Transcript Document

Microscopic particle-vibration coupling
models
G. Colò
YIPQS Long- term workshop Dynamics and Correlations in Ex otic Nuclei (DCEN2011)
YIPQS Long-term workshop
Dynamics and Correlations in Exotic Nuclei (DCEN2011)
20th September - 28th October, 2011
Yukawa Institute for Theoretical Physics, Kyoto, Japan
9/ 20/ 11 3:34 AM
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
E   Hˆ    Hˆ eff   Eˆ 
 Slater determinant  ˆ 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
Skyrme effective force
short-range repulsion
(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.
NPA 553, 297c (1993)
M. Stoitsov et al., PRC 82, 054307
(2010).
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 potential is not energy-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:
In the Dyson equation
ε + <Σ(ε)>
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
• APPROXIMATIONS AND PHENOMENOLOGICAL INPUTS HAVE
BEEN OFTEN INTRODUCED IN THESE THEORIES.
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)
• QUALITATIVELY, ALL THE CALCULATIONS HAVE PREDICTED A
REDUCTION OF THE S.P. GAP E.G. IN 208Pb. (m*/m from ≈ 0.7 to 1).
• 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
• Few !
microscopic Vph
• 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).
• 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.
• Veff ?
• For electron systems it is possible to start from the bare Coulomb force:
+
…
+
G
=
W
Phys. Stat. Sol. 10, 3365 (2006)
• In the nuclear case, the bare VNN does not describe well vibrations !
P. Papakonstantinou
et al., Phys. Rev. C
75, 014310 (2006)
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).
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%.
208Pb
(neutron states) – SLy5
HF+ t ensor (SLy5)
PVC (SLy5)
PVC (T44)
PVC (T44+ t ensor)
σ
1.51
1.21
1.09
1.03
• Further steps: re-fitting of the force and/or study of
higher-order processes.
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.
1st order
2nd order
direct
exchange
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)