Ab initio effective interactions and operators from IM-SRG

Size: px

Start display at page:

Download "Ab initio effective interactions and operators from IM-SRG"

Isabel Boyd
5 years ago
Views:

Canada s national laboratory for particle and nuclear physics

nucleaire et en physique des particules U = eη b initio

Ragnar Stroberg TRIUMF Double Beta Decay Workshop U OU = O +

1 Canada s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucleaire et en physique des particules U = eη b initio effective interactions and operators from IM-SRG dh ds = [η, H] Ragnar Stroberg TRIUMF Double Beta Decay Workshop U OU = O + [η, O] +... Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

2 Outline Conceptual introduction to valence space IM-SRG Targeted normal ordering Ensemble reference states Effective valence space operators Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

3 Introduction Starting point: non-relativistic Schrödinger equation with nucleons as our degrees of freedom. H Ψ = E Ψ We don t know this This is hard to solve Effective theory H is scheme and scale dependent. Strongly-interacting system highly correlated hard to solve. The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. Paul Dirac, 1929 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

4 Many-body approaches Microscopic NCSM, GFMC, etc Use realistic H, solve directly Works well for light systems Operators treated consistently Basis dimension grows rapidly Phenomenological SM, RP, IBM, DFT, etc. Make the problem tractable Missing physics adjust H Much larger reach in How to adjust other operators? Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

5 Many-body approaches Microscopic NCSM, GFMC, etc Use realistic H, solve directly Works well for light systems Operators treated consistently Basis dimension grows rapidly Phenomenological SM, RP, IBM, DFT, etc. Make the problem tractable Missing physics adjust H Much larger reach in How to adjust other operators? Microscopic/Effective Lee-Suzuki, MBPT, IM-SRG Systematically treat missing physics Consistently transform other operators Does the expansion converge? Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

6 IM-SRG Effective Interaction Goal: Find a unitary transformation U such that H = UHU P H Q = Q H P = 0 Ψ i ˆP H ˆP Ψ i = Ψ i H Ψ i Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

7 IM-SRG U may always be written as U = e η, for some generator η ( ) 0 θ For two-level system, η = θ 0 ( ) For our Hamiltonian, take η = 1 2 atan 2Hod h.c. Perform multiple rotations: U N = e η N... e η 2 e η 1 Iterate until η N = 0 Infinitessimal rotation of angle ds dh(s) ds = [η(s), H(s)] White 2002; Tsukiyama, Bogner, and Schwenk 2011; Morris, Parzuchowski, and Bogner 2015 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

8 IM-SRG Why In-Medium? To deal with the problem of induced many-body forces e η = 1 + η + 1 2! η = ll terms beyond two-body operators are too expensive to handle Define states with respect to a reference Φ 0 (Normal Ordering) If Φ 0 is a reasonable approximation of Ψ, then many-body terms are less important Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

9 Valence space IM-SRG Excluded configurations treated with IM-SRG (definition of H od ) Valence configurations treated explicitly with standard shell model diagonalization In following, all calculations use SRG evolved E&M N 3 LO NN + local N 2 LO 3N (kindly provided by ngelo Calci) core valence excluded decouple decouple Entem and Machleidt 2003; Navrátil 2007; Tsukiyama, Bogner, and Schwenk 2012 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

10 Valence space IM-SRG: Ground states (c) O (Z=8) Bogner et al Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

11 Valence space IM-SRG: Ground states MR-IMSRG SCGF CCSD(T) CRCC IT-NCSM (c) O (Z=8) Bogner et al. 2014; Cipollone, Barbieri, and Navrátil 2013; Hergert et al Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

12 Valence space IM-SRG: Ground states What are we missing? core valence excluded decouple decouple The other methods use the target nucleus as Φ 0, while we use the core Other methods better capture effect 3N forces between valence nucleons Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

13 Valence space IM-SRG: Ground states What are we missing? core valence excluded decouple decouple core valence excluded decouple decouple Strategy: s valence neutrons are added, 22 O becomes a better reference than 16 O, so use that But still decouple the full sd shell Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

14 Valence space IM-SRG: Ground states MR-IMSRG SCGF CCSD(T) CRCC IT-NCSM (c) O (Z=8) Bogner et al. 2014; Brown et al. 2006; Cipollone, Barbieri, and Navrátil 2013; Hergert et al Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

15 Valence space IM-SRG: Ground states (c) O (Z=8) MR-IMSRG SCGF CCSD(T) CRCC IT-NCSM Bogner et al. 2014; Brown et al. 2006; Cipollone, Barbieri, and Navrátil 2013; Hergert et al Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

