[PPT] - Correlated Reference States and E ff ective Hamiltonians in the PowerPoint Presentation

SLIDE 1

Heiko Hergert

Facility for Rare Isotope Beams & Department of Physics and Astronomy Michigan State University

Correlated Reference States and Effective Hamiltonians in the IMSRG Framework

SLIDE 2

H. H., Phys. Scripta, Phys. Scripta 92, 023002 (2017)
H. H., S. K. Bogner, T. D. Morris, A. Schwenk, and K. Tuskiyama, Phys. Rept. 621, 165 (2016)
H. H., S. Bogner, T. Morris, S. Binder, A. Calci, J. Langhammer, R. Roth, Phys. Rev. C 90,

041302 (2014)

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

(Multi-Reference) In-Medium Similarity Renormalization Group

SLIDE 3

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Transforming the Hamiltonian

reference state: single Slater

determinant

ε

ε

⇤

⇤ ⇤ ⇤ ⇥

p

h

pp

hh

ppp

hhh

excitations relative

to reference state: normal-ordering

ppp

hhh

pp

hh

p

h

SLIDE 4

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Decoupling in A-Body Space

aim: decouple reference state from excitations

⇤

⇤ ⇤ ⇤ ⇥

⇤

⇤ (∞) ⇤ ⇤ ⇥

⇥ ⇥

p

h

pp

hh

ppp

hhh

p

h

pp

hh

ppp

hhh

ppp

hhh

pp

hh

p

h

ppp

hhh

pp

hh

p

h

SLIDE 5

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Flow Equation

⇤

⇤ ⇤ ⇤ ⇥

⇤

⇤ (∞) ⇤ ⇤ ⇥

d dsH(s) =

η(s), H(s)
,

e.g., η(s) ≡

Hd(s), Hod(s)
p

h

pp

hh

ppp

hhh

p

h

pp

hh

ppp

hhh

ppp

hhh

pp

hh

p

h

ppp

hhh

pp

hh

p

h

Operators

truncated at two-body level - matrix is never constructed explicitly!

SLIDE 6

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Decoupling

ff-diagonal couplings

are rapidly driven to zero

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

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s E @MeVD 40Ca

E E+MBPTH2L

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10-5 10-4 10-3 10-2 10-1 100 101

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520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
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520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

, λ = . −, =

non-perturbative resummation of MBPT series (correlations)

SLIDE 7

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê Ê ‡ ‡ ‡ ‡ ‡ ‡‡‡‡‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡ ‡

10-5 10-4 10-3 10-2 10-1 100 101

600
580
560
540
520

s E @MeVD 40Ca

E E+MBPTH2L

, λ = . −, =

absorb correlations into RG-improved Hamiltonian
reference state is ansatz for transformed, less correlated

eigenstate:

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Decoupling

U(s)HU†(s)U(s)

n
= EnU(s)
n
U(s)HU†(s)U(s)
n
= EnU(s)
n
U(s)
n

! =

SLIDE 8

“standard” IMSRG: build correlations on top of Slater determinant (=independent-particle state)

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Correlated Reference States

! IMSRG(2) IMSRG(3) IMSRG(4) IMSRG(5) . . . Collective (aka static) correlations, e.g. due to intrinsic deformation:

SLIDE 9

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Correlated Reference States

! MR-IMSRG(2) . . . MR-IMSRG: build correlations on top of already correlated state (e.g., from a method that describes static correlation well) use generalized normal ordering with 2B,… densities

SLIDE 10

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

MR-IMSRG References States

Slater determinants (uncorrelated)
number-projected Hartree-Fock Bogoliubov vacua
Generator Coordinate Method (with projections)
small-scale No-Core Shell Model
clustered states, Density Matrix Renormalization Group,

tensor networks etc. available future

SLIDE 11

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

MR-IMSRG References States

Slater determinants (uncorrelated)
number-projected Hartree-Fock Bogoliubov vacua
Generator Coordinate Method (with projections)
small-scale No-Core Shell Model
clustered states, Density Matrix Renormalization Group,

tensor networks etc. available future

SLIDE 12

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Titanium Isotopes

E. Leistenschneider et al., PRL 120, 062503 (2018)

58

AME16 TITAN +

34

AME16 1.8/2.0(EM) VS-IMSRG NN+3N(lnl) GGF N2LOsat GGF N2LOsat MR-IMSRG

48 50 52 54 56

480
460
440
420
400

28 Mass Number

Binding Energy [MeV]

(a)

Neutron Number 26 32 34 30

SLIDE 13

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Titanium Isotopes

N=32 sub-shell closure too pronounced: combined effect of method & interaction !

