Operator Evolution Using SRG Flow Equations for Few-Body Systems

Operator Evolution Using SRG Flow Equationsfor Few-Body Systems

Eric R. Anderson

Department of PhysicsThe Ohio State University

October 17, 2009

In Collaboration with: S.K Bogner,R.J. Furnstahl, E.D. Jurgenson, &R.J. Perry

Work supported by NSF and UNEDF/SciDAC (DOE)

Anderson SRG Evolved Operators in Few-Body Systems

SRG: A Brief Review

The Similarity Renormalization Group (SRG)

→ provides a means to systematically evolve computationally difficultpotentials and operators

Os = UsOs=0U†s ⇐⇒

dOsds= [ηs ,Os ] = [[Trel ,Hs ],Os ]

Hamiltonian Operator → driven toward diagonal or decoupled form

3S1 N3LO 500MeV – Entem & Machleidt

Note:

λ = 1

s1/4fm−1

Unitarily evolve operators consistent with any initial potential

→ Number Operator, Electromagnetic Form Factors, etc.→ Deuteron and a 1D system of bosons provide “lab”

Issues: – Decoupling

– Induced few-body operators

– Convergence in harmonic oscillator basis


High and Low Momentum operators in the Deuteron

Integrand of 〈ψd |U†(Ua†qaqU

†)U |ψd〉 for q = 3.02 fm−1

Integrand for q = 0.34 fm−1

MomentumDistribution

0 1 2 3 4

q [fm−1

]

10−5

10−4

10−3

10−2

10−1

100

101

102

4π [u

(q)2 +

w(q

)2 ] [fm

3 ]

N3LO unevolved

Vs at λ = 2.0 fm−1

Vs at λ = 1.5 fm−1

a✝

qa

qdeuteron

High momentum components suppressedIntegrated value does not change, but nature of operator doesSimilar for other long distance operators:

⟨r2⟩, 〈Qd〉, &

⟨1r

⟩


Demonstration of Decoupling In Expectation Values

Evolve Hamiltonian & operators to λ in full space → TRUNCATE at Λ:

0 1 2 3 4 510

-8

10-6

10-4

10-2

100

102

q (fm -1 )

a+ a

Evolved to λ=6.0, Cut at Λ=2.5

UnevolvedSRG Evolved

0 1 2 3 4 510

-8

10-6

10-4

10-2

100

102

q (fm -1 )

a+ a



Momentum distribution

Λ = 2.5 fm−1

λ = 6.0 fm−1, 3.0 fm−1, and 2.0 fm−1

Decoupling for all q is successful

when λ < Λ

Calculated with AV18 potential.

0 1 2 3 4 510

-8

10-6

10-4

10-2

100

102

q (fm -1 )

a+ a




Many-Body evolution of Operators

Analysis of many-body evolution with operators normal ordered in the vacuum:

dOs

ds= [[Trel,Hs ], Os ] =⇒

∑

ij

Tij a†i aj ,

∑

i′ j′

Ti′ j′ a†i′aj′ +

1

2

∑

pqkl

Vpqkl|s a†p a†q al ak

, Os

→ Only one non-vanishing contraction in the vacuum: ai a†j = δij

A general operator O for an A-body system can be written as

O = O(1) + O(2) + O(3) + ∙ ∙ ∙+ O(A)

where the O(i) label the i = 1, 2, 3, ...,A-body components of the operator.

– SRG operator Os will have contributions at all levels of n so that O(n) 6= O(n)s .

Expanding commutators and making contractions, one finds:

→ Evolution of an operator is fixed in each n-body subspace→ E.g.,

* Evolve O in a 1-body space to Os =⇒ determines O(1)s for all A

– 1-body operators unchanged with this generator

* Evolve O in a 2-body space to Os =⇒ determines O(2)s for all A

* Evolve O in a 3-body space to O′s =⇒ determines O(3)s for all A, etc.


