Ab initio effective interactions and operators from...

Canada’s national laboratory for particle and nuclear physicsLaboratoire national canadien pour la recherche en physique nucleaire

et en physique des particules

Ab initio effective interactions andoperators from IM-SRG

Ragnar StrobergTRIUMF

Double Beta Decay Workshop

U = eη

dHds = [η,H]

UOU † = O + [η,O] + . . .

Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 1 / 27

Outline

Conceptual introduction to valence space IM-SRG

Targeted normal ordering

Ensemble reference states

Effective valence space operators


Introduction

Starting point: non-relativistic Schrodinger equation with nucleons as ourdegrees of freedom.

H|Ψ〉 = E|Ψ〉

We don’t know this

This is hard to solve

Effective theory → H is scheme and scale dependent.

Strongly-interacting system → highly correlated → hard to solve.

The underlying physical laws necessary for the mathematical theoryof a large part of physics and the whole of chemistry are thus

completely known, and the difficulty is only that the exact applicationof these laws leads to equations much too complicated to be soluble.

–Paul Dirac, 1929


Many-body approaches

NCSM, GFMC, etc

Use realistic H, solve directly

Works well for light systems

Operators treated consistently

Basis dimension grows rapidly

Microscopic

SM, RPA, IBM, DFT, etc.

Make the problem tractable

Missing physics → adjust H

Much larger reach in A

How to adjust other operators?

Phenomenological

Lee-Suzuki, MBPT, IM-SRG

Systematically treat missing physics

Consistently transform other operators

Does the expansion converge?

Microscopic/Effective


IM-SRG

Effective Interaction

Goal: Find a unitary transformation Usuch that

H = UHU †

〈P |H|Q〉 = 〈Q|H|P 〉 = 0

〈Ψi|P HP |Ψi〉 = 〈Ψi|H|Ψi〉


IM-SRG

U may always be written as U = eη, for some generator η

For two-level system, η =(

0 θ−θ 0

)For our Hamiltonian, take η = 1

2atan(

2Hod∆

)− h.c.

Perform multiple rotations: UN = eηN . . . eη2eη1

Iterate until ηN = 0

Infinitessimal rotation of angle ds → dH(s)ds = [η(s), H(s)]

White 2002; Tsukiyama, Bogner, and Schwenk 2011; Morris, Parzuchowski, and Bogner 2015


IM-SRG

Why “In-Medium”?⇒ To deal with the problem of induced many-body forces

eη = 1 + η +1

2!η2 + . . .

= 1 + + + + . . .

All terms beyond two-body operators are too expensive to handle

Define states with respect to a reference |Φ0〉 (Normal Ordering)

If |Φ0〉 is a reasonable approximation of |Ψ〉, then many-bodyterms are less important


Valence space IM-SRG

Excluded configurations treatedwith IM-SRG (definition of Hod)

Valence configurations treatedexplicitly with standard shell modeldiagonalization

In following, all calculations useSRG evolved E&M N3LO NN +local N2LO 3N (kindly provided byAngelo Calci)

Entem and Machleidt 2003; Navratil 2007; Tsukiyama, Bogner, and Schwenk 2012

core

vale

nce

excl

uded

decouple

decouple


Valence space IM-SRG: Ground states

15 20 25 30A

180

160

140

120

100

80

60

40Egs

(MeV

)

(c)

AO (Z=8)

ExperimentIMSRG(C=R)

Bogner et al. 2014

; Cipollone, Barbieri, and Navratil 2013; Hergert et al. 2014



15 20 25 30A

180

160

140

120

100

80

60

40Egs

(MeV

)

(c)

AO (Z=8)

ExperimentMR-IMSRGSCGFCCSD(T)CRCCIT-NCSMIMSRG(C=R)

Bogner et al. 2014; Cipollone, Barbieri, and Navratil 2013; Hergert et al. 2014



What are we missing?

core

vale

nce

excl

uded

decouple

decouple The other methods use the targetnucleus as |Φ0〉, while we use thecore

Other methods better captureeffect 3N forces between valencenucleons

Strategy:As valence neutrons are added, |22O〉 becomes a better reference than|16O〉, so use that

But still decouple the full sd shell



What are we missing?

core

vale

nce

excl

uded

decouple

decouple

core

vale

nce

excl

uded

decouple

decouple

Strategy:As valence neutrons are added, |22O〉 becomes a better reference than|16O〉, so use that

But still decouple the full sd shell



15 20 25 30A

180

160

140

120

100

80

60

40Egs

(MeV

)

(c)

