Equation of motion method for generating exact multiphonon states

1
Equation of motion method for generating exact
multiphonon states

• N. Lo Iudice
• Università di Napoli Federico II
• Trento07

2
From mean field to multiphonon approaches

• Anharmonic and multiphonon spectra Experimental
  evidence
• Necessity of going beyond mean field (or
  harmonic) approaches (RPA)
• Some successful phenomenological (IBM)
• and microscopic (QPM) multiphonon approaches
• A new (in principle exact) multiphonon method

3
Collective modes anharmonic features

• Within mean field
• Decay into single-particle motion (Landau
  damping)
• Beyond mean field
• Coupling to more complex configurations
  (doorway 2p-2h) (collisional damping) determining
  the spreading width

From T. Auman, P.F. Bortignon, H. Hemling, Ann.
Rev. Nucl. Part. Sc. 48, 351 (1998))
4
Multiphonon excitations Exp. evidence
(spherical nuclei)

• 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-phonon multiplets
• Q2 Q30gt, Q2Q2Q30gt
• Proton-neutron (F-spin) mixed-symmetry states
• (N. Pietralla et al. PRL 83, 1303 (1999))
• Q2(p) - Q2(n) (Q2(p) Q2(n)) N0gt,
  

5
Multiphonon excitations Exp. evidence (spherical
nuclei)

• High-energy
• (N. Frascaria, NP A482, 245c(1988) T. Auman,
  P.F. Bortignon, H. Hemling, Ann. Rev. Nucl. Part.
  Sc. 48, 351 (1998))
• Double
• D D 0gt
• and
• triple
• D D D 0gt
• dipole giant resonances

6
Inadequacy of mean field approaches

• Standard mean field approach RPA
• RPA is an harmonic approximations
• H,O ? h? O
• where
• O Sph X(ph) ap ah Y(ph) ah ap
  (closed shell nuclei)
• O Sµ? X(µ?) aµ a ? Y(µ?) aµ a? (open
  shell nuclei)
• As such, it is not adequate for describing
  anharmonic spectra

7
From mean field to multiphonon approaches

• For anharmonic effects, a multiphonon space is
  needed
• Ideal method Boson expansion
• bµ
  Sph cph apah ? B B B B B
  B ..
• and Fermion-Boson mapping
• Equivalent mapping routes
• Operator Mapping (S. T. Belyaev and V. G.
  Zelevinsky, Nuc. Phys. 39, 582 (1962))
• 
  OF ? OB
• State mapping (T. Marumori et al. Prog. Theor.
  Phys. 31, 1009 (1964))
• ngt bµ b? .b? 0gt ?
  n) Bi Bj ..Bk 0)
• HB
  Sn n n) ltnHngt (n
• Constraint ltnHngt
  (nHBn)

8
From mean field to multiphonon approaches

• Alternative to Boson expansion
• Microscopic approaches
• Nuclear Field theory (Bohr, Mottelsson, Bes,
  Broglia..)
• P.F. Bortignon, R. A. Broglia, D.R. Bes, and R.
  Liotta, Phys. Rep. 30, 305 (1977)
• Especially suitable for
• i) Damping of vibrational modes
• ii) Spreading widths of giant resonances
  
• Quasiparticle-phonon model (QPM)
• (V.G. Soloviev, Theory of Atomic Nuclei
  Quasiparticles and Phonons, Bristol, 1992)
• especially suitable for multiphonon excitations
• Phenomenological approaches
• Geometric collective model (GCM) (Greiner)
• Interacting Boson Model (IBM) (Arima and
  Iachello)

9
Two multiphonon approaches

• A microscopic one QPM
• (V. G. Soloviev, Theory of Atomic Nuclei
  Quasiparticles and Phonons, Bristol, 1992
• H Hsp Vpair VPP VFF
• Q? Sij X(ij) ai aj Y(ij) aiaj
  
• HQPM Si? ?i? Q? Q? Hvq
• ?? (JM) Sici Q?(i)
• Sij cij Q (i) Q(j)
• Sijk cijk Q(i) Q(j) Q(k)

• A phenomenological oneIBM
  
• (Arima Iachello)
• AJ0,2 S cij (ai ?aj)J0,2
• AJ0,2 (s, d)
• IBM) (s)ns (d) nd 0)
• IBM successful in
• low-energy
• spectroscopy

10
A successfull IBM and QPM study!

• p-? MS states
• First detected in 94Mo
• N. Pietralla et al. PRL 83, 1303 (1999)
• Since then, found in several nuclei
• Werner et al. PL B550 (02),
• C. Fransen et al. PRC 71 (06)

