert Roth – TU Darmstadt – 04/2013
chiral NN+3NN3LO + N2LO, triton-fit, 500 MeV
α = 0.000 fm4
Λ =∞ fm−1
Jπ =1
2
+, T =
1
2, ℏΩ = 28MeV
3B-Jacobi HO matrix elements
0 → E → 18 20 22 24 26 28(E, )
28
26
24
22
20
18
↓E′
↓
0
.
(E′ ,′)
NCSM ground state 3H
0 2 4 6 8 101214161820Nmx
-8
-6
-4
-2
0
2
.
E[MeV]
12
SRG Evolution in Three-Body Space
Robert Roth – TU Darmstadt – 04/2013
α = 0.320 fm4
Λ = 1.33 fm−1
Jπ =1
2
+, T =
1
2, ℏΩ = 28MeV
3B-Jacobi HO matrix elements
0 → E → 18 20 22 24 26 28(E, )
28
26
24
22
20
18
↓E′
↓
0
.
(E′ ,′)
NCSM ground state 3H
0 2 4 6 8 101214161820Nmx
-8
-6
-4
-2
0
2
.
E[MeV]
suppression ofoff-diagonal coupling= pre-diagonalization
significantimprovementof convergence
behavior
13
Hamiltonian in A-Body Space
Robert Roth – TU Darmstadt – 04/2013
evolution induces n-body contributions eH[n]α
to Hamiltonian
eHα = eH[1]α+ eH[2]
α+ eH[3]
α+ eH[4]
α+ . . .
truncation of cluster series inevitable — formally destroys unitarityand invariance of energy eigenvalues (independence of α)
SRG-Evolved Hamiltonians
NN only: start with NN initial Hamiltonian and keep two-bodyterms only
NN+3N-induced: start with NN initial Hamiltonian and keep two-and induced three-body terms
NN+3N-full: start with NN+3N initial Hamiltonian and keep two-and all three-body terms
α-variation provides adiagnostic tool to assessthe contributions of omittedmany-body interactions
14
Sounds easy, but...
Robert Roth – TU Darmstadt – 04/2013
➊ initial 3B-Jacobi HO matrix elements of chiral 3N interactions
• direct computation using Petr Navratil’s ManyEff code (N2LO)
• conversion of partial-wave decomposed moment-space matrix ele-ments of Epelbaum et al. (N2LO, N3LO,...)
➋ SRG evolution in 2B/3B space and cluster decomposition
• efficient implementation using adaptive ODE solver & BLAS;largest JT-block takes a few hours on single node
➌ transformation of 2B/3B Jacobi HO matrix elements into JT-coupledrepresentation
• transform directly into JT-coupled scheme; highly efficient implemen-tation; can handle E3mx = 16 in JT-coupled scheme
➍ data management and on-the-fly decoupling in many-body codes
• optimized storage scheme for fast on-the-fly decoupling; can keep allmatrix elements up to E3mx = 16 in memory; suitable for GPUs
15
Importance Truncated
No-Core Shell Model
Roth, Calci, Langhammer, Binder — in preparation (2013)
Roth, Langhammer, Calci et al. — Phys. Rev. Lett. 107, 072501 (2011)
Navrátil, Roth, Quaglioni — Phys. Rev. C 82, 034609 (2010)
Roth — Phys. Rev. C 79, 064324 (2009)
Roth, Gour & Piecuch — Phys. Lett. B 679, 334 (2009)
Roth, Gour & Piecuch — Phys. Rev. C 79, 054325 (2009)
Roth, Navrátil — Phys. Rev. Lett. 99, 092501 (2007)
16
No-Core Shell Model
Robert Roth – TU Darmstadt – 04/2013
Barrett, Vary, Navratil, Maris, Nogga, Roth,...
NCSM is one of the most powerful anduniversal exact ab-initio methods
construct matrix representation of Hamiltonian using a basis of HOSlater determinants truncated w.r.t. HO excitation energy NmxℏΩ
solve large-scale eigenvalue problem for a few extremal eigenvalues
all relevant observables can be computed from the eigenstates
range of applicability limited by factorial growth of basis with Nmx & A
adaptive importance truncation extends the range of NCSM by reduc-ing the model space to physically relevant states
17
Importance Truncated NCSM
Robert Roth – TU Darmstadt – 04/2013
Roth, PRC 79, 064324 (2009); PRL 99, 092501 (2007)
converged NCSM calcula-tions essentially restrictedto lower/mid p-shell
full Nmx = 10 calculationfor 16O very difficult(basis dimension > 1010)
0 2 4 6 8 10 12 14 16 18 20Nmx
-150
-140
-130
-120
-110
.