16 Valence space IM-SRG: Closed subshells Ψ Φ E/ (MeV) ME VS (C=R) VS (TNO) Single Ref. 12 C 14 C 22 O 24 O 28 O 28 Si 34 Si 32 S 40 Ca Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

17 Valence space IM-SRG: Ground state of 28 Si Energy (MeV) IM-SRG Shell Model Si Ψ Φ 0 2 IM-SRG Single Reference Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

18 side: Basis dependence of occupation numbers 3 N LO = 6 N max Λ=2.0 fm, hω=24 MeV a i a i = Ua i a i U 0h 11/2 0g 1d7/2 3/2 2s 1/2 1d 5/2 0g 9/2 0f 1p 5/2 1/2 1p 3/2 0f 7/2 0d 3/2 1s 1/2 0d 5/2 0p 1/2 0p 3/ Si Occupations: 1) IM-SRG basis 2) Oscillator basis 1 2 0s 1/ p 3/ p 1/ d 5/ s 1/ d 3/ neutron occupation 0 0s 1/ proton occupation Same calculation, different occupations! Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

19 What to do about 22 Na? core valence excluded decouple decouple core valence excluded decouple decouple Φ 0 = 16 O (Underestimate 3N) Φ 0 = 28 Si (Overestimate 3N) Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

20 Ensemble reference state (Equal filling) Replace Φ 0 with ensemble (mixed) state characterized by density matrix: ρ = Definition of normal ordering: i T r(ρ{a 1... a N}) = i α i Φ i Φ i α i Φ i {a 1... a N} Φ i = 0 Wick contraction: {a pa q } = i α i Φ i a pa q Φ i n p δ pq {a pa q } = n p δ pq, {a p a q} = (1 n p )δ pq, {a p a q } = {a pa q} = 0 Now n p can be fractional, which is exactly what we want! No N-representability problem. Thouless 1957; Gaudin 1960; Perez-Martin and Robledo 2008 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

21 Ensemble reference state (Equal filling) Replace Φ 0 with ensemble (mixed) state characterized by density matrix: ρ = Definition of normal ordering: i T r(ρ{a 1... a N}) = i α i Φ i Φ i α i Φ i {a 1... a N} Φ i = 0 Wick contraction: {a pa q } = i α i Φ i a pa q Φ i n p δ pq {a pa q } = n p δ pq, {a p a q} = (1 n p )δ pq, {a p a q } = {a pa q} = 0 Now n p can be fractional, which is exactly what we want! No N-representability problem. Thouless 1957; Gaudin 1960; Perez-Martin and Robledo 2008 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

22 Ensemble reference state (Equal filling) Replace Φ 0 with ensemble (mixed) state characterized by density matrix: ρ = Definition of normal ordering: i T r(ρ{a 1... a N}) = i α i Φ i Φ i α i Φ i {a 1... a N} Φ i = 0 Wick contraction: {a pa q } = i α i Φ i a pa q Φ i n p δ pq {a pa q } = n p δ pq, {a p a q} = (1 n p )δ pq, {a p a q } = {a pa q} = 0 Now n p can be fractional, which is exactly what we want! No N-representability problem. Thouless 1957; Gaudin 1960; Perez-Martin and Robledo 2008 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

23 Ensemble reference state (Equal filling) Replace Φ 0 with ensemble (mixed) state characterized by density matrix: ρ = Definition of normal ordering: i T r(ρ{a 1... a N}) = i α i Φ i Φ i α i Φ i {a 1... a N} Φ i = 0 Wick contraction: {a pa q } = i α i Φ i a pa q Φ i n p δ pq {a pa q } = n p δ pq, {a p a q} = (1 n p )δ pq, {a p a q } = {a pa q} = 0 Now n p can be fractional, which is exactly what we want! No N-representability problem. Thouless 1957; Gaudin 1960; Perez-Martin and Robledo 2008 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

24 Ensemble reference state (Equal filling) core valence excluded decouple decouple core valence excluded decouple decouple ρ = O 16 O O 22 O 18 O ρ = O 16 O Si 28 Si 22 Na Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

25 Valence 3N forces: 10 B, 22 Na, 46 V Ground state of 10 B is 3 + 3N forces are required to reproduce this without fitting E E 3 + (MeV) E E 3 + (MeV) 1 (a) B Λ 3N =400 MeV B Λ 3N =500 MeV B 8 He 4 He NCSM N max IMSRG reference Navrátil and Ormand 2002; Pieper, Varga, and Wiringa 2002; Gebrerufael, Calci, and Roth 2015 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

