Beyond Landau-Migdal theory

Transcription

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 to the sound mode D = 0.5 MeV Pseudo- Goldstone Sound mode

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

12 At twice saturation density

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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 The bulk part is taken from microscopic EOS The rest is standard key quantity

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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 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 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 Slope of the S.E. Incompressibility Skewness J = MeV L = MeV K = MeV 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 Fine tuning of E/A at saturation E/A = MeV E/A = MeV Optimal value E/A = MeV

28 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

29 Results for the extrapoleted masses (193, Audi s compilation)

30 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).

31 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

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

33 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)

34 Relevance of the details of the EOS Quadratic form of the interaction energy

35 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

36 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)

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

38 Summarizing The aim of the work is not to get an EDF that can present an accuracy better than the most performing EDF (e.g. HFB by Goriely, Chamel, et al.). We want to produce an EDF that is accurate in reproducing the smooth part of the binding energy throughout the mass table with a minimal number of parameters and keeping fixed the bulk part, extracted from the Nuclear Matter EOS. This should be a procedure that is more reliable for the extrapolation to very exotic nuclei (to be checked by comparing with other EDF obtained along the same lines) Indeed fluctuations of the masses beyond regular shell effects are associated with classical chaotic motion (Bohigas & Leboeuf, PRL 2002) and are essentially random. They should not be reproduced by the fitting procedure.

39 A section (schematic) of a neutron star Application of BCPM to Astrophysics

40 Structure of the Neutron Star crust The densities involved are typical nuclear densities Physical conditions quite different B.K. Sharma, M. Centelles, X. Vinas, G.F. Burgio, M.B. Astronomy & Astophysics 584, A103 (2015).

41 The constraint from the observed maximum mass.

42 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 1. Improving the EDF, eventually fitting also the single particle energies. 2. 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.

43 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.

44 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

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

46 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

47 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.

48 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)

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

50 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).

51 H = H 0 + V e X i " i a i a i + The many-body approach 2 H Equations for G, M, W,, can be derived The set of equations for these quantities has been derived in the famous paper(s) by L. Hedin for the Coulomb force. PVC is a well-defined approximation scheme (first iteration of the Hedin s equations). We have extended these equations in the case of density dependent interactions. The validity of this scheme when a functional is given can be discussed. L. Schiacchitano, M.Sc. Thesis, University of Milano (unpublished); M. Baldo et al., JPG 42, (2015)

52 (1, 2, 3) = (1, 2) (1, 3) + Hedin s equation G(1, 2) = G 0 (1, 2) + G 0 (1, 3)M(3, 4)G(4, 2) M(1, 2) = ig(1, 3) (3, 2, 4)W (4, 1) W (1, 2) = (1, 2) + (1, 3) (3, 4)W (4, 2) (1, 2) = ig(1, 3)G(4, 1) (3, 4, 2) M(1, 2) G(4, 5) G(4, 6)G(7, 5) (6, 7, 3) G = Green s function, M = self-energy, W = induced interaction, Π = polarization propagator, Γ = vertex function

53 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

54 At the second iteration. Dynamical part.

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

56 Microscopic structure of the phonon Symmetry factor Pauli principle

57 Skyrme force SV Substantial contribution of the bubble diagram Skyrme force Sly5

58 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?

59 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.

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

61 40 Ca hole states RED = M(ω) BLUE = M bubble (ω) YELLOW = M(ω) - M bubble (ω) Different behaviour for low and high angular momentum states. In the case of L = 2 states there seem to be convergence. " max p D. Rizzo, M.Sc. Thesis, University of Milano (unpublished)

62 40 Ca particle states RED = M(ω) BLUE = M bubble (ω) YELLOW = M(ω) - M bubble (ω) Different behaviour for low and high angular momentum states. In the case of L = 3 states there seem to be convergence. " max p

63 This is a general problem in performing calculations in the particle-vibration coupling If one adopts an effective interaction that includes zero-range terms, then the result is diverging The problem can be cured considering effective interaction with a finite range, as the version of the Gogny force developed in Bruyeres-le-Chatel (it should be compatible with NM EOS) Question : what is the meaning of the actual value of the range?

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