48 50 52 54 56 8 12 16 20 24 48 50 52 54 56

2 4 6 8

No-shell hyp. AME16 TITAN +

(c) (b)

Neutron Number 26 28 32 34 26 28 32 34 Neutron Number

AME16 1.8/2.0(EM) VS-IMSRG NN+3N(lnl) GGF N

2LOsat

GGF N

2LOsat

MR-IMSRG

30 30 S2n [MeV] Mass Number

No-shell hyp. residual:

∆2n [MeV] Mass Number

52 54 56

0.4

0.0 0.4

E. Leistenschneider et al., PRL 120, 062503 (2018)

SLIDE 14

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Calcium Isotopes

■

■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■

◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆

[]
+() λ= -

+() λ= -

◆

■

HH, in preparation

SLIDE 15

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Calcium Isotopes

[]
HH, in preparation

parabola explained by sd-pf configuration mixing in Shell model: static correlation

SLIDE 16

Excited States

N. M. Parzuchowski, T. D. Morris, S. K. Bogner, Phys. Rev. C

95, 044304 (2017)

S. R. Stroberg, A. Calci, H. H., J. D. Holt, S. K. Bogner, R. Roth,
A. Schwenk, PRL 118, 032502 (2017)
S. R. Stroberg, H. H., J. D. Holt, S. K. Bogner, A. Schwenk,
Phys. Rev. C 93, 051301(R) (2016)
S. K. Bogner, H. H., J. D. Holt, A. Schwenk, S. Binder, A. Calci,
J. Langhammer, R. Roth, Phys. Rev. Lett. 113, 142501 (2014)

SLIDE 17

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

MR-IMSRG References States

Slater determinants (uncorrelated)
number-projected Hartree-Fock Bogoliubov vacua
Generator Coordinate Method (with projections)
small-scale No-Core Shell Model
clustered states, Density Matrix Renormalization Group,

tensor networks etc. available future

SLIDE 18

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

EOM-IMSRG

use IMSRG Hamiltonian as

input for Equation-of-Motion approach

all nucleons active
currently include up to 2p2h

excitation operators

⇤

⇤ (∞) ⇤ ⇤ ⇥

p

h

pp

hh

ppp

hhh

ppp

hhh

pp

hh

p

h

EOM-IMSRG: N. M. Parzuchowski et al., PRC 95, 044304

SLIDE 19

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Valence Space Decoupling

construct non-empirical interactions (and other operators) for use in the nuclear configuration interaction method

2v-0h 2q-0h 3p-1h 4p-2h 4p-2h 3p-1h 2q-0h 2v-0h

⇤

⇤ ⇤ ⇤ ⇥

0p-0h ε ε

h v

q p

valence particle states hole states (core) non-valence particle states

SLIDE 20

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Valence Space Decoupling

change definition of off-diagonal Hamiltonian:

2v-0h 2q-0h 3p-1h 4p-2h 4p-2h 3p-1h 2q-0h 2v-0h 2v-0h 2q-0h 3p-1h 4p-2h 4p-2h 3p-1h 2q-0h 2v-0h

⇤

⇤ ⇤ ⇤ ⇥

⇤

⇤ (∞) ⇤ ⇤ ⇥

⇥

= {

, , , , , , }

SLIDE 21

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Ground-State Energies

(initial) normal ordering and IMSRG decoupling in the target

nucleus

consistent with (MR-)IMSRG ground state energies (and

CC, SCGF , …) for the same Hamiltonian

S. R. Stroberg, A. Calci, HH, J. D. Holt, S. K.Bogner, R. Roth, A. Schwenk, PRL 118, 032502 (2017)

SLIDE 22

Excitation Spectra

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018
S. K. Bogner et al., PRL113, 142501 (2014), S. R. Stroberg et al., PRC 93, 051301(R) (2016)

22O 23O 24O

Ex [ MeV]

CCEI IM-SRG USDB Exp. CCEI IM-SRG USDB Exp. CCEI IM-SRG USDB Exp.

