1. Beyond Landau-Migdal theory
M. Baldo
Istituto Nazionale di Fisica Nucleare, Sez. Catania, Italy
ECT*, May 2017

2. Plan of the talk . Landau theory in homogeneous systems.
. Microscopic realization in nuclear matter and
collective excitations.
. Finite nuclei : the Kohn-Sham method.
. Microscopic realization and collective excitations in nuclei.
. Beyond the Landau-Migdal theory : the single particle
structure and dynamics.
. Conclusions and prospects.

3. The jump at the Fermi surface
Free gas
Interacting gas
Expected
Actual calculations (BBG)
M.B. et al., PRC 41, 1748 (1990)

4. The basic formulae
imply the very existence of an energy functional E of the occupation numbers.
One of the main goal of Landau theory is the calculations of
the collective modes in terms of single particle energies and the
effective interaction

5. An explicit but simplified form of the functional is the Skyrme
functional, which depends only on density.
It is adjusted to reproduce the ground state binding.
rearrangement term

6. Microscopic realization of the functional : BHF

7. Comparing the interactions

8. Overview of superfluid gaps in homogeneous matter (below the crust)
We consider the region where neutron superfludity
can be neglected

9. Including the nuclear interaction and
neutrons in the normal phase
Nuclear interaction from BHF as Skyrme-like functional
monopolar approximation

10. From the pseudo-Goldstone
the sound mode
Pseudo-
Goldstone
Sound mode

11. Application to the collective excitations of homogeneous Neutron Star matter : Spectral functions

12. At twice saturation density

16. GOING TO FINITE SYSTEMS (NUCLEI)
NEW FEATURES
Finite size effects
Surface effects (gradient terms)
Long range correlations due to surface modes
Coupling between single particles and surface modes
ALL THAT CAN BE CONSIDERED BY IMPLEMENTIG
THE LANDAU – MIGDAL THEORY WITHIN THE
ENERGY DENSITY FUNCTIONAL METHOD

17. The simplest and most direct way to connect the infinite
homogeneous system to finite systems is to follow the
Kohn-Sham scheme.
One introduces a local density functional, that in the case of atoms and molecules reads

18. The approach is based on the Hohenberg-Kohn theorem.
One assumes
The exact functional can be approximated by a local one
The density can be written
Then
Iportant remark : the orbitals
are NOT the single particle levels
In particular the density matrix CANNOT be written as

19. A further assumption is that the correlation energy
is locally equal to the correlation energy of an electron
gas at the local density.
This is a local density approximation, which connects the
functional with the bare particle interaction and many-body
theory.
The approximation can be improved by adding gradient
corrections, then the functional will contain derivatives
of the density
In principle the expansion can be obtained from
microscopic calculations in non-homogeneous gas,
provided the density profile is smooth enough.
This is not the case in nuclei, and a more phenomenogical approach is advisable for the gradient expansion (or equivalent)

20. The nuclear Kohn-Sham functional can be then written
key quantity
The bulk part is taken from microscopic EOS
The rest is standard

22. Bulk part from microscopic nuclear matter EOS
The coefficients are NOT parameters to be fitted to nuclear data.
They are fixed by the microscopic EOS
M. Baldo, L.M. Robledo,P.Schuck and X.Vinas, Phys. Rev. C87, (2013)

23. The use of a polynomial for interpolating analytically the EOS
is convenient for at least two reasons
Integer powers of the density avoids some problem that one gets with fractional powers (self-interaction, regularization, ……)
The linear and quadratic terms of the EOS can be reproduced
directly by the surface term, thus reducing its free parameters
from 3 to 1 (which was taken as the range parameter )
and no subtraction term is necessary.
In practice the fit for symmetric nuclear matter was constrained to
get the saturation point exactly at
This requires only a small adjustement within the theoretical uncertainty

24. COMMENT
The reason of fine tuning the energy of the saturation
point is the extreme sensitivity of the results
on the precise value of this physical quantity.
One finds that it has to be fixed with an accuracy
substantially better than Ke V !
NO MICROSCOPIC THEORY can be so accurate
This sensitivity can be easily understood from the
fact that the energy/nucleon must multiplied by the
mass number A, that can be e.g
Finally notice that the effective mass was taken equal to 1,
as in the original Kohn-Sham formulation.
However modifications are possible

25. The other physical characteristics of the EOS as
obtained from the mcroscopic calculation.
Symmetry energy J = MeV
Slope of the S.E L = MeV
Incompressibility K = MeV
Skewness K’ = MeV
Second derivative of J KS = MeV
These values are compatible with the ‘experimental’
constraints, with the possible exception of KS, which however
is not well determined.

