George Papadimitriou Many-body methods for the description of bound weakly bound and unbound nuclear states Understanding nuclear structure and reactions microscopically, including the continuum. March 17-21, 2014, GANIL, France B. Barrett N. Michel, W. Nazarewicz, M.Ploszajczak, J. Rotureau, J. Vary, P. Maris.

Outline Nuclear Physics on the edge of stability  Experimental and Theoretical endeavors The Gamow Shell Model (GSM)  Applications on charge radii of Helium Halos and neutron correlations Alternative method for extracting resonance parameters:  The Complex Scaling Method in a Slater basis. Outlook, conclusions and Future Plans

New exotic resonant states: 7 H, 13 Li, 10 He, 26 O… PRC 87, , PRL , PRL , PRL 109, recently ) Metastable states embedded in the continuum are measured. Very dilute matter distribution Extreme clusterization close to particle thresholds. New decay modes: 2n radioactivity? Life on the edge of nuclear stability: Experimental highlights Shell structure revisited: Magic numbers disappear, other arise. Provide stringent constraints to theory But also: Theory is in need for predictions and supporting certain experimental aspects From: A.Gade Nuclear Physics News 2013 A.Spyrou et al Marques et al (conflicting experiment)

Fig: Bertsch, Dean, Nazarewicz, SciDAC review 2007 Dimension of the problem increases One size does not fit all!

Life on the edge of nuclear stability: Theory Weak binding and the proximity of the continuum affects bulk properties and spectra of nuclei. The very notion of the mean-field and shell structure is under question Nuclei are open quantum system and the openness is governed by the S n Dobaczewski et al Prog.Part.Nucl.Phys. 59, 432 (2007) Dobaczewski, Nazarewicz Phil. Trans. R.Soc. A 356 (1998) Especially the clusterization of matter is a generic property of the coupling to the continuum (or the impact of the open reaction channels). Clusterization does not depend on the specific characteristics of the NN interaction Okolowicz, Ploszajczak, Nazarewicz Fortschr. Phys. 61, 69 (2013)

Input Forces Many-body Methods techniques Open Channels Coupling to continuum Physics of nuclei close to the drip-line Life on the edge of nuclear stability: Theory Additionally, complementary to the above: a new aspect is quality control 1) Cross check of codes/benchmarking 2) Statistical tools to estimate errors of calculations… Recent Paradigms: DFT functionals, new chiral forces, new extrapolation techniques

Resonant and non-resonant states (how do they appear?) Solution of the one-body Schrödinger equation with outgoing boundary conditions and a finite depth potential Solutions with outgoing boundary conditions

The Berggren basis (cont’d) The eigenstates of the 1b Shrödinger equtaion form a complete basis, IF: T.Berggren (1968) NP A109, 265 are complex continuum states along the L + contour (they satisfy scattering b.c) In practice the continuum is discretized via a quadrature rule (e.g Gauss-Legendre): with The shape of the contour is arbitrary, and any state between the contour and the real axis can be expanded in such as basis (proof by T. Berggren) we also consider the L + scattering states

Berggren’s Completeness relation and Gamow Shell Model resonant states (bound, resonances…) Non-resonant Continuum along the contour Many-body basis Hermitian Hamiltonian The GSM in 4 steps N.Michel et.al 2002 PRL Hamiltonian diagonalized Hamiltonian matrix is built (complex symmetric): Many body correlations and coupling to continuum are taken into account simultaneously

GSM HAMILTONIAN “recoil” term coming from the expression of H in relative coordinates. No spurious states Y.Suzuki and K.Ikeda PRC 38,1 (1988) Hamiltonian free from spurious CM motion Appropriate treatment for proper description of the recoil of the core and the removal of the spurious CoM motion.  We assume an alpha core in our calculations.. V ij is a phenomelogical NN interaction, fitted to spectra of nuclei: Minnesota force is used, unless otherwise indicated.