E[MeV]
16ONN-only
α = 0.04 fm4
ℏΩ = 20MeV
ImportanceTruncation
reduce model spaceto the relevant basis
states using an a prioriimportance measurederived from MBPT
0 2 4 6 8 10 12 14 16 18 20Nmx
-150
-140
-130
-120
-110
.
E[MeV]
IT-NCSM
+ full NCSM
similar strategies havefirst been developed and
applied in quantum chemistry:
configuration-selectivemultireference CI
Buenker & Peyerimhoff (1974);
Huron, Malrieu & Rancurel (1973);
and others...
18
Importance Truncation: Basic Idea
Robert Roth – TU Darmstadt – 04/2013
starting point: approximation |Ψref,m⟩ for the target stateswithin a limited reference space Mref
|Ψref,m⟩ =∑
ν∈Mref
C(ref,m)ν
|ν⟩
measure the importance of individual basis state |ν⟩ /∈ Mref
via first-order multiconfigurational perturbation theory
κ(m)ν= −⟨ν |H |Ψref,m⟩
Δεν
construct importance-truncated space M(κmin) from all basisstates with |κ(m)
ν| ≥ κmin for any m
solve eigenvalue problem in importance truncated spaceMIT(κmin) and obtain improved approximation of target state
importance measure onlyprobes 2p2h excitations ontop of Mref for a two-body
Hamiltonianembed into iterativescheme to access full
model space
19
Importance Truncation: Iterative Scheme
Robert Roth – TU Darmstadt – 04/2013
property of Nmx-truncated space: step from Nmx to Nmx + 2requires 2p2h excitations at most
sequential calculation for a range of NmxℏΩ spaces:
do full NCSM calculations up to a convenient Nmx
use components of eigenstates with |C(m)ν| ≥ Cmin as initial |Ψref,m⟩
➊ consider all states |ν⟩ /∈ Mref from an Nmx + 2 space and addthose with |κ(m)
ν| ≥ κmin to importance-truncated space MIT
➋ solve eigenvalue problem in MIT
➌ use components of eigenstate with |C(m)ν| ≥ Cmin as new |Ψref,m⟩
➍ goto ➊
full NCSM space isrecovered in the limit(κmin, Cmin)→ 0
20
Threshold Extrapolation
Robert Roth – TU Darmstadt – 04/2013
-148.5
-148.0
-147.5
-147.0
-146.5
-146.0
.
[MeV]
Hint
0.00
0.01
0.02
0.03
.
[MeV] Hcm
0 2 4 6 8 10
κmin × 105
0
0.01
0.02
0.03
.
J
16ONN-only
α = 0.04 fm4
ℏΩ = 20MeV
Nmx = 8
repeat calculations for asequence of importancethresholds κmin
observables show smooththreshold dependence andsystematically approach thefull NCSM limit
use a posteriori extrapo-lation κmin → 0 of observ-ables to account for effect ofexcluded configurations
21
Threshold Extrapolation
Robert Roth – TU Darmstadt – 04/2013
-148.5
-148.0
-147.5
-147.0
-146.5
-146.0
.
[MeV]
Hint
0 2 4 6 8 10
κmin × 105
-155.0
-154.0
-153.0
-152.0
-151.0
-150.0
.
[MeV]
Hint
16ONN-only
α = 0.04 fm4
ℏΩ = 20MeV
Nmx = 8
Nmx = 12
repeat calculations for asequence of importancethresholds κmin
observables show smooththreshold dependence andsystematically approach thefull NCSM limit
use a posteriori extrapo-lation κmin → 0 of observ-ables to account for effect ofexcluded configurations
22
Constrained Threshold Extrapolation
Robert Roth – TU Darmstadt – 04/2013
16ONN-only, α = 0.04 fm4
ℏΩ = 20MeV, Nmx = 12
0 2 4 6 8 10κmin × 105
-158
-156
-154
-152
-150
.
