Ab Initio Electromagnetic Transitions with theIMSRG

Nathan Parzuchowski

Michigan State University

March 2, 2017

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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