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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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eH ⇡ E0 + eH(1) + eH(2) + eH(3)<latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit><latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit><latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit>
Unitary-Model-Operator Approach (UMOA)
✦ Cluster expansion
9
K. Suzuki and R. Okamoto, PTP 92, 1045 (1994). TM et al., PRC 96, 054312 (2017).
eH ⇡ E0 + eH(1) + eH(2) + eH(3)<latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit><latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit><latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit>
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
Unitary-Model-Operator Approach (UMOA)
✦ Actual calculation procedure
10
K. Suzuki and R. Okamoto, PTP 92, 1045 (1994). TM et al., PRC 96, 054312 (2017).
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.
UMOA - definition of P and Q✦ Choice of P and Q is crucial in actual calculations. ★ P and Q for S(1)
★ P and Q for S(2)
✦ For any S(n), P and Q can be determined systematically. 23
P (1) =X
aFermi level
|aiha|
Q(1) =X
a>Fermi level
|aiha|
Q(2) =1
2
X
ab>Fermi level
|abihab|
P (2) =1
2
X
abFermi level
|abihab|
Unitary-Model-Operator Approach (UMOA)
✦ Cluster expansion
24
K. Suzuki and R. Okamoto, PTP 92, 1045 (1994). TM et al., PRC 96, 054312 (2017).
eH ⇡ E0 + eH(1) + eH(2) + eH(3)<latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit><latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit><latexit sha1_base64="zpLrPQLCVCEfDpOuQod+ZUHf9HU=">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</latexit>
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
Unitary-Model-Operator Approach (UMOA)
✦ Perturbative evaluation for three-body term
25
K. Suzuki and R. Okamoto, PTP 92, 1045 (1994). TM et al., PRC 96, 054312 (2017).
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)