Applications of Renormalization Group Methods in Nuclear Physics 5 - - PowerPoint PPT Presentation
Applications of Renormalization Group Methods in Nuclear Physics 5 Dick Furnstahl Department of Physics Ohio State University HUGS 2014 Outline: Lecture 5 Lecture 5: New methods and IM-SRG in detail New methods with some applications
Dick Furnstahl
Department of Physics Ohio State University
HUGS 2014
Lecture 5: New methods and IM-SRG in detail
New methods with some applications In-Medium Similarity Renormalization Group
Lecture 5: New methods and IM-SRG in detail
New methods with some applications In-Medium Similarity Renormalization Group
NUclear Computational Low Energy Initiative Collaboration of physicists, applied mathematicians, and computer scientists = ⇒ builds on UNEDF project [unedf.org] US funding but many international collaborators See computingnuclei.org for highlights!
Validated ¡Nuclear ¡ Interac/ons ¡ Structure ¡and ¡Reac/ons: ¡ Light ¡and ¡Medium ¡Nuclei ¡ Structure ¡and ¡Reac/ons: ¡ Heavy ¡Nuclei ¡
Chiral ¡EFT ¡ Ab-‑ini/o ¡ Op/miza/on ¡ Model ¡valida/on ¡ Uncertainty ¡Quan/fica/on ¡
Neutron ¡Stars ¡ Neutrinos ¡and ¡ Fundamental ¡Symmetries ¡
Ab-‑ini/o ¡ RGM ¡ CI ¡ Load ¡balancing ¡ Eigensolvers ¡ Nonlinear ¡solvers ¡ Model ¡valida/on ¡ Uncertainty ¡Quan/fica/on ¡ ¡ ¡ DFT ¡ TDDFT ¡ Load ¡balancing ¡ Op/miza/on ¡ Model ¡valida/on ¡ Uncertainty ¡Quan/fica/on ¡ Eigensolvers ¡ Nonlinear ¡solvers ¡ Mul/resolu/on ¡analysis ¡ ¡
Stellar ¡burning ¡ fusion ¡
Neutron ¡drops ¡ ¡ EOS ¡ Correla/ons ¡
Fission ¡
Explosion of many-body methods using microscopic input
Ab initio (new and enhanced methods; microscopic NN+3NF) Stochastic: GFMC/AFDMC (new: with local EFT); lattice EFT Diagonalization: IT-NCSM Coupled cluster (CCSD(T), CR-CC(2,3), Bogoliubov, . . . ) IM-SRG (In-medium similarity renormalization group) Self-consistent Green’s function Many-body perturbation theory Shell model (usual: empirical inputs) Effective interactions from coupled cluster, IM-SRG Density functional theory Microscopic input, e.g., through density matrix expansion
Do we really need all of these methods?
Compare to lattice QCD: Are all the different lattice actions needed?
clover quarks on anisotropic lattices (mass spectrum) domain wall quarks (chiral symmetry) highly improved staggered quarks (high-precision extrapolations) and more!
Answer: yes! Complementary strengths Cross-check results Identify theory error bars
A frame from an animation illustrating the typical four-dimensional structure of gluon-field configurations used in describing the vacuum properties of QCD.
Oxygen chain with 3 methods [from H. Hergert et al. (2013)]
12 14 16 18 20 22 24 26 175 150 125 100 75 50 25
A E MeV⇥ AO E3 Max⇥14
⇤⇥1.9 fm1
IMSRG
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
12 14 16 18 20 22 24 26 175 150 125 100 75 50 25
A E MeV⇥ AO E3 Max⇥14
⇤⇥1.9 fm1
IMSRG ⇥ ITNCSM
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
⇤ ⇤ ⇤ ⇤
10 12 14 16 18 20 22 24 26 175 150 125 100 75 50 25
A E MeV⇥ AO E3 Max⇥14
⇤⇥1.9 fm1
IMSRG ⇥ ITNCSM
⇤
12 14 16 18 20 22 24 26 175 150 125 100 75 50 25
A E MeV⇥ AO E3 Max⇥14
⇤⇥1.9 fm1
IMSRG
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
12 14 16 18 20 22 24 26 175 150 125 100 75 50 25
A E MeV⇥ AO E3 Max⇥14
⇤⇥1.9 fm1
IMSRG ⇥ ITNCSM
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
⇤ ⇤ ⇤ ⇤
10 12 14 16 18 20 22 24 26 175 150 125 100 75 50 25
A E MeV⇥ AO E3 Max⇥14
⇤⇥1.9 fm1
IMSRG ⇥ ITNCSM
⇤
NN + 3N-full (400) NN + 3N-ind. exp.
