Localized form of Fock terms in nuclear covariant density functional theory

Transcription

1 Computational Advances in Nuclear and Hadron Physics September 21 October 3, 215, YITP, Kyoto, Japan Localized form of Fock terms in nuclear covariant density functional theory Haozhao Liang ùíî RIKEN Nishina Center, Japan October 21, 215 Original idea HZL, P.W. Zhao, P. Ring, X. Roca-Maza, J. Meng, PRC 86, 2132(R) (212)

2 Outline 1 Introduction 2 Theoretical Framework Relativistic Hartree-Fock theory Zero-range reduction Fierz transformation 3 Results and Discussion Coupling strengths in different channels Dirac mass splitting Spin-isospin resonances 4 A new fitting 5 Summary and Perspectives

3 Many-body systems and density functional theories Research on quantum mechanical many-body problems is essential in many areas of modern physics electrons in metal, atoms in molecule, electrons in atom, nucleons in nucleus... Density functional theories (DFT) Hohenberg & Kohn:1964 reducing the many-body problems formulated in terms of N-particle wave functions to the one-particle level with the local density distribution ρ(r) no other method achieves comparable accuracy at the same computational cost Kohn-Sham scheme Kohn & Sham:1965 for any interacting system, there exists a local single-particle (Kohn-Sham) potential v KS (r), such that the exact ground-state density of the interacting system can be reproduced by non-interacting particles moving in this local potential: ρ(r) = ρ KS (r) occ i φ i (r) 2 system energy density functional: E[ρ(r)] = T [ρ(r)] + E ext. [ρ(r)] + E H [ρ(r)] + E xc [ρ(r)]

4 Nuclear DFT Nuclear DFT is a promising tool for investigating the ground-state and excited state properties of nuclei throughout the nuclear chart. Since the 197s, lots of experience have been accumulated in implementing, adjusting, and using the DFT in nuclei. Petkov&Stoitsov:1991, Bender:23, Nakatsukasa:215,...

5 Covariant density functional theory RH theory The covariant version of DFT takes into account Lorentz symmetry. stringent restrictions on the number of parameters CDFT in Hartree level (RH/RMF theory) has received wide attention due to its successful description of lots of nuclear phenomena. Serot:1986, Ring:1996, Vretenar:25, Meng:26, Paar:27, Nikšić:211; Meng & Zhou, JPG 42, 9311 (215) spin-orbit splittings, pseudospin symmetry HZL, Meng, Zhou, Phys. Rep. 57, 1 84 (215) EoS in symmetric and asymmetric nuclear matter ground-state properties of finite spherical and deformed nuclei collective rotational and vibrational excitations low-lying spectra of transitional nuclei involving quantum phase transitions... Something more: the isovector channels Difficult to disentangle the isovector-scalar (δ) and isovector-vector (ρ) channels, unless a tuning is performed based on selected microscopic calculations. Roca-Maza:211 Nuclear spin-isospin resonances, e.g., GTR and SDR, cannot be described in a fully self-consistent way.

6 RH+RPA for spin-isospin resonances RH+RPA for spin-isospin resonances De Conti:1998, 2, Vretenar: 23, Ma:24, Paar:24, Nik sić:25 example: Gamow-Teller resonance (GTR) in 28 Pb ( S = 1, L =, J π = 1 + ) a. add π-meson b. fit g full self-consistency is missing

7 Covariant density functional theory RHF theory CDFT in Hartree-Fock level (RHF theory) several attempts to include the Fock term in the relativistic framework Bouyssy:1985,1987, Bernardos:1993, Marcos:24 DDRHF theory achieved quantitative descriptions of binding energies and radii Long, Giai, Meng, PLB 64, 15 (26); Long, Sagawa, Giai, Meng, PRC 76, (27); Long, Sagawa, Meng, Giai, EPL 82, 121 (28); Long, Ring, Giai, Meng, PRC 81, 2438 (21) effective mass splitting in asymmetric nuclear matter can be described naturally Long, Giai, Meng, PLB 64, 15 (26) nuclear spin-isospin resonances can be described in a fully self-consistent way HZL, Giai, Meng, PRL 11, (28); HZL, Giai, Meng, PRC 79, (29); HZL, Zhao, Meng, PRC 85, 6432 (212)

8 RHF+RPA for Gamow-Teller resonances Gamow-Teller resonances in 48 Ca, 9 Zr, and 28 Pb H F e x p. 4 8 C a G T R e x p. 9 Z r G T R R - ( 1 / M e V ) e x p. P K O 1 R - ( 1 / M e V ) H F e x p. P K O E ( M e V ) E ( M e V ) GTR excitation energies can be reproduced in a fully self-consistent way. cf. Skyrme functional SAMi 2 1 HZL, Giai, Meng, PRL 11, (28)

9 GTR excitation energies and strength GTR excitation energies in MeV and strength in percentage of the 3(N Z) sum rule within the RHF+RPA framework. Experimental and the RH+RPA results are given for comparison. HZL, Giai, Meng, PRL 11, (28) 48 Ca 9 Zr 28 Pb energy strength energy strength energy strength experiment ± ± RHF+RPA PKO PKO PKO RH+RPA DD-ME The pion is not included in PKO2.