Ex [ MeV]

CCEI IM-SRG USDB Exp. CCEI IM-SRG USDB Exp. 20Ne 24Mg

sd-shell spectra agree very well with experiment and USDA/B… … for NN+3N(400) with “wrong” cD = -0.2.

SLIDE 23

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Transitions

N. M. Parzuchowski, S. R. Stroberg et al., PRC 96, 034324;

converged VS-/EOM-IMSRG results consistent with NCSM

SLIDE 24

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Transitions

N. M. Parzuchowski, S. R. Stroberg et al., PRC 96, 034324; EOM-IMSRG: N. M. Parzuchowski et al., PRC 95, 044304
non-zero B(E2) from Shell model: VS-IMSRG induces

effective neutron charge

B(E2) much too small: effect of intermediate 3p3h, …

states that are truncated in IMSRG evolution

SLIDE 25

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Transitions

N. M. Parzuchowski, S. R. Stroberg et al., PRC 96, 034324
B(E2) much too small: effect of intermediate 3p3h, …

states that are truncated in IMSRG evolution

SLIDE 26

Capturing Static Correlations: IMSRG+GCM

J. M. Yao, C. F

. Jiao, L. J. Wang, J. Engel, H. H., in preparation

SLIDE 27

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

MR-IMSRG References States

Slater determinants (uncorrelated)
number-projected Hartree-Fock Bogoliubov vacua
Generator Coordinate Method (with projections)
small-scale No-Core Shell Model
clustered states, Density Matrix Renormalization Group,

tensor networks etc. available future

SLIDE 28

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Example: 20Ne

reference: particle-

number & angular- momentum projected HFB

range of deformed

reference states flow to the 20Ne ground state

deviation from Shell

model result: correlations beyond MR- IMSRG(2)

10

3

10

2

10

1

10

40
36
32

SM (0

+ 2): -33.735 MeV

β=0.0 β=0.1 β=0.2 β=0.3 β=0.4

E (MeV) Flow parameter s

SM (0

+ 1): -40.491 MeV

20Ne

USD

SLIDE 29

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Approximate MR-IMSRG(3)

approximate MR-IMSRG(3): induced 3B terms recover

bulk of missing correlation energy

size will be reference-state dependent

10

3

10

2

10

1

10 10

1

13
12
11

MR-IMSRG(2) MR-IMSRG(2) + induced-3B

12.185
12.334

E (MeV) flow parameter s

18O

SM: -12.171 MeV USD

40
30
20
10

18 20 22 24 26 28

0.1

0.0 0.1 0.2 0.3 0.4 0.5

E (MeV)

IMSRG(2), s=0 IMSRG(2), s=sMax IMSRG(2) +induced-3B SM (exact)

AO

NN: USD

Deviation (MeV) Mass number

IMSRG(2) IMSRG(2) + induced-3B

verbinding

SLIDE 30

B(E2) (e2fm4) 0+ 2+ 4+

eMax08 EXP . eMax06

63(5) 71(6) 42 56

0+ 2+ 4+

33 42

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

IMSRG+GCM for 20Ne

rotational band spread out
B(E2) significantly boosted, but still underestimated (2B

part of effective E2 not included yet, spectrum spread out)

hw=20 MeV

SLIDE 31

Merging IMSRG and NCSM

E. Gebrerufael, K. Vobig, HH and R. Roth, in preparation
E. Gebrerufael, K. Vobig, HH and R. Roth, PRL 118,

152503 (2017)