26 Valence 3N forces: 10 B, 22 Na, 46 V Ground state of 10 B is 3 + 3N forces are required to reproduce this without fitting Similar situation for 22 Na and 46 V Normal ordering with ensemble reference captures this E E 3 + (MeV) E E 3 + (MeV) 1 (a) B Λ 3N =400 MeV B Λ 3N =500 MeV B 8 He 4 He NCSM N max IMSRG reference 0.8 (b) Λ 3N = Λ 3N = First ab initio calculations MeV 400 MeV of 22 Na and V to obtain 0.2 correct ordering Na 0.6 Navrátil and Ormand 2002; Pieper, Varga, and Wiringa 2002; Exp. 22 Gebrerufael, Na 16 Calci, O 22 and Roth Na O IMSRG ref. IMSRG ref. E E 3 + (MeV) (c) Λ 3N = 400 MeV V Exp. 46 V 40 Ca IMSRG ref. Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

27 Ground states with ensemble reference (a) C (Z=6) CCSD(T) IT-NCSM (d) Na (Z=11) (b) N (Z=7) SCGF (e) Ca (Z=20) MR-IMSRG CRCC F (Z=9) (f) SCGF CCSD(T) Ni (Z=28) MR-IMSRG CRCC Cipollone, Barbieri, and Navrátil 2013; Binder et al. 2014; Hergert et al. 2014; Jansen et al. 2015; Roth(priv. comm.) Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

28 Ground states with ensemble reference CCSD(T) IT-NCSM SCGF CCSD(T) SCGF CCSD(T) (a) (b) C (Z=6) N (Z=7) F (Z=9) MR-IMSRG MR-IMSRG CRCC 400 CRCC (a) (d) 120 (e) 650 (f) C (Z=6) Na (Z=11) Ca (Z=20) Ni (Z=28) IT-NCSM 140 Cipollone, Barbieri, and Navrátil 2013; Binder et al. 2014; Hergert et al. 2014; Jansen et al. 2015; Roth(priv. comm.) Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

29 Ground states with ensemble reference (a) C (Z=6) CCSD(T) IT-NCSM (d) Na (Z=11) (b) N (Z=7) SCGF (e) Ca (Z=20) MR-IMSRG CRCC F (Z=9) (f) SCGF CCSD(T) Ni (Z=28) MR-IMSRG CRCC Cipollone, Barbieri, and Navrátil 2013; Binder et al. 2014; Hergert et al. 2014; Jansen et al. 2015; Roth(priv. comm.) Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

30 Ground states with ensemble reference (a) C (Z=6) CCSD(T) IT-NCSM (d) Na (Z=11) (f) (b) N (Z=7) 450 Ni (e) 500 (Z=28) SCGF MR-IMSRG CRCC F (Z=9) SCGF CCSD(T) Ca (Z=20) 700 Ni (Z=28) MR-IMSRG CRCC (f) MR-IMSRG CRCC Cipollone, Barbieri, and Navrátil 2013; Binder et al. 2014; Hergert et al. 2014; Jansen et al. 2015; Roth(priv. comm.) Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

31 Germanium/Selenium 0 + states E(0 + ) (MeV) Ge IM-SRG Exp E(0 + ) (MeV) 74 Ge IM-SRG Exp E(0 + ) (MeV) 76 Ge IM-SRG Exp E(0 + ) (MeV) Se IM-SRG Exp E(0 + ) (MeV) Se IM-SRG Exp E(0 + ) (MeV) Se IM-SRG Exp Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

32 Effective operators Effective operators Some very preliminary results Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

33 Effective operators Operators transform just like H: Õ = e Ω Oe Ω = O + [Ω, O] + 1 [Ω, [Ω, O]] Tensor operators require additional angular momentum coupling: Õ λ = e Ω O λ e Ω = O λ + [Ω, O λ ] λ [Ω, [Ω, Oλ ] λ ] λ +... Shell model expectation values then reflect full-space expectation values: Ψ O Ψ = Ψ SM Õ Ψ SM Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

34 Effective operators The deuteron Valence space: 0s shell No effects of induced many-body forces Bare quadrupole operator (λ = 2) gives identically zero 0s 1/2 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

35 Effective operators The deuteron E (MeV) Binding Energy Q (fm 2 ) Quadrupole Moment e max e max Energy correctly reproduced 0s 1/2 0s 1/2 Q 0s 1/2 0s 1/2 0 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