10 Physical mechanisms of GTR RH+RPA no contribution from isoscalar mesons (σ, ω), because exchange terms are missing. π-meson is dominant in this resonance. g has to be refitted to reproduce the experimental data RHF+RPA isoscalar mesons (σ, ω) play an essential role via the exchange terms. π-meson plays a minor role. g = 1/3 is kept for self-consistency. HZL, Giai, Meng, PRL 11, (28)

11 Covariant density functional theory RHF theory CDFT in Hartree-Fock level (RHF theory) several attempts to include the Fock term in the relativistic framework Bouyssy:1985,1987, Bernardos:1993, Marcos:24 DDRHF theory achieved quantitative descriptions of binding energies and radii Long, Giai, Meng, PLB 64, 15 (26); Long, Sagawa, Giai, Meng, PRC 76, (27); Long, Sagawa, Meng, Giai, EPL 82, 121 (28); Long, Ring, Giai, Meng, PRC 81, 2438 (21) effective mass splitting in asymmetric nuclear matter can be described naturally Long, Giai, Meng, PLB 64, 15 (26) nuclear spin-isospin resonances can be described in a fully self-consistent way HZL, Giai, Meng, PRL 11, (28); HZL, Giai, Meng, PRC 79, (29); HZL, Zhao, Meng, PRC 85, 6432 (212) RHF includes non-local potentials v HF (r, r ), the simplicity of KS scheme is lost. RHF is much more complicated than RH theory. The computational cost is too expensive for including paring, deformation, projection, cranking,...

12 To construct RH functionals from RHF scheme It is therefore highly desirable to stay within the conventional Kohn-Sham scheme in nuclear physics to find a covariant density functional based on only local potentials, yet keeping the merits of the exchange terms Possible/promising solution: construct RH functionals from RHF scheme take the constraints introduced by exchange terms of RHF scheme into account We start from an important observation: RHF functional PKO2 [Long:28] well describes the neutron-proton Dirac mass splitting in asymmetric nuclear matter and nuclear spin-isospin resonances only includes σ-, ω-, ρ-mesons, but not π-meson masses of mesons are heavy zero-range approximation is reasonable Fierz transformation: Fock terms local Hartree terms

13 In this work To construct RH functional from RHF scheme by the following procedure start with RHF parametrization PKO2 perform the zero-range reduction perform the Fierz transformation With such RH functional thus obtained, to investigate proton-neutron Dirac mass splitting in neutron matter Gamow-Teller and spin-dipole resonances Goal(s) To verify whether the important effects of exchange terms can be maintained by the mapping from RHF functional to RH functional.

14 Relativistic Hartree-Fock theory Covariant density functional theory RHF theory Effective Lagrangian density Bouyssy:1987, Long:26 ( L = [iγ ψ µ µ M g σ σ γ µ g ω ω µ + g ρ τ ρ µ + e 1 τ )] 3 A µ ψ µ σ µ σ 1 2 m2 σσ Ωµν Ω µν m2 ωω µ ω µ 1 4 R µν R µν m2 ρ ρ µ ρ µ 1 4 F µν F µν (1) System Hamiltonian H = T = L φ i φi L (2) Ground-state trial wave function Φ = a c a (3) Energy functional of the system E = Φ H Φ = E k + E D σ + E D ω + E D ρ + E D A +E E σ + E E ω + E E ρ + E E A (4)

15 Zero-range reduction Zero-range reduction Yukawa propagators of the mesons for m i q, D i (r, r ) = 1 e m i r r 4π r r, D i(q) = 1 m 2 i + q 2 (5) D i (q) 1 m 2 i q2 m 4 i + D i (r, r ) 1 m 2 i δ(r r ) (6) within the zero-order approximation. Zero-range reduction of meson-nucleon couplings α HF S = g σ 2, α HF mσ 2 V = g ω 2, α HF mω 2 tv = g ρ 2, (7) mρ 2