AO (Z=8)

ExperimentMR-IMSRGSCGFCCSD(T)CRCCIT-NCSMIMSRG(C=R)

Bogner et al. 2014; Brown et al. 2006; Cipollone, Barbieri, and Navratil 2013; Hergert et al. 2014



15 20 25 30A

180

160

140

120

100

80

60

40Egs

(MeV

)

(c)

AO (Z=8)

ExperimentMR-IMSRGSCGFCCSD(T)CRCCIT-NCSMIMSRG(C=R)IMSRG(TNO)

Bogner et al. 2014; Brown et al. 2006; Cipollone, Barbieri, and Navratil 2013; Hergert et al. 2014


Valence space IM-SRG: Closed subshells

12 C 14 C 22 O 24 O 28 O 28 Si 34 Si 32 S 40 Ca1211109876

E/A

(MeV

)

AMEVS (C=R)VS (TNO)Single Ref.

|⟨Ψ|Φ0

⟩|2

0.30 0.90 0.82 0.94 1.00 0.06 0.91 0.43 1.00


Valence space IM-SRG: Ground state of 28Si

IM-SRGShell Model

|⟨Ψ|Φ0

⟩|2 IM-SRG

Single Reference

252

250

248

246

244

242

240

238

Ener

gy (M

eV)

0+

0+

0+

0+

0+

0+

0.06

0.24

0.06

0.000.07

28 Si


Aside: Basis dependence of occupation numbers

a†iai = Ua′†i a′iU†

neutron occupation01234567

1.96

3.87

1.93

4.68

0.57

0.57

=24 MeVωh, -1=2.0 fmΛLO 3N

= 6maxN

1/20s

3/20p

1/20p

5/20d

3/20d

1/21s

7/20f

5/20f

3/21p

1/21p

9/20g

7/20g

5/21d

3/21d

1/22s

11/20h

proton occupation0 1 2 3 4 5 6 7

1.96

3.87

1.93

4.68

0.57

0.57

Si28

Occupations:1) IM-SRG basis

2) Oscillator basis

1 2

0s1/2 2.0 1.960p3/2 4.0 3.870p1/2 2.0 1.930d5/2 6.0 4.681s1/2 0.0 0.570d3/2 0.0 0.57. . . . . . . . .

Same calculation,

different occupations!


What to do about 22Na?co

reva

lenc

eex

clud

ed

decouple

decouple

|Φ0〉 = |16O〉

(Underestimate 3N)

core

vale

nce

excl

uded

decouple

decouple

|Φ0〉 = |28Si〉

(Overestimate 3N)


Ensemble reference state (Equal filling)

Replace |Φ0〉 with ensemble (mixed) state characterized by density matrix:

ρ =∑i

αi|Φi〉〈Φi|Definition of normal ordering:

Tr(ρa†1 . . . aN) =∑i

αi〈Φi|a†1 . . . aN|Φi〉 = 0

Wick contraction:

a†paq =∑i

αi〈Φi|a†paq|Φi〉 ≡ npδpq

a†paq = npδpq , apa†q = (1− np)δpq , apaq = a†pa†q = 0

Now np can be fractional, which is exactly what we want!

No N-representability problem.

Thouless 1957; Gaudin 1960; Perez-Martin and Robledo 2008


Ensemble reference state (Equal filling)co

reva

lenc

eex

clud

ed

decouple

decouple

ρ = 23 |

16O〉〈16O|+ 13 |

22O〉〈22O|

18O

core

vale

nce

excl

uded

decouple

decouple

ρ = 12 |

16O〉〈16O|+ 12 |

28Si〉〈28Si|

22Na


Valence 3N forces: 10B, 22Na, 46V

Ground state of 10B is 3+

3N forces are required toreproduce this withoutfitting

Similar situation for 22Naand 46V

Normal ordering withensemble referencecaptures this

First ab initio calculationsof 22Na and 46V to obtaincorrect ordering

Navratil and Ormand 2002; Pieper, Varga, and Wiringa 2002; Gebrerufael, Calci, and Roth 2015

-3-2-101

E−E

3+ (M

eV)

1+3+

Λ3N=400 MeV

(a)

10 B

4 6 8 4 He8 He10 B-3-2-101

E−E

3+ (M

eV) 1+

3+

Λ3N=500 MeV10 BNCSM Nmax IMSRG reference

Exp. 22 Na 16 O 22 Na 16 O0.60.40.20.00.20.40.60.8

E−E

3+ (M

eV) 1+

3+

22 Na

(b) Λ3N =500 MeV

Λ3N =400 MeV

IMSRG ref. IMSRG ref.Exp.46 V40 Ca

0.1

0.0

0.1

0.2 1+

3+

46 V

(c) Λ3N =400 MeV

IMSRG ref.


