news from the in-medium similarity renormalization group · h. hergert - the ohio state university...

Heiko HergertDepartment of Physics, The Ohio State University

News from the In-Medium Similarity Renormalization Group

H. Hergert - The Ohio State University - “Three-Body Forces: From Matter to Nuclei”, ECT*, Trento, 05/05/2014

Outline

• Similarity Renormalization Group

• In-Medium SRG for Closed-Shell Nuclei

• Open-Shell Nuclei from the Multi-Reference IM-SRG

• Ab Initio Shell-Model Hamiltonians

• Outlook

Similarity Renormalization Group

Review: S. Bogner, R. Furnstahl, and A. Schwenk, Prog. Part. Nucl. Phys. 65 (2010), 94

E. Anderson, S. Bogner, R. Furnstahl, and R. Perry, Phys. Rev. C82 (2011), 054001E. Jurgenson, P. Navratil, and R. Furnstahl, Phys. Rev. C83 (2011), 034301R. Roth, S. Reinhardt, and H. H., Phys. Rev. C77 (2008), 064003 H. H. and R. Roth, Phys. Rev. C75 (2007), 051001


Similarity Renormalization Group

• evolved Hamiltonian

• flow equation:

• choose to achieve desired behavior, e.g. decoupling of momentum or energy scales

• consistently evolve observables of interest

Basic Concept

continuous unitary transformation of the Hamiltonian to band-diagonal form w.r.t. a given “uncorrelated” many-body basis

( ) = ( ) †( ) ≡ + ( )

( ) =η( ), ( )

, η( ) =

( ) †( ) = −η†( )

η( )

In-Medium SRG for Closed-Shell Nuclei

H. H., S. K. Bogner, S. Binder, A. Calci, J. Langhammer, R. Roth, and A. Schwenk, Phys. Rev. C 87, 034307 (2013)

K. Tsukiyama, S. K. Bogner, and A. Schwenk, Phys. Rev. Lett. 106, 222502 (2011)


Decoupling in A-Body Space

excitations relative to reference state: normal-ordering


Decoupling in A-Body Space

aim: decouple reference state (0p-0h) from excitations

(∞)


Normal Ordering

• second quantization:

• particle- and hole density matrices:

• define normal-ordered operators recursively:

• algebra is simplified significantly because

• Wick’s theorem gives simplified expansions (fewer terms!) for products of normal-ordered operators

λ =

−→ δ , ∈ , ξ = λ − δ −→ − δ ≡ −( − )δ

...... = † . . . † . . .

...... = : ...

... : +λ : ...... : +

+λ λ − λ λ

: ...

... : + + . . .

: ...... :

=


Normal-Ordered Hamiltonian

Normal-Ordered Hamiltonian

= +

: : +

: : +

: :

E0 = + +

two-body formalism with in-medium contributions from

three-body interactions

f = + +

Γ = +

W =


Choice of Generator

=

: :: :

= − ¯

=

: :: :

∼

Off-Diagonal Hamiltonian & Generator

≡ + , ≡

: : + , ≡

:

: +


In-Medium SRG Flow Equations

0-body Flow

1-body Flow

= ＋

= ＋＋＋

~ 2nd order MBPT for H(s)



2-body Flow

= ＋－－

＋＋＋－

s channel t channel u channel

ladders rings

O(N6) scaling (before particle/hole distinction)


Decoupling

off-diagonal couplings are rapidly driven to zero

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

105 104 103 102 101 100 101600

580

560

540

520

s

EMeV

40CaEEMBPT2

, λ = . − , =

non-perturbative resummation of MBPT series

(correlations)


Hamiltonians

• NN: chiral interaction at N3LO (Entem & Machleidt)

• 3N: chiral interaction at N2LO (cD,cE fit to 3H energy & half-life)

Initial Hamiltonian

• NN + 3N-induced: start with initial NN Hamiltonian, keep two- and three-body terms

• NN + 3N-full: start with initial NN + 3N Hamiltonian, keep two- and three-body terms

SRG-Evolved Hamiltonians


He4 O16 O24 Ca40 Ca48 Ni48 Ni56

9

8

7

6EA

MeV

Λfm12.22.01.9

Results: Closed-Shell Nuclei

NN + 3N-ind.