11
p-? MS states in IBM and QPM

• Symmetric
• n, ?gts QSn 0 gt
• (Qp Qn)n 0 gt
• MS
• n, ?gtMS QAQS (n-1) 0 gt
• (Qp - Qn) (Qp Qn) (n-1) 0 gt
• Signature
• Preserving symmetry
• M(E2) ? QS n ? n-1
• Changing symmetry
• M(M1) ? Jn Jp n ? n

E2
n3
M1
E2
n2
E2
n2
n1
E2

M1
n2
MS
E2

Sym
12
0 in deformed nuclei the need for a microscopic
approach

• 0 states populated by (p,t) in large abundance
  for the first time in 158Gd
• (Lesher et al. PRC 65 (2002) )
• since then, in several deformed nuclei
• D.A. Meyer et al. PRC 76, 044309 (2006)
• D. Bucurescu et al. PRC 73, 064309 (2006)
• The (sdpf) IBM cannot account for all observed
  0 and their properties
• Zamfir et al. Phys. Rev. C 66 (2002)
• A microscopic approach is badly needed QPM
• N.L. A.Sushkov N. Yu. Shirikova, PRC 70 (04),
  PRC, 72 (05)
• ?n Si Ci (n) i, 0 gt Sij Cij (n)
  (?i ? ?j)0 gt
• QPM spectroscopic factors
• Sn(p,t) ?n (p,t) / ?0 (p,t) 2
• ?n (p,t) lt0n, N 2 P0 00,Ngt
  

13
Success and limitations of the QPM

• Fully microscopic and valid
• 
  at low and high energy.
• Suitable also for the
• double
  GDR (Ponomarev, Voronov)
• Limitations
• Valid for separable interactions
• Antisymmetrization enforced in the
• 
  quasi-boson approximation,
• Gs not explicitly correlated
• (Quasi-Boson approx)

14
Further attempts

• Multistep Shell model (MSM)
• 
  (R.J. Liotta and C.
  Pomar, Nucl. Phys. A382, 1 (1982))
• They expand and linearize
• ltaH, O,O0gt
  ( O Sph X(ph) a p ah )
• Multiphonon model (MPM)
• (M. Grinberg, R. Piepenbring et al. Nucl. Phys.
  A597, 355 (1996)
• Along the same lines
• Both MSM and (especially) MPM look involved
  

15
A new (exact) multiphonon approach

• Eigenvalue problem in a multiphonon space
• H ? ?
  gt E? ? ? gt
• ? ? gt ? H Sn ? Hn Hn ??n
  ßgt
• 
  ( n 0,1.....N )
• Generation of the n ßgt (basis states)
• An obvious (prohibitive!!) choice
• 
  n ßgt ?1, ?2, ?i ,?ngt
• where (TDA)
• 
  ?i gt Sph c ph(?i ) ap
  ah 0gt
• A more workable choice
• 
  n ßgt S a ph Caph ap ah n-1 a gt
• (suggested by TDA)
• n1
  n1 ßgt ?i gt Sph c ph(?i ) ap ah
  0gt
• But
• Also with this choice we run into a series of
  problems!!

16
Problem Overcompletness

• n ßgt S a ph Caph
  ap ah n-1 a gt
• The multiphonon states ap ah n-1 a gt
• are not fully antysymmetrized !!!
• 
  
• ap ah n-1 a gt
  ?
• 
  
• 
  p h
  p h
  
  
• The multiphonon states are
• not linearly
  independent
• and form an
• 
  overcomplete set.

17
Implications of the redundancy

• n ßgt S j Cj igt
• having put
• i gt ap
  ah n-1 a gt (not linearly independent )
• ?
• Eigenvalue problem
• Sj ltiHjgt - Ei ltijgt Cj 0
  of general form
• But (problems again!!)
• i. A direct calculation of ltiHjgt and ltijgt is
  prohibitive !!
• ii. The eigenstates would contain spurious
  admixtures!!
• How to circumvent these problems?

18
EOM Construction of the Equations

• Crucial ingredient
• lt n ß
  H, ap ah n-1 agt
• Preliminary step
• Derive
• lt n ß H, ap ah n-1 agt ( Eß(n)- Ea(n-1))
  lt n ß ap ah n-1 a gt
• (LHS)
  (RHS)
• It follows from
• request
• lt n ß H n a gt
  E a(n) daß
• property
• lt n ß apah n ? gt dn,n-1 lt n ß
  apah n-1 ? gt

19
Equations of Motion LHS

• Commutator expansion
• lt n ß H, ap ah n-1 a gt
• (ep- eh)
  lt n ß ap ah n-1 a gt. linear
• 1/2 Sijp Vhjpk lt n ß ap
  ah ai aj n-1 a gt not linear
• Linearization
  
• lt n ß H, ap ah n-1 a gt
• Sph? (ep- eh) d?ß 1/2 Sij Vhjpk lt
  n-1 ? ai aj n-1 a gt
• lt n
  ß ap ah n-1 ? gt
• Sph? Aa?(n)(phph) lt n ß ap ah
  n-1 ? gt

Î S ? n-1 ? gtlt n-1 ?
20
LHSRHS AX EX

• 
  
  
  
• where
• Aa?(n) ( ij)
  epeh dij (n-1) daß (n-1)
• VPH?H VHP?P VPP?P
  VHH?H aißj
• 
  
• n 1 (?P 0 ?H dhh)
• Tamm-Dancoff
• 
  
• Thus
• n1 n1 ßgt
  ?i gt Sph c ph(?i ) ap ah 0gt
  C X

?j? Aa?(n) ( ij) X? (ß) (j) (Eß(n) Ea (n-1)
) Xa(ß) (i)
?H lt n-1,?ahah n-1,agt ?P lt
n-1,?apap n-1,agt
Xa(ß) (i ) lt n ß ap ah n-1 agt
A(1) X (ß) (Eß (1) - E0(0) ) X (ß) A(1)
(ij) dijepeh V(phhp)
21
General Eigenvalue Problem

• Reminder
• 
  
  
  
  
• 
  
  X aß (ph) lt n ß ap ah n-1 a gt ? C aß
  (ph)
• Insert n ßgt Sa ph C aß
  (ph) ap ah n-1 a gt
• 
  
• 
  Xaß(n)(ph) lt n ß ap ah n-1 a
  gt
• X DC
  
• 
  
• where

?? Aa?(n) X?ß(n) (Eß(n) - Ea(n-1 ) )Xaß(n)
A X E X
(AD)C H C E DC
22
General Eigenvalue Problem

• Solution of problem i)
• Aa ß(n) ( ij) epeh dij (n-1) daß (n-1)
• VPH?H VHP?P VPP?P VHH?Haißj
• Dij ltijgt lt n-1 a( ah ap)( ap ah)
  n-1 ß gt
• 
  

(AD)C H C E DC
D ?H (n-1) ?P (n-1) ?H (n-1) ?P (n-1)
C (n-1) X (n-1) C (n-1) X (n-1) ?P (n-2)
recursive relations
Problem i) solved!!!!
23
General Eigenvalue Problem

• Solution of Problem ii) (redundancy)
  
• 
  
• Removal of redundancy Choleski
• D 0 ? D L LT
  L ?
• Determination of L ?lij follows from D
  L LT ?
• l211
  d11
• ? Recursive formulas l11 lj1 dj1
  
• 
  l2ii dii Sk1,i-1
  l2ik
• lii
  lji dji Sk1,i-1 lik ljk
  

(AD)C H C E DC
Det D 0
l11 0 0 0 0 0 l21 l22 0 0 0.. .0 l31
l32 l33 0 0.. .0 li1 ...lii 0....0 lN1
lN2 lNN-1 lNN
24
General Eigenvalue Problem

• Solution of Problem ii) (redundancy)
  
• D 0 ? D L LT
  L ?
• 
  
  
  
• Property
• DetD (DetL)2 l112 l222 ...lii2. ?21
  ?2i (D ?i gt ?i ?i gt )
• 
  
  
  
• ?
  ? lnn 0 ? ?n gt linearly dependent
  
• Numerical stability
• Sequence order lii ljj ? j
  gt i
• Once lnn 0 ? lii 0
  ? i gtn ? stop at the nth step

l11 0 0 0 0 0 l21 l22 0 0 0.. .0 l31
l32 l33 0 0.. .0 li1 ...lii 0....0 lN1
lN2 lNN-1 lNN
(AD)C H C E DC
Det D 0
Det D 0
Det L 0
X
25
General Eigenvalue Problem

• Solution of Problem ii) (redundancy)
  
• Choleski decomposition
• Matrix inversion
  
  
  
• Exact
  eigenvectors
• 
  
• n ßgt S aph Caph ap
  ah n-1 a gt ? H n (phys)

(AD)C H C E DC
D D
HC (D-1AD)C E C
26
Iterative generation of phonon basis

• Starting point 0gt
• Solve
  H(1) C(1) E(1) C(1)
• 
  
  n1, agt
• 
  X(1) ?(1)
• Solve
  H(2) C (2) E (2) C
  (2)
• 
  
  n2,agt
• 
  X(2) ?(2)
• 
  X(n-1) ?(n-1)
• Solve
  H(n) C (n) E
  (n) C (n)
• 
  X(n) ?(n)
  n,a.gt