SLIDE 32

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

MR-IMSRG References States

Slater determinants (uncorrelated)
number-projected Hartree-Fock Bogoliubov vacua
Generator Coordinate Method (with projections)
small-scale No-Core Shell Model
clustered states, Density Matrix Renormalization Group,

tensor networks etc. available future

SLIDE 33

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Concept

E. Gebrerufael, K. Vobig, HH and R. Roth, PRL 118, 152503 (2017)

NCSM

define reference state

IMSRG

evolve

perators

NCSM

extract

bservables
diagonalization in small model space
use eigenstate as reference
evolve Hamiltonian and observables with

MR-IMSRG

decoupling in A-body space
diagonalize evolved Hamiltonian
calculate eigenstates, observables

SLIDE 34

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

12C: Hamiltonian Matrix Evolution

Nmax=0 Slater determinants Slater determinants Nmax=2 Nmax=4 Nmax=4 Nmax=2 Nmax=0

figures by E. Gebrerufael

SLIDE 35

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

12C: Hamiltonian Matrix Evolution

Nmax=0 eigenstates Slater determinants Nmax=2 Nmax=4 Nmax=4 Nmax=2 Nmax=0

figures by E. Gebrerufael

Nmax=0,2,4 eigenvalues

(almost) identical due to decoupling…

… but IMSRG truncation

artifacts appear eventually (missing induced 3B+ terms)

SLIDE 36

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

for sufficiently large flow parameter

Evolution of the Hamiltonian Matrix
Nmax=0

eigenstates

figures by E. Gebrerufael

induced couplings

between reference and Nmax=0 states

E(s) does not track

lowest eigenvalue ➡ diagonalize H(s)

SLIDE 37

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Evolution of Ground-State Energies

strongly enhanced

convergence

plateau in flow
identify critical smax

at which induced many-body terms become relevant

Robert Roth - TU Darmstadt - March 2017

E [MeV]

95
90
85
80
75
70
65
60

s [MeV-1] E [MeV] 10-4 10-3 10-2 10-1 100

160
150
140
130
120
110

12C 20O

ch

Λ α=0 ħΩ e H N N

Nmax=0 Nmax=2 Nmax=4 Nmax=6 Nmax=8

EM 500/400 NO2B λ = 1.88 fm-1 ℏ" = 20 MeV Nmax = 0,2,4,6,8

IT-NCSM, full 3N

SLIDE 38

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

12C: Excitation Spectra

“uncertainty band”: flow parameter variation from smax/2

to smax

excellent agreement for converged states

NCSM IM-NCSM

EM 500/400

NO2B λ = 1.88 fm-1 ℏ" = 20 MeV

SLIDE 39

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

first 2 and 4 robust and well converged in IM-NCSM

NCSM

IM-NCSM

20O: Excitation Spectra

excellent agreement for converged states
predict 1+ state that has not yet been observed

experimentally

EM 500/400 NO2B λ = 1.88 fm-1 ℏ" = 20 MeV

SLIDE 40

Epilogue

SLIDE 41

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018

Where Do We Go from Here?

Revisit optical potentials (à la J. Rotureau et al., PRC 95,

024315)

MR-EOM / GCM / … to describe few-particle and

collective correlation

continuum coupling for exotic nuclei (see K. Fossez)
Use IMSRG-evolved Hamiltonians in RGM/NCSMC/…)
Utopia: Can we systematically connect many-body

system to few-body system via IMSRG (or other RG) methods?

SLIDE 42

S. K. Bogner, K. Fossez, J. Hill, M.

Hjorth-Jensen, J. M. Yao

NSCL/FRIB, Michigan State University

S. R. Stroberg

Reed College

T. D. Morris

UT Knoxville & Oak Ridge National Laboratory

J. D. Holt, P

. Navrátil

TRIUMF , Canada

E. Gebrerufael, K. Hebeler, S. König
R. Roth, A. Schwenk, C. Stumpf,
K. Vobig, R. Wirth

TU Darmstadt, Germany

R. J. Furnstahl, N. M. Parzuchowski

The Ohio State University

J. Engel

University of North Carolina - Chapel Hill

T. Duguet, V. Somà, A. Tichai

CEA Saclay, France

C. Barbieri
U. Surrey, UK
J. Simonis

Johannes Gutenberg University of Mainz, Germany

H. Hergert - FRIB Theory Alliance Workshop: “From Bound States to the Continuum”, Jun 12, 2018