Ab Initio Electromagnetic Transitions with the
IMSRG
Nathan Parzuchowski
Michigan State University
March 2, 2017
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Outline
IMSRG
IMSRG rotates the Hamiltonian into a coordinate system where
simple methods (e.g. Hartree-Fock) are approximately exact:
H̄(s) = U(s)HU † (s)
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EOM-IMSRG
Approaches for Excited States
Additional processing
needed for excited states.
GS-decoupling has softened
couplings between excitation
rank.
Equations-of-motion:
Approximately diagonalize
excitation block.
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EOM-IMSRG
Equations-of-Motion IMSRG
EOM-IMSRG equation, in terms of evolved operators:
[H̄(s), X̄ν† (s)]|Φ0 i = ων X̄ν† (s)|Φ0 i
IMSRG: No correlations between ground and excited states.
X̄ν† =
X
ph
x̄hp ap† ah +
1 X pp0 † †
x̄ 0 a a 0 ah0 ah
4 0 0 hh p p
pp hh
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) = e Ω(s)
IMSRG via Magnus expansion:
dΩ
1
= η + [Ω, η] − [Ω, [Ω, η]] + · · ·
ds
2
Effective operators from Baker-Campbell-Hausdorff:
1
Ō(s) = O + [Ω, O] + [Ω, [Ω, O]] + · · ·
2
T. Morris, N. Parzuchowski, S. Bogner, Phys. Rev. C 92, 034331 (2015).
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Effective Operators
Electromagnetic Observables:
12 Exp
E(2 +1 ) (MeV)
11
NCSM
14
C
Valence-space
IMSRG
ω=20MeV
emax = 4
emax = 6
emax = 8
emax =10
emax =12
emax =14
10
EOM
IMSRG
9
8
B(E2; 2 +1 → 0 +0 ) (e 2 fm 4 )
7
5
4
3
2
N3 LO(500) NN
+ N2 LO(400) 3N λsrg = 2. 0 fm−1
0 2 4 6 8 12
Nmax
16
20
24
ω (MeV)
28
14
12
PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (2016).
16
20
24
ω (MeV)
C
28
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Effective Operators
Electromagnetic Observables:
14 Exp
NCSM
E(1 +1 ) (MeV)
13
14
C
Valence-space
IMSRG
emax = 4
emax = 6
emax = 8
emax =10
emax =12
emax =14
12
EOM
IMSRG
11
10
ω=20MeV
B(M1; 1 +1 → 0 +0 ) (µN2 )
9
ω (MeV)
2
1
0
14
0 2 4 6 8 12
Nmax
16
20
24
ω (MeV)
28
PRELIMINARY Exp: Ajzenberg-Selove, Nuc. Phys. A 1523, 1 (1991).
12
16
20
24
ω (MeV)
C
28
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Effective Operators
Electromagnetic Observables:
7
6
Exp
22
O
Valence-space
IMSRG
EOM
IMSRG
E(2 +1 ) (MeV)
5
4
3
2
1
B(E2; 2 +1 → 0 +0 ) (e 2 fm 4 )
0
6
5
4
3
2
1
0
22
12
16
20
24
ω (MeV)
28
emax = 4
emax = 6
emax = 8
emax =10
emax =12
emax =14
O
12
16
PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (2016).
20
24
ω (MeV)
28
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Effective Operators
Electromagnetic Observables:
10
9
Exp
48
Ca
Exp
56
48
Ca,56,60 Ni
Ni
Exp
60
Ni
E(2 +1 ) (MeV)
8
7
6
5
4
3
2
B(E2; 2 +1 → 0 0+ ) (e 2 fm 4 )
1
150
100
50
0
10
15
20
ω (MeV)
25
30
10
15
20
ω (MeV)
25
30
PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (2016).
10
15
20
ω (MeV)
25
30
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Extensions/Applications
Perturbative Triples Correction: EOM-IMSRG({3},2)
EOM-IMSRG({3},2): O(N 7 ) :
X̄ν† = X1p1h + X2p2h
δEν =
X
†
2
|hΦabc
ijk |H̄ X̄ν |Φ0 i|
(0)
ijk
abc
abc
ων − hΦabc
ijk |H̄|Φijk i
N. Parzuchowski, T. Morris, S. Bogner, arXiv:1611.00661
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Extensions/Applications
EOM-IMSRG({3},2) for
14
C,
22
O
E.M. N3 L0(500) + Navratil N2 LO 3N(400) λ=2.0 fm−1
emax = 8 ~ω = 20 MeV
18
8
2+
2+
2+23-00+
1-
6
14 4+
1012
10
8
1+
2+
3+
4+
6
4
14 C
2
0 EOM-IMSRG(2,2) ({3},2)
16 2-
1+14+32+
Energy (MeV)
Energy (MeV)
2312 011+
10
Exp
PRELIMINARY Exp: NNDC ENDSF
4 3+
2+
2
0 EOM-IMSRG(2,2) ({3},2)
(0-,1-,2-)?
