Multi-Reference IMSRG for Deformed Nuclei · H. Hergert et al. , Phys. Rep. (2016) generator The IMSRG method: decoupling the reference state of H from the rest ... Lots of works

Jiangming YaoNational Superconducting Cyclotron Laboratory

& Facility for Rare Isotope Beams (FRIB)Michigan State University

Multi-Reference IMSRG for Deformed Nuclei

1

Progress in Ab Initio Techniques in Nuclear Physics, TRIUMF, Feb. 28, 2019

(H. Hergert, substituting for)

Jiangming YaoNational Superconducting Cyclotron Laboratory

& Facility for Rare Isotope Beams (FRIB)Michigan State University

Multi-Reference IMSRG for Deformed Nuclei

2

Progress in Ab Initio Techniques in Nuclear Physics, TRIUMF, Feb. 28, 2019

(H. Hergert, substituting for)

Progress in Ab Initio Techniques in Nuclear Physics, TRIUMF

Multi-reference IMSRG for deformed nuclei Jiangming Yao, Feb 28, 2019.

Introduction

Great potential combined with conventional many-body methods

Various ab Initio methods Applicable to Sn isotopes

Challenge for open-shell nuclei and heavier nuclei

SMCC (CC+SM), VS-IMSRG (IMSRG+SM), EOM-IMSRG, etc

Remarkable progress achieved in ab initio calculations of atomic nuclei

Collective correlations

Problem inherited from Shell model:Basis dimension grows quickly

dim ⇡✓

⌦⇡

N⇡

◆✓

⌦⌫

N⌫

◆

Limited to a small model space

3

see, e.g., S. R. Stroberg et al., arXiv:1902.06154 (2019); E. Gebrerufael et al., PRL 118, 152503 (2017)

Model space

H. Hergert et al., Phys. Rep. (2016)



The excitation energies of the first 2+ states show an overall agreement, but fail to present the features related to the onset of large deformation (island of inversion).

Introduction

0

2

4

6

2+ Ene

rgy

(MeV

)

8 10 12 14 16 18 200

2

4

6

8 10 12 14 16 18 200

2

4

6

8 10 12 14 16 18 20 8 10 12 14 16 18 20

Neutron Number N

0

2

4

6

2nd order3rd orderexp.

8O 10Ne 12Mg14Si

16S 18Ar 20Ca

J. Simonis, K. Hebeler, J. D. Holt, J. Menéndez, and A. Schwenk (2016)

chiral NN+3N interaction (MBPT)

The MBPT+SM for sd shell nuclei

4

Difficult for cross-shell excitations (requiring for a larger model space) See Takayuki Miyagi’s talk



Introduction

The E2 transition strengths from ground state to the first 2+ state are systematically underestimated.

N. M. Parzuchowski et al., ’17

The IMSRG(2)+SM/EOM for near spherical nuclei

5

Extension from the IMSRG(2) to IMSRG(3)? From single-reference to multi-reference state with built-in collective correlation?



6

The in-medium NCSM is able to reproduce the excitation energies of the first 2+ states in 30,32Mg.

E. Gebrerufael et al., ’17

0

1

2

3

4

5

E⇤

[MeV

]

0+

2+

2+4+

4+30Mg

0+

2+0+

(2+)

(4+)(4+)

5

10�5 10�4 10�3 10�2 10�1 100

s [MeV�1]

0

1

2

3

4

5

E⇤

[MeV

]

0+

2+

4+

32Mg

Exp.

0+

2+0+

4+

selected sodium isotopes as a function of the flow parameter obtained in themodel spaces Nmax = 0 (•, dotted line) and 2 (⌅, dashed line). In both cases,the reference state is the lowest eigenstate of the initial Hamiltonian obtained in

In a more sophisticated variant of IMSRG approach, called “in-medium NCSM” (MR-IMSRG+NCSM), the reference state is chosen as the ground state of NCSM calculation within a small model space.

NCSM Define

reference state

IMSRG Evolve

Operators

NCSM Extract

observables

The MR-IMSRG+NCSM for deformed nuclei

Introduction

The multi-reference framework can reproduce the excitation energy.

How about the E2 transition strengths?See talks & posters by Tobias Mongelli, Laura Mertes



7

Multi-reference state in GCM

The generator coordinate method (GCM)

Trial wave function

Slater determinant of (quasi)particle states

Determined by variational principle

Discretized form:

Ab initio GCM: Merge the mean-field and beyond techniques with the IMSRG

NCSM

GCM

Nmax

LR correlation

SR c

orre

latio

n Model Space

emax



8

Choice of Hamiltonian: NN+3N from Chiral EFT

H =X

i<j

(pi

� pj

)2

2mA+

X

i<j

V (2)ij

+X

i<j<k

V (3)ijk

X X

π π π

c1, c3, c4 cD cE

3N: N2LO

NN: N3LO Entem and Machleidt (’03)

Hebeler et al., PRC83, 031301(R) (2011)

0.8 1.0 1.2 1.4 1.6

kF [fm−1

]

−30

−25

−20

−15

−10

−5

0

Ene

rgy/

nucl

eon

[MeV

]

Λ = 1.8 fm−1

Λ = 2.8 fm−1

Λ = 1.8 fm−1

NN only

Λ = 2.8 fm−1

NN only

Vlow k NN from N3LO (500 MeV)

3NF fit to E3H and r4He Λ3NF = 2.0 fm−1

3rd order pp+hh

NN + 3N

NN only

The Hamiltonian



9

HFB Generate

reference state

Potential Energy Surface


Take into account static correlations (pairing, deformation) via symmetry breaking.



