1. Complex-energy shell model description of light nuclei
Witold Nazarewicz (UTK/ORNL)
Nuclear Many-Body Open Quantum Systems:
CONTINUUM AND CORRELATIONS IN LIGHT NUCLEI
ECT*, Trento, Italy 16-21, 2011
in collaboration with
N. Michel, G. Papadimitriou, M .Płoszajczak, J. Rotureau
R. IdBetan, and A.T. Kruppa

2. Nucleus as an open quantum system

3. Physics of rare isotopes is demanding
Interactions
Poorly-known spin-isospin components come into play
Long isotopic chains crucial
Interactions
Many-body
Correlations
Open
Channels
11Li
45Fe
298U
Open channels
Nuclei are open quantum systems
Exotic nuclei have low-energy decay thresholds
Coupling to the continuum important
Many body correlations
Mean-field concept often questionable
Asymmetry of proton and neutron Fermi surfaces gives rise to new couplings

4. Closed Quantum System Open quantum system
infinite well
scattering continuum
resonance
bound states
discrete states only
Identification of right degrees of freedom crucial !!!

5. Resonant (Gamow) states
outgoing
solution
complex pole
of the S-matrix
Humblet and Rosenfeld, Nucl. Phys. 26, 529 (1961)
Siegert, Phys. Rev. 36, 750 (1939)
Gamow, Z. Phys. 51, 204 (1928)
Also true in many-channel case!

6. Our answer: Complex-Energy Shell Model (Gamow Shell Model)
Open problems in the theory of nuclear open quantum systems
N. Michel et al., J. Phys. G 37, (2010)
What is the interplay between mean field and correlations in open quantum systems?
What are properties of many-body systems around the reaction threshold?
What is the origin of cluster states, especially those of astrophysical importance?
What should be the most important steps in developing the theory that will treat nuclear structure and reactions consistently?
What is Quantum Mechanics of open quantum systems?
How are effective interactions modified in open quantum systems?
Our answer: Complex-Energy Shell Model (Gamow Shell Model)
Formulation: Phys. Rev. Lett. 89, (2002); Phys. Rev. C 67, (2003); Phys. Rev. C 70, (2004); J. Phys. G 31, S1321 (2005); J. Phys. G (Topical Review) 36, (2009)
Density matrix renormalization group approach to OQS: Phys. Rev. Lett. 97, (2006); Phys. Rev. C 79, (2009)
Continuum coupling and s.o. splitting: Eur. Phys. J. A 25, s1.503 (2005)
Antibound states and halo formation: Phys. Rev. C 74, (2006)
Threshold effects and single-nucleon overlaps: Phys. Rev. C 75, (R) (2007); Nucl. Phys. A 794, 29 (2007)
Isospin mixing: Phys. Rev. C 82, (2010)
Charge radii in halo nuclei, in preparation (2011)

7. Berggren ensemble for a given jl channel:
One-body basis
Newton
Berggren ensemble for a given jl channel:
Berggren 1968

8. Gamow states and completeness relations
T. Berggren, Nucl. Phys. A109, 265 (1968); A389, 261 (1982)
T. Lind, Phys. Rev. C47, 1903 (1993)
Gamow Shell Model:
Based on the Rigged Hilbert Space formulation of quantum mechanics
Contour is identified and discretized for each partial wave
Many-body Slater determinants are built out of resonant and scattering states
GSM Hamiltonian matrix is computed (using external complex scaling) and Lanczos-diagonalized (the matrix is complex symmetric!)
The many-body Gamow states are identified
Expectation values of operators are computed
Discretization/truncation optimized my means of DMRG

9. Generalized Variational Principle
(a complex extension of the usual variational principle)
N. Moiseyev, P.R. Certain, and F. Weinhold, Mol. Phys. 36, 1613 (1978).
N. Moiseyev, Phys. Rep. 302, 212 (1998)
is stationary around any eigenstate
That is,
It should be noted that the complex variational principle is a stationary principle rather than an upper of lower bound for either the real or imaginary part of the complex eigenvalue. However, it can be very useful when applied to the squared modulus of the complex eigenvalue. Indeed,
Example: GSM+DMRG calculations for 7Li
J. Rotureau et al., Phys. Rev. C 79, (2009)

11. Experimental charge radii of 6He, 8He, 11Li, 11Be
L.B.Wang et al
P.Mueller et al
1.67fm
2.054(18)fm
2.068(11)fm
1.929(26)fm
9Li
11Li
R.Sanchez et al
2.217(35)fm
2.467(37)fm
10Be
High precision measurements based on laser spectroscopy
Charge radii were extracted from isotopic shift measurements with the help of atomic theory calculations
11Be
W.Nortershauser et al
2.357(16)fm
2.460(16)fm
L.B.Wang et al, PRL 93, (2004)
P.Mueller et al, PRL 99, (2007)
R.Sanchez et al PRL 96, (2006)
W.Nortershauser et al PRL 102, (2009)
GSM calculations for 6He, 8He by G. Papadimitriou et al.

