Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucleaire et en physique des particules Ab initio effective interactions and operators from IM-SRG Ragnar Stroberg TRIUMF Double Beta Decay Workshop U = e η dH ds =[η,H ] U OU † = O +[η, O]+ ... Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 1 / 27
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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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 2 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 3 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 4 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 4 / 27
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〉
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 5 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 6 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 7 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 8 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 9 / 27
Shell model expectation values then reflect full-space expectationvalues:
〈Ψ|O|Ψ〉 = 〈ΨSM |O|ΨSM 〉
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 22 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 23 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 23 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 24 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 25 / 27
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)
USD-bareUSD-effExpIMSRG-bareIMSRG-eff
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
)
USD-bareUSD-effExpIMSRG-bareIMSRG-eff
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 26 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 28 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 29 / 27
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
20 Neemax=4emax=6emax=8emax=10
bare1b
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 29 / 27
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
20 Neemax=4emax=6emax=8emax=10
bare1b1b+2b
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 29 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 30 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 32 / 27
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
Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, 2016 33 / 27
References I
Binder, Sven et al. (2014). “Ab initio path to heavy nuclei”. In: Phys. Lett. B 736, pp. 119–123. issn: 03702693. doi: