A new equation of motion method for multiphonon nuclear spectra .

1. A newequationofmotionmethodformultiphononnuclearspectra. N. Lo Iudice Università di Napoli Federico II Kazimierz08

2. Acknowledgments • J. Kvasil, F. Knapp, P. Vesely (Prague) • F. Andreozzi, A. Porrino (Napoli) • Also Ch. Stoyanov (Sofia) A.V. Sushkov (Dubna)

3. From mean field to multiphonon approaches Anharmonic features of nuclear spectra: Experimental evidence of multiphonon excitations Necessity of going beyondmean field approaches A successful microscopic(QPM) multiphononapproach A new (in principle exact) multiphonon method

4. Collective modes: anharmonic features Mean field:Landau damping • Beyond mean field: • * Spreading width • * *Multiphon excitations - High-energy(N. Frascaria, NP A482, 245c(1988); T. Auman, P.F.Bortignon, H. Hemling, Ann. Rev. Nucl. Part. Sc. 48, 351 (1998)) Double and tripledipolegiantresonances - Low-energy M. Kneissl. H.H. Pitz, and A. Zilges, Prog. Part. Nucl. Phys. 37, 439 (1996); M. Kneissl. N. Pietralla, and A. Zilges, J.Phys. G, 32, R217 (2006) : • Two- and three-phononmultiplets • Proton-neutronmixed-symmetrystates (N. Pietrallaet al. PRL 83, 1303 (1999))

5. A microscopicmultiphononapproach: QPMSoloviev, TheoryofAtomic Nuclei: Quasiparticles and Phonons, Bristol, (1992) H = Hsp + Vpair + Vpp + Vff H[(a†a), (a†a†),(aa)]  H[(α†α ),(α†α†),(αα ) ]  α†α†O†λ αα Oλ Oλ†=Σkl [Xkl(λ)α†k α†l–Ykl(λ)αkαl]  HQPM= Σnλωn (λ)Q†λ Qλ + Hvq  Ψν= ΣncnQ†ν(n) |0> + ΣijCijQ† (i) Q†(j) |0> + ΣijkCijkQ†(i) Q†(j) Q†(k)|0>

6. Symmetric |n, ν>s = QSn |0 > = (Qp + Qn)n |0 > MS |n, ν>MS = QAQS (n-1) |0 > = (Qp - Qn) (Qp + Qn) (n-1) |0 > Signature Preserving symmetry transitions M(E2)  QS n → n-1 Changing symmetry transitions M(M1)  Jn – Jp n → n π-νMixedSymmetry E2 n=3 E2 M1 n=2 E2 n=2 M1 E2 n=1 MS n=1 J+1 E2 M1 J J-1 Sym J Scissors multiplet S | n,J> = (Jp – Jn) |nJ> = ΣJ’ |n J’> <nJ’| S|nj> Bsc(M1) = ΣJ’ |<nJ’|M(M1)|nJ>|2 ~ 1.5 - 2 μN2

7. A QPM calculation: N=90 isotonesN. L, Ch. Stoyanov, D.Tarpanov, PRC 77, 044310 (2008) E2 M1

8. 4+ state in OsisotopesN. L. and A. V. Sushkovsubmittedto PRC 4+ Hexadecapoleone-phonon? Ψ  |n=1,4+> E4 S(t,α) Double-g ?  |> E2 E4 2 Eg E2 R4(E2) = B(E2,4+→ 2+)/B(E2,2+ → 0+) 2 4+ 2g+ 2g+ 0+ 0+ QPM Ψ  0.60 |n=1,4+> + 0.35 |>

