recent progress in the unitary-model-operator approach · recent progress in the...

Recent progress in the unitary-model-operator approach

Takayuki Miyagi Center for Nuclear Study, the University of Tokyo

Collaborators: T. Abe, M. Kohno, P. Navratil, R. Okamoto, T. Otsuka, N. Shimizu, and S. R. Stroberg

1

　　 Mar. 1, 2018

Introduction✦ One of the fundamental problems is to understand the nuclear structure and reactions based on underlying nuclear interaction.

Nuclear interaction

many-body problem

✦ Nuclear interaction：progress in chiral EFT ★ High-precision NN force ★ 3N force consistent with NN force

✦ Many-body problem：Ab initio calculation methods

major issues

Nuclear force from chiral EFT✦ Chiral effective field theory

3

Weinberg, van Kolck, Kaiser, Epelbaum, Glöckle, Meißner, Entem, Machleidt, …

SRG evolution - 1✦ Bare nuclear interactions are too “hard” to be applied for usual many-body methods.

✦ We usually need the “soft” interactions that provides the rapid convergence with respect to the model-space size.

4

S. K. Bogner, R. J. Furnstahl, and R. J. Perry, PRC 75, 061001 (2007).

dH(s)

ds= [⌘(s), H(s)] H(s) = U

†(s)HU(s)

dU(s)

ds= �⌘(s)U(s) s: resolution scale in unit of fm4

λ: momentum scale s-1/4

SRG evolution - 2✦ 2-body SRG flow (1S0 channel)

✦ 3-body SRG flow (JPT = 1/2+1/2)

5

λ=∞ fm-1 λ=2 fm-1 λ=1 fm-1

Normal Ordering Approximation✦ Explicit treatment of 3BF is difficult

✦ Rearrangement of 3BF w.r.t the ref. state

6

R. Roth et al., PRL 109 (2012). S. Binder et al., PRC 87 (2013). …

V3N =1

36

X

a1a2a3b1b2b3

V a1a2a3b1b2b3

Aa1a2a3b1b2b3

, V a1a2a3b1b2b3

= ha1a2a3|V3N |b1b2b3i, Aa1a2···b1b2··· = c†a1

c†a2· · · cb2cb1

Aa1a2···b1b2··· |0i = 0 |0> : nucleon vacuum

V3N = W +X

a1b1

W a1b1

eAa1b1

+1

4

X

a1a2b1b2

W a1a2b1b2

eAa1a2b1b2

+1

36

X

a1a2a3b1b2b3

W a1a2a3b1b2b3

eAa1a2a3b1b2b3

Important less importanteAa1a2···b1b2··· |�i = 0 |Φ>: reference state

If the reference state is sufficiently close to the true ground state, the effect of the residual 3BF should be small. HF reference state is employed in practical applications. Discard the residual 3BF term => NO2B approximation

eAa1a2···b1b2··· = Aa1a2···

b1b2··· � (all contracted terms)

Many-body methods✦ Ab initio calculation methods related with this work

✴ Few-body system (Faddeev equation, Faddeev-Yakubovsky equation, etc.)

✴ Green’s Function Monte Carlo

✴ No-Core Shell Model

✴ Nuclear Lattice Effective Field Theory

✴ Coupled-Cluster Method

✴ Self-Consistent Green’s Function Method

✴ In-Medium Similarity Renormalization Group Approach

✴ Unitary-Model-Operator Approach (UMOA)

✴ …

7

Unitary-Model-Operator Approach (UMOA)

✦ Many-body Schrödinger equation

✦ Unitary operator U

✦ Cluster expansion

8

H| i = E| i

eH|�i = E|�i

(U†HU)(U †| i) = E(U†| i)

K. Suzuki and R. Okamoto, PTP 92, 1045 (1994). TM et al., PRC 96, 054312 (2017).

U = eS(1)

eS(2)

S(1) =AX

i

si, S(2) =AX

i<j

sij

Uncorrelated single-Slater determinant

0p0h 1p1h 2p2h 0p0h 1p1h 2p2h

( )perturbative treatment

H = E0 +AX

i

ti +AX

i<j

vij

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9



eH(1) =X

i

ehi; ehi = e�si(ti + wi)e

si

eH(2) =X

i<j

evij �X

i

ewi

evij = e�sije�(si+sj)(ti + wi + tj + wj + vij)esi+sjesij � (ehi + ehj)

ewi = e�siwiesi

eH(3) =X

i<j<k

evijk �X

i<j

ewij

( )

ha| ew|bi =X

c⇢F

hac|ev|bci


✦ Actual calculation procedure

10


Input wi Determine si (OLS in 1B space)

Determine sij (OLS in 2B space)

Get ~wi (Normal Ordering)

New wi

Observables

Calculation setup★ Interaction:

✤ NN: chiral N3LO (Λ = 500 MeV/c)

✤ 3N: chiral N2LO local form (Λ = 400 MeV/c)

★ NO2B approximation through the Hartree-Fock calculations

✤ 3-body matrix elements e3max = 14.

★ UMOA calculations are done with the emax truncation.