Eλ[M
eV]
1.0
0.75
0.5
0.25
λ = 0
for free: importance selectiongives perturbative energy cor-rection Δexcl(κmin) accountingfor excluded states
formal property
Δexcl(κmin)→ 0 for κmin→ 0
auxiliary parameter λ defininga family of energy sequences
Eλ(κmin) = E(κmin) + λΔexcl(κmin)
simultaneous extrapolationfor family of λ-values with con-straint Eλ(0) = Eextrap
proposed byBuenker & Peyerimhoff (1975):“Energy Extrapolation in CI
Calculations”
23
Uncertainty Quantification
in the IT-NCSM
24
Uncertainty Quantification
Robert Roth – TU Darmstadt – 04/2013
Importance Truncation
use sequence of (Cmin, κmin)-truncated model spaces
extrapolate to κmin → 0 us-ing poynomial ansatz or morerefined constrained extrapola-tion scheme
uncertainty estimate derivedfrom extrapolation protocol
systematic uncertainty ab-sent in full NCSM
Model-Space Truncation
use sequence of Nmx-trunc-ated model spaces
extrapolate to Nmx → ∞using exponential ansatz ormore elaborate extrapolationschemes
uncertainty estimate derivedfrom extrapolation protocol
same extrapolation uncer-tainties as in full NCSM
25
Comment on Cmin Truncation
Robert Roth – TU Darmstadt – 04/2013
-148.5
-148.0
-147.5
-147.0
-146.5
-146.0
.
E[M
eV]
0 2 4 6 8 10
κmin × 105
-155.0
-154.0
-153.0
-152.0
-151.0
-150.0
.
E[M
eV]
16ONN-only
α = 0.04 fm4
ℏΩ = 20MeV
Nmx = 8
Nmx = 12
Cmin = 1× 10−4
Î Cmin = 2× 10−4
Cmin = 3× 10−4
Cmin = 5× 10−4
truncation of reference state tocomponents with |Cν | ≥ Cmin
technical reason: importanceselection phase scales with(dimMref)
2
typically Cmin = 2× 10−4
practically noinfluence on thresholdextrapolated energies
26
Protocol: Simple κmin Extrapolation
Robert Roth – TU Darmstadt – 04/2013
-148.5
-148.0
-147.5
-147.0
-146.5
-146.0
.
E[M
eV]
0 2 4 6 8 10
κmin × 105
-155.0
-154.0
-153.0
-152.0
-151.0
-150.0
.
E[M
eV]
16ONN-only
α = 0.04 fm4
ℏΩ = 20MeV
Nmx = 8
Nmx = 12
perform IT-NCSM calculations forrange of κmin-values, typicallyκmin = 3,3.5, ...,10× 10−5
extrapolation κmin → 0 using poly-nomial Pp(κmin) fit to full κmin-set,typically of order p = 3
generate uncertainty band fromset of alternative extrapolations
• Pp−1 and Pp+1 extrapolations us-ing full κmin-range
• Pp extrapolations with lowest andlowest two κmin-points dropped
quote standard deviation as nomi-nal uncertainty
27
Protocol: Constrained κmin Extrapolation
Robert Roth – TU Darmstadt – 04/2013
16ONN-only, α = 0.04 fm4
ℏΩ = 20MeV, Nmx = 12
0 2 4 6 8 10κmin × 105
-157
-156
-155
-154
-153
-152
-151
.
Eλ[M
eV]
select a few λ-values to get sym-metrical approach towards com-mon Eextrap = Eλ(κmin = 0)
constrained simultaneous extrap-olation κmin → 0 using polynomialPp(κmin), typically of order p = 3
generate uncertainty band fromset of constrained extrapolations
• Pp−1 and Pp+1 extrapolations us-ing full κmin-range
• Pp extrapolations with lowest andlowest two κmin-points dropped
• Pp extrapolations with smallestand largest λ-set dropped
std. deviation gives uncertainty
28
Characterization of SRG-Evolved
NN+3N Hamiltonians
Roth, Calci, Langhammer, Binder — in preparation (2013)
Roth, Langhammer, Calci et al. — Phys. Rev. Lett. 107, 072501 (2011)
29
4He: Ground-State Energies
Robert Roth – TU Darmstadt – 04/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
2 4 6 8 10 12 14 16 ∞Nmx
-29
-28
-27
-26
-25
-24
-23
.
E[MeV]
NN+3N-induced
2 4 6 8 10 12 14 ∞Nmx
Exp.
NN+3N-full
2 4 6 8 10 12 14 ∞Nmx
Î
α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4 α = 0.16 fm4
Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1 Λ = 1.58 fm−1
ℏΩ = 20MeV
30
12C: Ground-State Energies
Robert Roth – TU Darmstadt – 04/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
2 4 6 8 10 12 14 ∞Nmx
-110
-100
-90
-80
-70
-60
.
