Microscopic particle-vibration coupling models

G. Colò

Co-workers

• K. Moghrabi, M. Grasso, N. Van Giai (IPN-Orsay, France)

• H. Sagawa (University of Aizu, Japan)

• L. Cao (Institute of Modern Physics, Chinese Academy of Science, Lanzhou, China)

• X. Roca-Maza, P.F. Bortignon (University of Milano, Italy)

See also next talk by K. Mizuyama …

Energy density functionals (EDFs) for nuclei

EE effHH

Slater determinant 1-body density matrix• 8-10 free parameters (typically). Skyrme/Gogny vs. RMF/RHF.

• Large domain of applicability, up to the case of uniform matter/neutron stars (g.s. energies, nuclear vibrations and rotations).

attraction

short-range repulsion

Skyrme effective force

(Some) limitations of EDFs• Single-particle states and their spectroscopic factors (S) - They do not belong to the DFT framework (by definition).

• Widths of GRs and other excited states.

M. Stoitsov et al., PRC 82, 054307 (2010).

NPA 553, 297c (1993)

EDF = the potential is not energy-dependent

The equation for the self-energy (Dyson equation) reads

and the exact expression for the one-body Green’s function is

A set of closed equations for G, Π, W, Σ, Γ can be written (v12 given). They can be found e.g. in the famous paper(s) by L. Hedin in the case of the Coulomb force – they hold more generally. (Open question: density-dependent two-body forces ?).

EDFs vs. many-body approaches

In the Dyson equation

we assume the self-energy is given by the coupling with RPA vibrations

In a diagrammatic way

2nd order PT:

ε + <Σ(ε)>

+ + … =

Particle-vibration coupling

Particle-vibration coupling (PVC) for nuclei

Density vibrations are the most prominent feature of the low-lying spectrum of spherical systems

• APPROXIMATIONS AND PHENOMENOLOGICAL INPUTS HAVE BEEN OFTEN INTRODUCED IN THESE THEORIES.

• QUALITATIVELY, ALL THE CALCULATIONS HAVE PREDICTED A REDUCTION OF THE S.P. GAP E.G. IN 208Pb. (m*/m from ≈ 0.7 to 1).

E.g., in the original Bohr-Mottelson model, the phonons are treated as fluctuations of the mean field δU and their properties are taken from experiment.

C. Mahaux et al., Phys. Rep. 120, 1 (1985)

• Microscopic calculations are now feasible. One starts from Hartree or Hartree-Fock with Veff, by assuming this includes short-range correlations, and add PVC on top of it. All is calculated using the same Hamiltonian or EDF consistently.

RPA

microscopic Vph• Few !

• RMF + PVC calculations have been done first by E. Litvinova and P. Ring. More results along this line have been presented in this workshop by A. Afanasjev.

•Pioneering Skyrme calculation by V. Bernard and N. Van Giai in the 80s (neglect of the velocity-dependent part of Veff in the PVC vertex).

• Veff ?

P. Papakonstantinou et al., Phys. Rev. C 75, 014310 (2006)

• For electron systems it is possible to start from the bare Coulomb force:

• In the nuclear case, the bare VNN does not describe well vibrations !

Phys. Stat. Sol. 10, 3365 (2006)

+ … + =

W

G

We have implemented a version of PVC in which the treatment of the coupling is exact, namely we do not wish to make any approximation in the vertex.

The whole phonon wavefunction is considered, and all the terms of the Skyrme force enter the p-h matrix elements

A consistent study within the Skyrme framework

Our main result: the (t1,t2) part of Skyrme tend to cancel quite significantly the (t0,t3) part.

We compare perturbation theory and full diagonalization of H0 + HPVC.

GC, H. Sagawa, P.F. Bortignon, PRC 82, 064307 (2010).

Diagonalization of the PVC Hamiltonian

We start from the basis made up with particles (or holes) around a core, and with vibrations of the same core (i.e., phonons).

H on this basis

Beyond the second order approximation.

Relationship with the shell-model (or configuration-interaction) formulation. GCM ?

40Ca (neutron states) – SLy5 • The tensor contribution is in this case negligible, whereas the PVC provides energy shifts of the order of MeV.

• The r.m.s. difference between experiment and theory is:

σ(HF+tensor) = 0.95 MeV

σ(PVC) = 0.59 MeV• If we express the average of the absolute values of the difference with experiment:

Δ(HF+tensor) = 1.07 MeV

Δ(PVC) = 0.50 MeV

• The reproduction of the experimental properties of the low lying vibrations is, of course, crucial. In some cases, SLy5 gives ≈ 30% discrepancies. Some of the interactions by T. Lesinski et al. are accurate at the level of 10-20%.

• Further steps: re-fitting of the force and/or study of higher-order processes.

208Pb (neutron states) – SLy5

Zero-range forces and ultraviolet divergences

+ + … =

We start from the divergences of “prototype” diagrams, corresponding to the second-order corrections to the energy.

We consider, from now on, the case of uniform systems (momentum labels). We study E/A = E/A(HF) + ΔE/A.

direct exchange

1st order

2nd order

Aim of our work: renormalizing this divergence

We include a momentum cutoff Λ among the parameters of the interaction, and we show that for every value of Λ the remaining parameters can be determined in such a way that the total energy of the system remains the same.

The idea is similar to that of renormalization.

Formulas are general. Useful for atomic gases !

Numerical application is for nuclear matter.

A simplified Skyrme force is employed (t0,t3).

Our benchmark is the EOS obtained with the set SkP.

• The divergence is studied in detail for different densities.

• For every Λ we build a new SkPΛ such that the EOS does not change.

• Note that the interest is not only for Skyrme practitioners. The Gogny force has also a contact term. Even with genuine finite range forces, one may be interested in considering second-order effects in a more limited space than that implied by the natural cutoff.

• Note also that the present technique is different from the one employed in the case of the pairing channel.

Conclusions

• Microscopic particle-vibration coupling calculations are now available - based on the self-consistent use of nonrelativistic or covariant functionals.

• Results for single-particle states are improved compared to mean-field.

• It would be reasonable to re-fit coupling parameters if one introduces particle-vibration coupling (or, more generally, if one goes beyond mean-field).

• In the case of zero-range interactions, one has first to handle divergences !

• We have developed a strategy for the renormalization of these divergences. It may help solving various problems (e.g., RPA correlation energies also diverge).

Higher-order terms in DFT to mimick PVC ?

ωn

Since the phonon wavefunction is associated to variations (i.e., derivatives) of the denisity, one could make a STATIC approximation of the PVC by inserting terms with higher densities in the EDF.

Extra slides

A reminder on effective mass(es)

E-mass: m/mE k-mass: m/mk

A+Σ(E) B

-B -A-Σ*(-E)

Σphp’h’ (E) = Σα Vph,α(E-Eα+iη)-1Vα,p’h’

The state α is not a 2p-2h state but 1p-1h plus one phonon

Σphp’h’(E) =

Pauli principle !

Re and Im Σ

cf. G.F.Bertsch et al., RMP 55 (1983) 287

N. Paar, D. Vretenar, E. Khan, G.C., Rep. Prog. Phys. 70, 691 (2007)