Ab Initio Electromagnetic Transitions with theIMSRG
Nathan Parzuchowski
Michigan State University
March 2, 2017
1 / 21
Outline
IMSRG
IMSRG rotates the Hamiltonian into a coordinate system wheresimple methods (e.g. Hartree-Fock) are approximately exact:
H(s) = U(s)HU†(s)
2 / 21
EOM-IMSRG
Approaches for Excited States
Additional processingneeded for excited states.
GS-decoupling has softenedcouplings between excitationrank.
Equations-of-motion:Approximately diagonalizeexcitation block.
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EOM-IMSRG
Equations-of-Motion IMSRG
EOM-IMSRG equation, in terms of evolved operators:
[H(s), X †ν (s)]|Φ0〉 = ωνX†ν (s)|Φ0〉
IMSRG: No correlations between ground and excited states.
X †ν =∑ph
xph a†pah +
1
4
∑pp′hh′
xpp′
hh′ a†pa†p′ah′ah
Truncation to two-body ladder operators: EOM-IMSRG(2,2).
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Effective Operators
Effective Operators in the IMSRG
IMSRG unitary transformation can be explicitly constructed:
U(s) = eΩ(s)
IMSRG via Magnus expansion:
dΩ
ds= η + [Ω, η]− 1
2[Ω, [Ω, η]] + · · ·
Effective operators from Baker-Campbell-Hausdorff:
O(s) = O + [Ω,O] +1
2[Ω, [Ω,O]] + · · ·
T. Morris, N. Parzuchowski, S. Bogner, Phys. Rev. C 92, 034331 (2015).
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Effective Operators
Electromagnetic Observables: 14C
7
8
9
10
11
12E
(2+ 1)
(MeV
)Exp NCSM
ω=20MeV
Valence-spaceIMSRG
EOMIMSRG
emax= 4
emax= 6
emax= 8
emax=10
emax=12emax=14
2
3
4
5
B(E
2;2
+ 1→
0+ 0)
(e2fm
4)
0 2 4 6 8Nmax
12 16 20 24 28ω (MeV)
N3LO(500) NN
+ N2LO(400) 3N λsrg = 2. 0 fm−1
12 16 20 24 28ω (MeV)
14C
PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (2016).6 / 21
Effective Operators
Electromagnetic Observables: 14C
9
10
11
12
13
14E
(1+ 1)
(MeV
)Exp NCSM
ω=20MeV
ω (MeV)
Valence-spaceIMSRG
EOMIMSRG
emax= 4
emax= 6
emax= 8
emax=10
emax=12emax=14
0
1
2
B(M
1;1
+ 1→
0+ 0)
(µ2 N)
0 2 4 6 8Nmax
12 16 20 24 28ω (MeV)
12 16 20 24 28ω (MeV)
14C
PRELIMINARY Exp: Ajzenberg-Selove, Nuc. Phys. A 1523, 1 (1991).7 / 21
Effective Operators
Electromagnetic Observables: 22O
0
1
2
3
4
5
6
7
E(2
+ 1)
(MeV
)Exp Valence-space
IMSRGEOM
IMSRG
0
1
2
3
4
5
6
B(E
2;2
+ 1→
0+ 0)
(e2fm
4)
12 16 20 24 28ω (MeV)
12 16 20 24 28ω (MeV)
22O
emax= 4
emax= 6
emax= 8
emax=10
emax=12emax=14
PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (2016).8 / 21
Effective Operators
Electromagnetic Observables: 48Ca,56,60Ni
1
2
3
4
5
6
7
8
9
10
E(2
+ 1)
(MeV
)
Exp 48Ca Exp 56Ni Exp 60Ni
0
50
100
150
B(E
2;2
+ 1→
0+ 0
) (e
2fm
4)
10 15 20 25 30ω (MeV)
10 15 20 25 30ω (MeV)
10 15 20 25 30ω (MeV)
PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (2016). 9 / 21
Extensions/Applications
Perturbative Triples Correction: EOM-IMSRG(3,2)
EOM-IMSRG(3,2): O(N7) :
X †ν = X1p1h + X2p2h
δEν =∑ijkabc
|〈Φabcijk |HX †ν |Φ0〉|2
ω(0)ν − 〈Φabc
ijk |H|Φabcijk 〉
N. Parzuchowski, T. Morris, S. Bogner, arXiv:1611.00661
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Extensions/Applications
EOM-IMSRG(3,2) for 14C, 22O
E.M. N3L0(500) + Navratil N2LO 3N(400) λ=2.0 fm−1
emax = 8 ~ω = 20 MeV
EOM-IMSRG(2,2) (3,2) Exp0
2
4
6
8
10
12
Ene
rgy
(MeV
)
0+
4+
3-
3-
3-
2+
2+
2+
2+
1+1+
1-
1-1-
2-0-
0-
2-
14C
EOM-IMSRG(2,2) (3,2) Exp0
2
4
6
8
10
12
14
16
18
Ene
rgy
(MeV
)
(0-,1-,2-)?