Hergert&et&al.,&PRL&110,&242501&(2013)&&
&
In-medium SRG, importance-truncated NCSM, coupled cluster Same Hamiltonian = ⇒ test for consistency between methods Impact of three-nucleon force (3NF) on dripline Need precision experiment and theory
Hoyle state from lattice chiral EFT [E. Epelbaum et al.]
Triple-α resonance in 12C Low-resolution (coarse) lattice Suited to adjust to clusters Order-by-order improvement: LO = ⇒ NLO = ⇒ N2LO
[Also high-precision GFMC!]
n L ∼ 10 . . . 20 fm
a ∼ 1 . . . 2 fm Experiment NNLO [O(Q3)] IB + EM [O(Q2)] NLO [O(Q2)] LO [O(Q0)]
E (MeV) 0+
1
0+
2
2+
1, Jz = 0
2+
1, Jz = 2
Hoyle state from lattice chiral EFT [E. Epelbaum et al.]
Triple-α resonance in 12C Low-resolution (coarse) lattice Suited to adjust to clusters Order-by-order improvement: LO = ⇒ NLO = ⇒ N2LO
[Also high-precision GFMC!]
n L ∼ 10 . . . 20 fm
a ∼ 1 . . . 2 fm
“Survi
0 0.02 0.04 0.06 0.08 0.1 0.12 t (MeV-1) LO [A] LO [B] LO [!]
0.12
0 0.02 0.04 0.06 0.08 0.1 0.12 t (MeV-1) LO [C] LO [D] LO ["] E(t) (MeV)
Probing α cluster structure of 0+ states How does the triple-α reaction rate depend
Much more! Most recent:
16O structure
Spectral functions from self-consistent Green’s function
[figures from C. Barbieri]
neutron+ removal neutron+ addi2on sca5ering
56Ni ectral!func.on):
rth:Jensen,!Pys.!Rev.!C79,!064313!(2009);!CB,!Phys.!Rev.!LeI.!103,!202502!(2009)]!
Sh
ab(!) = 1
π Im gab(!)
One-body Greens function (or propagator) gab(ω) describes the motion of quasi-particles and quasi-holes Contains all the structure information probed by nucleon transfer Imaginary (absorptive) part of gab(ω) is the spectral function
A+1 A:1
E
Confronting theory and experiment to both driplines
Precision mass measurements test impact of chiral 3NF Proton rich [Holt et al. (2012)] Neutron rich [Gallant et al. (2012)] Many new tests possible!
AN collaboration + Holt, Menendez, Schwenk, submitted.
16 17 18 19 20 21 22 23 24 Mass Number A
Ground-State Energy (MeV)
NN NN+3N NN+3N (sdf7/2p3/2) AME2011 IMME
N=8
Shell model description using chiral potential evolved to Vlow k plus 3NF fit to A = 3, 4 Excitations outside valence space included in 3rd order MBPT
Confronting theory and experiment to both driplines
Precision mass measurements test impact of chiral 3NF Proton rich [Holt et al. (2012)] Neutron rich [Gallant et al. (2012)] Many new tests possible!
AN collaboration + Holt, Menendez, Schwenk, submitted.
