Ab Initio Electromagnetic Transitions with the IMSRG Nathan Parzuchowski Michigan State University March, 017 1 / 1
Outline IMSRG IMSRG rotates the Hamiltonian into a coordinate system where simple methods (e.g. Hartree-Fock) are approximately exact: H(s) = U(s)HU (s) / 1
EOM-IMSRG Approaches for Excited States Additional processing needed for excited states. GS-decoupling has softened couplings between excitation rank. Equations-of-motion: Approximately diagonalize excitation block. 3 / 1
EOM-IMSRG Equations-of-Motion IMSRG EOM-IMSRG equation, in terms of evolved operators: [ H(s), X ν (s)] Φ 0 = ω ν X ν (s) Φ 0 IMSRG: No correlations between ground and excited states. X ν = ph x p h a pa h + 1 x pp hh a pa p a h a h pp hh Truncation to two-body ladder operators: EOM-IMSRG(,). / 1
Effective Operators Effective Operators in the IMSRG IMSRG unitary transformation can be explicitly constructed: IMSRG via Magnus expansion: U(s) = e Ω(s) dω ds = η + [Ω, η] 1 [Ω, [Ω, η]] + Effective operators from Baker-Campbell-Hausdorff: Ō(s) = O + [Ω, O] + 1 [Ω, [Ω, O]] + T. Morris, N. Parzuchowski, S. Bogner, Phys. Rev. C 9, 03331 (015). 5 / 1
Effective Operators Electromagnetic Observables: 1 C E( + 1 ) (MeV) 1 11 10 9 8 Exp NCSM ω=0mev Valence-space IMSRG emax= emax= 6 emax= 8 emax=10 emax=1 emax=1 EOM IMSRG B(E; + 1 0 + 0 ) (e fm ) 7 5 3 N 3 LO(500) NN + N LO(00) 3N λsrg =. 0 fm 1 1 C 1 16 0 8 0 6 8 N max 1 16 0 8 PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (016). 6 / 1
Effective Operators Electromagnetic Observables: 1 C E(1 + 1 ) (MeV) 1 Exp NCSM 13 1 11 10 ω=0mev Valence-space IMSRG e max= e max= 6 e max= 8 e max=10 e max=1 e max=1 EOM IMSRG B(M1; 1 + 1 0 + 0 ) (µ N) 9 1 0 0 6 8 N max 1 16 0 8 1 C 1 16 0 8 PRELIMINARY Exp: Ajzenberg-Selove, Nuc. Phys. A 153, 1 (1991). 7 / 1
Effective Operators Electromagnetic Observables: O E( + 1 ) (MeV) 7 6 5 3 Exp Valence-space IMSRG EOM IMSRG B(E; + 1 0 + 0 ) (e fm ) 1 0 6 5 3 1 0 1 16 0 8 O emax= emax= 6 emax= 8 emax=10 emax=1 emax=1 1 16 0 8 PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (016). 8 / 1
Effective Operators Electromagnetic Observables: 8 Ca, 56,60 Ni E( + 1 ) (MeV) 10 Exp 9 8 Ca Exp 56 Ni Exp 60 Ni 8 7 6 5 3 1 B(E; + 1 0 + 0 ) (e fm ) 150 100 50 0 10 15 0 5 30 10 15 0 5 30 10 15 0 5 30 PRELIMINARY Exp: Pritychenko et. al., Nucl. Data Tables 107, 1 (016). 9 / 1
Extensions/Applications Perturbative Triples Correction: EOM-IMSRG({3},) EOM-IMSRG({3},): O(N 7 ) : X ν = X 1p1h + X ph δe ν = ijk abc Φ abc ijk H X ν Φ 0 ω (0) ν Φ abc ijk H Φ abc ijk N. Parzuchowski, T. Morris, S. Bogner, arxiv:1611.00661 10 / 1
