FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance

FIXED POINTS OF THE SIMILARITYRENORMALIZATION GROUP AND THE

NUCLEAR MANY BODY PROBLEM

E. Ruiz Arriola (with Sergio Szpigel and Varese S. Timoteo)

Departmento de Fısica Atomica, Molecular y NuclearUniversidad de Granada (Spain)

22nd European Conference onFew Body Problems in Physics

Krakow, Poland9 - 13 September 2013

Enrique Ruiz Arriola SRG Fixed

References

Implicit vs Explicit Renormalization and EffectiveInteractionse-Print: arXiv:1307.1231Long distance symmetries for nuclear forces and thesimilarity renormalization groupAIP Conf.Proc. 1520 (2013) 346-348Nuclear Symmetries of the similarity renormalization groupfor nuclear forcesPoS CD12 (2013) 106Symmetries of the Similarity Renormalization Group forNuclear ForcesPhys.Rev. C86 (2012) 034002


Introduction

How much do we need to know light nuclei to predict heavy nuclei ?

Nucleon size a ∼ 1fm

Nuclear Force ∼ 1/mπ = 1.4fm

Nuclear matter (interparticle distance)

ρnm = 0.17fm−3 =1

(1.8fm)3

Fermi Momentum

kF = 270MeV λF = π/kF = 2.3fm� 1/√

mπMN = 0.5fm

1 Can we ignore explicit core and explicit (and/or chiral) pions ? → R. NavarroPerez

2 What are the errors in the interaction→J. E. Amaro


Nuclear many body Hamiltonian H

H =∑

i

Ti +∑i<j

V2,ij +∑

i<j<k

V3,ijk +∑

i<j<k<l

V4,ijkl + . . .

NN: V2,ij (deuteron+NN scattering data)

3N: Triton+ N-deuteron scattering

4N: α−particle, dd ,tp etc, scattering

Chiral hierarchy of few body multipionic forces (Weinberg)

Typical Range of multinucleon forces e−mπd ∼ 0.2

VNN ∼ e−mπd VNNN ∼ e−2mπd VNNNN ∼ e−3mπd

Typical NN wavelengths ≥ 1/√

mπMN ∼ 0.5fm

→ Few wavelengths within a range(Coarse grained Effective interactions)


The off-shell problem

Two-body NN Interactions are not uniquely determined by perfect scatteringdata, or spectrum.

How large is the ambiguity ?

Polyzou-Glockle (Few Body System 1990)1 “Different off-shell extensions of two-body forces can be

equivalently realized as three-body interactions”2 “There are no experiments measuring only three-body

binding energies and phase shifts that can determine ifthere are no three-body forces in a three-body system.”

3 “There may be some systems for which it is possible to finda representation in which three-body forces are notneeded.”

Linear correlation (Tjon line) between triton and α particle binding energykeeping two body scattering fixed

Bα = aBt + b


Isospectral flow in SRG

Wilson-Glazek generator is unitary

dVs

ds= [[T ,Hs],Hs] = [[T ,Vs],T + Vs]→ TrHn

s = TrHn0

Convergence in Frobenius norm and metric (potentials can be compared)

||V ||2 ≡ TrV 2 d(A,B) ≡ ||A− B||

Monotonous decrease

dds

TrV 2s = 2Tr[T ,Vs]2 = −2Tr[T ,Vs]†[T ,Vs] ≤ 0

s0 < s 0 < TrV 2s ≤ TrV 2

s0

Limiting Potential is the smallest possible with the same spectrum

lims→∞

TrV 2s = min

VTrV 2

∣∣∣T +V =UH0U†

High energy states are enhanced by Frobenius norm

1 =2π

∫ ∞0

p2dp|p〉〈p| → TrV 2 =

(2π

)2 ∫ ∞0

p2dp∫ ∞

0k2dk |V (p, k)|2


Integrating out vs Similarity Renormalization Group

Λ0

Λ1

Λ2

k’

k

λ0

λ1

λ2

k’

k

Vlowk → Scattering reproduced until the cut-off.

δlowk(k ,Λ) = δ(k)θ(Λ− k)

VSRG Scattering reproduced at ALL eneries.

