In-medium Similarity Renormalization Groupfor nuclear many-body systems	

Koshiroh Tsukiyama (CNS/ U. Tokyo)	

Dec. 3-5 2011 	

HPCI project “Large-scale quantum many-body calculations for nuclear property
and it’s application”	
In collaboration with 	Achim Schwenk (GSI/EMMI, TUD) and Scott K. Bogner (NSCL/MSU)	

1

Description of nuclei from NN/NNN	Methods should be controlled and improvable
Medium-mass nuclei

Density functional theory	

• Systematic	

• computationally efficient	• Universal EDF unknown	

Light nuclei

configuration interaction	(v) (v)
Heff= (E Ec )NN/NNN(v)shell evolution, deformation,
double-bata decay,..	

ab initio	

binding energy, radius	

accuracy, feasibility	

2Λ	V(k,k’)

QCD	

AV18	

H( ) = Trel + V2N ( ) + V3N ( ) + V4N ( ) + · · ·χEFT	

pion-exchange and short-range contact	

power counting (still open)	

-systematic expansion	-many-body forces automatically 	

hard	

RG evolution to lower resolution/Λ	still hard 	

NN observables kept unchanged	

Low-momentum interactions	Vlow-k, SRG	Bogner-Schwenk-Furnstahl, PPNP, 65, 94 (2010)	

3Similarity Renormalization Group 	Glazek and Wilson, Phys. Rev. D48, 5863(1993), or Wegner, Ann. Phys. (Leipzig) 3, 77 (1994)	

The unitary evolution of the Hamiltonian via flow equation	

dH(s) = [ , H(s)]dsdU (s) †(s) =U (s)can arbitrarily be defined	

dsdod 2od0
Tr H(s) = [H(s), H (s)] by F. Wegner	

ds
Flexibility of choosing the Hd(s) for a particular problem. 	

lower λcutoff λ≡ s-1/4 as evolution variable	

Block diagonalization using SRG

Bogner et al, PRC75, 061001(R) (2007)

with block-diagonal flow operator Bogner et al. (2008)Anderson et al, PRC77, 037001 (2008)
low-momentum blocks very similar to Vlow k4

thetialocaltricayromhemn inracieldforromnty
ncetterody
forarelotn ision,foreV.
NN
systingExpt.
"8.61234 5 6 7"1! [fm ]1020

RG and many-body interactions	FIG. 1: (Color online) Ground-state energy of 3 H as a function of the SRG evolution parameter, λ. See Table I for thenomenclature of the curves.

Free-space SRG, evolving consistent 3N interactions => exact method	

"244Ground-State Energy [MeV]

ternts.
ubbamaarethe. If
notapares isntrile Ie to"25He• NN only => λ-dependent	

• + induced NNN => almost λ-independent	

3N LO (500 MeV)

Jurgenson, Furnstahl and Navratil PRL103, 082501(2009)	

"26"27"28"29

same trend for heavier systems

NN-only
NN+NNN-induced+NNN-initial

Roth et al., PRL107, 07201(2011)	

Expt.

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34 5"11020! [fm ]FIG. 2: (Color online) Ground-state energy of 4 He as a function of the SRG evolution parameter, λ. See Table I for thenomenclature of the curves.

In-medium SRG• Defined in many-body system (finite density)	

energy calculations will only be approximately unitary
for A > 2 and the ground-state energy will vary with λ•(squares). Keeping the induced NNN yields a flat line
(circles), which implies an exactly unitary transformation; the line is equally flat if an initial NNN is included(diamonds). Note that the net induced three-body is• to the initial NNN contribution and thus iscomparableof natural size.
In Fig. 2, we examine the SRG evolution in λ for 4 He
(2)
(3)
with !Ω = 36 MeV. The !Vλ " and !Vλ " matrix elements were evolved in A = 2 and A = 3 with Nmax = 28

Approximate evolution of 3-, .. A-body operators within 2bmachinery.	

Different SRG evolutions for different mass regions.	

K.T., S. Bogner and A. Schwenk, PRL106, 222502(2011) 	

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Normal-ordered Hamiltonian	†Tij ai aj +Ĥ =ij1
2!2(2) † †Vijkl ai aj al ak +ijkl

13!2ijklmn(3)† † †Vijklnm ai aj ak an am al + · · ·Normal order w.r.t. a finite-density Fermi vacuum |Φ〉, e.g. HF.	

H = E0 +ij

fij {a†i aj } +12!2ijkl† †{aijkli aj al ak } +13!2ijklmn

Wijklmn {a†i a†j a†k an am al }where coefficients of normal-ordered operators are given by	

E0 = ⇥| H |⇥⇥ =k1
1(2)Tkk nk +Vijij ni nj +
2 ij
61(2)
Vikjk nk +fij = Tij +2k1(2)
ijkl = Vijkl +4 m| {Ai Aj · · · } |=0(3)Vijkijk ni nj nkijk(3)Vikljkl nk nlkl(3)Vijmklm nm(3)

Wijklmn = Vijklmn

3-body and higher-body interactions through density- dependent coefficients.	