Electromagnetic Form Factors - 1-body Initial Current

Elastic electron-deuteronscattering

Calculation of GC , GQ , GM

1-body versus 2-bodyevolution?

⇐⇒ bare versus evolved0 1 2 3 4 5

10-3

10-2

10-1

100

q (fm-1 )

GC

λ=1.5

UnevolvedSRG EvolvedSRG Evolved wfwith Bare Operator

0 1 2 3 4 510

-3

10-2

10-1

100

q (fm-1 )

GQ

λ=1.5


0 1 2 3 4 510

-3

10-2

10-1

100

q (fm-1 )

MG

M/M

dλ=1.5


Wave function is derived from the NNLO 550/600 MeV – Epelbaum et al.potential and the evolution is run to λ = 1.5 fm−1.


Momentum Distribution in Few-Body Model Space

A Test Bed for 3D NCSM calculations:Embed in 1D few-body HO space (code from E.D. Jurgenson)

→ System of A bosons interacting via a model potential

→ Here we have: 〈NAiA| a†kak |NAiA〉

NA = Symmetric Jacobi states, iA =degeneracies

0 5 10k

10−5

10−4

10−3

10−2

10−1

100

n(k)

/A

λ = ∞λ = 10.00λ = 5.00λ = 3.00

0 5 10k

0 5 10k

A = 2 A = 3 A = 4

Nmax

= 40

High momentum components suppressedAnderson SRG Evolved Operators in Few-Body Systems

High and Low Momentum Operators in Oscillator BasisNumber operator at: q = 15.0 & q = 0.5

0 10 20 30 400

1

2

3

4

5x 10

-5

Ncut

n(k)

/A

λ=100λ=3λ=2

0 10 20 30 400.35

0.4

0.45

0.5

0.55

0.6

Ncut

n(k)

/A

λ=100λ=3λ=2

A=3 System

Evolve Hamiltonian & operators toλ at large Nmax→Truncate model space at Ncut−→ Poor convergence of long rangeoperators

ΛUV ∼√mNmax~ω; ΛIR ∼

√m~ω

Nmax

−→Can this be corrected?

⟨r2⟩Value

0 10 20 30 40

0.2

0.25

0.3

0.35

0.4

0.45

0.5

‹ r2 ›

Ncut

λ=100λ=3λ=2λ=1.5λ=1


An alternative generator - diagonal in the basis

The harmonic oscillator Hamiltonian Hho is diagonal in the basis

Considerηs = [Hho,Hs ]

〈r2〉 for A=3 Many Contributions to Binding Energy

Greatly Improved convergence

Generates spurious deep bound states . . .


Controlled IR and UV renormalization I

Consider Trel + αr2, where α is a parameter which can be adjusted

to optimize the renormalization (here, α = 1), so that

ηs = [Trel + αr2,Hs ]

〈r2〉 for A=3 Many Contributions to Binding Energy

Convergence improves with decreasing λ

No spurious deep bound states. Is hierarchy of many-body forcesunder control?


Controlled IR and UV renormalization II

Convergence of High (q=15.0) and Low (q=0.5) number operators

with ηs = [Trel + αr2,Hs ]

0 10 20 30 400

1

2

3

4

5x 10

-5

Ncut

n(k)

/A

λ=100λ=3λ=2λ=1

0 10 20 30 400.35

0.4

0.45

0.5

0.55

0.6

Ncut

n(k)

/A

λ=100λ=3λ=2λ=1

Convergence ofbinding energy →


Conclusion/Outlook

Summary:

Nuclear operators can be consistently evolved and calculatedwith the SRG

Many-body contributions can be extracted at each order

HO Basis is a pathway to few body operators &electromagnetic interactions

Alternative generators in 1D NCSM

Outlook:

Variational few-body calculations

Study alternative generators in 3D NCSM

Higher order electromagnetic interactions in few-body nuclei


The End


Operator Evolution Using SRG Flow Equations for Few-Body Systems

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