Valence 3N forces: 10B, 22Na, 46V

Ground state of 10B is 3+

3N forces are required toreproduce this withoutfitting

Similar situation for 22Naand 46V

Normal ordering withensemble referencecaptures this

First ab initio calculationsof 22Na and 46V to obtaincorrect ordering

Navratil and Ormand 2002; Pieper, Varga, and Wiringa 2002; Gebrerufael, Calci, and Roth 2015

-3-2-101

E−E

3+ (M

eV)

1+3+

Λ3N=400 MeV

(a)

10 B

4 6 8 4 He8 He10 B-3-2-101

E−E

3+ (M

eV) 1+

3+

Λ3N=500 MeV10 BNCSM Nmax IMSRG reference

Exp. 22 Na 16 O 22 Na 16 O0.60.40.20.00.20.40.60.8

E−E

3+ (M

eV) 1+

3+

22 Na

(b) Λ3N =500 MeV

Λ3N =400 MeV

IMSRG ref. IMSRG ref.Exp.46 V40 Ca

0.1

0.0

0.1

0.2 1+

3+

46 V

(c) Λ3N =400 MeV

IMSRG ref.Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 18 / 27

Ground states with ensemble reference

8 10 12 14 16 18 20 22 24A

120

100

80

60

40

Egs

(MeV

)

(a)AC (Z=6)

ExperimentCCSD(T)IT-NCSMIMSRG(C=R)IMSRG(TNO)

10 15 20 25A

160

140

120

100

80

60

40

20

0

Egs

(MeV

)

(b)AN (Z=7)

ExperimentSCGFIMSRG(C=R)IMSRG(TNO)

15 20 25 30A

220

200

180

160

140

120

100

80

60

Egs

(MeV

)

AF (Z=9)

ExperimentSCGFCCSD(T)IMSRG(C=R)IMSRG(TNO)

20 25 30 35 40A

260

240

220

200

180

160

140

120

100

Egs

(MeV

)

(d)ANa (Z=11)

ExperimentIMSRG(C=R)IMSRG(TNO)

35 40 45 50 55 60A

550

500

450

400

350

300

250

200

Egs

(MeV

)

(e)ACa (Z=20)

ExperimentMR-IMSRGCRCCIMSRG(C=R)IMSRG(TNO)

50 55 60 65 70 75 80A

750

700

650

600

550

500

450

400

350

300

Egs

(MeV

)

(f)

ANi (Z=28)


8 10 12 14 16 18 20 22 24A

120

100

80

60

40

Egs

(MeV

)

(a)AC (Z=6)


50 55 60 65 70 75 80A

750

700

650

600

550

500

450

400

350

300

Egs

(MeV

)

(f)

ANi (Z=28)


Cipollone, Barbieri, and Navratil 2013; Binder et al. 2014; Hergert et al. 2014; Jansen et al. 2015; Roth(priv. comm.)



8 10 12 14 16 18 20 22 24A

120

100

80

60

40

Egs

(MeV

)

(a)AC (Z=6)


10 15 20 25A

160

140

120

100

80

60

40

20

0

Egs

(MeV

)

(b)AN (Z=7)


15 20 25 30A

220

200

180

160

140

120

100

80

60

Egs

(MeV

)

AF (Z=9)


20 25 30 35 40A

260

240

220

200

180

160

140

120

100

Egs

(MeV

)

(d)ANa (Z=11)


35 40 45 50 55 60A

550

500

450

400

350

300

250

200

Egs

(MeV

)

(e)ACa (Z=20)


50 55 60 65 70 75 80A

750

700

650

600

550

500

450

400

350

300

Egs

(MeV

)

(f)

ANi (Z=28)


8 10 12 14 16 18 20 22 24A

120

100

80

60

40Egs

(MeV

)

(a)AC (Z=6)


50 55 60 65 70 75 80A

750

700

650

600

550

500

450

400

350

300

Egs

(MeV

)

(f)

ANi (Z=28)





8 10 12 14 16 18 20 22 24A

120

100

80

60

40

Egs

(MeV

)

(a)AC (Z=6)


10 15 20 25A

160

140

120

100

80

60

40

20

0

Egs

(MeV

)

(b)AN (Z=7)


15 20 25 30A

220

200

180

160

140

120

100

80

60

Egs

(MeV

)