9. 4+: QPM versus EXP

10. Successes and limitations of theQPM Successes Itisfullymicroscopicand valid atlowand highenergy. includingthe double GDR(Ponomarev, Voronov) Limitations: Valid for separable interactions Antisymmetrization enforced in the quasi-boson approximation, Ground state not explicitly correlated (QBA) Temptativeimprovements MultistepShellmodel (MSM) (R.J.Liotta and C. Pomar, Nucl. Phys. A382, 1 (1982)) Theyexpand and linearize <α|[[H, O†],O†]|0> ( O†= ΣphX(ph)a†pah ) Multiphononmodel (MPM) (M. Grinberg, R. Piepenbringet al. Nucl. Phys. A597, 355 (1996) Along the samelines BothMSM and (especially) MPM look involved

11. A new (exact) multiphonon approach Eigenvalue problem in a multiphonon space |Ψν > H = Σn HnHn |n; β> ( n= 0,1.....N ) H | Ψν > = Eν| Ψν > Generationof the|n; β> (basis states) An obvious (but prohibitive!!) choice |n; β> = | ν1, ν2,… νi ,…νn> where (TDA) | νi > = Σphcph(νi )a†p ah|0> A workable choice|n=1; β> = | νi > = Σphcph(νi )a†p ah|0> ( TDA)  |n; β> =ΣαphC(n)αpha†p ah| n-1; α >

12. EOM: Construction of the Equations <n; β |[H, a†p ah]| n-1; α> Crucialingredient Preliminary step: <n; β |[H, a†p ah]| n-1; α> = ( Eβ(n) – Eα(n-1))<n; β |a†p ah| n-1; α > (LHS) (RHS) It follows from * request < n; β|H|n; α > = Eα(n)δαβ ** property <n; β|a†pah|n’; γ > = δn’,n-1< n; β |a†pah| n-1; γ >

13. Equations of Motion: LHS Commutatorexpansion < n; β | [H, a†p ah] | n-1; α >= (εp- εh) <n; β |a†p ah| n-1; α >. + linear 1/2 Σijp’Vhjpk<n; β|a†p’ aha†iaj| n-1; α > + not linear Î = Σγ|n-1; γ >< n-1; γ| Linearization < n; β | [H, a†p ah] | n-1; α >= = Σp’h’γ Aαγ(n)(ph;p’h’)<n; β|a†p’ ah’| n-1; γ>

14. LHS=RHS A(n ) X(n)= E(n)X(n) Aαγ(n) (ij)= [(εp–εh ) + Eα(n-1) ] δij(n-1)δαβ(n-1)+ [VPHρH+ VHPρP+VPPρP+ VHHρH]αiβj Xα(β) (i ) = < n; β |a†p ah| n-1; α> ρH ≡{< n-1,γ|a†hah’|n-1,α>} ρP ≡ {< n-1,γ|a†pap’|n-1,α>} n =1 TDA A(1)X (1) = E (1)X (1) A(1)(ij) = δij[(εp–εh )+ E(0)] + V(p’hh’p) n=1 |n=1; β> = | νi > = Σphcpha†p ah|0> n> 1 |n; β> =ΣαiCα(β) (i ) a†pah| n-1; α > n=1 C = X n> 1 X= DC Dαα’ (ij) = < n-1; α’|(a†h’ap’)( a†p ah)| n-1; α> overlap matrix

15. General Eigenvalue Problem A(n)X(n)= E(n)X(n) X(n)=D(n)C(n) (AD)C= H C= E DC Eigenvalue Equation Dαα’ (ij) = < n-1; α’|(a†h’ap’)( a†p ah)| n-1; α> overlap matrix Problem i)how to compute D Problem ii) redundancyDetD = 0

16. General Eigenvalue Problem (AD)C= H C= E DC • Solution of problem i) Aαγ(n) (ij)= [εp–εh + Eα(n-1)]δij(n-1)δαβ(n-1) + [VPHρH+ VHPρP+VPPρP+ VHHρH]αiβj Dαα’ (ij) = < n-1; α’|(a†h’ap’)( a†p ah)| n-1; α> overlap matrix D = ρH (n-1) – ρP (n-1)ρH (n-1) ρP(n-1) = C (n-1) X (n-1) – C (n-1) X (n-1)ρP (n-2) recursive relations Problem i) solved!!!!

