FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit...
Transcript of FIXED POINTS OF THE SIMILARITY …efb22.if.uj.edu.pl/talks/RuizArriola.pdf · Implicit vs Explicit...
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
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
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)
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
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)
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
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 + . . .
Enrique Ruiz Arriola SRG Fixed
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′)
Enrique Ruiz Arriola SRG Fixed
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 + . . .
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
SRG evolution (Wilson generator)
Enrique Ruiz Arriola SRG Fixed
SRG evolution (Wegner generator)
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
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
f r e ea s c . o r d .
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
f r e ea s c . o r d .
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
f r e ea s c . o r d .
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
f r e ea s c . o r d .
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
f r e ea s c . o r d .
E i (MeV
)
i
1 S 0 - 1 0 0 p t s
Enrique Ruiz Arriola SRG Fixed
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
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 - 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
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 - 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
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 - 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
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 - 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
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 0 p t s
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
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 = infinityλ = 2 fm-1
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 = infinityλ = 2 fm-1
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 = infinityλ = 2 fm-1
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 = infinityλ = 3 fm-1
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 = infinityλ = 3 fm-1
λ = 2 fm-1
λ = 1 fm-1
ExpGFMCUCOM
4He
Enrique Ruiz Arriola SRG Fixed
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 = infinityλ = 3 fm-1
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 = infinityλ = 3 fm-1
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 = infinityλ = 3 fm-1
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 = infinityλ = 2 fm-1
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 = infinityλ = 2 fm-1
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 = infinityλ = 2 fm-1
λ = 1 fm-1
AFDMC
nuclear matter
Enrique Ruiz Arriola SRG Fixed
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
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Linear correlations in two regimes
∆Bα/∆Bt ∼ 2(λ→ 0) ∆Bα/∆Bt ∼ 4(λ ∼ 1)
Enrique Ruiz Arriola SRG Fixed
The on-shell limit
Wilson and Wegner generator results (N=50)
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On-shell results
limλ→0
Et (λ) = −32
Bd limλ→0
Eα(λ) = −3Bd
Enrique Ruiz Arriola SRG Fixed
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)
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed
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
Enrique Ruiz Arriola SRG Fixed