Hartree-Fock and Hartree-Fock-Bogoliubov with Modern Effective Interactions

Transcription

1 Hartree-Fock and Hartree-Fock-Bogoliubov with Modern Effective Interactions Heiko Hergert Institut für Kernphysik, TU Darmstadt

2 Overview Motivation Modern Effective Interactions Unitary Correlation Operator Method Similarity Renormalization Group Few-Body Systems Many-Body Systems Hartree-Fock, Many-Body Perturbation Theory & Beyond Hartree-Fock-Bogoliubov Conclusions H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

3 From QCD to Nuclear Structure Nuclear Structure Low-Energy QCD H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

4 From QCD to Nuclear Structure Nuclear Structure chiral interactions: consistent NN & N interaction derived within χeft Realistic Nuclear Interactions Low-Energy QCD traditional NN-interactions: Argonne V8, CD Bonn, reproduce experimental NN phaseshifts with high precision induce strong short-range central & tensor correlations H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

5 From QCD to Nuclear Structure Nuclear Structure Exact / Approx Many-Body Methods exact solution of the many-body problem for light and intermediate masses (GFMC, NCSM, CC,) controlled approximations for heavier nuclei (HF & MBPT,) rely on restricted model spaces of tractable size not suitable for the description of short-range correlations Realistic Nuclear Interactions Low-Energy QCD H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

6 From QCD to Nuclear Structure Nuclear Structure Exact / Approx Many-Body Methods Modern Effective Interactions Realistic Nuclear Interactions adapt realistic potential to the available model space tame short-range correlations improve convergence behavior conserve experimentally constrained properties (phase shifts) generate new realistic interaction provide consistent effective interaction & effective operators unitary transformations most convenient Low-Energy QCD H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

7 Deuteron: Manifestation of Correlations Argonne V8 Deuteron Solution M S = ( + ) ρ () M S = ±, r φ L 8 6 L = L =,M S ( r) 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

8 Deuteron: Manifestation of Correlations Argonne V8 Deuteron Solution M S = ( + ) ρ () M S = ±, r φ L 8 6 L = L =,M S ( r) 6 8 r [fm] short-range repulsion supresses wavefunction at small distances r central correlations H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

9 Deuteron: Manifestation of Correlations Argonne V8 Deuteron Solution M S = ( + ) ρ () M S = ±, r φ L 8 6 L = L =,M S ( r) 6 8 r [fm] short-range repulsion supresses wavefunction at small distances r central correlations tensor interaction generates D-wave admixture in the ground state tensor correlations H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

10 Modern Effective Interactions I Unitary Correlation Operator Method (UCOM)

11 Unitary Correlation Operator Method Correlation Operator define an unitary operator C to describe the effect of short-range correlations [ C = exp[ i G] = exp i ] g ij i<j H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

12 Unitary Correlation Operator Method Correlation Operator define an unitary operator C to describe the effect of short-range correlations [ C = exp[ i G] = exp i ] g ij i<j Correlated States imprint short-range correlations onto uncorrelated many-body states ψ = C ψ Correlated Operators adapt Hamiltonian and all other observables to uncorrelated many-body space Õ = C O C ψ O ψ = ψ C O C ψ = ψ Õ ψ H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

13 Unitary Correlation Operator Method explicit ansatz for the correlation operator motivated by the physics of short-range central and tensor correlations Central Correlator C r radial distance-dependent shift in the relative coordinate of a nucleon pair g r = [ s(r) qr + q r s(r) ] q r = [ r r q + q r r ] Tensor Correlator C Ω angular shift depending on the orientation of spin and relative coordinate of a nucleon pair g Ω = ϑ(r)[ ( σ q Ω )( σ r) + ( r q Ω ) ] q Ω = q r r q r s(r) and ϑ(r) for given potential determined by energy minimization in the two-body system (for each S, T ) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

14 Correlated States: The Deuteron r φ L = r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

15 Correlated States: The Deuteron L = r φ r Cr φ r [fm] central correlations [fm] s(r) r [fm] r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