36 rms charge radius (fm) Into the sd shell: Neon Radii 3.1 IM-SRG (Scaled) IM-SRG Skyrme HF Ne IM-SRG (targeted) Marinova et al. 2011; Brown 1998 Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

37 Tensor Operators Moments N S µ(2 + ) (µ N ) O 20 F 20 Ne USD-bare USD-eff Exp IMSRG-bare IMSRG-eff 20 Na 20 Mg Q(2 + ) (efm 2 ) O USD-bare USD-eff Exp IMSRG-bare IMSRG-eff 20 F 20 Ne 20 Na 20 Mg Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

38 Tensor Operators Transitions N S B(M1; ) (µ 2 N ) O USD-bare USD-eff Exp IMSRG-bare IMSRG-eff 20 F 20 Ne 20 Na 20 Mg B(E2; ) (e 2 fm 4 ) O 20 F 20 Ne USD-bare USD-eff Exp IMSRG-bare IMSRG-eff 20 Na 20 Mg Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

Summary b initio methods provide a means to calculate matrix elements where fitting to data is not possible Effective valence-space approach enables consistent treatment of

Evolved tensor operators produce some renormalization more work to be done. MEC corrections to operators can be handled without additional trouble Collaborators:. Calci, J.

39 Summary b initio methods provide a means to calculate matrix elements where fitting to data is not possible Effective valence-space approach enables consistent treatment of excited states, transitions, open-shell/deformed systems Targeted normal ordering with an ensemble reference provides a reasonable approximation of valence 3N forces Evolved tensor operators produce some renormalization more work to be done. MEC corrections to operators can be handled without additional trouble Collaborators:. Calci, J. Holt, P. Navrátil NSCL/MSU S. Bogner, H. Hergert, T. Morris, N. Parzuchowski TU Darmstadt. Schwenk, J. Simonis Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

40 ppendix Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

41 Tensor Operators Moments N S bare 20 Ne bare emax=4 emax=6 emax=8 emax=10 20 Ne µ(2 + ) (µ N ) Q 0 (2 + ) (efm 2 ) ω (MeV) ω emp ω (MeV) ω emp Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

42 Tensor Operators Moments N S bare 1b 20 Ne bare 1b emax=4 emax=6 emax=8 emax=10 20 Ne µ(2 + ) (µ N ) Q 0 (2 + ) (efm 2 ) ω (MeV) ω emp ω (MeV) ω emp Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

43 Tensor Operators Moments N S bare 1b 1b+2b 20 Ne bare 1b 1b+2b emax=4 emax=6 emax=8 emax=10 20 Ne µ(2 + ) (µ N ) Q 0 (2 + ) (efm 2 ) ω (MeV) ω emp ω (MeV) ω emp Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

44 Tensor Operators Transitions N S B(M1; ) (µ 2 n ) bare 1b 1b+2b emax=4 emax=6 emax=8 emax=10 20 F ω (MeV) B(E2; ) (e 2 fm 4 ) ω (MeV) ω emp emax=4 emax=6 emax=8 emax=10 20 Ne bare 1b 1b+2b Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

45 How to choose ˆΩ? toy problem: ( ) ( ) ( ) ɛ1 h Ĥ = od 0 θ cos θ sin θ, ˆΩ =, eˆω = h od ɛ 2 θ 0 sin θ cos θ ( eˆω Ĥe ˆΩ ɛ1 cos = 2 θ + ɛ 2 sin 2 ) θ + h sin 2θ h od cos 2θ + ɛ2 ɛ1 2 sin 2θ h od cos 2θ + ɛ2 ɛ1 2 sin 2θ ɛ 2 cos 2 θ + ɛ 1 sin 2 θ h sin 2θ ( ) h od 0 θ = 1 2 tan 1 2hod ɛ 1 ɛ 2 θ 1 θ h od ɛ 1 ɛ 2 S. R. White, J. Chem Phys. 117, 7472 (2002) Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

46 IM-SRG pplication to 16 O: Φ 0 = η H od h.c. H od is any term that connects Φ 0 to any other configuration s is the total angle rotated Ground state energy given by a single matrix element: Φ 0 H Φ 0 E 0 (MeV) η O N2LO ST ω=22 MeV 13 shells s O N2LO ST ω=22 MeV 13 shells Tsukiyama, Bogner, and Schwenk 2011; Ekström et al s Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

terms of a i a i = U (a i )U X X = Ci a i + Cj a j + Cjkl a j a k al +.