AF (Z=9)


20 25 30 35 40A

260

240

220

200

180

160

140

120

100

Egs

(MeV

)

(d)ANa (Z=11)


35 40 45 50 55 60A

550

500

450

400

350

300

250

200

Egs

(MeV

)

(e)ACa (Z=20)


50 55 60 65 70 75 80A

750

700

650

600

550

500

450

400

350

300

Egs

(MeV

)

(f)

ANi (Z=28)


8 10 12 14 16 18 20 22 24A

120

100

80

60

40

Egs

(MeV

)

(a)AC (Z=6)


50 55 60 65 70 75 80A

750

700

650

600

550

500

450

400

350

300

Egs

(MeV

)

(f)

ANi (Z=28)




Germanium/Selenium 0+ states

IM-SRG Exp

0

5

10

15E

(0+

) (M

eV)

72 Ge

-740.98 -628.69

IM-SRG Exp2

0

2

4

6

E(0

+) (

MeV

)

74 Ge

-756.72 -645.66

IM-SRG Exp1

0

1

2

3

E(0

+) (

MeV

)

76 Ge

-772.90 -661.60

IM-SRG Exp0.50.00.51.01.52.02.5

E(0

+) (

MeV

)

72 Se

-738.10 -622.40

IM-SRG Exp202468

10E

(0+

) (M

eV)

74 Se

-758.87 -642.89

IM-SRG Exp

0.50.00.51.01.52.02.53.0

E(0

+) (

MeV

)

76 Se

-778.51 -662.07


Effective operators

Effective operators

Some very preliminary results


Effective operators

Operators transform just like H:

O = eΩOe−Ω = O + [Ω,O] +1

2[Ω, [Ω,O]] + . . .

Tensor operators require additional angular momentum coupling:

Oλ = eΩOλe−Ω = Oλ + [Ω,Oλ]λ +1

2[Ω, [Ω,Oλ]λ]λ + . . .

Shell model expectation values then reflect full-space expectationvalues:

〈Ψ|O|Ψ〉 = 〈ΨSM |O|ΨSM 〉


Effective operators

The deuteron

Valence space: 0s shell

No effects of induced many-body forces

Bare quadrupole operator (λ = 2) gives identically zero

0s1/2


Effective operators

The deuteron

0 2 4 6 8 10emax

2

1

0

1

E (M

eV)

BindingEnergy

0 2 4 6 8 10emax

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Q (f

m2

) QuadrupoleMoment

Energy correctly reproduced

〈0s1/20s1/2|Q|0s1/20s1/2〉 6= 0


Into the sd shell: Neon Radii

18 20 22 24 26 28 30A

2.5

2.6

2.7

2.8

2.9

3.0

3.1rm

s ch

arge

radi

us (f

m)

ANe

IM-SRG

IM-SRG (Scaled)

Skyrme HF

IM-SRG (targeted)

Marinova et al. 2011; Brown 1998


Tensor Operators

Moments

N

S

20 O 20 F 20 Ne 20 Na 20 Mg2

1

0

1

2

3

µ(2

+) (µN

)

USD-bareUSD-effExpIMSRG-bareIMSRG-eff

20 O 20 F 20 Ne 20 Na 20 Mg30

25

20

15

10

5

0

5

10

Q(2

+) (ef

m2

)USD-bareUSD-effExpIMSRG-bareIMSRG-eff


Tensor Operators

Transitions

N

S

20 O 20 F 20 Ne 20 Na 20 Mg0.0

0.1

0.2

0.3

0.4

0.5

B(M

1;3

+→

2+

) (µ

2 N)


20 O 20 F 20 Ne 20 Na 20 Mg

0

10

20

30

40

50

60

70

80

B(E

2;2

+→

0+

) (e2

fm4

)



Summary

Ab initio methods provide a means to calculate matrix elementswhere fitting to data is not possibleEffective valence-space approach enables consistent treatment ofexcited states, transitions, open-shell/deformed systemsTargeted normal ordering with an ensemble reference provides areasonable approximation of valence 3N forcesEvolved tensor operators produce some renormalization – more workto be done.MEC corrections to operators can be handled without additionaltrouble

Collaborators:

A. Calci, J. Holt, P. Navratil

NSCL/MSU S. Bogner, H. Hergert, T. Morris, N. Parzuchowski

TU Darmstadt A. Schwenk, J. SimonisRagnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 27 / 27

Appendix


Tensor Operators

Moments

N

S

12 16 20 24 28ω (MeV)

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

µ(2

+) (µN

)