δSRG(k , λ) = δ(k)


Operator space

In NN system most states are continuum states (except deuteron)

Equations need discretization and cut-off in momentum space

pn (n = 1, . . . ,N)→ ∆pn ≡ wn → pmax = Λ

Closure relation

1 =2π

N∑n=1

wnp2n |pn〉〈pn|

Standard matrix multiplication

Anm =2π

pn√

wnAnmpm√

wm → 〈A,B〉 =N∑

n,m=1

A∗nk Bkn

SRG equations

dVnm

ds= −(en − em)2Vnm +

∑k

(en + em − 2ek )Vnk Vkm


Fixed points and stability analsis

Fixed points (Wilson)

dds

∑nm|Vnm|2 = −

∑nm|Vnm|2(εn − εm)2 = 0→ Vnk = Vnδnk

Energy eigenvaluesHψn = Enψn ≡ (εn + Vn)ψn

Perturbation around the equilibrium point

Vnk = Vnδnk + ∆nk → ∆V ′nk = −∆Vnk (εn − εm)(En − Em)

Only ordered as free ones are asymptotically stable (crossing forbbiden)

Hnm(s) = Enδn,m + Cnme−(εn−εm)(En−Em)s + . . .


LS equation on the grid

Rij = Vij +∑k 6=i

2π

wk p2k

Rik Vkl

p2i − p2

k

Phase shifts

Rnn = −tan δLS

n

pn≡ Vn

Limiting potential has no off-shellness

limλ→0

Vnm(λ) = −tan δLS

n

pnδnm

However, the LS phase shifts are not independent of λ in a finite grid

δ(pn, λ) 6= δ(pn, λ′)


Wegner generator

Evolution equation

dHds

= [[HD ,H],H] HD = diagH

dds

Tr(H − HD)2 = 2Tr[HD ,H]2 = −2Tr[HD ,H]†[HD ,H] ≤ 0

so that ||H − HD || → 0

lims→∞

H = HD = minH=UH0U† ||H − HD ||

Wilson generator and Wegner generators provide the same final fixed points uptp permutations

Wegner generator (all points are stable, crossing allowed)

Hnm(s) = Enδn,m + Cnme−(En−Em)2s + . . .


Toy model for S-waves

Separable interaction

Vα(p, p′) = Cαe−(p2+p′2)/L2α α =1 S0,

3 S1 (1)

0.0 0.5 1.0 1.5 2.0 2.5 3.00

50

100

150

p Hfm-1L

∆HpLHdegreesL

Parameter α0 r0 C LUnits (fm) (fm) (fm) (fm−1)1S0 -23.74 2.77 -1.9158 0.69133S1 5.42 1.75 -2.3006 0.4151


SRG evolution (Wilson generator)


SRG evolution (Wegner generator)


Diagonal Matrix Elements Evolution

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5

- 1 0

- 5

0

5V ii

(fm)

λ ( f m - 1 )

( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 )

1 S 0 - W i l s o n - 1 0 p t s

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5

- 1 0

- 5

0

5

V ii (fm

)

λ ( f m - 1 )

( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 )

1 S 0 - W e g n e r - 1 0 p t s

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5

- 1 0

- 5

0

5

V ii (fm

)

λ ( f m - 1 )

( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 ) ( 8 , 8 ) ( 9 , 9 ) ( 1 0 , 1 0 )

1 S 0 - W i l s o n - 2 0 p t s

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5

- 1 0

- 5

0

5

V ii (fm

)

λ ( f m - 1 )

( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 ) ( 8 , 8 ) ( 9 , 9 ) ( 1 0 , 1 0 )

1 S 0 - W e g n e r - 2 0 p t s


0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0

1 0

V ii (fm

)

λ ( f m - 1 )

( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 )

3 S 1 - W i l s o n - 1 0 p t s

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5

- 1 0

- 5

0

5

1 0

1 5

V ii (fm

)

λ ( f m - 1 )

( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 )

3 S 1 - W e g n e r - 1 0 p t s

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 1 5 0

- 1 0 0

- 5 0

0

5 0

V ii (fm

)

λ ( f m - 1 )

( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 ) ( 8 , 8 ) ( 9 , 9 ) ( 1 0 , 1 0 )

3 S 1 - W i l s o n - 2 0 p t s

0 , 0 0 , 2 0 , 4 0 , 6 0 , 8 1 , 0- 3 0

- 2 0

- 1 0

0

1 0

2 0

V ii (fm

)

λ ( f m - 1 )

( i , i ) ( 1 , 1 ) ( 2 , 2 ) ( 3 , 3 ) ( 4 , 4 ) ( 5 , 5 ) ( 6 , 6 ) ( 7 , 7 ) ( 8 , 8 ) ( 9 , 9 ) ( 1 0 , 1 0 )

3 S 1 - W e g n e r - 2 0 p t s


Eigenvalues ordering

1 2 3 4 5- 5

0

5

1 0

1 5

2 0

2 5

3 0

3 5

f r e ea s c . o r d .