E[MeV]
NN+3N-induced
2 4 6 8 10 12 ∞Nmx
Exp.
NN+3N-full
2 4 6 8 10 12 ∞Nmx
Î
α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4 α = 0.16 fm4
Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1 Λ = 1.58 fm−1
ℏΩ = 20MeV
31
16O: Ground-State Energies
Robert Roth – TU Darmstadt – 04/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
2 4 6 8 10 12 14 ∞Nmx
-180
-160
-140
-120
-100
-80
.
E[MeV]
NN+3N-induced
2 4 6 8 10 12 ∞Nmx
Exp.
NN+3N-full
2 4 6 8 10 12 ∞Nmx
Î
α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4 α = 0.16 fm4
Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1 Λ = 1.58 fm−1
ℏΩ = 20MeV
in progress:explicit inclusion of
induced 4N
(→ A. Calci)clear signature of
induced 4N originatingfrom initial 3N
32
16O: Lowering the Initial 3N Cutoff
Robert Roth – TU Darmstadt – 04/2013
standard
500 MeV
2 4 6 8 1012141618Nmx
-150
-145
-140
-135
-130
-125
-120
-115
-110
.
E[MeV]
cD= −0.2cE = −0.205
reduced 3N cutoff(cE refit to 4He binding energy)
450 MeV
2 4 6 8 1012141618Nmx
cD= −0.2cE = −0.016
400 MeV
2 4 6 8 1012141618Nmx
cD= −0.2cE = 0.098
350 MeV
2 4 6 8 1012141618Nmx
cD= −0.2cE = 0.205
Î
α = 0.04 fm4 α = 0.05 fm4 α = 0.0625 fm4 α = 0.08 fm4
Λ = 2.24 fm−1 Λ = 2.11 fm−1 Λ = 2.00 fm−1 Λ = 1.88 fm−1NN+3N-full
ℏΩ = 20MeV
lowering theinitial 3N cutoff
suppresses induced4N terms
33
Spectroscopy of 12C
Robert Roth – TU Darmstadt – 04/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
0
2
4
6
8
10
12
14
16
18
.
E[MeV]
NN+3N-induced
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
NN+3N-full
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
12CℏΩ = 16MeV
α = 0.04 fm4 α = 0.08 fm4
Λ = 2.24 fm−1 Λ = 1.88 fm−1
34
Spectroscopy of 12C
Robert Roth – TU Darmstadt – 04/2013
Roth, et al; PRL 107, 072501 (2011)
NN only
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
0
2
4
6
8
10
12
14
16
18
.
E[MeV]
NN+3N-induced
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
NN+3N-full
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 0
2+ 1
4+ 0
2 4 6 8 Exp.Nmx
12CℏΩ = 16MeV
α = 0.04 fm4 α = 0.08 fm4
Λ = 2.24 fm−1 Λ = 1.88 fm−1
spectra largelyinsensitive toinduced 4N
35
Ab Initio IT-NCSM Calculations
for p- and sd-Shell Nuclei
36
Spectroscopy of Carbon Isotopes
Robert Roth – TU Darmstadt – 04/2013
0+ 1
2+ 1
2+ 1
2 4 6 8 Exp.0
2
4
6
8
.
E[MeV]
10C
0+ 0
0+ 0
0+ 0
1+ 0
1+ 1
2+ 0
2+ 02+ 1
4+ 0
2 4 6 8 Exp.0
2
4
6
8
10
12
14
16
12C
0+ 1
0+ 1
0+ 1
1+ 1
2+ 1
2+ 1
2+ 14+ 1
2 4 6 8 Exp.0
2
4
6
8
10
14C
0+ 2
0+ 2
2+ 2
2+ 23? 24+ 2
2 4 6 Exp.Nmx
0
1
2
3
4
.
E[MeV]
16C
0+ 3
2+ 3
2+ 3
2 4 6 Exp.Nmx
0
1
2
3
4
18C
0+ 4
2+ 4
2 4 6 Exp.Nmx
0
1
2
3
4
20C
NN+3N-full
Λ3N = 400MeV
α = 0.08 fm4
ℏΩ = 16MeV
37
Spectroscopy of Carbon Isotopes
Robert Roth – TU Darmstadt – 04/2013
PRELI
MINARY
2 4 6 80
1
2
3
4
5
6
7
8
.
