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Microscopic particle-vibration coupling models

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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 …

1-body density matrix

Slater determinant

Energy density functionals (EDFs) for nuclei

- 8-10 free parameters (typically). Skyrme/Gogny vs. RMF/RHF.
- Large domain of applicability, up to the case of uniformmatter/neutronstars (g.s. energies, nuclearvibrations and rotations).

Skyrme effective force

attraction

short-range repulsion

NPA 553, 297c (1993)

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

(Some) limitations of EDFs

- Single-particlestates and theirspectroscopicfactors (S) - They do notbelong to the DFT framework (by definition).
- Widths of GRs and otherexcitedstates.

EDFs vs. many-body approaches

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

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

EDF = the potentialisnotenergy-dependent

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 ?).

+ + … =

Particle-vibration coupling (PVC) for nuclei

2nd order PT:

ε + <Σ(ε)>

In the Dyson equation

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

In a diagrammatic way

Particle-vibration coupling

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

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

- 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.

RPA

microscopic Vph

- Microscopiccalculations are nowfeasible. Onestarts from Hartree or Hartree-Fock with Veff, by assumingthisincludes short-rangecorrelations, and add PVC on top of it. Alliscalculatedusing the sameHamiltonian or EDF consistently.

- Few !

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

- RMF + PVC calculationshavebeendone first by E. Litvinova and P. Ring. More resultsalongthis line havebeenpresented in this workshop by A. Afanasjev.

- Veff ?

+ … + =

G

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

W

- 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)

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A consistent study within the Skyrme framework

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

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).

H on this basis

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).

Beyond the secondorderapproximation.

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

- 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

40Ca (neutron states) – SLy5

- 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%.

208Pb (neutron states) – SLy5

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

1st order

2nd order

direct

exchange

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.

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

Our benchmark is the EOS obtained with the set SkP.

Aim of our work: renormalizing this divergence

We include a momentum cutoff Λamong the parameters of the interaction, and we show that for every value of Λthe 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.

- 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 interestisnotonly for Skyrmepractitioners. The Gogny force hasalso a contactterm. Even with genuine finite rangeforces, onemay be interested in consideringsecond-ordereffects in a more limitedspacethanthatimplied by the naturalcutoff.
- Note alsothat the presenttechniqueisdifferent from the oneemployed in the case of the pairingchannel.

Conclusions

- Microscopicparticle-vibrationcouplingcalculations are nowavailable - based on the self-consistent use of nonrelativistic or covariantfunctionals.
- Results for single-particlestates are improvedcompared to mean-field.
- Itwould be reasonable to re-fitcouplingparametersifoneintroducesparticle-vibrationcoupling (or, more generally, ifonegoesbeyondmean-field).
- In the case of zero-rangeinteractions, onehas first to handledivergences !
- Wehavedeveloped a strategy for the renormalization of thesedivergences. Itmay help solvingvariousproblems (e.g., RPA correlationenergiesalso diverge).

Higher-order terms in DFT to mimick PVC ?

ωn

Since the phononwavefunctionisassociated to variations (i.e., derivatives) of the denisity, onecouldmake a STATICapproximation of the PVC by insertingterms with higherdensities in the EDF.

Extra slides

A reminder on effective mass(es)

E-mass: m/mE

k-mass: m/mk

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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)