16 Correlated States: The Deuteron r φ r Cr φ r CΩCr φ L = r [fm] L = r [fm] L = r [fm] central correlations tensor correlations [fm] r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 8 6 s(r) ϑ(r) r [fm]

17 Correlated States: The Deuteron r φ r Cr φ r CΩCr φ L = r [fm] L = r [fm] L = r [fm] central correlations tensor correlations [fm] r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 8 6 s(r) ϑ(r) only short-range tensor r [fm] correlations treated by C Ω

18 Correlated Interaction: V UCOM S S D V AV8 6 q(ls)jt q (L S)JT q [fm ] q [fm ] 6 V UCOM q [fm ] q [fm ] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

19 Correlated Interaction: V UCOM S S D V AV8 6 q(ls)jt q (L S)JT pre-diagonalization of Hamiltonian q [fm ] q [fm ] 6 V UCOM q [fm ] q [fm ] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

20 Modern Effective Interactions II Similarity Renormalization Group (SRG)

21 Similarity Renormalization Group unitary transformation of the Hamiltonian to a band-diagonal form with respect to a given uncorrelated many-body basis Flow Equation for Hamiltonian evolution equation for Hamiltonian H(α) = C (α) H C(α) d dα H(α) = [ η(α), H(α) ] dynamical generator defined as commutator with the operator in whose eigenbasis H shall be diagonalized η(α) = [ T int, H(α) ] B [ = µ q, H(α) ] [Bogner et al, PRC7 6(R) (7); Hergert & Roth, PRC7 (R) (7)] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

22 The SRG Generator: A Closer Look typical N N interaction operators: O p {½, σ σ, l, l s, s ( r, r), } {½, τ τ, } Radial Kinetic Energy η r () [ q r, V] = p [ q r, v p(r)o p ] = p ( ) q r v p (r)o p + O p v p (r)q r Angular Kinetic Energy η Ω () [ l, V ] = [ l, v t (r)s ( r, r) ] = iv t (r)s ( r, q Ω ) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

23 The SRG Generator: A Closer Look typical N N interaction operators: O p {½, σ σ, l, l s, s ( r, r), } {½, τ τ, } Radial Kinetic Energy η r () [ q r, V] = p [ q r, v p(r)o p ] = p ( ) q r v p (r)o p + O p v p (r)q r Angular Kinetic Energy η Ω () [ l, V ] = [ l, v t (r)s ( r, r) ] = iv t (r)s ( r, q Ω ) η() has the same structure as the UCOM generators g r and g Ω H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

24 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 Argonne V8 r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

25 SRG Evolution: The Deuteron S S D q q 6 strong q off-diagonal contributions q VSRG(q, q ) VSRG(q, q ) 8 6 Argonne V8 r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG short-range central & tensor correlations

26 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

27 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

28 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

29 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

30 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

31 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

32 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

33 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG

34 SRG Evolution: The Deuteron S S D q q 6 suppression q of off-diagonal contributions q VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG elimination of shortrange correlations

35 SRG Evolution: The Deuteron S S D q q q q 6 VSRG(q, q ) VSRG(q, q ) 8 6 α = fm r φ L= 6 8 r [fm] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 SRG r φ L= SRG extract UCOM correlation functions s(r) and ϑ(r)

36 Few-Body Systems

37 He: Convergence AV8 He V UCOM 6 N max 6 6 E [MeV] - E AV8 8 6 E [MeV] ω [MeV] 6 8 ω [MeV] NCSM code by P Navrátil [PRC 6, ()] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

38 He: Convergence AV8 He V UCOM 6 N max 6 E [MeV] - E AV E [MeV] - residual state-dependent long-range correlations 6 8 ω [MeV] 6 8 ω [MeV] NCSM code by P Navrátil [PRC 6, ()] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

39 He: Convergence AV8 He V UCOM 6 N max E [MeV] - E AV8 8 6 E [MeV] ω [MeV] 6 8 ω [MeV] NCSM code by P Navrátil [PRC 6, ()] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