B(E2,2+→
0+)[e2fm
4] 10C
2 4 6 80
1
2
3
4
5
6
7
8
12C
2 4 6 80
1
2
3
4
5
6
7
8
14C
2 4 6Nmx
0
0.5
1
1.5
2
2.5
3
.
B(E2,2+→
0+)[e2fm
4] 16C
2 4 6Nmx
0
0.5
1
1.5
2
2.5
3
18C
2 4 6Nmx
0
1
2
3
4
5
20C
NN+3N-full
Λ3N = 400MeV
α = 0.08 fm4
ℏΩ = 16MeV
B(E2,2+1→ 0+
gs)
B(E2,2+2 → 0+gs)
NEW
EXPE
RIME
NT
ANL,Mc
Cutch
anetal.
NEW
EXPE
RIME
NT
NSCL, Petri e
t al.
NEW
EXPE
RIME
NT
NSCL, Voss e
t al.
NEW
EXPE
RIME
NT
NSCL, Petri e
t al.
38
Ground States of Oxygen Isotopes
Robert Roth – TU Darmstadt – 04/2013
Hergert, Binder, Calci, Langhammer, Roth; arXiv:1302.7294
NN+3N-induced(chiral NN)
2 4 6 8 10 12 14 16 18Nmx
-160
-140
-120
-100
-80
-60
-40
.
E[MeV]
12O
14O
16O18O20O22O24O26O
NN+3N-full(chiral NN+3N)
2 4 6 8 10 12 14 16 18Nmx
12O
14O
16O18O20O22O26O24O
Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ
39
Ground States of Oxygen Isotopes
Robert Roth – TU Darmstadt – 04/2013
Hergert, Binder, Calci, Langhammer, Roth; arXiv:1302.7294
NN+3N-induced(chiral NN)
12 14 16 18 20 22 24 26
A
-180
-160
-140
-120
-100
-80
-60
-40
.
E[MeV]
NN+3N-full(chiral NN+3N)
12 14 16 18 20 22 24 26
A
AO
experiment
IT-NCSM
Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ
parameter-freeab initio calculations with
full 3N interactionshighlights predictive power
of chiral NN+3N Hamiltonians
40
Ground States of Oxygen Isotopes
Robert Roth – TU Darmstadt – 04/2013
Hergert, Binder, Calci, Langhammer, Roth; arXiv:1302.7294
NN+3N-induced(chiral NN)
12 14 16 18 20 22 24 26
A
-180
-160
-140
-120
-100
-80
-60
-40
.
E[MeV]
NN+3N-full(chiral NN+3N)
12 14 16 18 20 22 24 26
A
AO
experiment
IT-NCSM IM-SRG
(→ H. Hergert)
Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ
41
Ground States of Oxygen Isotopes
Robert Roth – TU Darmstadt – 04/2013
Hergert, Binder, Calci, Langhammer, Roth; arXiv:1302.7294
NN+3N-induced(chiral NN)
12 14 16 18 20 22 24 26
A
-180
-160
-140
-120
-100
-80
-60
-40
.
E[MeV]
NN+3N-full(chiral NN+3N)
12 14 16 18 20 22 24 26
A
AO
experiment
IT-NCSM IM-SRG
(→ H. Hergert)
È CCSDÎ Λ-CCSD(T)
(→ S. Binder)
Λ3N = 400 MeV, α = 0.08 fm4, E3mx = 14, optimal ℏΩ
different many-bodyapproaches using sameNN+3N Hamiltonian give
consistent results minor differences are
understood (NO2B, E3mx,...)
42
Multi-Reference
Normal-Ordering Approximation
43
Motivation: Normal Ordering
Robert Roth – TU Darmstadt – 04/2013
avoid formal and computationalchallenges of including explicit 3Nterms in many-body calculations
circumvent formal extension of many-body method to includeexplicit 3N interactions
avoid the increase of computational cost caused by inclusionof explicit 3N interactions
normal-ordered two-body approximation works very well forclosed-shell systems (→ S. Binder)
can we do the same for open-shell nuclei?