16 Fierz transformation Fierz transformation (I) Sixteen Dirac matrices form a complete system O S = 1, O V = γ µ, O T = σ µν, O PS = γ 5, O PV = γ 5 γ µ so that any one can be expressed as a linear superposition of variants with a changed sequence of spinors, (āo i d)( co i b) = k c ik (āo k b)( co k d), (8) with the coefficients c ik in the so-called Fierz table Fierz:1937, Okun:1982, Sulaksono:23 S 1 4 S V T PS PV V T PS PV (9) For the isospin coefficients, δ qa q d δ qc q b = 1 2 [ δqa q b δ qc q d + q a τ q b q c τ q d ], (1a) q a τ q d q c τ q b = 1 2 [ 3δqa q b δ qc q d q a τ q b q c τ q d ]. (1b)

17 Fierz transformation Fierz transformation (II) Fierz transformation: from α HF to α H α H S = αhf S 4 8 αhf V 12 8 αhf tv (11a) α H ts = 1 8 αhf S 4 8 αhf V αhf tv (11b) α H V = 1 8 αhf S αhf V αhf tv (11c) α H tv = 1 8 αhf S αt H = 1 16 αhf S αtt H = 1 16 αhf S α H PS = 1 8 αhf S αhf V αhf V αhf tv αhf tv (11d) (11e) (11f) (11g) α H tps = 1 8 αhf S αhf V 4 8 αhf tv (11h) α H PV = αhf S αhf V αhf tv (11i) α H tpv = αhf S αhf V 2 8 αhf tv (11j)

18 Coupling strengths in different channels Nucleon-meson coupling strengths of PKO2 g Starting point: nucleon-meson coupling strengths g σ, g ω, and g ρ of PKO2 P K O g P K O g P K O Long, Sagawa, Meng, Giai, EPL 82, 121 (28) Zero-range reduction α HF S = g σ 2, α HF mσ 2 V = g ω 2, α HF mω 2 tv = g ρ 2, mρ 2 Fierz transformation α H j = k c jk α HF k,

19 Coupling strengths in different channels RHF equivalent zero-range coupling strengths (I) The RHF equivalent parametrization derived from PKO2 is called PKO2-H S V P K O 2 -H D D -M E 2 S ( f m 2 ) P K O 2 -H D D -M E 2 V ( f m 2 ) 1 t S ( f m 2 ) ts P K O 2 -H t V ( f m 2 ) tv P K O 2 -H D D -M E αs H and αh V are consistent with those of DD-ME2 Lalazissis:25 αts H appears proton-neutron Dirac mass splitting in asymmetric nuclear matter is modified for another delicate balance between ts and tv channels α H tv

20 Dirac mass splitting Dirac mass and its isospin splitting M * D /M P K O 2 - H p r o to n n e u tr o n M * D /M ( M * D, p - M * D, n ) / M.1 D B H F. 5 P K O 2 -H P K O DBHF: van Dalen, et al., EPJA 31, 29 (27) It is found that MD,p > M D,n. The splitting behavior in neutron matter is quantitatively consistent with the prediction of the Dirac-Brueckner-Hartree-Fock (DBHF) calculations. The constraints introduced by the Fock terms of the RHF scheme into the ts channel of the present density functional are straight forward and quite robust.

21 Spin-isospin resonances RHF equivalent zero-range coupling strengths (II) T P K O 2 -H 1 8 P S P K O 2 -H 1.5 P V P K O 2 -H T ( f m 2 ) 1..5 P S ( f m 2 ) P V ( f m 2 ) tt P K O 2 -H 1 8 tp S P K O 2 -H 1.5 tp V P K O 2 -H t T ( f m 2 ) 1..5 t P S ( f m 2 ) t P V ( f m 2 ) α(t)t H, αh (t)ps, and αh (t)pv are explicitly determined by exchange effects of RHF scheme (t)ps and (t)pv channels vanish in ground-state descriptions due to the parity conservation, but crucial for spin-isospin resonances

22 Spin-isospin resonances Particle-hole residual interactions in charge-exchange channel Particle-hole (ph) residual interactions ts channel: V ts (1, 2) = αts[γ H τ] 1 [γ τ] 2 δ(r 1 r 2 ), (12a) tv channel: V tv (1, 2) = αtv H [γ γ µ τ] 1 [γ γ µ τ] 2 δ(r 1 r 2 ), (12b) tt channel: V tt (1, 2) = αtt[γ H σ µν τ] 1 [γ σ µν τ] 2 δ(r 1 r 2 ), (12c) tps channel: V tps (1, 2) = αtps[γ H γ 5 τ] 1 [γ γ 5 τ] 2 δ(r 1 r 2 ), (12d) tpv channel: V tpv (1, 2) = αtpv H [γ γ 5 γ µ τ] 1 [γ γ 5 γ µ τ] 2 δ(r 1 r 2 ). (12e) For the charge-exchange spin-flip modes, the tt and tpv channels are expected to play the dominant roles, where the operator [σ τ] [σ τ] is sandwiched by the large components of wave functions.