40 He: Convergence AV8 He V UCOM 6 N max E [MeV] - E AV8 6 8 ω [MeV] 8 6 E [MeV] omitted higher-order cluster contributions 6 8 ω [MeV] NCSM code by P Navrátil [PRC 6, ()] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

41 Three-Body Interactions Strategies Correlated Hamiltonian in Many-Body Space H = C (T + V NN + V N ) C = T [] + ( T [] + Ṽ[] NN ) + ( T [] + Ṽ[] NN + Ṽ[] N ) + = T + V UCOM + V [] UCOM + H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

42 Three-Body Interactions Strategies Correlated Hamiltonian in Many-Body Space H = C (T + V NN + V N ) C = T [] + ( T [] + Ṽ[] NN ) + ( T [] + Ṽ[] NN + Ṽ[] N ) + = T + V UCOM + V [] UCOM + strategies for treating the three-body contributions: ➊ include full V [] UCOM consisting of genuine and induced N terms ➋ replace V [] UCOM by phenomenological three-body force ➌ minimize V [] UCOM by proper choice of unitary transformation H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

43 Three-Body Interactions Tjon Line - - AV8 Nijm II Nijm I Tjon-line: E( He) vs E( H) for phase-shift equivalent NNinteractions E( He) [MeV] Exp CD Bonn V NN + V N E( H) [MeV] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

44 Three-Body Interactions Tjon Line - - AV8 Nijm II Nijm I Tjon-line: E( He) vs E( H) for phase-shift equivalent NNinteractions E( He) [MeV] Exp CD Bonn increasing C Ω -range E( H) [MeV] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

45 Three-Body Interactions Tjon Line - - AV8 Nijm II Nijm I Tjon-line: E( He) vs E( H) for phase-shift equivalent NNinteractions E( He) [MeV] Exp CD Bonn increasing C Ω -range -9 - increasing α E( H) [MeV] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

46 Three-Body Interactions Tjon Line - - AV8 Nijm II Nijm I Tjon-line: E( He) vs E( H) for phase-shift equivalent NNinteractions E( He) [MeV] Exp CD Bonn increasing α increasing C Ω -range use α / range of C Ω to test dependence of V α or V UCOM tune contributions of net N force E( H) [MeV] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

47 Three-Body Interactions Tjon Line - - AV8 Nijm II Nijm I Tjon-line: E( He) vs E( H) for phase-shift equivalent NNinteractions E( He) [MeV] Exp CD Bonn increasing C Ω -range use α / range of C Ω to test dependence of V α or V UCOM tune contributions of net N force -9 V UCOM E( H) [MeV] min net N force: I ϑ = 9 fm with phen N force: I ϑ = fm H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

48 Many-Body Systems

49 Hartree-Fock with V UCOM E = Ψ Hint Ψ - E/A [ MeV] Rch [ fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment I ϑ [ fm ]: 9 H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

50 Hartree-Fock with V UCOM E/A [ MeV] Rch [ fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment I ϑ [ fm ]: 9 8 H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

51 Hartree-Fock with SRG Potentials E/A [ MeV] Rch [ fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment α[ fm ]: H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

52 Hartree-Fock with SRG Potentials E/A [ MeV] need N force for saturation Rch [ fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment α[ fm ]: H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

53 Hartree-Fock with V UCOM E/A [ MeV] Rch [ fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment HF H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

54 Hartree-Fock with V UCOM - E/A [ MeV] long-range correlations are missing Rch [ fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment HF H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

55 Perturbation Theory with V UCOM E/A [MeV] Rch [fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment HF HF + PT H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

56 RPA Ring Summation with V UCOM E/A [MeV] Rch [fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment HF HF + PT HF + ring sum H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

57 RPA Ring Summation with V UCOM E/A [MeV] long-range correlations are perturbative tractable within PT, BHF, CC, RPA, Rch [fm] indication for N force He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd experiment HF HF + PT HF + ring sum H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

58 N Forces: Energies & Radii E/A [ MeV] Rch [ fm] He 6 O O Si Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd exp (I ϑ [ fm ], C N [ GeV fm 6 ]): (9, ) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