4+
2+
?
0+
3+
2+
22 O
Exp
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Conclusion
Summary/Outlook
Spectra and observables are now available with EOM- and
VS-IMSRG
Results are consistent with NCSM.
E2 strengths consistently under-predicted (except
14
C).
Moving Forward...
EOM-IMSRG is systematically improvable, perturbative
corrections are possible.
Next: multi-reference-EOM-IMSRG (Heiko’s talk) for static
correlations and open shells.
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Conclusion
Thank you!
IMSRG at MSU:
Scott Bogner
Heiko Hergert
Fei Yuan
IMSRG at ORNL:
Titus Morris
IMSRG at TRIUMF:
Ragnar Stroberg
Jason Holt
Also thanks to:
Petr Navrátil (NCSM results)
Gaute Hagen
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Equations-of-Motion (EOM) Methods for Excited States
Define a ladder operator Xν† such that:
|Ψν i = Xν† |Ψ0 i
Eigenvalue problem re-written in terms of X † :
Ĥ|Ψν i = Eν |Ψν i → [H, Xν† ]|Ψ0 i = (Eν − E0 )Xν† |Ψ0 i
Approximations are made for Xν† , |Ψ0 i.
RPA :
|Ψ0 i ≈ |ΦHF i
Xν† =
X
[xhp ap† ah + yhp ah† ap ]
ph
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Method Comparison in 2D Quantum Dots
2.2
ω = 1. 0
RMS Error
0.095
0.066
0.031
1.9
Excitation Energy (Hartree)
Method
(2,2)
({3},2)-MP
({3},2)-EN
1.6
1.3
1.0
Line width: 1p1h content
0.7
FCI
EOM(2,2)
EOM({3},2)
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Solving the IM-SRG equations
d H̄
= [η(s), H̄(s)]
ds
η(s) ∝ H̄ OD (s)
Transform operators via flow equation.
requires very precise ODE solver
small step sizes in s needed for convergence
Construct the unitary transformation explicitly.
Magnus expansion
U(s) = e −Ω(s)
dΩ
1
1
= η(s) − [Ω(s), η(s)] + [Ω(s), [Ω(s), η(s)]] + . . .
ds
2
12
less precision needed, larger step sizes
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Corrections to IM-SRG(2) with 4 He
Ground State Energy (MeV)
-27
emax= 3 (Magnus(2/3))
emax= 5
emax= 7
emax= 9
emax= 9 (Magnus(2) [3])
emax= 9 (Magnus(2) )
CCSD
Λ-CCSD(T)
-27.5
-28
-28.5
20
25
_
h ω (MeV)
30
CC results: G. Hagen et. al., PRC 82 034330 (2010).
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Center of Mass Treatment
Hcm
1
3
2
Hcm = Tcm + mAΩ2 Rcm
− ~Ω
2
2
is evolved as an effective operator in IM-SRG:
dHcm (s)
= [η(s), Hcm (s)]
ds
CoM frequency Ω calculated in the manner of Hagen et. al.
r
2
4
4
~Ω = ~ω + Ecm (ω, s) ±
(Ecm (ω, s))2 + ~ωEcm (ω, s)
3
9
3
G. Hagen, T. Papenbrock, and D. J. Dean,
Phys. Rev. Lett. 103, 062503 (2009).
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CoM diagnostic for
16
O 3- state
E.M. N3LO Λ=500 NN at λSRG =2.0 fm−1
10
10
Ecm (ω)
Ecm(Ω+)
Ecm(Ω-)
6
4
Ground State
2
0
10
Ecm (ω)
Ecm ( Ω+)
Ecm ( Ω-)
8
Ecm (MeV)
Ecm (MeV)
8
6
4
3- State
2
15
20
_25
hω
30
35
40
0
10
15
20
_25
hω
30
35
40
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Lawson CoM Treatment: H = Hint + βCM HCM (Ω)
50
3+
Energy (MeV)
40
2+
30
0+
20
13-
10
0
0.0
1.0
2.0
3.0
βCM
4.0
5.0
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C
102
Egs (MeV)
14
β =0
β =1
β =3
β =5
103
104
105
106
15
20
25
15
20
25
20
25
10
B(E2; 21+ → 0gs+ ) (e2 fm4 ) E(21+ ) (MeV)
Lawson for
9
8
7
6
5
4
3
2
15
ω (MeV)
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