17. General Eigenvalue Problem (AD)C= H C= E DC Solution of Problem ii) (redundancy) *Choleski decomposition ** Matrixinversion Exact eigenvectors DĎ HC = (Ď-1AD)C = E C |n; β>=ΣαphCα(β) (i )a†p ah |n-1; α>  Hn (phys)

18. Iterative generation of phonon basis Starting point|0> Solve Ĥ(1)C(1) = E(1)C(1) |n=1, α> X(1) ρ(1) Solve Ĥ(2)C (2) = E (2)C (2) |n=2,α> X(2) ρ(2) ……… X(n-1) ρ(n-1) Solve Ĥ(n)C (n) = E (n)C (n) X(n) ρ(n) |n,α.> The multiphonon basis is generated !!!

19. H: Spectral decomposition, diagonalization H= Σ nαE α(n)|n; α><n;α| + (diagonal) + Σnα β|n; α><n;α| H |n’;β><n’;β| (off-diagonal) n’= n ±1, n±2 Off-diagonalterms: Recursive formulas < n; β | H| n-1; α> = Σphγϑαγ(n-1) (ph)Xγ(β) (ph) < n; β | H| n-2; α > =Σ V pp’hh’Xγ(β) (ph) Xγ(α) (p’h’) |Ψν> = Σnα Cα(ν) (n) | n;α> |n;α> = Σγ Cγ(α)a†p ah | n-1;γ> Outcome H |Ψν> = Eν|Ψν>

20. C3 | λ1 λ2 λ3> + C1 | λ1> C2 | λ1 λ2> + C0 |0> Ψ0Ψν(n) EMPM E.m. response W.F. |Ψν> = Σn{λ} C{λ}(ν) (n) | n;{λ1λ2.λn }> |λ> = Σph cph (λ) a†p ah|0> e.m. operator Мλμ= rλ Yλμ Strength Function S(Eλ) = Σ Bn (Eλ) δ(E- En) Bn (Eλ) =|<Ψnν|| Мλ ||Ψ0>|2

21. 16O: TDA (CM free) response and SMspacedimensions

22. EMPM : Exact implementation in 16O for N=4 SM space All particle-hole (p-h) configurations up to3ħω (2s,1d,0g) (1p,0f) (1s,0d) 0p 0s Free ofCM spuriousadmixtures

23. Meanfield versus EMPM E response

24. NEW runningsums

25. EW runningsums

26. CM motion Hamiltonian H = H0 + V = ΣihNils(i)+ Gbare ( VBonnA  Gbare) CM motion ( F. Palumbo Nucl. Phys. 99 (1967)) H H+Hg Hg= g [P2/(2Am) + (½) mAω2 R2] For g>>1 E CM >> Eintr

27. CMmotion in TDA: Isoscalar E1

28. CMmotion in EMPM

29. Spectra

30. Perspectives: New formulation < n; β |[H, a†p ah]| n-1; α>  < n; β |[H, O†λ]| n-1; α> O†λ= Σph cph(λ )a†pah λ’γAαγ(n) (λ,λ’) X(β)γ λ’= Eβ(n)X (β)γ λ X(β)αλ= < n; β |O†λ| n-1; α > Aαγ(n) (λ,λ’) = [ Eλ + Eα(n-1) ] δλλ’δαγ+ ρλλ’Vραγ(n-1) ρλλ’ (kl)≡<λ’| a†kal|λ>ραγ(n-1) (kl) = < n-1,γ| a†kal|n-1,α> |n; β> =ΣαλC(β)αλO†λ| n-1; α > = ΣC(β) {λi} |λ1,.…λi….λN > C (β) {λi} = Σ C(β)αλ1C(α)γλ2C (γ)δλ3