59 N Forces: Energies & Radii E/A [ MeV] Rch [ fm] He O Ca 8 Ni 68 Ni 88 Sr Sn Sn 8 Pb 6 O Si 8 Ca 6 Ni 78 Ni 9 Zr Sn 6 Gd exp (I ϑ [ fm ], C N [ GeV fm 6 ]): (9, ) (, ) (, ) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

60 N Forces: HF Single-Particle Energies 6 O protons neutrons ε[mev] I ϑ [ fm ] 9 Exp 9 Exp C N [ GeV fm 6 ] occupied unoccupied H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

61 N Forces: HF Single-Particle Energies 8 Pb protons neutrons ε[mev] I ϑ [ fm ] 9 Exp 9 Exp C N [ GeV fm 6 ] occupied unoccupied H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

62 N Forces: HF Single-Particle Energies 8 Pb protons neutrons ε[mev] I ϑ [ fm ] C N [ GeV fm 6 ] 9 Exp 9 N force Exp improves level density occupied unoccupied H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

63 Beyond Hartree-Fock perturbation theory, Brueckner-HF, RPA Hartree-Fock (Mean-Field) Hartree-Fock- Bogoliubov Long-Range Correlations Pairing Correlations H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

64 HFB Theory Overview Bogoliubov Transformation HFB Densities & Fields β k = q U qk c q + V qkc q ρ kk Ψ c k c k Ψ = (V V T ) kk where β k = q U qk c q + V qk c q { }! βk, β k = { β k, }! β k = { βk, β k }! = δ kk κ kk Ψ ck c k Ψ = (V U T ) kk Γ kk = ( ) rel + v A t qq kk = ( ) rel + v A t qq kq,k q ρ qq kk,qq κ qq H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

65 HFB Theory Overview Bogoliubov Transformation HFB Densities & Fields β k = q U qk c q + V qkc q ρ kk Ψ c k c k Ψ = (V V T ) kk where β k = q U qk c q + V qk c q { }! βk, β k = { β k, }! β k = { βk, β k }! = δ kk κ kk Ψ ck c k Ψ = (V U T ) kk Γ kk = ( ) rel + v A t qq kk = ( ) rel + v A t qq kq,k q ρ qq kk,qq κ qq Energy E[ρ, κ, κ ] = Ψ H Ψ Ψ Ψ (tr Γρ tr κ ) HFB Equations ( ) U (H λn) V ( Γ λ ) ( ) U Γ + λ V = E ( ) U V H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

66 Particle Number Projection Projected Energy E(N ) = Ψ HPN Ψ Ψ P N Ψ = π P N π dφ Ψ He iφ(n N ) Ψ Flocard & Onishi, Ann Phys, 7 (997, approx PNP); Sheikh et al, Phys Rev C66, 8 (, exact PNP) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

67 Particle Number Projection Projected Variation of Energy Projected Energy Ψ HPN Ψ π E(N ) = Ψ P N Ψ = π δe(n π dφ Ψ He iφ(n N ) Ψ ) = π dφ e iφ(n N ) { δ H (E(N φ P N P ) H ) δ log e iφn } φ N H φ He iφn / e iφn Flocard & Onishi, Ann Phys, 7 (997, approx PNP); Sheikh et al, Phys Rev C66, 8 (, exact PNP) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

68 Particle Number Projection Projected Variation of Energy Projected Energy Ψ HPN Ψ E(N ) = Ψ P N Ψ = π π dφ Ψ He iφ(n N ) Ψ π P δe(n δ ) = π dφ e iφ(n N ) { δ H (E(N φ P ) H ) δlog e iφn } φ N N H φ He iφn / e iφn Lipkin-Nogami + PAV power series expansion expansion coefficients not varied indeterminate / numerically unstable at shell closures exact PNP after variation Flocard & Onishi, Ann Phys, 7 (997, approx PNP); Sheikh et al, Phys Rev C66, 8 (, exact PNP) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