26. FITTING
* We considered the 579 nuclei of the Audi and Wapstrad
compilation, NPA 729, 337 (2003)
* Extrapolated mass were not included
* The only non-mean field corrections included in the fit is the rotational energy correction
* The quadrupole zero-point energy corrections
was NOT included.
We expect that, as with Gogny force, its inclusion
would reduce substantially the average deviation.

27. Sharp dependence on the range parameter
Connection with surface energy
These results suggest
that the optimal value
of the range parameter
is determined by the
right balance between
bulk and surface energies

28. Weak dependence on the spin-orbit parameter
Characterization of the parameters
Range parameter essentially free
Spin-orbit strength can be taken at standard values
Saturation point : fine tuned around the theoretical one
The pairing strength is not optimized but just taken at a fixed value
corresponding to the Gogny force, renormalized for the different
effective mass (the bare one in our case).

29. Relevance of deformations
Well deformed nuclei are ‘more mean field’.
Spherical nuclei need additional correlations beyond mean field,
in particular long range correlatios due to surface modes

30. Radii calculations
Once the bindings have been fitted, one can calculate the radius of each nucleus.
Radii are NOT fitted.

31. Some references
M. Baldo, P. Schuck and X. Vinas, Phys. Lett. B663, 390 (2008)
L.M. Robledo, M. Baldo, P. Schuck and X. Vinas, Phys. Rev. C77, (2008)
X. Vinas, L.M. Robledo, P. Schuck and M. Baldo, Int. J. of Mod. Phys. E18, 935 (2009)
L.M. Robledo, M. Baldo, P. Schuck and X. Vinas, Phys. Rev. C81, (2010)
M. Baldo, L.M. Robledo,P. Schuck and X. Vinas, J. of Phys. G 37, (2010)
M. Baldo, L.M. Robledo,P.Schuck and X.Vinas, Phys. Rev. C87, (2013)
B.K. Sharma, M. Centelles, X. Vinas, G.F. Burgio and M. Baldo, A&A 584, A103 (2015)
M. Baldo, L. Robledo, P. Schuck and X. Vinas, Phys. Rev. C95, (2017)

32. Estimating the monopole and quadrupole GR energies
From the functional one can derive the effective force and
calculate the excited states, which is one of the main goal of the
Landau-Migdal approach, in particular GMR and GQR.
Alternatively the centroids of GMR and GQR can be estimated from the energy weighted sum rules.
Since RPA fulfills the sum rules, these are equivalent
to RPA calculations.
In turn, the sum rules can be calculated by scaling or constrained
HF calculations
The two estimates give comparable results

33. The energy of the quadrupole giant resonance looks
slightly underestimated.
It is possible that this is due to the effective mass, which is taken at the bare value.
The effective mass can be introduced in the functional
without affecting the EOS
In this way to the interaction energy one adds
the kinetic energy correlation.
The effective mass can be taken from nuclear matter
calculations. It will be density dependent.
M.B., L. Robledo, P. Schuck and X. Vinas, PRC95, (2017)

34. GMR and GQR centroids
Overall trend
Comparing with data
Anomalous soft monopole in the
Sn isotopes region

35. BEYOND THE EFFECTIVE MEAN FIELD
A general problem common to EDF is the distribution
of the single particle energies, which shows clear
discrepancy with the phenomenologicalal data, beyond the
experimental uncertainty.
Two strategies are possible
Improving the EDF, eventually fitting also the single particle energies.
Go beyond the mean field, introducing correlations
in some many-body scheme.
The approach has a further motivation : one can
describes fragmentation and width of the single
particle states and of e.g. Giant Resonances.
Following line 2.

36. COMMENT
A generic EDF can be minimized through the Kohn-Sham
orbitals, which then satisfy Hartree-like equations.
All correlations are embodied in the EDF, and therefore
the orbitals cannot represent the physical single particle
states.
One can try to introduce correlations in the single
particle states going beyond the effective mean field.
Warning : if some correlations are introduced in the
single particle states, then the effective force (or EDF)
must be modified. In practice this means that it has
to be refitted.
This is not an easy task.