31. Vertices Aαγ(n) (λ,λ’) = [ Eλ + Eα(n-1) ] δλλ’δαγ+ ρλλ’Vραγ(n-1) TDA (n=1) MPEM (n=3) γ -------- --------- p’ λ’ h’ p h λ α Vph’hp’ ρλλ’Vραγ(n-1)

32. THANK YOU

33. EW sum rule SEW (E ) = ½ Σμ <[M (E μ),[H, M (E μ)]> = [(2  + 1)2/16π ](ħ2/2m) A < r2  -2> SEW (E 1, τ = 0) = [(2  + 1)2]/16π (ħ2/2m) x A[11< r4> -10 R2<r2> + 3 R4]

34. Ground state |Ψ0> = C0(0) |0> + ΣλCλ(0) |λ, 0> + Σλ1λ2Cλ1λ2(0) |λ1 λ2, 0> 1 = < Ψ0|Ψ0> = P0 + P1 + P2

35. 16O negative parity spectrum • Up to three phonons

36. IVGDR Мλμ= τ3 r Y1μ≈ Rπ - Rν TDA |1->IV ~ |1 (p-h) (1ħω)>(TDA)

37. ISGDR Мλμ= r Y1μ≈ RCM !!! Мλμ= r3 Y1μ Toroidal |1->IS ~ |1(p-h) (3 ħω)> + |2 (p-h) (1ħω + 2ħω)> + |3(p-h) (1ħω )>

38. Octupole modes Мλμ= r3 Y1μ Low-lying |3->IS ~ |1(p-h) (3 ħω)> + |2(p-h) (1ħω + 2ħω)> + |3(p-h) (1ħω )>

39. Effect of CM motion

40. Effect of the CM motion

41. Concluding remarks • The multiphonon eigenvalue equations - have a simple structure • yield exact eigensolutions of a general H • The 16O test shows that • an exact calculation in the full multiphonon space is feasible at least up to 3 phonons and 3 ħω. • To go beyond • Truncation of the space needed!!! • Truncation is feasible (the phonon states are correlated). • Ariformulation for an efficient truncation is in progress

42. Implications of the redundancy |n; β> =ΣjCj |i> where |i > = a†p ah| n-1; α >(not linearly independent ) Eigenvalue problemofgeneral form Σj [<i|H|j> - Ei <i|j>]Cj = 0 But (problems again!!) i.A direct calculation of <i|H|j>and<i|j> is prohibitive !! ii.The eigenstates would contain spurious admixtures!! How to circumvent these problems?

43. Problem: Overcompletness for n>1 |n; β> =ΣαphCαpha†p ah| n-1; α > a†p ah | n-1; α > are not fully antysymmetrized !!!  p h p h a†p ah| n-1; α > ≡ The multiphononstates are not linearly independent and form an overcompleteset.

44. 16O as theoretical lab Structure of16O: Atheoretical challenge Pioneering work: First excited 0+as deformed 4p-4h excitations G. E. Brown, A. M. Green, Nucl. Phys. 75, 401 (1966) (TDA) IBM (includes up to 4 TDA Bosons) H. Feshbach and F. Iachello, Phys. Lett. B 45, 7 (1973); Ann. Phys. 84, 211 (194) SM up to 4p-4h and 4 ħω W.C. Haxton and C. J. Johnson, PRL 65, 1325 (1990) E.K. Warbutton, B.A. Brown, D.J. Millener, Phys. Lett. B293,7(1992) No-core SM (NCSM) Huge space!!! Symplectic No-core SM (SpNCSM) a promising tool for cutting the SM space T. Dytrych, K.D. Sviratcheva, C. Bahri, J. P. Draayer, and J.P. Vary, PRL 98, 162503 (2007) Self-consistent Green function (SCGF) (extends RPA so as to include dressed s.p propagators and coupling to two-phonons) C. Barbieri and W.H. Dickhoff, PRC 68, 014311 (2003); W.H. Dickhoff and C. Barbieri, Pro. Part. Nucl. Phys. 25, 377 (2004)