69 Particle Number Projection Projected Variation of Energy Projected Energy Ψ HPN Ψ E(N ) = Ψ P N Ψ = π π dφ Ψ He iφ(n N ) Ψ π P δe(n δ ) = π dφ e iφ(n N ) { δ H (E(N φ P ) H ) δlog e iφn } φ N N H φ He iφn / e iφn Lipkin-Nogami + PAV power series expansion expansion coefficients not varied indeterminate / numerically unstable at shell closures exact PNP after variation VAP higher (but managable) computational effort implement with care: subtle cancellations between divergences of direct, exchange, and pairing terms Flocard & Onishi, Ann Phys, 7 (997, approx PNP); Sheikh et al, Phys Rev C66, 8 (, exact PNP) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

70 Particle Number Projection Projected Variation of Energy Projected Energy Ψ HPN Ψ E(N ) = Ψ P N Ψ = π π dφ Ψ He iφ(n N ) Ψ π P δe(n δ ) = π dφ e iφ(n N ) { δ H (E(N φ P ) H ) δlog e iφn } φ N N H φ He iφn / e iφn Lipkin-Nogami + PAV power series expansion expansion coefficients not varied indeterminate / numerically unstable at shell closures exact PNP after variation VAP higher (but managable) computational effort implement with care: subtle cancellations between divergences of direct, exchange, and pairing terms Structure of HFB equations is preserved by both methods! Flocard & Onishi, Ann Phys, 7 (997, approx PNP); Sheikh et al, Phys Rev C66, 8 (, exact PNP) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

71 Sn Isotopes: Binding & Pairing Energies - E/A [ MeV] Epair [ MeV] A Sn A Sn exp HFB H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

72 Sn Isotopes: Binding & Pairing Energies - E/A [ MeV] Epair [ MeV] A Sn A Sn exp HFB PAV LN+PAV VAP H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

73 Density-Dependent Force: Pairing Epair [ MeV] HFB PAV LN+PAV VAP linear density dependence, t = 6 C N: V ρ = t ( + P σ ) ρ ( ( r + r ) ) δ ( r r ) mixed density for PAV/VAP: ρ( R) ρ φp,φ n ( R) phenomenological VAP calculations: E pair MeV (Stoitsov et al, nucl-th/66; Anguiano et al, Phys Lett B (), 6) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

74 Density-Dependent Force: Pairing Epair [ MeV] HFB PAV LN+PAV VAP linear density dependence, t = 6 C N: V ρ = t ( + P σ ) ρ ( ( r + r ) ) δ ( r r ) mixed density for PAV/VAP: ρ( R) ρ φp,φ n ( R) phenomenological VAP calculations: E pair MeV correct order of magnitude with realistic NN int (Stoitsov et al, nucl-th/66; Anguiano et al, Phys Lett B (), 6) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

75 Density-Dependent Force: Pairing E p pair [ MeV] E n pair [ MeV] HFB PAV LN+PAV VAP H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

76 Density-Dependent Force: Pairing E p pair [ MeV] E n pair [ MeV] systematics: more realistic HFB PAV LN+PAV VAP N force H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

77 Conclusions

78 Modern Effective Interactions Status treatment of short-range central and tensor correlations by unitary transformations: Unitary Correlation Operator Method Similarity Renormalization Group universal phase-shift equivalent correlated interaction V UCOM Outlook connections between UCOM and SRG inclusion & treatment of N Forces, in particular chiral interactions H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

79 HF, HFB, and Beyond perturbation theory, Brueckner-HF, RPA, Mean-Field (Hartree-Fock) Hartree-Fock- Bogoliubov Long-Range Correlations Pairing Correlations QRPA H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

80 HF, HFB, and Beyond Status fully consistent HFB calculations with particle number projection inclusion of N-forces: contact & finite range matrix elements for HF, density-dependent force for HFB, RPA, Like-particle- & pn-qrpa (benchmarked) Outlook pn pairing, Isobaric Analog & Gamow-Teller Resonances deformation and symmetry restoration by projection (isospin, parity, angular momentum) caveat: analytic structure of density-dependent forces is very important in symmetry-projected HFB (J Dobaczewski, arxiv: 78) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