E i (MeV

)

i

1 S 0 - 1 0 p t s

1 2 3 4 5 6 7 8 9- 5

0

5

1 0

1 5

2 0

2 5

3 0


E i (MeV

)

i

1 S 0 - 2 0 p t s

1 2 3 4 5 6 7 8 9 1 0 1 1- 5

0

5

1 0

1 5

2 0


E i (MeV

)

i

1 S 0 - 3 0 p t s

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5- 5

0

5

1 0

1 5

2 0


E i (MeV

)

i

1 S 0 - 4 0 p t s

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7- 2

0

2

4

6

8

1 0

1 2


E i (MeV

)

i

1 S 0 - 5 0 p t s

1 5 1 0 1 5 2 0 2 5 3 0- 2

0

2

4

6

8


E i (MeV

)

i

1 S 0 - 1 0 0 p t s


Eigenvalues ordering

1 2 3 4 5- 1 0

0

1 0

2 0

3 0

4 0

5 0

6 0

f r e ew e g . o r d .a s c . o r d .K u k u l i n e t . a l p r e s c .

E i (MeV

)

i

3 S 1 - 1 0 p t s

1 2 3 4 5 6 7 8 9- 5

0

5

1 0

1 5

2 0

2 5

3 0

3 5


E i (MeV

)

i

3 S 1 - 2 0 p t s

1 2 3 4 5 6 7 8 9 1 0 1 1- 5

0

5

1 0

1 5

2 0


E i (MeV

)

i

3 S 1 - 3 0 p t s

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5- 5

0

5

1 0

1 5

2 0


E i (MeV

)

i

3 S 1 - 4 0 p t s

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7- 5

0

5

1 0

1 5


E i (MeV

)

i

3 S 1 - 5 0 p t s

1 5 1 0 1 5 2 0 2 5 3 0- 4

- 2

0

2

4

6

8

1 0


E i (MeV

)

i

3 S 1 - 1 0 0 p t s


Binding Energies

Mean field Slater Determinant

ψ(~p1, . . . , ~pA) = A[φn1,l1,s,ms1,t,mt1

(~p1) . . . φnA,lA,s,msA,t,mtA(~pA)

]. (2)

Single particle states (Harmonic oscillator)

Pnl (p) = Nnl e− 1

2 b2p2(bp)l L

l+ 12

n−1

(b2p2

)(3)

Two body interaction (Talmi-Moshinsky)

〈V2〉A =∑nlJS

gnlJS〈nl|V JST |nl〉 , (4)

Nuclei: Shell model (mean field)

d : (1s)2 t : (1s)3 4He : (1s)4 ,

16O : (1s)4(1p)12 40Ca : (1s)4(1p)12(2s)4(1d)20


Binding Energies

0.5 1 1.5 2 2.5 3b (fm)

-10

-5

0

5

10

15

20

B (M

eV)

λ = infinityλ = 3 fm-1

λ = 2 fm-1

λ = 1 fm-1

3H

1 1.5 2 2.5 3 3.5 4rrms (fm)

-20

-10

0

10

20

30

B /

A (M

eV)


λ = 1 fm-1

Exp

40Ca

Binding Energies - AV18

1 1.5 2 2.5 3 3.5 4rrms (fm)

-20

-10

0

10

20

30

B /

A (M

eV)


λ = 1 fm-1

ExpCCBHF

16O

0 0.5 1 1.5 2 2.5 3 3.5kF (fm-1)

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

B /

A (M

eV)


λ = 1 fm-1

AFDMC

nuclear matter

0.5 1 1.5 2 2.5 3b (fm)

-2

-1

0

1

2

3

4

5

6

B (M

eV)


λ = 2 fm-1

λ = 1 fm-1

d

0.5 1 1.5 2 2.5 3rrms (fm)

-40

-30

-20

-10

0

10

20

30

40

B (M

eV)


λ = 2 fm-1

λ = 1 fm-1

ExpGFMCUCOM

4He


Binding Energies

Binding Energies - N3LO

0.5 1 1.5 2 2.5 3b (fm)

-2

-1

0

1

2

3

4

5

6

B (M

eV)


λ = 2 fm-1

λ = 1 fm-1

d

0.5 1 1.5 2 2.5 3b (fm)

-10

-5

0

5

10

15

20

B (M

eV)


λ = 2 fm-1

λ = 1 fm-1

3H

0.5 1 1.5 2 2.5 3rrms (fm)

-40

-30

-20

-10

0

10

20

30

40

B (M

eV)


λ = 2 fm-1

λ = 1 fm-1

ExpGFMCUCOM

4He

1 1.5 2 2.5 3 3.5 4rrms (fm)