47 IM-SRG Where did all the correlations go? Original single particle basis: φi i = a i 0i The transformed H is implicitly in terms of a i a i = U (a i )U X X = Ci a i + Cj a j + Cjkl a j a k al +... j6=i jk The single-particle orbits are now much more complicated! U Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

48 References I Binder, Sven et al. (2014). b initio path to heavy nuclei. In: Phys. Lett. B 736, pp issn: doi: /j.physletb url: Bogner, S. K. et al. (2014). Nonperturbative shell-model interactions from the in-medium similarity renormalization group. In: Phys. Rev. Lett , p issn: doi: /PhysRevLett arxiv: url: Brown, B.. (1998). New Skyrme interaction for normal and exotic nuclei. In: Phys. Rev. C 58.1, pp issn: doi: /PhysRevC url: Brown, B.. et al. (2006). Tensor interaction contributions to single-particle energies. In: Phys. Rev. C 74.6, p issn: doi: /PhysRevC url: Cipollone,., C. Barbieri, and P. Navrátil (2013). Isotopic Chains round Oxygen from Evolved Chiral Two- and Three-Nucleon Interactions. In: Phys. Rev. Lett , p issn: doi: /PhysRevLett url: Ekström,. et al. (2015). ccurate nuclear radii and binding energies from a chiral interaction. In: Phys. Rev. C 91.5, p issn: doi: /PhysRevC url: Entem, D. R. and R. Machleidt (2003). ccurate charge-dependent nucleon-nucleon potential at fourth order of chiral perturbation theory. In: Phys. Rev. C 68.4, p issn: doi: /PhysRevC url: Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

49 References II Gaudin, Michel (1960). Une démonstration simplifiée du théorème de wick en mécanique statistique. In: Nucl. Phys. 15, pp issn: doi: / (60) url: Gebrerufael, Eskendr, ngelo Calci, and Robert Roth (2015). Open-Shell Nuclei and Excited States from Multi-Reference Normal-Ordered Hamiltonians. In: p. 6. arxiv: url: Hergert, H. et al. (2014). b initio multireference in-medium similarity renormalization group calculations of even calcium and nickel isotopes. In: Phys. Rev. C 90.4, p issn: doi: /PhysRevC url: Jansen, G. R. et al. (2015). Deformed sd-shell nuclei from first principles. In: arxiv: url: Marinova, K. et al. (2011). Charge radii of neon isotopes across the s d neutron shell. In: Phys. Rev. C 84.3, p issn: doi: /PhysRevC url: Morris, T. D., N. M. Parzuchowski, and S. K. Bogner (2015). Magnus expansion and in-medium similarity renormalization group. In: Phys. Rev. C 92.3, p issn: doi: /PhysRevC arxiv: url: Navrátil, P. (2007). Local three-nucleon interaction from chiral effective field theory. In: Few-Body Syst , pp issn: doi: /s url: Navrátil, Petr and W Erich Ormand (2002). b initio shell model calculations with three-body effective interactions for p-shell nuclei. In: Phys. Rev. Lett , p issn: doi: /PhysRevLett url: Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

50 References III Perez-Martin, Sara and L. M. Robledo (2008). Microscopic justification of the equal filling approximation. In: Phys. Rev. C 78.1, p issn: doi: /PhysRevC url: Pieper, S., K. Varga, and R. Wiringa (2002). Quantum Monte Carlo calculations of =9,10 nuclei. In: Phys. Rev. C 66.4, p issn: doi: /PhysRevC url: Thouless, D. J. (1957). Use of Field Theory Techniques in Quantum Statistical Mechanics. In: Phys. Rev , pp issn: X. doi: /PhysRev url: Tsukiyama, K., S. K. Bogner, and. Schwenk (2011). In-Medium Similarity Renormalization Group For Nuclei. In: Phys. Rev. Lett , p issn: doi: /PhysRevLett url: (2012). In-medium similarity renormalization group for open-shell nuclei. In: Phys. Rev. C 85.6, p issn: doi: /PhysRevC url: White, Steven R. (2002). Numerical canonical transformation approach to quantum many-body problems. In: J. Chem. Phys , p issn: doi: / url: Ragnar Stroberg (TRIUMF) Valence space IM-SRG May 13, / 27

The shell model as an ab initio tool: effective interactions and operators from IM-SRG

Canada s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucleaire et en physique des particules U = eη The shell model as an ab initio