Phys. Rev. C 87, 034307 (2013), arXiv: 1212.1190 [ nucl-th]

He4 O16 O24 Ca40 Ca48 Ni48 Ni5610

9

8

7

6

EAMeV Λfm1

2.22.01.9

NN + 3N-full (400)

experiment

validate chiralHamiltonians

Open-Shell Nuclei from theMulti-Reference IM-SRG

H. H., S. Binder, A. Calci, J. Langhammer, and R. Roth, Phys. Rev. Lett 110, 242501 (2013)


10 12 14 16 18 20 22 24 26175

150

125

100

75

50

25

0

A

EMeV

AOE3Max14Λ1.9 fm1

IMSRG

10 12 14 16 18 20 22 24 26175

150

125

100

75

50

25

0

A

EMeV

AOE3Max14Λ1.9 fm1

IMSRG

ITNCSM

10 12 14 16 18 20 22 24 26175

150

125

100

75

50

25

0

A

EMeV

AOE3Max14Λ1.9 fm1

IMSRG

ITNCSM

CCSD

10 12 14 16 18 20 22 24 26175

150

125

100

75

50

25

0

A

EMeV

AOE3Max14Λ1.9 fm1

IMSRG

ITNCSM

CCSD

CCSDT

10 12 14 16 18 20 22 24 26175

150

125

100

75

50

25

0

AEMeV

AOE3Max14Λ1.9 fm1

IMSRG

10 12 14 16 18 20 22 24 26175

150

125

100

75

50

25

0

AEMeV

AOE3Max14Λ1.9 fm1

IMSRG

ITNCSM

10 12 14 16 18 20 22 24 26175

150

125

100

75

50

25

0

AEMeV

AOE3Max14Λ1.9 fm1

IMSRG

ITNCSM

CCSD

10 12 14 16 18 20 22 24 26175

150

125

100

75

50

25

0

AEMeV

AOE3Max14Λ1.9 fm1

IMSRG

ITNCSM

CCSD

CCSDT

NN + 3N-full (400)

Phys. Rev. Lett. 110, 242501 (2013)

Results: Oxygen Chain

• reference state: number-projected Hartree-Fock-Bogoliubov vacuum (pairing correlations)

• consistent results from different many-body methods

NN + 3N-ind.

no refit of3N interaction

exp.


12 14 16 18 20 22 24 26

175

150

125

100

75

50

A

EMeV

AO3NMeV ΛSRGfm1350 1.9 ... 2.2400450

Variation of Scales

• variation of initial 3N cutoff only

• diagnostics for chiral interactions

• dripline at A=24 is robust under variationsNN + 3N-full (400)

Phys. Rev. Lett. 110, 242501 (2013)


34 36 38 40 42 44 46 48 50 52 54 56 58 60 62500

450

400

350

300

250

A

EMeV

ACa

Λfm12.21.92.21.92.21.9

MRIMSRG

CCSD

CRCC2,3

NN + 3N-full (400)

exp.

Calcium and Nickel Isotopes

48505254565860626466687072747678808284868890

700

600

500

400

A

EMeV

ANi

Λfm12.21.92.21.92.21.9

MRIMSRG

CCSD

CRCC2,3

NN + 3N-full (400)

P r e

l i m

i n

a r y

!

• improved free-space 3N SRG evolution in higher partial waves necessary (S. Binder et al., arXiv:1312.5685 [nucl-th])

• calculations for pf-shell nuclei in progress, heavier nuclei in reach

= , =


38 40 42 44 46 48 50 52 54 56 58 60 6210

0

10

20

30

40

A

S 2nMeV

ACa

Λfm12.21.9


38 40 42 44 46 48 50 52 54 56 58 60 6210

0

10

20

30

40

A

S 2nMeV

ACa

Λfm12.21.9

Gor'kov prelim. 2.0

NN + 3N-full (400)

exp.

= , =IM-SRG:

50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 9010

0

10

20

30

40

50

AS 2

nMeV

ANi

Λfm12.21.9

P r e l i

m i n a r y

!NN + 3N-full (400)

deformation (?)

• 2n separation energies too high (consistent with growing over-binding)

• (sub-)shell closures too pronounced - 3N tensor/spin-orbit ?



