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Lawrence Livermore National Laboratory
Physical and Life Sciences/Physics
Many-Nucleon Interactions via the SRGTrento, Italy
July 13, 2011
Eric JurgensonCollaborators: P. Maris, R. Furnstahl, P. Navratil, E. Ormand, J. Vary
Prepared by LLNL under Contract DE-AC52-07NA27344
Performance Measures x.x, x.x, and x.x
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2Option:UCRL# S&T – PLS/Physics
Ab initio: from the begining
Start with χEFTs SRG is well suited to the many-
body hierarchy
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3Option:UCRL# S&T – PLS/Physics
What is the Similarity Renormalization Group (SRG) ?
PRC 75,061001 (2007) [arXiv: nucl-th/0611045]
€
Hs = UsHUs+ ≡ Trel + Vs
€
s = 1λ4( )
€
ηs = [Gs,Hs] ⇒ dHs
ds= T,Hs[ ],Hs[ ]
Unitary Transformations:
€
dHs
ds= [η s,Hs] where
€
ηs = dUs
dsUs
+ = −η s+
Implement as flow equations
With Gs any Hermitian operator
€
Gs = Trel ,Hdiag ,Hosc,HBD,T +VNN ,exp[T],L
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4Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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5Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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6Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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7Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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8Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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9Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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10Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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11Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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12Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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13Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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14Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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15Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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16Option:UCRL# S&T – PLS/Physics
SRG evolves Hamiltonians unitarily
€
Hs = UsHUs+ ⇒ dHs
ds= T,Hs[ ],Hs[ ]
€
s = 1λ4( )
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17Option:UCRL# S&T – PLS/Physics
Many-Body Forces (ANFs)
ANFs arise from eliminating DoF’s Omitting them leads to model
dependence (Tjon line) 3NF help saturate nuclear matter All many-body methods must deal
with them SRG induces ANFs! :
€
dHs
ds= a+a, a+a+aa
2−body1 2 3 ∑∑
⎡
⎣ ⎢
⎤
⎦ ⎥, a+a + a+a+aa
2−body1 2 3 ∑
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥=K + a+a+a+aaa
3−body1 2 4 3 4 +K
RG flows extend consistently to many-body spaces State of the art now includes evolved 3NFs!
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18Option:UCRL# S&T – PLS/Physics
Triton
SRG improves convergence
PRL 103, 82501 (2009) [arXiv: 09005.1873]
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19Option:UCRL# S&T – PLS/Physics
Helium
SRG induces many-body forces – how big are they?
PRL 103, 82501 (2009) [arXiv: 09005.1873]
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20Option:UCRL# S&T – PLS/Physics
Convergence in 6Li
Increased NA2max to 300 and NA3max to 40 Simple extrapolations show spread in λ Example here for one ħΩ – need optimal for each λ
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21Option:UCRL# S&T – PLS/Physics
Lithium
Optimal ħΩ shifts with evolution
Extrapolations use this info Error bars are consistent
with previous work λ dependence reduced from 4 to <1 MeV
PRC, to appear [arXiv: 1011.4085]
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22Option:UCRL# S&T – PLS/Physics
Carbon
Difference is subtle here but trackable - plot difference of λ=1.5 - 2.2 vs. parameters
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23Option:UCRL# S&T – PLS/Physics
Nmax/ħΩ dependence
Interesting spread behavior in the basis space Clues to the source of convergence artifacts
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24Option:UCRL# S&T – PLS/Physics
New Generators
Scale c3 and c4
Plot the effect on the spread
Many choices of Gs
HO basis complicates the choice
More to come…
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25Option:UCRL# S&T – PLS/Physics
Hierarchy Preservation: A=3
Show hierarchy and suppression of high momentum strength
Expectation values of contributions to evolution
€
d V3
dλ= V 2,V2[ ] − V 3,V3[ ]
PRC, to appear [arXiv: 1011.4085]
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26Option:UCRL# S&T – PLS/Physics
Hierarchy Preservation: A=4
Repeat for A=4 Well defined hierarchy No positive feedback loops
€
d V4
dλ= V 2,V3[ ] + V 3,V2[ ] + V 3,V3[ ] − V 4 ,V4[ ]
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27Option:UCRL# S&T – PLS/Physics
Hierarchy Preservation: induced vs. initial 3NF
Compare Initial with Induced Similar composition of contributions
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28Option:UCRL# S&T – PLS/Physics
Carbon Revisited
Induced ANFs are as expected from existing work
Extrapolation in ħΩ should show reduction in spread
Larger systems are needed to determine 3/4NF needs…
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29Option:UCRL# S&T – PLS/Physics
Adding 3NFs to the many-body problem…
Coupled Cluster (CC)
Density Functional Theory (DFT)
NCSM/RGM (reactions)
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30Option:UCRL# S&T – PLS/Physics
Conclusion
Recap SRG improves convergence Variational and model space
independent Truncations are well
understood/controlled Induced forces are of natural
size, though details still to be investigated
Great freedom in SRG form for such work
Outlook Monitor hierarchy of induced
higher-body forces (MFDn, Bigstick, IT-NCSM)
Coupled cluster results with SRG evolved 3NF inputs
Apply these 3NFs to reactions (NCSM+RGM)
Explore various ideas about the choice of Gs