81 Perspectives Modern Effective Interactions treatment of short-range central and tensor correlations by unitary transformations: UCOM, SRG, Lee-Suzuki, phase-shift equivalent correlated interaction V UCOM universal input for Innovative Many-Body Methods No-Core Shell Model, Importance Truncated NCSM, Coupled Cluster Method, Hartree-Fock plus MBPT, Padé Resummed MBPT, BHF, HFB, RPA, Fermionic Molecular Dynamics, H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

82 Last Words My Collaborators R Roth, P Papakonstantinou, A Zapp, P Hedfeld, S Reinhardt Institut für Kernphysik, TU Darmstadt T Neff, H Feldmeier Gesellschaft für Schwerionenforschung (GSI) N Paar Department of Physics Faculty of Science, University of Zagreb, Croatia References H Hergert, R Roth, Phys Rev C7, (R) (7) N Paar, P Papakonstantinou, H Hergert, and R Roth, Phys Rev C7, 8 (6) R Roth, P Papakonstantinou, NPaar, H Hergert, T Neff, and H Feldmeier, Phys Rev C7, (6) H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

83 Appendix

84 B: Hallmark of a N Interaction? 7 6 V UCOM Exp + E E+ [MeV] ω ω ω 6 ω 8 ω ω = 8MeV H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

85 B: Hallmark of a N Interaction? 7 6 V UCOM Exp CDBonn Chiral NN NN+NNN + E E+ [MeV] ω ω ω 6 ω 8 ω ω = 8MeV ω -6 6 ω H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

86 B: Hallmark of a N Interaction? 7 V6 UCOM gives correct level ordering without any N interaction E E+ [MeV] - - V UCOM Exp CDBonn Chiral + NN NN+NNN ω ω ω 6 ω 8 ω 8 ω 6 ω -68 ω = 8MeV -6 H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

87 RPA + Realistic Interactions RPA, ERPA & SRPA + Matrix Elements of Correlated Realistic Interaction V UCOM fully self-consistent RPA based on the Hartree-Fock orbits using the same V UCOM recovering sum rules with high precision, spurious center-ofmass mode fully decoupled at kev Extended-RPA and Second-RPA to include effects of ground state correlations and complex configurations H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

88 RPA with V UCOM R [fm /MeV] R [e fm 6 /MeV] R [e fm /MeV] 7 Ca ISM 8 Pb ISM IVD IVD ISQ ISQ [MeV] E ex E ex [MeV] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

89 RPA with V UCOM R [fm /MeV] R [e fm 6 /MeV] R [e fm /MeV] 7 Ca ISM IVD ISQ [MeV] E ex 8 Pb ISM 6 ISGMR in good agreement with experiment IVD E ex [MeV] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7 8 6 IVGDR & ISGQR centroid energies too high ISQ need for additional correlations & three-nucleon force

90 SRPA: Complex Configurations R [e fm 6 /MeV] R [e fm /MeV] Ca PRELIMINARY IVD ISQ RPA (full ph space) SRPA (full ph+ph space) ( major shells, Γ = MeV) complex configurations have significant impact on response E ex [MeV] H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

91 SRPA: Complex Configurations R [e fm 6 /MeV] R [e fm /MeV] Ca PRELIMINARY IVD ISQ E ex [MeV] RPA (full ph space) SRPA (full ph+ph space) ( major shells, Γ = kev) complex configurations have significant impact on response possibility to investigate fine structure H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7

92 Outlook: RPA with Three-Body Forces R [fm /MeV] R [e fm /MeV] R [e fm /MeV] 7 9 Zr PRELIMINARY ISM IVD [MeV] E ex ISQ long-range tensor correlator & repulsive three-body contact interaction systematic improvement of rms-radii single-particle spectra strength distributions standard V UCOM V UCOM with long-range tensor & three-body contact interaction H Hergert TU Darmstadt INT Program on Nuclear Many-Body Approaches for the st Century, /7