Extensions/Applications EOM-IMSRG({3},) for 1 C, O E.M. N 3 L0(500) + Navratil N LO 3N(00) λ=.0 fm 1 e max = 8 ω = 0 MeV Energy (MeV) 1 10 8 6-3- 0-1- 1+ + 1+ 1- + 3- + + + - 3-0- 0+ 1-1 C Energy (MeV) 18 16 1 + 1-0- 1 10 8 6-1+ + 3+ + 3+ + (0-,1-,-)? + +? 0+ 3+ + O 0 EOM-IMSRG(,) ({3},) Exp 0 EOM-IMSRG(,) ({3},) Exp PRELIMINARY Exp: NNDC ENDSF 11 / 1
Conclusion Summary/Outlook Spectra and observables are now available with EOM- and VS-IMSRG Results are consistent with NCSM. E strengths consistently under-predicted (except 1 C). Moving Forward... EOM-IMSRG is systematically improvable, perturbative corrections are possible. Next: multi-reference-eom-imsrg (Heiko s talk) for static correlations and open shells. 1 / 1
Conclusion Thank you! IMSRG at MSU: Scott Bogner Heiko Hergert Fei Yuan IMSRG at ORNL: Titus Morris IMSRG at TRIUMF: Ragnar Stroberg Jason Holt Also thanks to: Petr Navrátil (NCSM results) Gaute Hagen 13 / 1
Equations-of-Motion (EOM) Methods for Excited States Define a ladder operator X ν such that: Ψ ν = X ν Ψ 0 Eigenvalue problem re-written in terms of X : Ĥ Ψ ν = E ν Ψ ν [H, X ν ] Ψ 0 = (E ν E 0 )X ν Ψ 0 Approximations are made for X ν, Ψ 0. RPA : Ψ 0 Φ HF X ν = [x p h a pa h + y p h a h a p] ph 1 / 1
Method Comparison in D Quantum Dots. ω = 1. 0 Method RMS Error (,) 0.095 ({3},)-MP 0.066 ({3},)-EN 0.031 Line width: 1p1h content Excitation Energy (Hartree) 1.9 1.6 1.3 1.0 0.7 FCI EOM(,) EOM({3},) 15 / 1
Solving the IM-SRG equations d H ds = [η(s), H(s)] η(s) H OD (s) Transform operators via flow equation. requires very precise ODE solver small step sizes in s needed for convergence Construct the unitary transformation explicitly. Magnus expansion U(s) = e Ω(s) dω ds = η(s) 1 [Ω(s), η(s)] + 1 [Ω(s), [Ω(s), η(s)]] +... 1 less precision needed, larger step sizes 16 / 1
Corrections to IM-SRG() with He Ground State Energy (MeV) -7-7.5-8 e max = 3 (Magnus(/3)) e max = 5 e max = 7 e max = 9 e max = 9 (Magnus() [3]) e max = 9 (Magnus() ) CCSD Λ-CCSD(T) -8.5 0 5 30 h _ CC results: G. Hagen et. al., PRC 8 03330 (010). 17 / 1
Center of Mass Treatment H cm = T cm + 1 maω R cm 3 Ω H cm is evolved as an effective operator in IM-SRG: dh cm (s) ds = [η(s), H cm (s)] CoM frequency Ω calculated in the manner of Hagen et. al. Ω = ω + 3 E cm(ω, s) ± 9 (E cm(ω, s)) + 3 ωe cm(ω, s) G. Hagen, T. Papenbrock, and D. J. Dean, Phys. Rev. Lett. 103, 06503 (009). 18 / 1
CoM diagnostic for 16 O 3- state E.M. N3LO Λ=500 NN at λ SRG =.0 fm 1 10 10 8 E cm (ω) E cm (Ω+) E cm (Ω-) 8 E cm (ω) E cm ( Ω+) E cm ( Ω-) E cm (MeV) 6 Ground State E cm (MeV) 6 3- State 0 10 15 0 5 30 35 0 _ h ω 0 10 15 0 5 30 35 0 _ h ω 19 / 1
Lawson CoM Treatment: H = H int + β CM H CM (Ω) 50 Energy (MeV) 0 30 0 10 3+ + 0+ 1-3- 0 0.0 1.0.0 3.0.0 5.0 β CM 0 / 1
Lawson for 1 C E gs (MeV) 10 103 10 105 106 β =0 β =1 β =3 β =5 15 0 5 9 8 7 E( + 1 ) (MeV)10 B(E; + 1 0 + gs ) (e fm ) 6 5 3 15 0 5 15 0 5 1 / 1