0+
4+
4+
4+
2-
2+
2+
2+
2+
3+3+
3+
?
1+
1-0-
22O
PRELIMINARY Exp: NNDC ENDSF 11 / 21
Conclusion
Summary/Outlook
Spectra and observables are now available with EOM- andVS-IMSRG
Results are consistent with NCSM.E2 strengths consistently under-predicted (except 14C).
Moving Forward...
EOM-IMSRG is systematically improvable, perturbativecorrections are possible.Next: multi-reference-EOM-IMSRG (Heiko’s talk) for staticcorrelations and open shells.
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Conclusion
Thank you!
IMSRG at MSU:Scott BognerHeiko HergertFei Yuan
IMSRG at ORNL:Titus Morris
IMSRG at TRIUMF:Ragnar StrobergJason Holt
Also thanks to:Petr Navratil (NCSM results)Gaute Hagen 13 / 21
Equations-of-Motion (EOM) Methods for Excited States
Define a ladder operator X †ν such that:
|Ψν〉 = X †ν |Ψ0〉
Eigenvalue problem re-written in terms of X †:
H|Ψν〉 = Eν |Ψν〉 → [H,X †ν ]|Ψ0〉 = (Eν − E0)X †ν |Ψ0〉
Approximations are made for X †ν , |Ψ0〉.
RPA : |Ψ0〉 ≈ |ΦHF 〉 X †ν =∑ph
[xph a†pah + yph a
†hap]
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Method Comparison in 2D Quantum Dots
Method RMS Error
(2,2) 0.095(3,2)-MP 0.066(3,2)-EN 0.031
Line width: 1p1h content
FCI EOM(2,2) EOM(3,2)
0.7
1.0
1.3
1.6
1.9
2.2
Exc
itat
ion
Ene
rgy
(Har
tree
)
ω= 1. 0
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Solving the IM-SRG equations
dH
ds= [η(s), H(s)] η(s) ∝ HOD(s)
Transform operators via flow equation.
requires very precise ODE solversmall step sizes in s needed for convergence
Construct the unitary transformation explicitly.
Magnus expansionU(s) = e−Ω(s)
dΩ
ds= η(s)− 1
2[Ω(s), η(s)] +
1
12[Ω(s), [Ω(s), η(s)]] + . . .
less precision needed, larger step sizes
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Corrections to IM-SRG(2) with 4He
20 25 30h_ ω (MeV)
-28.5
-28
-27.5
-27
Gro
und
Stat
e E
nerg
y (M
eV)
emax
= 3 (Magnus(2/3))e
max= 5
emax
= 7e
max= 9
emax
= 9 (Magnus(2) [3])e
max= 9 (Magnus(2) )
CCSDΛ-CCSD(T)
CC results: G. Hagen et. al., PRC 82 034330 (2010). 17 / 21
Center of Mass Treatment
Hcm = Tcm +1
2mAΩ2R2
cm −3
2~Ω
Hcm is evolved as an effective operator in IM-SRG:
dHcm(s)
ds= [η(s),Hcm(s)]
CoM frequency Ω calculated in the manner of Hagen et. al.
~Ω = ~ω +2
3Ecm(ω, s)±
√4
9(Ecm(ω, s))2 +
4
3~ωEcm(ω, s)
G. Hagen, T. Papenbrock, and D. J. Dean,Phys. Rev. Lett. 103, 062503 (2009).
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CoM diagnostic for 16O 3- state
E.M. N3LO Λ=500 NN at λSRG=2.0 fm−1
10 15 20 25 30 35 40h_ω
0
2
4
6
8
10
Ecm(M
eV)
Ecm (ω)Ecm ( Ω+)Ecm ( Ω-)
10 15 20 25 30 35 40h_ω
0
2
4
6
8
10
Ecm(M
eV)
Ecm (ω)Ecm(Ω+)Ecm(Ω-)
Ground State 3- State
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Lawson CoM Treatment: H = Hint + βCMHCM(Ω)
0.0 1.0 2.0 3.0 4.0 5.0βCM
0
10
20
30
40
50E
nerg
y (M
eV)
0+
2+
3+
3-1-
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Lawson for 14C
15 20 25
106
105
104
103
102
E gs (
MeV
)
β=0
β=1
β=3
β=5
15 20 25
7
8
9
10
E(2
+ 1) (
MeV
)
15 20 25ω (MeV)
2
3
4
5
6
B(E2
; 2+ 1→
0+ gs
) (e2
fm4
)
21 / 21