-20

-10

0

10

20

30

B /

A (M

eV)


λ = 1 fm-1

ExpCCBHF

16O

1 1.5 2 2.5 3 3.5 4rrms (fm)

-20

-10

0

10

20

30

B /

A (M

eV)


λ = 1 fm-1

Exp

40Ca

0 0.5 1 1.5 2 2.5 3 3.5kF (fm-1)

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

B /

A (M

eV)


λ = 1 fm-1

AFDMC

nuclear matter


SRG Correlations

The Wilson and Wegner binding energy results for SRG evolved forces

{−Bt ,−Bα} = minb

[(A− 1)〈

p2

2M〉+

A(A− 1)

212〈V1S0,λ + V3S1,λ〉

] ∣∣∣A=3,4

Ê

ÊÊÊ

Ê

ÊÊÊ

‡

‡‡‡‡‡

‡‡

Ï

ÏÏÏÏÏ

ÏÏ

Ú

ÚÚÚÚÚÚ

Ú

Ù

ÙÙÙÙÙÙ

Ù

-8 -6 -4 -2 0-30-25-20-15-10-50

BtHMeVL

BhHMeVL

Ê

ÊÊ

Ê

ÊÊÊÊ

‡

‡‡‡

‡‡‡‡

Ï

ÏÏÏ

Ï

ÏÏÏ

Ú

ÚÚÚÚ

ÚÚÚ

Ù

ÙÙÙÙ

ÙÙÙ

-8 -6 -4 -2 0-30-25-20-15-10-50

BtHMeVLBhHMeVL

Linear correlations in two regimes

∆Bα/∆Bt ∼ 2(λ→ 0) ∆Bα/∆Bt ∼ 4(λ ∼ 1)


The on-shell limit

Wilson and Wegner generator results (N=50)

ÊÊÊÊÊÊ

Ê

‡‡‡‡‡‡

‡

Ï

Ï

Ï

ÏÏÏ

Ï

0.0 0.1 0.2 0.3 0.4 0.5 0.6-15

-10

-5

0

l Hfm-1L

BHMeVL

Ê Deuteron‡ TritonÏ Helium

ÊÊÊÊ

ÊÊÊ

‡‡‡‡

‡‡‡

Ï

Ï

Ï

Ï

ÏÏÏ

0.0 0.1 0.2 0.3 0.4 0.5 0.6-15

-10

-5

0

l Hfm-1L

BHMeVL

Ê Deuteron‡ TritonÏ Helium

On-shell results

limλ→0

Et (λ) = −32

Bd limλ→0

Eα(λ) = −3Bd


SRG view of off-shellness and three-body force

Isospectral transformations

dVij

ds=

[[Tij ,Vij

],Tij + Vij

], (5)

dV123

ds= [[T12,V12] ,V13 + V23 + V123]

+ [[T13,V13] ,V12 + V23 + V123]

+ [[T23,V23] ,V12 + V13 + V123]

+ [[Trel,V123] ,Hs] . (6)

What is the initial condition ?

Final condition is unique

[T12,V12] = 0 [Trel,V123] = 0 (7)

Diagonal potential in momentum space (no off-shellness)


Correlations with on-shell 3-body forces

The on-shell triton (3 doublets) and α ( 6 doublets) binding

−Bt = −32

Bd︸︷︷︸3.3MeV

+ 〈t |V3|t〉︸︷︷︸off−shellness

−Bα = − 3Bd︸︷︷︸6.6MeV

+ 〈α|V3|α〉︸︷︷︸off−shellness

Taking 〈α|V3|α〉 = 4〈t |V3|t〉 ( 4 triplets )

Bα = 4Bt − 3Bd

= 4× 8.482− 3× 2.225 = 27.53 (exp.28.296) MeV

BΑ= 4Bt -3BdBΑ= 4Bt -3Bd

++Exp.Exp.

CD-BonnCD-Bonn

AV18AV18

NijmINijmINijmIINijmII

Vlowk HAV18LVlowk HAV18L

SRG HN3LOLSRG HN3LOL

6 7 8 9 1020

22

24

26

28

30

BtHMeVL

BΑ

HMeV

L


Conclusions

1 SRG methods allow to reduce off-shell ambiguitycompletely

2 Only measurable two-body information is needed3 Simple explanation of the observed linear correlations

(Tjon line)4 On-shell 3-body forces are large and 4-body forces are

moderate5 Extension to other nuclei, neutron and nuclear matter is

possible


Documents

FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit Renormalization and Effective Interactions e-Print: arXiv:1307.1231 Long distance