37. The distribution of the single particle strength can be identified
in general with the spectroscopic factor
which will be fragmented in different A+1 states at energies
for a given choice of the quantum numbers of j .
The spectroscopic factors can be extracted from ‘ab initio’
calculations, in particular shell model calculations with realistic
interactions, e.g. Kuo & Brown. The results can be compared
with data on direct trasfer reactions.
A pure single particle state is characterized by a strength
concentrated at the energy

38. Shell model calculations , 48 Ca
f 7/2
p 1/2
f 5/2
p 3/2
Martinez-Pinedo et al. , PRC 55, 187 (1997)

39. The observed shifts and fragmentations are typical
effects of the particle-vibration coupling.
Indeed the effective interaction or EDF should
include to a large extent the short range correlations
that are present in the nuclear system, what is
missing are mainly the long-range correlations
induced by the presence of the surface.
In fact the collective nuclear excitations involve the whole
nucleus and are therefore they are of long-range character.
Particle and phonon
degrees of freedom

40. These correlations are consistent with the RPA correlations
in the ground state. In fact the RPA scheme assumes that the
occupation numebers are different from 0 and 1, i.e. at least two particle-two hole excitations are contained in the g.s.

41. Effects of the coupling with phonons on the single particle levels
Fayans functional
The phonon is considered as a separate additional degrees of freedom
E.E. Saperstein, M.B., S.S. Pankratov and S.V. Tolokonnikov, JETPL 104, 609 (2016),
JETPL 104, 763 (2016).
E.E. Saperstein, M.B., N.V. Gnedzilov and S.V. Tolokonnikov, PRC 93, (2016)

42. Fragmentations of the proton single particle states in 204 Pb
Is it compatible with Landau-Migdal approach ?

43. A general method based on the functional derivative method.
The phonon is introduced microscopically from the
density-density propagator
M.B., P.F. Bortignon, G. Colo’, D. Rizzo and L. Sciacchitano,
JPG 42, (2015).

44. We are interested on the single particle self-energy
It has a static and a dynamical contribution
From the Dyson equation
the static part can be expressed also in terms of self-energy

45. At the second iteration. Dynamical part.

46. Static modification of the mean field
Variation of the density matrix
U =
‘ TADPOLE ‘
Gnezdilov et al., PRC 89, (2014)
Strong compensation ?

47. Microscopic structure of the phonon
Symmetry factor
Pauli principle

48. Skyrme force SV
Substantial contribution
of the bubble diagram
Skyrme force Sly5

49. CRITICAL POINTS
To go beyond the mean field enriches the nuclear structure
studies (fragmentation, spectroscopic factors, ….),
however it poses serious problems within the EDF approach,
in particular the refitting of the functional
In the particle-vibration coupling approach particular care
must be taken for the Pauli principle and statistical factors.
With zero range forces the problem of convergence arises
* Finite range. How to choose ?
* Maybe additional diagrams can help. Tadpole ?

50. OUTLOOK Improvement of the Landau-Migdal theory is possible
within the particle-vibration scheme
The approach introduces corrections to the single paricle levels
and their fragmentation
Even at lowest order both tadpole and bubble correction to the
one-phonon diagram are necessary
A general scheme to include fragmentation in the description of the
collective modes is still missing.

51. THANKS !!! Madrid L. Robledo COLLABORATORS Barcelona X. Vinas
B.K. Sharma
M. Centelles
Orsay P. Schuck
Moskow E.E. Saperstein
S.V. Tolokonnikov
S.S. Pankratov
N.V. Gnedzilov
Milano P.F. Bortignon
G. Colo’
L. Sciacchitano
D. Rizzo
COLLABORATORS
THANKS !!!

52. 40Ca – hole states RED = M(ω) BLUE = Mbubble(ω)
YELLOW = M(ω) -Mbubble(ω)
Different behaviour for low and high angular momentum states.
In the case of L = 2 states there seem to be convergence.
D. Rizzo, M.Sc. Thesis, University of Milano (unpublished)

53. 40Ca – particle states RED = M(ω) BLUE = Mbubble(ω)
YELLOW = M(ω) -Mbubble(ω)
Different behaviour for low and high angular momentum states.
In the case of L = 3 states there seem to be convergence.