A simple effective interaction for 9 He, and Gamow-SRG

Transcription

1 A simple effective interaction for 9 He, and Gamow-SRG Kévin Fossez February 28, 218 FRIB, MSU FRIB, MSU - Kévin Fossez Work supported by: DOE: DE-SC1336 (Michigan State University) DOE: DE-SC17887 (Michigan State University) DOE: DE-SC811 (NUCLEI SciDAC-3 collaboration) NSF: PHY

2 Physics of neutron-rich helium isotopes Few-body, effective scale separation, continuum couplings, exotic states... Two- and four-body halos ( 6,8 He). Broad resonances (1/2 in,7 He). Many th. results, high experimental interest. T. AL KALANEE et al. PHYSICAL REVIEW C 88, 3431 (213) Uncertain case of 9 He. Very little known on 1 He. FIG.. Summary of all experimental results for 9 He, up to MeV excitation energy. Solid lines represent states with well defined resonance. Dashed lines or hashed areas represent low-lying structures described by virtual s-wave states (see text for details). T. Al Kalanee et al., Phys. Rev. C 88, 3431 (213) FRIB, MSU - Kévin Fossez 2 the presence of a state in 9 He very close ( 2 kev) to calculated absolute cross section are observed as a function of

3 What are the options to describe 9 He? Practical vs. fundamental: Practical? shell model Explicit 3-body forces halo-eft ab initio Fundamental Well bound core of 4 He. Core approximation justified (SM, EFT). Dilute neutron matter above the core. Residual interaction genuinely residual Decent CSM/GSM descriptions of 1 He available (small spaces, truncations). Different phenomenological descriptions are in agreement, it must be a miracle or there is a good reason behind! Can we find a practical alternative, without explicit 3-body forces, and beyond the SM (with continuum) for He isotopes? FRIB, MSU - Kévin Fossez 3

4 What are the options to describe 9 He? Practical vs. fundamental: Practical? shell model Explicit 3-body forces halo-eft ab initio Fundamental Well bound core of 4 He. Core approximation justified (SM, EFT). Dilute neutron matter above the core. Residual interaction genuinely residual Decent CSM/GSM descriptions of 1 He available (small spaces, truncations). Different phenomenological descriptions are in agreement, it must be a miracle or there is a good reason behind! Can we find a practical alternative, without explicit 3-body forces, and beyond the SM (with continuum) for He isotopes? FRIB, MSU - Kévin Fossez 3

5 Effective interactions inspired from EFT (preliminary) A simple model: Phase-shift (deg) Core potential fitted on n- 4 He phase-shifts. Contact 2-body central term (3 Gaussian functions) for (L even, S = ) channels s 1/2 p 1/2 p 3/2 1 Energy (MeV) Only a prefactor V c in the interaction to fit! (New: just (L =, S = ) works too) Energy (MeV) Exp He J π = + 6 He J π = He 7 He J π = 3/2 J π = / He 8 He -.4 J π = + J π = V c (MeV) FRIB, MSU - Kévin Fossez 4

6 Effective interactions inspired from EFT (preliminary) UQ beyond a sensitivity analysis: (mean), σ (standard deviation). The uncertainty on the energy is given by: V (opt) c E = 1 2 E(V (opt) c + σ) E(V (opt) c σ). Questions: Why does the core need to be fitted on phase-shifts? Is there a proper EFT for all He isotopes behind this simple scheme? -3 + Can it be generalized to other isotopic chains? Parity inversion in 9 He, structure information on 8 1 A He. Powerful approach with full continuum couplings, tens of kev uncertainties. E (MeV) /2 4 He 1/2 (Γ =.9 MeV) /2 3/2 DMRG 1/2 (Γ = 2.1 MeV) FRIB, MSU - Kévin Fossez 2 + /2 + 1/2 1/2 + Exp +

7 Gamow-SRG Similarity renormalization group: Hamiltonian: Ĥ SRG evolution: dĥ(s) = [ˆη(s), Ĥ(s)] ds (flow equation) Ĥ(s) = Û(s)ĤÛ 1 (s) HO Flow generator: ˆη(s) = dû(s) Û 1 (s) ds Berggren Hermitian Non-Hermitian ˆη(s) anti-hermitian ˆη(s) Û(s) unitary Û(s) similarity FRIB, MSU - Kévin Fossez 6

8 Gamow-SRG The Berggren basis: Single particle basis including bound states, decaying resonances and scattering states. Im(k) bound states decaying resonances Re(k) discretized continuum in momentum space u l (k n) ũ l (k n) n (b,d) + u l (k i) ũ l (k i) w ki ˆ1 l,j. i Discretization: u l (k i) w ki u l (k i), u l (k n) ũ l (k n) ˆ1 l,j. n (b,d,i) FRIB, MSU - Kévin Fossez 7

9 Gamow-SRG Proof of principle: 1. Re (H) Im (H) > 1. Re(E) (MeV) Im(E) (MeV) White generator PRELIMINARY s Various generator tested. Consistent with observations on Hermitian matrices. It works best for a Berggren basis with selected scattering states s =. s =.94 s = 2.43 s = 6.3. Promising for IM-SRG in the Berggren basis < 1. FRIB, MSU - Kévin Fossez 8

10 Gamow-SRG Some technical observations: Non-Hermitian Hamiltonian: Ĥ = Ĥ h + Ĥ ah = 1 2 (Ĥ + Ĥ ) (Ĥ Ĥ ) Wegner flow generator for a non-hermitian Hamiltonian: ˆη W,cx = [Ĥ h,d + Ĥ ah,d, Ĥ h,od + Ĥ ah,od ] (unstable) ˆη G = [Ĥ h,d, Ĥod] = [Ĥ h,d, Ĥh,od + Ĥah,od] Wegner flow generator for the real part only: ˆη G,h = [Ĥ h,d, Ĥ h,od ] Not yet clear how to extract the anti-hermitian part (key for continuum). ˆη G,h = [Ĥ h,d, Ĥah,od] (not similarity) Re (H) s =. s =.94 s = 2.43 s = 6.3 Im (H) FRIB, MSU - Kévin Fossez 9 > 1.. < 1.

11 Thank you for your attention! Michigan State University: J. Rotureau. H. Hergert. S. Bogner. W. Nazarewicz. FRIB, MSU - Kévin Fossez

12 (NC)GSM vs DMRG s.p. poles s.p. scatt. {SD (N) } (pole space) {SD (N) 1 } (full space) H H 1 Ψ (pivot) Davidson (2D) Ψ 1 (No-Core) Gamow Shell Model (N. Michel) H s.p. pole (Complex-symmetric Hamiltonian matrices) {SD (), SD(1),..., SD(N) } H Ψ (pivot) Density Matrix Renormalization Group (J. Rotureau) P (s.p. poles/scatt.) {SD () 1, SD(1) 1,..., SD(N) 1 } H 1 Davidson Ψ 1 ρ 1 (j, j ) = Ψ j,h Ψ j,h h {Φ () 1, Φ(1) 1,..., Φ(N) 1 } select ε > 1 8 {φ () 1, φ(1) 1,..., φ(n) 1 } FRIB, MSU - Kévin Fossez {SD () 2, SD(1) 2,..., SD(N) 2 } H 2 Davidson Ψ 2 etc.