12. GSM HAMILTONIAN
We assume a core + valence nucleons (neutrons and protons)
Cluster Orbital Shell Model coordinates
GSM Hamiltonian translationally invariant
Antisymmetrization with respect to the core
residual interaction: schematic SDI, SGI, or finite-range MN
WS +S.O. field or GHF field
recoil term
Y.Suzuki and K.Ikeda
PRC 38,1 (1988)
Saito’s Orthogonality Condition Model:

13. Recoil term and GSM Interaction treatment
Recoil term and finite-range interactions expanded in HO basis:
Greek letters are oscillator states
Latin are Berggren states
No complex scaling is involved
Gaussian fall-off of HO states provides convergence
In the expansion we use up to n ~ HO radial nodes. (N = 2n + l).
For n ~ there is no dependence on the choice of the oscillator length.
Benchmarked by Complex Scaling calculations

14. Benchmarked against Complex Scaling (A. Kruppa et al.)
Calculations for 6He carried out in a large Slater basis 24
Operators must be transformed, e.g.,

15. GSM calculations for 6,8He nuclei
Im[k] (fm-1)
Re[k] (fm-1)
3.27
0p3/2 B
(0.17,-0.15) A
(2.0,0.0)
Model space: p-sd
0p3/2 resonance
i{p3/2} complex non-resonant part
i{s1/2}, i{p1/2}, i{d3/2}, i{d5/2} real continua
with i=1,…Nsh
Nsh=30 for p3/2 contour and Nsh=20 for each real continuum
111 shells total.
For 6He we used the s.p WS potential fitted to 5He s.p energies.
For 8He we used the spherical GHF+MMN basis
Modified Minnesota Interaction (MMN) (NPA 286, 53)
The strength parameters are adjusted to the g.s. binding energies of 6,8He.

16. Charge radii in COSM coordinates
Generalization to n-valence particles is straightforward:

17. The charge radii in Helium halos depend on:
Is the α-particle in the “sea” of valence neutrons, still an α-particle?
R.B.Wiringa and S.C Pieper, Annu.Rev.Nucl.Part.Sci. 51, 53 (2001) 4% 8%
→ “Swelling” of the core is not negligible
The point proton radii are converted to charge radii through:
Darwin-Foldy term (=0.033 fm2)
The spin-orbit density could have a non-negligible effect on the charge radius:
W. Bertozzi et al , PLB 41, 408 (1972); H.Esbensen et al PRC 76, (2007); A.Ong et al PRC 82, (2010)
The charge radii in Helium halos depend on:
The orbital motion of the core around the center of mass of the nucleus
The polarization of the core by the valence neutrons
The s.o contribution caused by the anomalous magnetic moment of the neutron

18. GSM+SDI for 6He ground state

19. Pairing Antihalo Effect
If pairing is present, the naïve
picture of halo changes:
K. Bennaceur et al., Phys. Lett. B496, 154 (2000)
pairing
no pairing
for odd-N
for even-N

21. Hagino and Sagawa: http://arxiv.org/abs/1105.5469v1

22. n 6He ground state Two-nucleon density distribution: total r q S=0
V. Kukulin et al., Nucl.Phys.A 453, 365 (1986).
G. Bertsch and H. Esbensen, Ann.Phys.(N.Y.) 209, 327 (1991).
M. V. Zhukov et al., Phys. Rep. 231, 151 (1993).
E. Nielsen et al., Phys. Rep. 347, 373 (2001).
K. Hagino and H. Sagawa, Phys. Rev. C 72, (2005).
total
S=0
S=1 n r q
6He ground state
Complex Scaling
Y. Kikuchi et al., Phys. Rev. C 81, (2010).

23. 6He ground state Full GSM (p-sd) p3/2 only Truncated GSM (p)
Two-neutron peak enhanced by the continuum coupling

24. 6He 2+ state GSM: [0.851, 0.109] MeV EXP: [0.822(25), 0.113(20)] MeV
extreme
s.p.
GSM, total
S=0
S=1

25. 6He Using Harmonic Oscillator states for high-j partial waves
Energies and radial properties are equivalent in both representations
The combination of Gamow states for low values of angular momentum and HO for higher, captures the relevant physics while keeping the size of the basis manageable
Applicable only with finite range forces (MMN)

26. GSM+DMRG+MMN application to 8He radius
(total GSM m-scheme dimension: 19,304,868)
Each contour discretized with 21 points; 10 HO states for both d waves:
84 shells total

27. 6He and 8He g.s. radii summary
significant
contribution
2.95 fm2 for 6He
2.32 fm2 for 8He

28. C.Bertulani and M.S. Hussein
Charge radii of 6,8He with MMN interaction
8He
Correlation angle for 6He
To be compared with
C.Bertulani and M.S. Hussein
PRC 76, (2007)
Angles estimated
from the available
B(E1) data and the
average distances between
neutrons.