38 40 42 44 46 48 50 52 54 56 58 60 6210

0

10

20

30

40

A

S 2nMeV

ACa

Λ2.2 ... 1.9 fm1

3N MeVc350400

50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 9010

0

10

20

30

40

50

AS 2

nMeV

ANi

Λ2.2 ... 1.9 fm1

3N MeVc350400

NN + 3N-full

exp. P r e l i

m i n a r y

!NN + 3N-full

= , =IM-SRG:

• 2n separation energies too high (consistent with growing over-binding)

• (sub-)shell closures too pronounced - 3N tensor/spin-orbit ?

IM-SRG + Shell Model

S. K. Bogner, H. H., J. D. Holt, A. Schwenk, S. Binder, A. Calci, J. Langhammer, R. Roth, arXiv:1402:1407 [nucl-th]

K. Tsukiyama, S. K. Bogner, and A. Schwenk, Phys. Rev. C 85, 061304(R) (2012)

H. Hergert - The Ohio State University - ESNT Program “Radioactive Ion Beam Experiments and Three-Nucleon Forces”, CEA Saclay, 04/07/14

Valence Space Decoupling

valence particle states

hole states(core)

non-valence particle states


Valence Space Decoupling

• use White-type generator with off-diagonal Hamiltonian

(∞)

= , , , ,

,

,


MBPT NN+3N

IM-SRG NN

IM-SRG NN+3N

Expt.

0

1

2

3

4

5

6

Ener

gy (

MeV

)

23O

5/2+

1/2+

3/2+

1/2+

5/2+

5/2+

1/2+

3/2+

3/2+

1/2+

5/2+

3/2+

(5/2+)

1/2+

(3/2+)

From Oxygen...

MBPT NN+3N

IM-SRG NN

IM-SRG NN+3N

Expt.0

1

2

3

4

5

6

7

8

Ener

gy (

MeV

)

22O

0+

2+

2+

2+

0+

0+

(2+)

2+

2+

(0+)0

+

0+

4+

4+ (4

+)

4+

2+

0+

2+

0+

3+

3+

3+

3+

0+

ℏΩ variation

arXiv: 1402.1407 [nucl-th], [figures by J. Holt]

• 3N forces crucial

• IM-SRG improves on finite-order MBPT effective interaction

• competitive with phenomenological calculations


MBPT USDb IM-SRG NN+3N-full

Expt.

0

1

2

3

4

5

Ener

gy (M

eV)

0+ 0+

2+

4+

0+

2+

2+

2+

2+

0+ 0+ 0+ 0+

4+

2+

2+

4+2+

26Ne


Expt.

0

1

2

3

4

Ener

gy (M

eV)

5/2+

9/2+

5/2+5/2+

9/2+

7/2+

5/2+

5/2+

3/2+

5/2+

3/2+

3/2+ 3/2+

1/2+ 1/2+ 1/2+ (1/2+)

7/2+

3/2+

7/2+

1/2+3/2+

(5/2+)

(3/2+)

25Ne

... Into the sd-Shell


Expt.

0

1

2

3

4

5

Ener

gy (M

eV)

5/2+5/2+

1/2+

5/2+

7/2+

7/2+

3/2+

1/2+

1/2+

9/2+

5/2+

9/2+3/2+

1/2+ 5/2+ 5/2+ (5/2+)

9/2+

1/2+

3/2+ 3/2+

3/2+

(1/2+)

(9/2+)(3/2+)

(3/2+)

(5/2+)

25F


Expt.

0

1

2

3

4

Ener

gy (M

eV)

3+

4+

1+

4+

4+

3+

2+

2+2+

1+ 1+ 1+ 1+

3+

4+2+

2+1+

3+

2+

3+

1+

26F

Conclusions


Conclusions & Outlook

• IM-SRG is an efficient new Ab-initio framework, suitable for closed- and open-shell, medium-mass & heavy nuclei

• new method for the derivation of shell-model interactions

• easy access to spectra, odd nuclei, intrinsic deformation

• new perspectives for old (?) problems: evolution of long-range correlations, construction of density functionals...

efficient extension to observables (Magnus expansion)

study of medium-mass & heavy isotopic chains with chiral Hamiltonians

excited states


Three-Nucleon Force Wishlist

• efficiency improvements for production codes

• cutoff variation

• basis sizes & oscillator parameters (e.g., for extrapolation)