Applications of the Berggren basis –Spectra- Helium isotopic chain ( 4 He core plus valence neutrons in the p-shell) Schematic NN force

L.B.Wang et al, PRL 93, (2004) P.Mueller et al, PRL 99, (2007) M. Brodeur et al, PRL 108, (2012) 6,8 He charge radiiApplications  M.Brodeur et al 4 He 6 He 8 He L.B.Wang et al 1.67fm2.054(18)fm 1.67fm RMS charge radii 2.059(7)fm 1.959(16)fm Very precise data based on Isotopic Shifts measurements Extraction of radii via Quantum Chemistry calculations with a precision of up to 20 figures! (Hyllerraas basis calculations) Model independence of results Z.-T.Lu, P.Mueller, G.Drake,W.Nörtershäuser, S.C. Pieper, Z.-C.Yan Rev.Mod.Phys. 2013, 85, (2013). “Laser probing of neutron rich nuclei in light atoms” 6 He: 2 as a strong correlated pair 8 He: 4  are distributed more symmetrically around the charged core Other effects also… Can we calculate and quantify these correlations? Stringent test for the nuclear Hamiltonian

6 He 8 He G. Papadimitriou et al PRC 84, s.o density and radii also calculated by S. Bacca et al PRC 86,

Radii (and other operators different than Hamiltonian) are challenging Example: Courtesy of P. Maris 1)How to reliably extrapolate radial operators to the infinite basis? Sid Coon et al, Furnstahl et al methods? 2)Renormalized operators? 3)Different basis?

Neutron correlations in 6 He ground state  Probability of finding the particles at distance r from the core with an angle θ nn Halo tail See also I. Brida and F. Nunes NPA 847,1 (2010) and P. Navratil talk

Coupling to the continuum crucial for clusterization In the absence of continuum p 1/2 -sd states the neutrons show no preference S=0 component (spin-antiparallel) dominant  Manifestation of the Pauli effect G. Papadimitriou et al PRC 84, Average opening angle calculated from the density: θ nn = 68 o Full continuum Only p3/2

Neutron correlations in 6 He 2+ excited state and spectroscopy  2+ neutrons almost uncorrelated… G.P et al PRC(R) 84, , 2011 Constructing an effective interaction in GSM in the p and sd shell. Effective interactions depend on the position of thresholds… : [4.13, 3.17] MeV : [4.75, 8.6] MeV : [4.4, 5.5 ] MeV : [1.82, 0.1] MeV GSM MN force fitted just to the g.s. energy of 6,8 He Fig. from

Additional tools in our arsenal Bound state technique to calculate resonant parameters and/or states in the continuum (see also talks by Lazauskas, Bacca, Orlandini) Prog. Part. Nucl. Phys. 74, 55 (2014) and 68, 158 (2013) (reviews of bound state methods) The complex scaling Belongs to the category of: Nuttal and Cohen PR 188, 1542 (1969) Lazauskas and Carbonell PRC (2005) Witala and Glöeckle PRC (1999) Aoyama et al PTP 116, 1 (2006) Horiuchi, Suzuki, Arai PRC 85, (2012) Nuclear Physics Chemistry Moiseyev Phys. Rep (1998) Y. K. Ho Phys. Rep. 99 1, (1983)

Additional tools in our arsenal Complex Scaling Method in a Slater basis A.T.Kruppa, G.Papadimitriou, W.Nazarewicz, N. Michel PRC (2014)  Powerful method to obtain resonance parameters in Quantum Chemistry  Involves L 2 square integrable functions.  Can (in general) be applied to available bound state methods techniques (i.e. NCSM, Faddeev, CC etc) 1) Basic idea is to rotate coordinates and momenta i.e. r  re iθ Hamiltonian is transformed to H(θ) = U(θ)H original U(θ) -1 H(θ)Ψ(θ) = ΕΨ(θ) complex eigenvalue problem The spectrum of H(θ) contains bound, resonances and continuum states. 2) Slater basis or Slater Type Orbitals (STOs): Basically, exponential decaying functions

Some results Comparison between CS Slater and CSM 0 + g.s, st excited Force Minnesota, α-n interaction KKNN Test the HO expansion of the NN force in GSM for the unbound 2 + state. In GSM the force is expanded in a HO basis: Talmi-Moshinsky transformation Numerical effort: Overlaps between HO and Gamow states. Very weak dependence of results on b n nmax.