40 41 42 43 44 45 46 47 48 Mass Number A
Ground-State Energy (MeV)
NN NN+3N NN+3N (pfg9/2) Exp and AME2011 extrapolation IMME
N=20
Shell model description using chiral potential evolved to Vlow k plus 3NF fit to A = 3, 4 Excitations outside valence space included in 3rd order MBPT
Solving the Nuclear Many-Body Problem
Assume filled core Active nucleons occupy valence space
Interaction and energies of valence space orbitals from original Vlow k This alone does not reproduce experimental data
0s 0p 0f,1p 0g,1d,2s 0h,1f,2p
Nuclei understood as many-body system starting from closed shell, add nucleons
Solving the Nuclear Many-Body Problem
Assume filled core Active nucleons occupy valence space
Interaction and energies of valence space orbitals from original Vlow k This alone does not reproduce experimental data – allow explicit breaking of core Strong interactions with core generate effective interaction between valence nucleons
Hjorth-Jensen, Kuo, Osnes (1995)
0s 0p 0f,1p 0g,1d,2s 0h,1f,2p
Nuclei understood as many-body system starting from closed shell, add nucleons
Solving the Nuclear Many-Body Problem
Assume filled core Active nucleons occupy valence space
Interaction and energies of valence space orbitals from original Vlow k This alone does not reproduce experimental data – allow explicit breaking of core Strong interactions with core generate effective interaction between valence nucleons
Hjorth-Jensen, Kuo, Osnes (1995)
Effective two-body matrix elements Single-particle energies (SPEs)
0s 0p 0f,1p 0g,1d,2s 0h,1f,2p
Nuclei understood as many-body system starting from closed shell, add nucleons
Normal-ordered 3N: contribution to valence neutron interactions
3N Forces for Valence-Shell Theories
O core
16
O core
16
Effective two-body Effective one-body
Combine with microscopic NN: eliminate empirical adjustments
ab V3N,eff a'b' = "ab
" =core
V3N "a'b'
a V3N,eff a' = 1 2 "#a
"# =core
V3N "#a'
GFMC: Calculating observables in light nuclei
Green’s Function Monte Carlo (GFMC) energies are accurate but lowest-order theory of one-body currents (blue) disagrees with experiment (black) Including two-nucleon currents based
Magne&c(Moments(
Electromagne,c-Transi,ons-
Note: not fully consistent yet!
Combining structure and reactions [P. Navratil et al.]
Resonating Group Method + NCSM:
r r
10 100 1000
Ekin [keV]
5 10 15 20 25 30 35
S-factor [MeV b]
BR51 AR52 CO52 AR54 He55 GA56 BA57 GO61 KO66 MC73 MA75 JA84 BR87 d+9d*+5d'*
3H(d,n) 4He
SRG-N
3LO Λ=1.45 fm
Potential to address unresolved fusion research related questions:
3H(d,n)4He fusion with polarized deuterium and/or tritium, 3H(d,n)4He bremsstrahlung,
Electron screening at very low energies …
P.N., S. Quaglioni, PRL 108, 042503 (2012)
10 100 1000
Ekin [keV]
5 10 15 20
S-factor [MeV b]
Bo52 Kr87 Sch89 Ge99 Al01 Al01 Co05 0d* 1d*+1d'* 3d*+3d'* 5d*+5d'* 7d*+5d'* 9d*+5d'*
d+
3He → p+ 4He
(b)
Ab initio fusion! In progress: SRG-evolved NNN interactions
Combining structure and reactions [P. Navratil et al.]
1, 5/2- 2
7Be(p8B radiative capture
7Be
p
The first ever ab initio calculations of 7Be(p8B
16 8B 2+ g.s. bound by
136 keV (expt. 137 keV) S(0) ~ 19.4(0.7) eV b Data evaluation: S(0)=20.8(2.1) eV b arXiv:1105.5977 [nucl-th]
Lecture 5: New methods and IM-SRG in detail
New methods with some applications In-Medium Similarity Renormalization Group
Choose a basis and a reference state |Φ0
The basis could be harmonic oscillators or Hartree-Fock or . . . Anti-symmetric wave functions: A-particle Slater determinants Use second-quantization formalism: creation/destruction
a†
p1ah1|Φ0
a†
p2a† p1ah2ah1|Φ0
The reference state is filled, so no particles or holes: 0p–0h If one particle moved to a higher level, leaves hole behind: 1p–1h Complete basis: Slater determinants from all 1p–1h, 2p–2h, . . .
[slides from H. Hergert]
Consider SRG with 0p–0h reference state (instead of vacuum)
reference state and 1p–1h, 2p–2h basis states
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of correlations into zeroth order E0!
0p-0h 1p-1h 2p-2h 3p-3h 3p-3h 2p-2h 1p-1h 0p-0h
A new ab-initio structure method that can be applied directly and to generate shell-model effective interactions!
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG decoupling for 40Ca
[slides from H. Hergert]
IM-SRG: decouples reference state (0p–0h) from excitations
= ⇒ Resummation of MBPT correlations into zeroth order E0!