• propagation of uncertainties

• input matrix elements: N3LO (...), Δ-full chiral EFT

• proper understanding of consistent NN/3N cutoffs/regularization


Acknowledgments

S. Bogner, T. MorrisNSCL, Michigan State University

S. Binder, A. Calci, K. Hebeler, J. D. Holt, J. Langhammer, R. Roth, A. SchwenkTU Darmstadt, Germany

R. J. Furnstahl, S. Koenig, S. More, R. J. PerryThe Ohio State University

P. PapakonstantinouIBS / Rare Isotope Science Project, S. Korea

H. Hergert - NSCL, Michigan State University - TRIUMF Theory Seminar, 20/01/11

Acknowledgments

Thanks to my collaborators:

R. Roth, P. Papakonstantinou, A. Günther, S. Reinhardt, S. Binder, A. Calci, J. LanghammerInstitut für Kernphysik, TU Darmstadt

S. BognerNSCL, Michigan State University

Acknowledgments

H. Hergert – NSCL, Michigan State University — Research Discussion, 09/09/2010

Thanks to my collaborators on this work:

• R. Roth, P. Papakonstantinou, A. Gunther, S. Reinhardt,S. Binder, A. Calci, J. LanghammerInstitut fur Kernphysik, TU Darmstadt

• T. Neff, H. FeldmeierGesellschaft fur Schwerionenforschung (GSI)

42

Wednesday, January 19, 2011

Supplements


Normal-Ordering & Wick’s Theorem

• define elementary contractions of a one-body operator w.r.t. a given reference state as

• define normal-ordered operators recursively through all possible internal contractions:

• Wick’s Theorem: products of normal-ordered operators can be expanded in terms of external contractions alone

...... = : ...

... : +λ : ...... : +

+λ λ − λ λ

: ...

... : + + . . .

: ...... :: ...

... : = (− ) − λ : ... ...... ... :

+ (− ) − ξ : ... ...... ... : + . . .

≡ † , λ ≡

, ξ ≡ λ − δ


Choice of Generator

• Wegner:

• White: (J. Chem. Phys. 117, 7472)

: approx. 1p1h, 2p2h excitation energies

• “imaginary time”: (Morris, Bogner)

• off-diagonal matrix elements are suppressed like (Wegner), (White), and (imaginary time)

• g.s. energies ( ) differ by

η =

,

−

→ ∞

η =

( ) : : +

( )

:

: +

η =

: : +

: : +

,

−−| |


Generalized Normal Ordering

• generalized Wick’s theorem for arbitrary reference states(Kutzelnigg & Mukherjee)

• define irreducible n-body density matrices of reference state:

. . .

• irreducible densities give rise to additional contractions:

. . .

ρ = λ + λ λ − λ λ

ρ = λ + λ λ + λ λ λ +

: ...... ::

...... : −→ λ

: ...... ::

...... : −→ λtwo-body flow unchanged,

O(N6) scaling preserved



0-body Flow

=

( − )η − η

+

η − η

¯ ¯

1-body Flow

=

η − η+

η − η( − )

+

η − η( ¯ ¯ + ¯ )

~ 2nd order MBPT for H(s)



2-body Flow

only linked diagrams contribute, IM-SRG size-extensive

=

η + η − η − η − η − η + η + η

+

η − η( − − )

+

( − )

η − η−η − η


In-Medium SRG Flow: Diagrams

(δ ) ∼

( δ ) ∼


In-Medium SRG Flow: Diagrams

& manymore...

non-perturbative resummation

(δ ) ∼

( δ ) ∼


Particle-Number Projected HFB State

• HFB ground state is a superposition of states with different particle number:

• calculate one- and two-body densities (project only once):

• work in natural orbitals (= HFB canonical basis):

=

= , ± ,...

,

≡

≡

π

π

φ φ(ˆ− )

λ =

, λ =

− λ λ + λ λ

λ = δ= δ

, ≤ ≤


20 25 30 35 40750

745

740

735

730

725

MeVEMeV

48Ca

Results

converged g.s. energies between CCSD and -CCSD(T)

need 3N forces

[gray lines: CCSD by S. Binder, ]= , =

20 25 30 35 40

596

594

592

590

588

586

584

582

MeV

EMeV

40Ca

VUCOM as a Pairing Force: Sn Isotopes

H. Hergert – NSCL, Michigan State University — CEA Bruyeres-le-Chatel, 27/05/2010

50 54 58 62 66 70 74 78 82N

0

0.5

1

1.5

.