29. Summary
Very precise measurements of 6,8He charge radii provide a valuable test of the configuration mixing and effective interactions in dripline nuclei
GSM applied to weakly bound or unbound nuclei. Many technical innovations, related to representation of non-resonant continua and two-body matrix elements, tested.
GSM results benchmarked against Complex Scaling with Slater basis
Using a finite range force (MMN) and adjusting the strengths to the g.s energies of 6,8He we provide a good description of measured charge radii.
Charge radii in 6,8He are primarily sensitive to the p3/2 occupation and S2n. Therefore, these data do not provide a “stringent test of ab-initio models”
The core polarization by the valence neutrons is small but NOT negligible. The same is true for the s.o contribution to the charge radii. Interestingly, thes two effects greatly cancel each other.
Two-neutron distribution functions computed for g.s and 2+ resonance in 6He have very different character. Continuum coupling changes relative weights of di-nuclear and cigar peaks in g.s. distribution.
Pairing antihalo effect seen in GSM calculations

30. Work in progress
Investigate the properties of Asymptotic Normalization Coefficients
Study exceptional points within the GSM
Develop the effective GSM finite-range interaction in the p-sd shell model interface
Develop the GSM-based reaction-theoretical framework
Use GSM to dipole and quadrupole molecules
Use the GSM formalism in ab-initio approaches

31. Backup

32. Rigged Hilbert Space: the natural framework to formulate
quantum mechanics
In mathematics, a rigged Hilbert space (Gel’fand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They can bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.
Mathematical foundations in the 1960s by Gel’fand et al. who combined Hilbert space
with the theory of distributions. Hence, the RHS, rather than the Hilbert space alone, is
the natural mathematical setting of Quantum Mechanics
I. M. Gel’fand and N. J. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic
Press, New York, 1964.
The resonance amplitude associated with the Gamow states is proportional to the complex delta function and such amplitude can be approximated in the near resonance region by the Breit-Wigner amplitude (Nucl. Phys. A812, 13 (2008)):
For a pedagogical description, see R. de la Madrid, Eur. J. Phys. 26, 287 (2005)

33. Density Matrix Renormalization Group
From the main configuration space all the |k>A are built (in J-coupled scheme)
Succesivelly we add states from the non resonant continuum state and construct
states |i>B
In the {|k>A|i>B}J the H is diagonalized
ΨJ=ΣCki {|k>A|i>B}J is picked by the overlap method
From the Cki we built the density matrix and the N_opt states are corresponding to the maximum eigenvalues of ρ.

34. Pairing and separation energy
Two-neutron separation energy S2n
One-neutron separation energy S1n
for odd-N
for even-N

35. The single-particle field characterized by l, determined by the p-h component of the effective interaction, and the pairing field D determined by the pairing part of the effective interaction are equally important when S1n is small.

36. The Density Matrix Renormalization Group (DMRG)
S.R White PRL 69 (1992) 2863
Open System extension: J.Rotureau et al : PRL 97, (2006); PRC 79, (2009)
Truncation Method applied to lattice models, spin chains, atomic nuclei….
Basic idea:
where A and B are partitions of the system.
 Approximate
in terms of m < M basis states (truncation)
 These m states are eigenstates of the density matrix or
 The difference between the exact and the approximated , has the minimal norm.
The partition of the system has to be decided by the practitioner.
In GSM+DMRG we
optimize the number
of non-resonant states
along the scattering
contours.

37. GSM+DMRG+MMN application to 8He radius
 Convergence properties of the DMRG are met for both radius and energy.
 DMRG converges on the right value. We compare a 2p-2h calculation with a full (4p-4h).
 The differences depict the model space effect on the observables (energy/radius).
The energy in DMRG is more attractive and the radius is smaller compared to the 2p-2h.
ε = 10-8

38. ε = 10-8 Density Matrix Renormalization Group application to 8He
p-sd shells (5 partial-waves) , 47 shells total.
Edmrg = MeV, EGSM = MeV  the system gained energy in DMRG as a result
of the larger model space (4p-4h). The difference is ~150keV.
Remember that EGSM (2p-2h) is the experimental value (MN was fitted in this way.)
Truncation in the DMRG sector is governed by the trace of the density matrix 
dim GSM =
ε = 10-8
dim DMRG = 3859

39. 0p3/2 state only Comment for slide 26:
This is the correlation density contour in the pole approximation (only 0p3/2)
In slide 26 the case that you mention as 0p3/2 state only is not the pole approximation.
Is a case that there is no 2-body interaction present. So, since no configuration is induced C2p3/2 = 1.
But this is a total amplitude i.e
(2 particles in 0p3/2 + 2 particles in non-resonant p3/2 continua + 1 in 0p3/2 and1 in non resonant p3/2 continua)

40. 6He g.s.

41. Effects of the space truncation on observables
For 8He we applied the DMRG algorithm to cope with huge dimensions
It is feasible to do GSM calculations allowing only 2 particles being excited (2p-2h truncation).
We fit the interaction to reproduce the g.s energy of 6,8He for the 2p-2h case:
MeV fm
(DMRG)
~ 150 keV difference in energy
0.004fm difference in radius
 We slightly renormalized the strength of the interaction in the full
calculation to take into account the effect of the increased model-space.