23 Spin-isospin resonances Gamow-Teller resonances R - ( 1 / M e V ) R - ( 1 / M e V ) e x p t. 4 8 C a G T R u n p e r. P K O 2 P K O 2 - H E ( M e V ) 2 8 P b G T R u n p e r. P K O 2 P K O 2 - H e x p t E ( M e V ) R - ( 1 / M e V ) Z r G T R u n p e r. P K O 2 P K O 2 - H e x p t E ( M e V ) GTR excitation energies can be well reproduced by the ph residual interactions of PKO2-H. The present results are similar as those by the original RHF+RPA. The difference is due to the zero-range approximation. RHF+RPA: HZL, Giai, Meng, PRL 11, (28) localized RHF+RPA: HZL, Zhao, Ring, Roca-Maza, Meng, PRC 86, 2132(R) (212)

24 Spin-isospin resonances Physical mechanisms of GTR d i a g o n a l m a t r i x e l e m e n t s ( M e V ) t o t a l σ ω ρ t o t a l t S t V t T t P S t P V 2 8 P b G T R ( ν 1 h 1 1 /2 ) - 1 ( π 1 h 9 /2 ) 1 t o t a l ρ M i g d a l π P V P K O 2 P K O 2 -H D D -M E 2 In RHF+RPA (PKO2), σ- and ω-mesons play the most important role in determining the properties of GTR via the exchange terms. In the RHF equivalent RPA (PKO2-H), tt and tpv channels are most important, their coupling strengths are intrinsically determined by Fierz transformation. In conventional RH+RPA (DD-ME2), the free π residual interaction is attractive and the Migdal term is repulsive by fitting to experimental data.

25 Spin-isospin resonances Physical mechanisms of GTR by PKO2-H d i a g o n a l m a t r i x e l e m e n t s ( M e V ) t o t a l σ ω ρ t o t a l t S t V t T t P S t P V 2 8 P b G T R ( ν 1 h 1 1 /2 ) - 1 ( π 1 h 9 /2 ) 1 t o t a l ρ M i g d a l π P V P K O 2 P K O 2 -H D D -M E 2 The dominant ph interactions are the operator [σ τ] [σ τ] sandwiched by large components of w.f., i.e., [σ ij τ] [σ ij τ] 2αtT H and [γ 5γ i τ] [γ 5 γ i τ] αtpv H The net contribution is then proportional to 2αtT H αtpv H = 1 ( ) α H 3 S + αv H Total ph strengths are determined by the delicate balance between S and V channels.

26 Spin-isospin resonances Spin-dipole resonances R - ( f m 2 / M e V ) Z r S D R e x p t s u m R - ( f m 2 / M e V ) Z r S D R e x p t s u m 5 P K O E ( M e V ) 5 P K O 2 - H E ( M e V ) expt: Yako, Sagawa, Sakai, PRC 74, 5133 (26) Not only the total strengths but also the individual contribution from different spin-parity J π components are almost identical. The energy hierarchy E(2 ) < E(1 ) < E( ) can be obtained naturally in the present RPA calculations in the local scheme. The constraints introduced by the Fock terms of the RHF scheme into the ph residual interactions are also straight forward and quite robust.

27 Spin-isospin resonances Physical mechanisms of SDR σ ω ρ ν π π ρ ω σ ρ ν π π ρ In RHF+RPA, the balance between σ- and ω-mesons are most important. In the Hartree equivalent RPA, the tt and tpv channels are most important. In conventional RH+RPA, the balance between the free π and g terms are changed in different J π components, this leads to E( ) < E(1 ).

28 Strategy map Ground-state properties mass, radii, M* isospin splitting, Succeed Fail To be done fail in M* splitting totally fail RHF σ, ω, ρ, π finite-range RHF to RH 1 channels zero-range RH refitted 1 chs. by 3 zero-range RH σ, ω, ρ finite-range succeed in GTR, SDR Excited state properties low-lying, GDR, GTR, SDR, fail in GTR, SDR Zhaoxi Li (Beihang) RIKEN IPA project ( )

29 Summary and Perspectives Summary A new method is proposed to take into account the Fock terms in local covariant density functionals. The advantages of existing RH functionals can be maintained, while the problems in the isovector channel can be solved. The neutron-proton Dirac mass splitting in asymmetric nuclear matter is in a very good agreement with the prediction of DBHF. The properties of GTR and SDR can be reproduced in a natural way. Perspectives This opens a new door for the development of nuclear local covariant density functionals with proper isoscalar and isovector properties in the future.