!(3

)n

[MeV

]

! [ fm4]

! 0.03! 0.04" 0.06# 0.10

Gogny D1S +VUCOM

" stability of gaps for wide range of !: stable 1S0 matrix elements

" residual reduction of gaps through contributions from higher partial waves

39

eMax

81012

,λ = . −


Isotopic “Chains”

• “closed-shell” Ni and Sn isotopes can give insight into isovector interaction...

• ... but complete isotopic chains would be preferable, i.e., devise an approach to open-shell nuclei

20 25 30 35 40660

650

640

630

MeV

EMeV

48Ni

20 25 30 35 40900

890

880

870

860

MeV

EMeV

56Ni

20 25 30 35 401320

1300

1280

1260

1240

1220

1200

MeV

EMeV

78Ni

VUCOM as a Pairing Force: Sn Isotopes

H. Hergert – NSCL, Michigan State University — CEA Bruyeres-le-Chatel, 27/05/2010

50 54 58 62 66 70 74 78 82N

0

0.5

1

1.5

.

!(3

)n

[MeV

]

! [ fm4]

! 0.03! 0.04" 0.06# 0.10

Gogny D1S +VUCOM

" stability of gaps for wide range of !: stable 1S0 matrix elements

" residual reduction of gaps through contributions from higher partial waves

39

eMax

81012

,λ = . − , NN only


CCSD/ -CCSD(T), , G. Hagen et al., PRL 109, 032502 (2012) -CCSD(T), , S.Binder et al., arXiv:1211.4748 [nucl-th] & PRL 109, 052501 (2012)

λ = ∞λ = . − . −

6 8 10 12 14

370

360

350

340

330

320

310

eMax

EMeV

Ca40E3Max14

28 MeVΛfm12.22.01.9


NN + 3N-ind.


6 8 10 12 14

500

450

400

350

eMax

EMeV

Ca40E3Max14

28 MeV

NN + 3N-full (400)

6 8 10 12 14

500

450

400

350

eMax

EMeV

Ca40E3Max12

28 MeV

NN + 3N-full (500)


6 8 10 12 14

500

450

400

350

eMax

EMeV

Ca40E3Max14

28 MeV

Λfm12.22.01.9

NN + 3N-ind.

validate chiral Hamiltonians

CCSD/ -CCSD(T), , G. Hagen et al., PRL 109, 032502 (2012) -CCSD(T), , S.Binder et al., arXiv:1211.4748 [nucl-th] & PRL 109, 052501 (2012)

λ = ∞λ = . − . −


Extrapolation

20 25 30 35377

376

375

374

MeV

EMeV

40CaE3Max14

=

( + / )

=

( + / )/

max./UV momentum:

radial extent:

simultaneous ultraviolet & infrared extrapolation:

( , ) = ∞ + exp− /

+ exp (− ∞ )

20 25 30 35380

360

340

320

300

280

260

MeV

EMeV

40CaE3Max14

eMax68101214

(R. Furnstahl, G. Hagen & T. Papenbrock, PRC 86,031301 (2012))


Multi-Reference Flow Equations

0-body flow:

1-body flow:

=

η − η+

η − η( − )

+

η − η( ¯ ¯ + ¯ )

+

η − ηλ +

η − η

λ

−

η − ηλ +

η − η

λ

=

( − )η − η

+

η − η

¯ ¯

+

λ +

η − ηλ


Multi-Reference Flow Equations

=

η + η − η − η − η − η + η + η

+

η − η( − − )

+

( − )

η − η−η − η

2-body flow:

2-body flowunchanged


Decoupling

∼ ,

λ ,

λ , . . .

∼

,

λ ,

λ

,

λ

, . . .

∼ . . .

• truncation in irreducible density matrices

• number of correlated vs. total pairs, triples, ... (caveat: highly collective reference states)

• perturbative analysis (e.g. for shell-model like states)

• verify for chosen multi-reference state when possible

news from the in-medium similarity renormalization group · h. hergert - the ohio state university...

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