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥
10⇥4 10⇥3 10⇥2 10⇥1 100 101 ⇥600 ⇥580 ⇥560 ⇥540 ⇥520
s E MeV⇥ 40Ca
E EMBPT⇤2⌅
(), λ = . −, =
IM-SRG equations: Flow equations
IM-SRG equations: Flow equation 2-body Flow
＋ ＋ ＋ － s channel t channel u channel ladders rings
IM-SRG iteration: Nonperturbative resummation of MBPT
, 02/19/2014
(δ) ∼ (δ) ∼
IM-SRG iteration: Nonperturbative resummation of MBPT
, 02/19/2014
& many more... (δ) ∼ (δ) ∼
IM-SRG results for closed-shell nuclei
[slides from H. Hergert]
He4 O16 O24 Ca40 Ca48 Ni48 Ni56 9 8 7 6
E⇤A MeV⇥
⇥fm1⇥ ⇤ 2.2 2.0 1.9
⌅ ⇥
NN + 3N-ind.
experiment
IM-SRG results for closed-shell nuclei
[slides from H. Hergert]
He4 O16 O24 Ca40 Ca48 Ni48 Ni56 9 8 7 6
E⇤A MeV⇥
⇥fm1⇥ ⇤ 2.2 2.0 1.9
⌅ ⇥
NN + 3N-ind.
He4 O16 O24 Ca40 Ca48 Ni48 Ni56 10 9 8 7 6
E⇤A MeV⇥
⇥fm1⇥ 2.2 2.0 1.9
⇥
NN + 3N-full (400) experiment
IM-SRG results for closed-shell nuclei
[slides from H. Hergert]
He4 O16 O24 Ca40 Ca48 Ni48 Ni56 9 8 7 6
E⇤A MeV⇥
⇥fm1⇥ ⇤ 2.2 2.0 1.9
⌅ ⇥
NN + 3N-ind.
He4 O16 O24 Ca40 Ca48 Ni48 Ni56 10 9 8 7 6
E⇤A MeV⇥
⇥fm1⇥ 2.2 2.0 1.9
⇥
NN + 3N-full (400) experiment
Multi-reference IM-SRG results for Oxygen chains
Reference state: number-projected Hartree-Fock-Bogoliubov vacuum (pairing correlations)
12 14 16 18 20 22 24 26 175 150 125 100 75 50 25
A E MeV⇥ AO E3 Max⇥14
⇤⇥1.9 fm1
IMSRG
12 14 16 18 20 22 24 26 175 150 125 100 75 50 25
A E MeV⇥ AO E3 Max⇥14
⇤⇥1.9 fm1
IMSRG
NN + 3N-ind. exp.
Multi-reference IM-SRG results for Oxygen chains
Variation of initial 3N cutoff only Diagnostics for chiral EFT interactions Dripline at A = 24 is robust under variations
⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤
⇧ ⇧ ⇧ ⇧ ⇧ ⇧ ⇧
12 14 16 18 20 22 24 26 175 150 125 100 75 50
A E MeV⇥
AO
⇥3NMeV⇥ ⇤SRGfm1⇥ 350 1.9 ... 2.2 400 450
⇤⇤⌅ ⇤⇥ ⇧⇤⌃
NN + 3N-full (400)
IM-SRG results for Calcium and Nickel chains [preliminary]
Reference state: number-projected Hartree-Fock-Bogoliubov vacuum (pairing correlations)
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ 34 36 38 40 42 44 46 48 50 52 54 56 58 60 500 450 400 350 300 250
A E MeV⇥
ACa
⇥fm1⇥ 2.2 1.9
MRIMSRG
⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤
⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅
⇥ ⇥ ⇥ ⇥ ⇧ ⇧ ⇧ ⇧ ⇧ ⌃ ⌃ ⌃ ⌃ ⌃ 34 36 38 40 42 44 46 48 50 52 54 56 58 60 500 450 400 350 300 250
A E MeV⇥
ACa
⇥fm1⇥ 2.2 1.9 2.2 1.9 2.2 1.9
⇤ ⌅
⇧ ⌃ MRIMSRG CCSD CRCC⇤2,3⌅
NN + 3N-full (400) exp.
⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤ ⇤
⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅ ⌅
⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇧ ⇧ ⇧ ⇧ ⇧ ⇧ ⇧ ⌃ ⌃ ⌃ ⌃ ⌃ ⌃ ⌃ 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 700 600 500 400
A E MeV⇥
ANi
⇥fm1⇥ 2.2 1.9 2.2 1.9 2.2 1.9
⇤ ⌅
⇧ ⌃ MRIMSRG CCSD CRCC⇤2,3⌅
NN + 3N-full (400)
= , =
IM-SRG valence-space decoupling
[slides from H. Hergert]
particle states hole states (core) non-valence particle states
IM-SRG valence-space decoupling
[slides from H. Hergert]
IM-SRG shell-model effective interaction
[slides from H. Hergert]
MBPT NN+3N IM-SRG NN IM-SRG NN+3N Expt. 1 2 3 4 5 6 Energy (MeV) 23O
1/2
+
5/2
+
5/2
+
1/2
+
3/2
+
3/2
+
1/2
+
5/2
+
3/2
+
(5/2
+)
1/2
+
(3/2
+)
MBPT NN+3N IM-SRG NN IM-SRG NN+3N Expt. 1 2 3 4 5 6 7 8 Energy (MeV) 22O
+
2
+
2
+
2
+ + +
(2
+)
2
+
2
+
(0
+) + +
4
+
4
+
(4
+)
4
+
2
+ +
2
+ +
3
+
3
+
3
+
3
+ +
ℏΩ variation
arXiv: 1402.1407 [nucl-th], [figures by J. Holt]
3N forces crucial IM-SRG improves on finite-order MBPT effective interaction Competitive with phenomenological calculations
IM-SRG shell-model effective interaction
[preliminary!]
MBPT USDb IM-SRG NN+3N-full Expt.
1 2 3 4 5
Energy (MeV)
+ +
2
+
4
+ +
2
+
2
+
2
+
2
+ + + + +
4
+
2
+
2
+
4
+
2
+
26Ne MBPT USDb IM-SRG NN+3N-full Expt. 1 2 3 4
Energy (MeV)
5/2
+
9/2
+
5/2
+
5/2
+
9/2
+
7/2
+
5/2
+
5/2
+
3/2
+
5/2
+
3/2
+
3/2
+
3/2
+
1/2
+
1/2
+
1/2
+
(1/2
+)
7/2
+
3/2
+
7/2
+
1/2
+
3/2
+
(5/2
+)
(3/2
+)
25Ne MBPT USDb IM-SRG NN+3N-full Expt. 1 2 3 4 5
Energy (MeV)
5/2
+
5/2
+
1/2
+
5/2
+
7/2
+
7/2
+
3/2
+
1/2
+
1/2
+
9/2
+
5/2
+
9/2
+
3/2
+
1/2
+
5/2
+
5/2
+
(5/2
+)
9/2
+
1/2
+
3/2
+
3/2
+
3/2
+
(1/2
+)
(9/2
+)
(3/2
+)
(3/2
+)
(5/2
+)
25F MBPT USDb IM-SRG NN+3N-full Expt. 1 2 3 4
Energy (MeV)
3
+
4
+
1
+
4
+
4
+
3
+
2
+
2
+
2
+
1
+
1
+
1
+
1
+
3
+
4
+
2
+
2
+
1
+
3
+
2
+
3
+
1
+
26F
IM-SRG normal ordering
[slides from H. Hergert]
, 02/19/2014
Normal-Ordered Hamiltonian
= +
: +
: +
:
E0 = + + f = + + Γ = + W =
IM-SRG normal ordering
[slides from H. Hergert]
, 02/19/2014
Normal-Ordered Hamiltonian
= +
: +
: +
:
E0 = + + f = + + Γ = + W =
IM-SRG normal ordering
[slides from H. Hergert]
, 02/19/2014
Normal-Ordered Hamiltonian
= +
: +
: +
:
E0 = + + two-body formalism with in-medium contributions from three-body interactions f = + + Γ = + W =
IM-SRG normal ordering
[slides from H. Hergert]
, 02/19/2014
Normal-Ordered Hamiltonian
= +
: +
: +
:
E0 = + + two-body formalism with in-medium contributions from three-body interactions f = + + Γ = + W =
IM-SRG equations: Choice of generator
:: :
:: :
≡ + , ≡
: + ,
≡
: +