ωem

p

20 Nebare

12 16 20 24 28ω (MeV)

10

9

8

7

6

5

4

3

2

Q0(2

+) (ef

m2

)

ωem

p

20 Neemax=4emax=6emax=8emax=10

bare


Tensor Operators

Moments

N

S

12 16 20 24 28ω (MeV)

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

µ(2

+) (µN

)

ωem

p

20 Nebare1b

12 16 20 24 28ω (MeV)

10

9

8

7

6

5

4

3

2

Q0(2

+) (ef

m2

)

ωem

p


bare1b


Tensor Operators

Moments

N

S

12 16 20 24 28ω (MeV)

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

1.16

µ(2

+) (µN

)

ωem

p

20 Nebare1b1b+2b

12 16 20 24 28ω (MeV)

10

9

8

7

6

5

4

3

2

Q0(2

+) (ef

m2

)

ωem

p


bare1b1b+2b


Tensor Operators

Transitions

N

S

12 16 20 24 28ω (MeV)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

B(M

1;3

+→

2+

) (µ

2 n)

emax=4emax=6emax=8emax=10

20 Fbare1b1b+2b

12 16 20 24 28ω (MeV)

0

5

10

15

20

25

B(E

2;2

+→

0+

) (e2

fm4

)

ωem

p

emax=4emax=6emax=8emax=10

20 Ne bare1b1b+2b


How to choose Ω?

A toy problem:

H =

(ε1 hodhod ε2

), Ω =

(0 θ−θ 0

), eΩ =

(cos θ sin θ− sin θ cos θ

)

eΩHe−Ω =

(ε1 cos2 θ + ε2 sin2 θ + h sin 2θ hod cos 2θ + ε2−ε1

2 sin 2θhod cos 2θ + ε2−ε1

2 sin 2θ ε2 cos2 θ + ε1 sin2 θ − h sin 2θ

)

h′od → 0 ⇒ θ = 12 tan−1

(2hodε1−ε2

)θ 1 ⇒ θ ≈ hod

ε1−ε2

S. R. White, J. Chem Phys. 117, 7472 (2002)Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 31 / 27

IM-SRG

Application to 16O:

|Φ0〉 =

η ∼ Hod∆ − h.c.

Hod is any term that connects|Φ0〉 to any other configuration

s is the total “angle” rotated

Ground state energy given by asingle matrix element: 〈Φ0|H|Φ0〉

Tsukiyama, Bogner, and Schwenk 2011; Ekstrom et al. 2015

0 1 2 3 4 5s

130

120

110

100

90

80

70

60

50

40

E0 (M

eV)

N2LOSAT

ω=22 MeV

13 shells

16 O

Experiment

0 5 10 15 20 25 30 35s

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

||η||

N2LOSAT

ω=22 MeV

13 shells

16 O


IM-SRG

Where did all the correlations go?

Original single particle basis: |φi〉 = a†i |0〉The transformed H is implicitly in terms of a†i

a†i = U †(a†i )U

= Cia†i +∑j 6=iCja†j +

∑jk

Cjkla†ja†kal + . . .

The single-particle orbits are now much more complicated!

⇒U


References I

Binder, Sven et al. (2014). “Ab initio path to heavy nuclei”. In: Phys. Lett. B 736, pp. 119–123. issn: 03702693. doi:

10.1016/j.physletb.2014.07.010. url: http://www.sciencedirect.com/science/article/pii/S0370269314004961.

Bogner, S. K. et al. (2014). “Nonperturbative shell-model interactions from the in-medium similarity renormalization group”.

In: Phys. Rev. Lett. 113.14, p. 142501. issn: 0031-9007. doi: 10.1103/PhysRevLett.113.142501. arXiv: 1402.1407. url:http://link.aps.org/doi/10.1103/PhysRevLett.113.142501.

Brown, B. A. (1998). “New Skyrme interaction for normal and exotic nuclei”. In: Phys. Rev. C 58.1, pp. 220–231. issn:

0556-2813. doi: 10.1103/PhysRevC.58.220. url: http://link.aps.org/doi/10.1103/PhysRevC.58.220.

Brown, B. A. et al. (2006). “Tensor interaction contributions to single-particle energies”. In: Phys. Rev. C 74.6, p. 061303.

issn: 0556-2813. doi: 10.1103/PhysRevC.74.061303. url: http://link.aps.org/doi/10.1103/PhysRevC.74.061303.

Cipollone, A., C. Barbieri, and P. Navratil (2013). “Isotopic Chains Around Oxygen from Evolved Chiral Two- and

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