44
Normal Ordering of 3N Interaction
Robert Roth – TU Darmstadt – 04/2013
starting point: three-body operator in second-quantized formwith respect to the zero-body vacuum |0⟩
V3N =1
36
∑
bc,bc
Vbc
bcAbcbc
Vbc
bc= ⟨bc|V3N |bc⟩ Abc
bc= †
†b
†ccb
single-reference normal ordering: assume reference state|SR⟩ given by a single Slater determinant
• standard toolbox: Wick theorem, contractions, etc.
multi-reference normal ordering: assume reference state|MR⟩ given by a superposition of Slater determinants
• generalized Wick theorem and n-tupel contractions proposed byMukherjee & Kutzelnigg (1997)
45
Multi-Reference Normal Ordering
Robert Roth – TU Darmstadt – 04/2013
three-body operator in normal-ordered form with respect tomulti-reference state |MR⟩
V3N =W +∑
c,c
WcceAcc+1
4
∑
bc,bc
Wbc
bceAbcbc+
1
36
∑
bc,bc
Wbc
bceAbcbc
where eA
indicates multi-reference normal ordered string of cre-
ation and annihilation operators (abstract concept)
matrix elements of normal-ordered n-body contributions in-volve one-, two- and three-body density matrices for |MR⟩
W =1
36
∑
bc,bc
Vbc
bcρbcbc
Wcc=1
4
∑
b,b
Vbc
bcρbb
Wbc
bc=∑
,
Vbc
bcρ
Wbc
bc= Vbc
bc
46
Multi-Reference Normal Ordering
Robert Roth – TU Darmstadt – 04/2013
discard normal-ordered three-body contribution to define thenormal-ordered two-body (NO2B) approximation
VNO2B =W +∑
c,c
WcceAcc+1
4
∑
bc,bc
Wbc
bceAbcbc
converted back into vacuum normal order with respect to |0⟩
VNO2B = V +∑
c,c
VccAcc+1
4
∑
bc,bc
Vbc
bcAbcbc
with new matrix elements
V =1
36
∑
bc,bc
Vbc
bc
ρbcbc− 18 ρ
ρbcbc+ 36 ρ
ρbbρcc
Vcc=1
4
∑
b,b
Vbc
bc
ρbb− 4 ρ
ρbb
Vbc
bc=∑
,
Vbc
bcρ
47
Single-Reference Normal Ordering
Robert Roth – TU Darmstadt – 04/2013
single-reference normal ordering is recovered by pugging indensity matrices for a single Slater-determinant
ρ= nδ
ρbb= ρ
ρbb− ρb
ρb
ρbcbc= ...
three-body operator in single-reference NO2B approximationconverted back into vacuum representation
VNO2B = V +∑
c,c
VccAcc+1
4
∑
bc,bc
Vbc
bcAbcbc
with simplified matrix elements
V =1
6
∑
bc
Vbcbc
nnbnc Vcc= −
1
2
∑
b
Vbcbc
nnb Vbc
bc=∑
Vbc
bcn
48
IT-NCSM with MR-NO2B Approximation
Robert Roth – TU Darmstadt – 04/2013
➊ perform NCSM with explicit 3N interaction for small Nmx
• ground state defines the reference state |MR⟩
• no explicit information on excited states enters
➋ compute zero-, one- and two-body matrix elements of MR-NO2Bapproximation
• density matrices for |MR⟩ can be precomputed and stored
• three-body density matrix is not need explicitly
➌ perform NCSM or IT-NCSM calculation up to large Nmx using MR-NO2B approximation
• same computational cost as a simple NN-only calculation
• larger model spaces become accessible
49
Benchmark: 6Li
Robert Roth – TU Darmstadt – 04/2013
0+
1+
2+
3+
-32
-30
-28
-26
-24
-22
-20
-18
.
E[MeV]
NN+3N-induced
0+
1+
2+
3+
2 4 6 8 10 12 Exp.Nmx
0
1
2
3
4
5
6
.
E[MeV]
0+
1+
2+
3+
-32
-30
-28
-26
-24
-22
-20
-18NN+3N-full
0+
1+
2+
3+
2 4 6 8 10 12 Exp.Nmx
0
1
2
3
4
5
6
6LiΛ3N = 500 MeV
α = 0.08 fm4
ℏΩ = 20 MeV
explicit 3N
50
Benchmark: 6Li
Robert Roth – TU Darmstadt – 04/2013
0+
1+
2+
3+
-32
-30
-28
-26
-24
-22
-20
-18
.
E[MeV]
NN+3N-induced
0+
1+
2+
3+
2 4 6 8 10 12 Exp.Nmx
0
1
2
3
4
5
6
.
