Beyond Landau-Migdal theory

1. Beyond Landau-Migdal theory M. Baldo Istituto Nazionale di Fisica Nucleare, Sez. Catania, Italy ECT*, 22-26 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 to 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 mostdirect way to connect the infinite homogeneoussystem to finite systemsis to follow the Kohn-Shamscheme. Oneintroduces a localdensityfunctional, that in the case of atoms and moleculesreads

18. The approachisbased on the Hohenberg-Kohntheorem. • Oneassumes • The exactfunctional can be approximated by a localone • 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 furtherassumptionisthat the correlationenergy islocallyequal to the correlationenergy of an electron gas at the localdensity. Thisis a localdensityapproximation, whichconnects the functional with the bare particleinteraction and many-body theory. The approximation can be improved by addinggradient corrections, then the functionalwillcontainderivatives of the density In principle the expansion can be obtained from microscopiccalculations in non-homogeneous gas, provided the densityprofileissmoothenough. Thisisnot the case in nuclei, and a more phenomenogicalapproachisadvisable for the gradientexpansion (or equivalent)

20. The nuclearKohn-Shamfunctional can be thenwritten key quantity The bulk part istaken from microscopic EOS The restis standard

22. Bulk part from microscopicnuclearmatter 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, 064305 (2013)

23. The use of a polynomial for interpolatinganalytically the EOS • isconvenient for atleasttworeasons • Integerpowers of the densityavoids some problemthatonegets with fractionalpowers (self-interaction, regularization, ……) • The linear and quadraticterms of the EOS can be reproduced • directly by the surfaceterm, thusreducingits free parameters • from 3 to 1 (whichwastakenas the rangeparameter ) • and no subtractiontermisnecessary. In practice the fit for symmetricnuclearmatterwasconstrained to get the saturationpointexactlyat Thisrequiresonly a small adjustementwithin the theoreticaluncertainty

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 100 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. 200. Finallynoticethat the effective mass wastakenequal to 1, as in the originalKohn-Shamformulation. Howevermodifications are possible

25. The otherphysicalcharacteristics of the EOS as obtained from the mcroscopiccalculation. Symmetryenergy J = 31.90 MeV Slope of the S.E. L = 52.96 MeV Incompressibility K = 212.4 MeV Skewness K’ = 879.6 MeV Second derivative of J KS = -96.75 MeV Thesevalues are compatible with the ‘experimental’ constraints, with the possibleexception of KS, whichhowever isnotwelldetermined.

26. FITTING * Weconsidered the 579 nuclei of the Audi and Wapstrad compilation, NPA 729, 337 (2003) * Extrapolated mass werenotincluded * The only non-meanfieldcorrectionsincluded in the fitis the rotationalenergycorrection * The quadrupole zero-pointenergycorrections was NOT included. Weexpectthat, as with Gogny force, itsinclusion would reduce substantially the averagedeviation.

27. Sharp dependence on the range parameter Connection with surfaceenergy Theseresultssuggest that the optimalvalue of the rangeparameter isdetermined by the right balance between bulk and surfaceenergies

28. Weak dependence on the spin-orbit parameter • Characterization of the parameters • Rangeparameteressentially free • Spin-orbitstrength can be takenat standard values • Saturationpoint : fine tunedaround the theoreticalone The pairingstrengthisnotoptimizedbut just takenat a fixedvalue corresponding to the Gogny force, renormalized for the different effective mass (the bare one in our case).

29. Relevance of deformations Welldeformed nuclei are ‘more meanfield’. Spherical nuclei needadditionalcorrelationsbeyondmeanfield, in particular long rangecorrelatios due to surfacemodes

30. Radii calculations Once the bindingshavebeenfitted, one can calculate the radius of eachnucleus. 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, 051301 (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, 034315 (2010) M. Baldo, L.M. Robledo,P. Schuck and X. Vinas, J. of Phys. G 37, 064015 (2010) M. Baldo, L.M. Robledo,P.Schuck and X.Vinas, Phys. Rev. C87, 064305 (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, 014318 (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 giantresonancelooks slightlyunderestimated. Itispossiblethatthisis due to the effective mass, whichistakenat the bare value. The effective mass can be introduced in the functional withoutaffecting the EOS In this way to the interactionenergyoneadds the kineticenergycorrelation. The effective mass can be taken from nuclearmatter calculations. Itwill be densitydependent. M.B., L. Robledo, P. Schuck and X. Vinas, PRC95, 014318 (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 particleenergies, which shows clear • discrepancy with the phenomenologicalal data, beyond the • experimentaluncertainty. • Twostrategies are possible • Improving the EDF, eventuallyfittingalso the single particleenergies. • Go beyond the meanfield, introducingcorrelations • in some many-body scheme. • The approachhas a furthermotivation : one can • describesfragmentation and width of the single • particlestates and of e.g. GiantResonances. Following line 2.

36. COMMENT A generic EDF can be minimizedthrough the Kohn-Sham orbitals, whichthensatisfyHartree-likeequations. Allcorrelations are embodied in the EDF, and therefore the orbitalscannotrepresent the physical single particle states. One can try to introduce correlations in the single particlestatesgoingbeyond the effectivemeanfield. Warning : if some correlations are introduced in the single particlestates, then the effective force (or EDF) must be modified. In practicethismeansthatithas to be refitted. Thisisnot 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 observedshifts and fragmentations are typical effects of the particle-vibrationcoupling. Indeed the effectiveinteraction or EDF should include to a large extent the short rangecorrelations that are present in the nuclearsystem, whatis missing are mainly the long-rangecorrelations induced by the presence of the surface. In fact the collectivenuclearexcitations involve the whole nucleus and are thereforethey are of long-rangecharacter. 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, 034312 (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, 085109 (2015).

44. We are interested on the single particle self-energy Ithas a static and a dynamicalcontribution From the Dyson equation the static part can be expressedalso in terms of self-energy

45. At the second iteration. Dynamical part.

46. Static modification of the mean field Variation of the densitymatrix U = + . . . . . . ‘ TADPOLE ‘ Gnezdilov et al., PRC 89, 034304 (2014) Strong compensation ?

47. Microscopicstructure of the phonon Symmetry factor Pauli principle

48. Skyrme force SV Substantialcontribution of the bubblediagram Skyrme force Sly5

49. CRITICAL POINTS To go beyond the meanfieldenriches the nuclearstructure studies (fragmentation, spectroscopicfactors, ….), howeveritposesseriousproblemswithin the EDF approach, in particular the refitting of the functional In the particle-vibrationcouplingapproachparticular care must be taken for the Pauli principle and statisticalfactors. With zero rangeforces the problem of convergencearises * Finite range. How to choose ? * Maybeadditionaldiagrams 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.