09/02/2009 SRG to the shell model, K. Tsukiyama 1

Similarity Renormalization Group

to the nuclear shell model

Koshiroh Tsukiyama

University of Tokyo

ENNEN09 on Feb. 9th 2009 at GSI

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Introduction

One of the major goals of nuclear physics

To calculate many-body observables quantitatively starting from microscopic NN or NNN interactions.

=>To understand nucleonic interactions from more fundamental degrees of freedom.

We don’t know exactly the nuclear forces.

If nucleonic interactions would be given, it is still NOT easy to calculate many-body observable.

Short-distance repulsion in nuclear forces as well as a strong tensor force.

Highly non-perturbative few- and many-body system

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Effective interactions for Nuclear Many-body System

UCOM

Usually one overcomes this problem by

Introducing the Brueckner G-matrix

=> Re-summation of in-medium pp scattering.

Using RG approach, Vlowk

Using unitary transformations

Unitary transformation

RG equations

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Two Types of RG transformations AV18 3S1

Vlow-k lowers a cutoff in (k, k’)

SRG drives Hamiltonian towards the band-diagonal

Both decouple the high momentum modes, leaving low energy NN

observables unchanged.

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UCOM and SRG and Vlowk

SRG vs. Vlowk

UCOM vs. SRG

S. K. Bogner, R. J. Furnstahl and R .J. PerryPRC75, 061001(R), 2007

H. Hergert and R. RothPRC75, 051001(R), 2007

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Similarity Renormalization Group

Consider a Hamiltonian that we split into diagonal and off-diagonal parts

The SRG provides a prescription to construct the appropriate unitary transformation.

Taking d/ds on both sides then gives

generator of the transformation

We can formally write the unitary transformation as an s-ordered exponential,

Instead of giving one unitary transformation, infinitesimal representations are given in SRG.

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Wegner’s Choice

The idea is to drive a Hamiltonian towards the energy diagonal

Design ηfor different things as s

H(s) driven towards diagonal in k-space

H(s) driven towards block diagonal

is positive semi-definite

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Free Space SRGIn each partial waves with and

Taken from http://www.physics.ohio-state.edu/~ntg/srg/

For AV18, 3S1

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Free Space SRG

Taken by arXiv:0708.3754v2

Full configuration interaction (FCI) calculation results

As λreduced (s increased) => low-k and high-k decoupled

Convergence is accelerated

But λ-dependent results.

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SRG and Many-body forces

SRG generates A unitary transformed Hamiltonian

Exactly the same value for observable

Only if the entire Hamiltonian is kept.

But many-body interactions are induced in the flow.

In principle up to A-body term is generated.

Bare H also have this structure, for instance, HEFT=H1b+H2b+H3b+H4b+…

NN flow is only approximately unitary

For SRG to be useful for nuclear structure, the point is

Maintaining the hierarchy of many-body forces

OK if the induced terms are of natural size.

Normal-ordering and truncation may be sufficient.

Calculate 3-body sector in the flow . (an advantage of SRG)

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Error estimation in CCSD

3N Interaction and Normal-Ordering

from AV18

Fitted to the data of 3H and 4He

Very small residual 3NF

Normal-ordering can preserve the hierarchy of Many-body forces.

In NCSM, density-dependent two-body terms are the most significant contributions of effective 3NF

Navrátil and Ormand in PRL88, 152502(2002)

Hagen, Papenbrock, Dean, Schwenk.. PRC76, 034302 (2007)

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In-medium SRG

3N, 4N.. forces appear as density dependence in expectation value of a core, s.p.e’s and TBME’s

Choose generator, following Wegner’s idea

Truncate flow equation up to Two-body normal-ordered operators

Rewrite a Hamiltonian taking normal-ordering

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Flow equations of In-medium SRG

Generator is written as

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Flow equations of In-medium SRG

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Flow equations of In-medium SRG

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Flow equations of In-medium SRG

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Perturbative content of SRG

In momentum basis, g is diagonal Flow eqs. can be simplified

Example: homogeneous system like NM

The flow equations at leading order are

Solution for E0 up to 2nd order in the bare coupling.

As s increases, correlation energy is shuffled into non-interacting VEV (HF).

Correlation energy

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Preliminary calculations

Wegner’s choice for the generator

s=0 s

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E0 (4He) flow (solved in small spaces) [N3LO(500MeV)]

0 0.1−30

−20

−10

0SRG, spsdpf−space

s [MeV]2

E0

[MeV

]

=1.5 fm−1

=1.0 fm−1

=2.0 fm−1

We need lager space to make the flow more reliable.

0 0.1

−20

−10

0G−matrix

spsd−shell

spsdpf−shell

s [MeV]2

E0

(4H

e) f

low

ω =-5MeV

VNNG-matrix

Free space SRG= as an input of In-medium SRG

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Observation

0 0.2 0.4−15

−10

−5

0

Off diagonal TBMEs

spsdpfg−shell

Diagonal TBME

s [MeV]2

ab

cd

G−matrix from N3LO(500 MeV)

Flow eqs are solved in

2<0s2|V|0s2>Tz=0,J=0

0 0.2 0.4−15

−10

−5

0

Off diagonal TBMEs

spsdpf−shell

Diagonal TBME

s [MeV]2

ab

cd

G−matrix from N3LO(500 MeV)

Flow eqs are solved in

2<0s2|V|0s2>Tz=0,J=0

0 0.2 0.4−15

−10

−5

0

Off diagonal TBMEs

spsd−shell

Diagonal TBME

s [MeV]2

ab

cd

G−matrix from N3LO(500 MeV)

Flow eqs are solved in

2<0s2|V|0s2>Tz=0,J=0

Energy off-diagonal matrix elements vanish after the flow.

if

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Observations

Energy off-diagonal matrix elements vanish after the flow.

if

There are some cross-shell TBME’s after the flow

Initial value-dependent (due to the small space flow?)

We don’t have to solve cluster problems.

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Summary

We have derived the flow equations for In-medium SRG in M- and J-scheme basis.

Current choice of the generator can get stable flow.

Enegy off-diagonal M.E’s are well suppressed after the flow.

HF covers the dominant part of the correlation energy Shell Model

There are some cross-shell TBME’s after the flow physical meaning

We need large space flow to see the convergence.

direct connection between bare nuclear forces and the nuclear shell model.

3NF in the initial Hamiltonian

Non-perturbative path to the nuclear shell model.

3NF in the normal-ordered Hamiltonian the magnitude of the induced 3NF

In-medium SRG for effective operator Oeff as well as Heff.

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Collaborators

Achim Schwenk (TRIUMF)

Scott Bogner (MSU)

Takaharu Otsuka (Tokyo)