Some results 6 He 0 + g.s. Valence neutrons radial density Phenomenological NN Minnesota interaction Correct asymptotic behavior

Some results 2 + first excited state in 6 He The 2+ state is a many-body resonance (outgoing wave) GSM exhibits naturally this behavior  but CS is decaying for large distances, even for a resonance state This is OK. The solution Ψ(θ) is known to “die” off (L 2 function)

Solution  Perform a direct back-rotation. What is that? In the case of the density this becomes: Back-rotation

The CS density has the correct asymptotic behavior (outgoing wave) 2+ densities in 6 He (real and imaginary part) Back rotation is very unstable numerically. An Ill posed inverse problem. Long standing problem in the CS community (in Quantum Chemistry as well) The problem lies in the analytical continuation of a square integrable function in the complex plane. We are using the theory of Fourier transformations and a regularization process (Tikhonov) to minimize the ultraviolet numerical noise of the inversion process.

Conclusions/Future plans  Berggren basis appropriate for calculations of weakly bound/unbound nuclei. GSM calculations provided insight behind the charge differences of Helium halo nuclei.  Construct effective interaction in the p and sd shell. Use realistic effective interactions for GSM calculations that stem from NCSM with a core, or Coupled Cluster or IM-SRG…  GSM is the Shell Model technique to: i) study 3N forces effects and continuum coupling for the detailed spectroscopy of heavy drip line nuclei. ii) exact treatment of many body correlations and coupling to continuum  Complementary method to describe resonant states: Complex Scaling in a Slater basis L 2 integrable basis formulation. Slater basis  correct asymptotic behavior Back rotation inverse problem solved. Apart from complex arithmetics the computational expense is as “tough” or as “easy” as for the solution of the bound state.  Explore complex scaling in more depth

Back up

Solution Back rotation is very unstable numerically. Unsolved problem in the CS community (in QC as well) The problem lies in the analytical continuation of a square integrable function in the complex plane. We are using the theory of Fourier transformations and Tikhonov regularization process to obtain the original (GSM) density To apply theory of F.T to the density, it should be defined in (-∞,+∞)  Now defined from (-∞,+∞)  F.T Value of (1) for x+iy (analytical continuation)   Tikhonov regularization x = -lnr, y = θ

Last slide before conclusions/future plans  NN force: JISP16 (A. Shirokov et al PRC79, ) and NNLO opt (A. Ekstrom et al PRL 110, )  Quality control: Verification/Validation, cross check of codes  MFDn/NC-GSM + computer scientists at LBNL (Ng, Yang, Aktulga), collaboration Goal: Scalable diagonalizations of complex symmetric matrices MFDn: Vary, Maris NCGSM: G.P, Rotureau, Michel…

Dimension comparison  Lanczos: “brute” force diagon of H.  DMRG: Diagon of H in the space where only the most important degrees of “freedom” are considered

 Similar treatment by Caprio, Vary, Maris in Sturmian basis

Complex Scaling

 construction of a block in :  construction of a superblock : superblock block Construct all many-body states associated with the pole space P Construct all many-body states associated with the space of the discrete continua C. Create many-body basis by coupling states in P and C.

 truncation with the density matrix : N opt states that correspond to the largest eigenvalues of the density matrix are kept truncation “up” truncation “down” The process is reversed… In each step (shell added) the Hamiltonian is diagonalized and N opt states are kept. Iterative method to take into account all the degrees of freedom in an effective manner. In the end of the process the result is the same (within keV) with the one obtained by “brute” force diagonalization of H. Sweep-down Sweep-up

Results: 4 He against Fadeev-Yakubovsky  2 neutrons  2 protons  Pole space A:0s1/2 (p/n)  Continuum space B: p3/2,p1/2,s1/2 real energy continua d5/2-d3/2 f5/2-f7/2 H.O states g7/2-g9/2  156 s.p. states total Dim for direct diagon: 119,864,088 E ab-initio = MeV E FY = MeV G.P., J.Rotureau, N. Michel, M.Ploszajczak, B. Barrett arXiv:

Neutron correlations in 8 He ground state G.Papadimitriou PhD thesis

Neutron correlations in 6 He 2+ excited state  2+ neutrons almost uncorrelated… G.P et al PRC(R) 84, , 2011