E[MeV]
0+
1+
2+
3+
-32
-30
-28
-26
-24
-22
-20
-18NN+3N-full
0+
1+
2+
3+
2 4 6 8 10 12 Exp.Nmx
0
1
2
3
4
5
6
6LiΛ3N = 500 MeV
α = 0.08 fm4
ℏΩ = 20 MeV
explicit 3N
©MR-NO2BNrefmx
= 2,4,6
51
Benchmark: 12C
Robert Roth – TU Darmstadt – 04/2013
0+
0+
0+
1+
2+
4+
-90
-85
-80
-75
-70
-65
-60
-55
-50
.
E[MeV]
NN+3N-induced
0+
0+
0+
1+
2+
4+
2 4 6 8 10 Exp.Nmx
02468
101214
.
E[MeV]
0+
0+
0+
1+
2+
4+
-95
-90
-85
-80
-75
-70
-65
NN+3N-full
0+
0+
0+
1+
2+
4+
2 4 6 8 10 Exp.Nmx
02468
101214
12CΛ3N = 500 MeV
α = 0.08 fm4
ℏΩ = 20 MeV
explicit 3N
©MR-NO2BNrefmx
= 2,4
MR-NO2B approximationworks surprisingly well forground states as well as
excited states
robust with respect tovariations of the reference
state, i.e. Nrefmx
52
Challenge: 10B
Robert Roth – TU Darmstadt – 04/2013
-54
-52
-50
-48
-46
-44
-42
-40
-38
.
E[MeV]
NN+3N-induced
0+
1+
1+
3+
2 4 6 8 10 Exp.Nmx
0
0.5
1.0
1.5
2.0
2.5
.
E[MeV]
0+
1+
1+
3+-64
-62
-60
-58
-56
-54
-52
-50
-48
NN+3N-full
0+
1+
1+
3+
2 4 6 8 10 Exp.Nmx
0
0.5
1.0
1.5
2.0
2.5
10BΛ3N = 500 MeV
α = 0.08 fm4
ℏΩ = 16 MeV
explicit 3N
©MR-NO2BNrefmx
= 2,4
53
Reflections
54
Questions I
Robert Roth – TU Darmstadt – 04/2013
How can we extend current ab-initio methods to describe open-shell and deformed nuclei?
• IT-NCSM describes open and closed-shell nuclei on the same footing
• MR-IM-SRG is a true open-shell approach for medium/heavy masses(→ H. Hergert), others are following
How can we include the effects of three-nucleon forces in a com-putationally efficient manner?
• explicit 3N interactions are used in IT-NCSM very efficiently
• CCSD and ΛCCSD(T) is available with explicit 3N (→ S. Binder)
• single- and multi-reference normal ordering provide robust approxima-tions at reduced cost
• this does not imply that any kind of ‘summation over the third parti-cle’ is accurate
55
Questions II
Robert Roth – TU Darmstadt – 04/2013
How can we describe the onset of pairing in nuclei within variousab-initio frameworks?
• any ab initio approach for open shells has to describe pairing
Can we develop reliable theoretical error estimates?
• defining element of any ab initio approach
• uncertainty quantification case by case within the many-body ap-proach, not just guessing
• uncertainty quantification also necessary for the chiral EFT inputs
How can we bridge structure and reactions in a consistent fashion?
• NCSM/RGM and NCSMC with 3N are on their way (→ P. Navratil)
How can we generate reliable predictions for the drip-lines?
• simply do all of the above...
56
Epilogue
Robert Roth – TU Darmstadt – 04/2013
thanks to my group & my collaborators
• S. Binder, A. Calci, B. Erler, E. Gebrerufael,H. Krutsch, J. Langhammer, S. Reinhardt, S. Schulz,C. Stumpf, A. Tichai, R. Trippel, R. WirthInstitut für Kernphysik, TU Darmstadt
• P. NavrátilTRIUMF Vancouver, Canada
• J. Vary, P. MarisIowa State University, USA
• S. Quaglioni, G. HupinLLNL Livermore, USA
• P. PiecuchMichigan State University, USA
• H. Hergert, K. HebelerOhio State University, USA
• P. PapakonstantinouIPN Orsay, F
• C. ForssénChalmers University, Sweden
• H. Feldmeier, T. NeffGSI Helmholtzzentrum
LENPICLow-Energy NuclearPhysics International
Collaboration
JUROPA LOEWE-CSC HOPPER
COMPUTING TIME
57