many-nucleon interactions via the srg trento, italy july 13, 2011

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

2Option:UCRL# S&T – PLS/Physics

Ab initio: from the begining

Start with χEFTs SRG is well suited to the many-

body hierarchy


What is the Similarity Renormalization Group (SRG) ?

PRC 75,061001 (2007) [arXiv: nucl-th/0611045]

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Hs = UsHUs+ ≡ Trel + Vs

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s = 1λ4( )

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ηs = [Gs,Hs] ⇒ dHs

ds= T,Hs[ ],Hs[ ]

Unitary Transformations:

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dHs

ds= [η s,Hs] where

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ηs = dUs

dsUs

+ = −η s+

Implement as flow equations

With Gs any Hermitian operator

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Gs = Trel ,Hdiag ,Hosc,HBD,T +VNN ,exp[T],L


SRG evolves Hamiltonians unitarily

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Hs = UsHUs+ ⇒ dHs

ds= T,Hs[ ],Hs[ ]

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s = 1λ4( )



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Hs = UsHUs+ ⇒ dHs

ds= T,Hs[ ],Hs[ ]

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s = 1λ4( )


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! :

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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!


Triton

SRG improves convergence

PRL 103, 82501 (2009) [arXiv: 09005.1873]


Helium

SRG induces many-body forces – how big are they?

PRL 103, 82501 (2009) [arXiv: 09005.1873]


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


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]


Carbon

Difference is subtle here but trackable - plot difference of λ=1.5 - 2.2 vs. parameters


Nmax/ħΩ dependence

Interesting spread behavior in the basis space Clues to the source of convergence artifacts


New Generators

Scale c3 and c4

Plot the effect on the spread

Many choices of Gs

HO basis complicates the choice

More to come…


Hierarchy Preservation: A=3

Show hierarchy and suppression of high momentum strength

Expectation values of contributions to evolution

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d V3

dλ= V 2,V2[ ] − V 3,V3[ ]

PRC, to appear [arXiv: 1011.4085]


Hierarchy Preservation: A=4

Repeat for A=4 Well defined hierarchy No positive feedback loops

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d V4

dλ= V 2,V3[ ] + V 3,V2[ ] + V 3,V3[ ] − V 4 ,V4[ ]


Hierarchy Preservation: induced vs. initial 3NF

Compare Initial with Induced Similar composition of contributions


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…


Adding 3NFs to the many-body problem…

Coupled Cluster (CC)

Density Functional Theory (DFT)

NCSM/RGM (reactions)


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

many-nucleon interactions via the srg trento, italy july 13, 2011

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