=> may be efficient truncation scheme	

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Decoupling (schematic picture)	

dH(s) = [ , H(s)]
ds

Φ0

Φp

η(s) is determined s as to eliminate Hod(s)Φ0Φpdecouples ground state	

Hod = fph + Γpphh	vertices connecting reference
state and np-nh excited states

Φ0 Φv

Φq

decouples valence space	

Hod = fph + Γpphh	+ fvq + fvh + Γvv’qq’ + Γvhpp’ + Γvv’v’’h
vertices connecting valence
space and outside of it.
7In-medium SRG flow equation	

dH(s)= [ , H(s)] = [ (1) +ds+(2)(3)+ ··· ,f ++ W + ···]

commutator form => no unlinked diagram 	=> size extensive: energy scales linearly w/ # of particles	

IM-SRG(2): n6, IM-SRG(3): n8	

Flow eqns.	

dE0 (s) = 2ds1(1)na n̄b ab fba +1+18ddsijkl (s) =2ab(2)abcd cdab (s)na nb n̄c n̄dabcd(3)abcdef Wdef abc na nb nc n̄d n̄e n̄fabcdef(1)(1Pij )( ia(1na(nanb ) (1(nn )

ajkl

(2)

fia ajkl )(1(1)

Pkl )( aiijala

1+2(2)

nb )( ijab abkl

(2)
ijab abkl )ab(2)Pij )(1Pkl ) bjal aibkfW

ab+pp or hh	

(3)

8 (1)

ph	(2)

fai ijal )Non-Perturbative feature: Schematic	

The flow equation can essentially be seen as	Γ[n+1]=	+	+	

Γ[n]	With the initial condition	Γ[0]=	=: V(bare two-body coupling)	

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Non-Perturbativeness of IM-SRG: Schematic	+	Γ[1]=	

Γ[2]=	Solving the flow equation step by step	+	V	Correlations to all order	+	+	+	+	+	+	+	+	+	

10+	+...	

IM-SRG(2), 95%	Flow equation by perturbative analysis	

Ė0 (s) = [ (2) , ][2] +[ (1) , f ][4] + [ (3) , W ][4]IM-SRG(3), 99%	f˙(s) = [ (1) , f ][2] + [ (2) , ][2] +[ (1) , ][3] + [ (2) , f ][3] + [ (2) , W ][3] + [ (3) , ][3] +[ (3) , W ][4]˙ (s) = [ (2) , f ][1] + [ (2) , ][2] +[ (1) , ][3] + [ (2) , W ][3] + [ (3) , ][3]+[ (1) , W ][4] + [ (3) , f ][4] + [ (3) , W ][4]Ẇ (s) = [ (3) , f ][2] + [ (2) , ][2] +[ (2) , W ][3] + [ (3) , ][3] +[ (1) , W ][4] + [ (3) , f ][4]• IM-SRG(2): 3rd-order exact for GS energy and 2nd-order exact for Veff.	

• IM-SRG(3): 4th-order exact for GS energy and 3rd-order exact for Veff.	

• IM-SRG is controlled and improvable method	

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Numerical calculations	

N3LO (Λ=500MeV) from χEFT Entem-Machleidt, PRC 68, 041001(R) (2003)	

Free-space SRG evolved version Vsrg (λ) Bogner-Perry-Furnstahl, PRC75, 061001 (2008)	

N=emax	

N=2	

N=1	

N=0	

N=emax	

N=2	

N=1	

N=0	12emax	

# SP	dim (f)	4	5	6	7	8	10	30	

42	56	

72	

90	

130	

3.4×104	

1.5×105	

4.7×105	

1.4×106	

3.5×106	

2.0×107	

4He with two different generators	

Input Vsrg λ=2.0fm-1	

95%
99%

•Truncation up to normal-ordered 2-body level is a good approximation. 	

13CCSD	Λ-CCSD(T)	

•Agrees well with CCSD (95% of correlation)	

•MBPT(2,3) break down	•IM-SRG(2) work for 16O and 40Ca	14hhpppp

hhpp

hhhhpps=0 MeV-2-3s=10 MeV-2

Hod gets suppressed.	

15-1

s=10 MeV-2

Evolution of operators	

Arbitrary operator evolved on equal footing	dH(s) = [ , H(s)]dsdOr (s) = [ , Or (s)]dsE.g., RMS radius	Or (s) = Or(0) (s) + Or(1) (s) + Or(2) (s) · · ·1Or (0) ⇥(ri Rcm )2A ir =| Or (0) | = lims(0)Or (s)

Joint benchmark is ongoing (NCFC, IT-NCSM,CCM,MBGF,UMOA,IM-SRG)	

Results agree within uncertainty
Next step from H. Kamada et al., PRC64:044001(2001)16

Ground-state convergence in 6Li (4He+”2” vs 6)	

Effective 2-body problem

176-body problem

6Li Spectra: IM-SRG vs NCSM	

Effective 2-body problem

Works for 18-body as well186-body problem

Summary	Summary	" We introduced SRG evolution of Hamiltonianin many-body medium (IM-SRG).	

" We numerically demonstrated the features of in-medium SRG.	

" Decoupling of a Hamiltonian, Size-extensivity, Non-perturbative feature.	

" Radius (arbitral operators can be evolved).	

" Contamination of center of mass excitation is very small.	

" Shell-model effective interactions for valence nucleons (p and sd).	

Work in Progress	

" Derivation of effective operator.	

" Systematic improvement; 3-body flow equations.	

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