10

| {z }

angular-momentum restoration operator

P̂JMK =

2J + 116⇡2

Z 4⇡

0

d↵

Z ⇡

0

d� sin(�)

Z 2⇡

0

d� D⇤JMK (↵,�, �)| {z }

Wigner function

rotation in real spacez }| {R̂(↵,�, �)

particle-number projector

P̂N0 =12⇡

Z 2⇡

0

d�N e�i�NN0

| {z }weight

rotation in gauge spacez }| {e i�NN̂

HFB Generate

reference state

Projection Compute

different kernels

Multi-reference state in GCMDynamic correlations via symmetry restoration.



11

GCM Solve

Hill-Wheeler equationX

qb

⇥H J

qa

,qb

� EJ

↵

N J

qa

,qb

⇤fJ

↵

(qb

) = 0 .

HFB Generate

reference state

Projection Compute

different kernels




12

GCM Solve

Hill-Wheeler equation

HFB Generate

reference state

Projection Compute

different kernels




13

GCM calculation for 32Mg

The 3NF makes the nucleus easier to deform. Coexistence of spherical vibrational excitations and prolate deformed rotational excitations.



Collectivity evolution in Mg isotopes

Data from NNDC Dataset



Collectivity evolution in Mg isotopes

Data from NNDC Dataset

Exp. from S. Watanabe et al., PRC89, 044610 (2014)


Multi-reference IMSRG for deformed nuclei Jiangming Yao, Feb 28, 2019.17

�� ph� �� pp�

hh�

� �� pp�p��

hh�h��

�

� �pp

� p��

hh� h

��

��

pp�

hh�

��

p h�

� �� p

h� �� pp�

hh�

� �� pp�p��

hh�h��

�

� �pp

� p��

hh� h

��

��

pp�

hh�

��

p h�

� ��

building correlations into H(s)

…

H. Hergert et al., Phys. Rep. (2016)

generator

The IMSRG method: decoupling the reference state of H from the rest states using a continuous unitary transformation

IMSRG evolution



IMSRG+GCM calculation

IMSRG



IMSRG+GCM calculation

Convergence with respect to the number of natural states (NoS)



20

IMSRG+GCM calculation for 32Mg

The prolate deformed configurations gained more energy than the weakly deformed one via IMSRG flow (targeting states or groups/bands of states)The dominant configuration is more concentrated around large deformed region, which enhances the quadrupole collectivity in 32Mg.

2018 Research DiscussionJ. M. Yao

The IMSRG+GCM gives significantly increased B(E2) values for 32,34 Mg, compared to the pure GCM calculation.

21

Deficiency of the adopted interaction.

The rms matter radii are somewhat improved, but still underestimated (~6%).

IMSRG+GCM: Magnesium isotopes

- The induced E2(2B) operator has a negligible contribution (<1%)

- Mainly caused by shifting the dominant component to largely deformed configurations.

NUCLEI2018J. M. Yao

Configurations in 32Mg

Neutron Proton

88

20 20

22

NUCLEI2018J. M. Yao

Configurations in 32Mg

Neutron Proton

88

20 20

22

NUCLEI2018J. M. Yao

Predominant configuration in ground-state of 32Mg

23

The ground state is dominated by the configuration with two neutrons excited from 0d3/2 across N=20 to 0f7/2 and 1p3/2, respectively!

pf

sd

20

configura+on

Caution: This is NOT an observable !



Summary The GCM on top of HFB calculation with the “magic” chiral interaction can

reproduce roughly the systematics in the excitation energies Ex(2+) of magnesium isotopes, but still underestimates the E2 transition strengths around 32Mg.

The MR-IMSRG evolution drives significantly the configuration having a large overlap with the reference state down in energy and shifts the mixing weight into this configuration for 32Mg in the second GCM calculation, which improves the description of the binding energy, charge radius and in particular the E2 transition strength.

In short, the GCM+MR-IMSRG provides a new efficient “no-core” method to study deformed nuclei.

Next: Lots of works are to be done to explore its capability, such as for shape-coexistence nuclei, the study of which may need the use of the ensemble technique.

Thank you for your attention !



25

Collaborators

J. Engel, L. J. Wang University of North Carolina at Chapel Hill

H. Hergert, R. Wirth, S. Bogner Michigan State University

R. A. Basili, J. P. Vary Iowa State University

Supported in part by the NUCLEI SciDAC Collaboration under Grant No. DE-SC0008641, the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Grants No. DE-SC0017887, No. DE-FG02-97ER41019, No. DE-SC0004142, and No. DE-SC0015376 (DBD Topical Theory Collaboration).

DOE Topical CollaborationNuclear Theory for ββ-decay and

Fundamental Symmetries

…

Documents

Multi-Reference IMSRG for Deformed Nuclei · H. Hergert et al. , Phys. Rep. (2016) generator The IMSRG method: decoupling the reference state of H from the rest ... Lots of works