11

e3max = max(2n1+l1+2n2+l2+2n3+l3)

D. R. Entem and R. Machleidt, Phys. Rev. C C68, 041001 (2003).

R. Roth, et al., Phys. Rev. Lett. 109, 052501 (2012).

emax = max(2n+l)

Convergence (NN+3N-full λ=1.88 fm-1)

12

CCSD: S. Binder, J. Langhammer, A. Calci, P. Navrátil, and R. Roth, Phys. Rev. C 87, 021303 (2013). IM-SRG: H. Hergert, S. K. Bogner, S. Binder, A. Calci, J. Langhammer, R. Roth, and A. Schwenk, Phys. Rev. C 87, 034307 (2013).

Converged results are found.

UMOA results are close to the CCSD and IM-SRG results.

Accuracy of UMOA calc.

13

E0 <H(1)> <H(2)> <H(3)> Etot |<H(3)>/Etot|

4He -0.40 -107.15 80.46 -0.71 -27.81 0.03

16O 32.42 -514.12 357.59 -3.04 -127.16 0.0240Ca 173.85 -1440.6 905.42 -7.08 -368.44 0.02

eH ⇡ E0 + eH(1) + eH(2) + eH(3).

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perturbative estimation

Cluster expansion works well <H(3)>/Etot can be thought as the error of UMOA calc.

@ emax = 14, hw = 25 MeV, NN+3N-full (λ=1.88 fm-1)

λ-dependence

14

Ground-state energies for oxygen isotopes

15

@emax=14 hw = 25 MeV

NN+3N-ind

NN+3N-full

Consistent with recent ab initio results

λ=1.88 fm-1λ=2.0 fm-1λ=2.24 fm-1Exp.

Exp.: AME2012

Root-mean-squared radii

16

Converged results are found.

Results are significantly smaller than experimental data.

Experimental data are estimated from charge radii.

SRG evolution of radius operator

17

Hamiltonian

Radius

SRG flow

3-body SRG flow

SRG evolution of radius operator

18

Reduction of λ dependence M. D. Schuster, et al., Phys. Rev. C 90, 011301 (2014). Radii are still small

NN+3N-full, emax=14, hw=25 MeV

Summary✦ UMOA works well similar to the other ab initio calculation methods.

✦ Radii are still small even if the 3NF effect is taken into account.

✦ SRG evolution gives minor modification for the radius operator.

✦ Extension of the UMOA framework for direct treatment of 3NF.

19

Future works

Backup

20

SRG evolution - 3✦ SRG transformation induces many-body forces

21

2NF 2NF 2NF 3NF

SRG SRG SRG

2NF 2NF 2NF 3NF3NF

NN-only NN+3N-ind NN+3N-full

E. D. Jurgenson, P. Navratil, and R. J. Furnstahl, PRL 103, 082501 (2009). E. D. Jurgenson, P. Navratil, and R. J. Furnstahl, PRC 83, 034301 (2011).

…

3NF

✦ Procedure for S ✴ Solve eigenvalue problem

✴ P and Q are the projection operator ✴ Decomposition of eigenvector into P and Q component

✴ Formal solution of ω

✦ Decoupling condition is satisfied

UMOA - determination of S

22

(P +Q)H(P +Q)| ki = ✏k| ki

|�ki = P | ki, !|�ki = Q| ki

! =dX

k=1

Q| kihe�k|P

S = arctanh(! � !†)

Q eHP = P eHQ = 0

eH = e�S

HeS

H

P Q P Q

eH = e�S

HeS

Okubo, PTP 1954. Lee and Suzuki, PLB 1980.



24



eH(1) =X

i

ehi; ehi = e�si(ti + wi)e

si

eH(2) =X

i<j

evij �X

i

ewi

evij = e�sije�(si+sj)(ti + wi + tj + wj + vij)esi+sjesij � (ehi + ehj)

ewi = e�siwiesi

eH(3) =X

i<j<k

evijk �X

i<j

ewij

evijk = e�(sij+sjk+ski)e�(si+sj+sk)(ti + wi + tj + wj + tk + wk + vij + vjk + vki)esi+sj+skesij+sjk+ski

� (ehi + ehj + ehk + evij + evjk + evki)

ewij = e�sije�(si+sj)(wi + wj)esi+sjesij


✦ Perturbative evaluation for three-body term

25


h eH(3)i ⇡ 1

4

X

ab>⇢F

X

ijkl⇢F

hij|ev|klihkl|s|abihij|s|abi

+X

abc>⇢F

X

ijk⇢F

hia|ev|jbihkj|s|caihik|s|cbi

Radii for oxygen isotopes

26

λ=1.88 fm-1λ=2.0 fm-1λ=2.24 fm-1

Radii are significantly small

NN+3N-ind

NN+3N-full

Exp.

Exp.: V. Lapoux, et al, Phys. Rev. Lett. 117, 052501 (2016).

@emax=14 hw = 25 MeV

Extension of UMOA Framework

✦ Ground-state energy for 4He (NN-only @ λ=2 fm-1)

27

more accurateless accurate

U ⇡ eS(1)

eS(2)

eH ⇡ eH(1) + eH(2)

U ⇡ eS(1)

eS(2)

eH ⇡ eH(1) + eH(2) + ( eH(3))

U ⇡ eS(1)

eS(2)

eH ⇡ eH(1) + eH(2) + eH(3)

U ⇡ eS(1)

eS(2)

eS(3)

eH ⇡ eH(1) + eH(2) + eH(3)

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