27
H Spectral decomposition, diagonalization

• H S na E a(n) n
  agtltna (diagonal)
• S na ß n
  agtltna H nßgtltnß (off-diagonal)
• 
  n
  n 1, n2
• Off-diagonal terms Recursive formulas
• lt n ß H n-1 a gt
  Sph? ?a? (n-1)(ph) X?(ß) (ph)
• lt n ß H n-2 a
  gt S Vpphh X?(ß) (ph) X?a(a) (ph)
• Outcome of diagonalization H ??gt E? ??gt

??gt Sna Ca(?) (n) nagt
nagt S? C?(a) ap ah n-1?gt
28
16O as theoretical lab

• Structure of 16O A theoretical 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 h?
• 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)
• 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)
• No-core SM (NCSM) Huge space!!!
• Symplectic No-core SM (SpNCSM) a promising tool
  for cutting the SM space

29
Numerical test A 16

• Calculation up to 3-phonons and 3h?
• Hamiltonian
• H H0 V Si hNils(i) Gbare
  ( VBonnA ? Gbare)
• CM motion (F. Palumbo Nucl. Phys. 99 (1967))
• H ?
  H Hg
• Hg g
  P2/(2Am) (½) mA ?2 R2
• Consistent choice of ph space It must
  includes all ph configurations up to 3h?
• H ? H??(int) Fn(CM) (E? En(CM)) ??(int)
  Fn(CM)
• For ggtgt1 E(ngt0)(CM) E0(CM) gtgt (Ev E0(CM))
• 
  ??(int) F(ngt0)(CM)

X
30
Ground state

• ?0gt C(0)0 0gt
• S? C?(0) ?, 0gt
• S ?1?2 C?1?2 (0) ? 1 ? 2, 0 gt
• ?, 0gt ? 1 ? 2, 0 gt
• 1 lt ?0?0gt P0 P1 P2

31
16O negative parity spectrum

• Up to three phonons

32
E.m. response
??gt Sn? C?(?) (n) n?1?2. ?n gt

• 
  
• C3 ?1 ?2 ?3 gt
• C1 ?1 gt
• C2 ?1 ?2gt
• C0 0gt
• gs
• MPEM

• ?gt Sph c ph(?) ap ah0gt
• ?gt
• 0gt
• gs
• TDA

33
IVGDR

• 1-gtIV 1(p-h) (1h?)gt
• 
  

34
ISGDR

• 1-gtIS 1(p-h) (3 h?)gt 2(p-h) (1h? 2h?)gt
  3(p-h) (1h? )gt
• Toroidal

35
Octupole modes

• 3-gtIS 1(p-h) (3 h?)gt 2(p-h) (1h? 2h?)gt
  3(p-h) (1h? )gt
• Low-lying

36
Effect of CM motion
37
Effect of the CM motion
38
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 h?.
• To go beyond
• Truncation of the
  space needed !!!
• Truncation is feasible (the phonon states are
  correlated).
• A riformulation for an efficient truncation is
  in progress

39
Perspectives New formulation

• ap ah n-1 a gt ? O? n-1 a gt
  O?
  Sph cph(? ) ap ah 0gt
• lt n ß H, ap ah
  n-1 agt ? lt n ß H, O? n-1 agt
• 
  
• 
  
• 
  
• 
  
  
• 
  
  

??? Aa?(n) (?,?) X(ß)? ? Eß(n) X (ß)? ?
X(ß)a? lt n ß O? n-1 a gt
??? (kl) lt ? akal ?gt
?a?(n-1) (kl) lt n-1,? akal n-1,agt
Aa?(n) (?,?) E? Ea(n-1) d? ? da?
??? V ?a?(n-1)
n ßgt Sa? C(ß)a? O? n-1 a gt
SC(ß) ?i
?1,.?i.?N gt C (ß) ?i S C(ß)a?1
C(a) ??2 C (?)d?3
40
Vertices

• TDA (n1) MPEM (n3)

?
--------
---------
p
?
h
p
h
?
a
Vphhp
??? V ?a?(n-1)
41
Acknowledgments

• J. Kvasil, F. Knapp (Prague)
• F. Andreozzi, A. Porrino (Napoli)

42

• THANK YOU

43
E2 response up to 3 h? Running sum

• Sn Sn (En E0(0) ) Bn(E2,0?2n)
• Rn Sn/SEW(E2)
• It is necessary to enlarge the space!!

44
E2 response up to 3 h?

• S(?,E2) Sn Bn(E2,0?2n) ??(?-?n)
• ??(x) (?/2p) / x2 (?/2)2
• M(E2µ) S(p)ep rp2 Y 2µ

45
Effect of CM on the E2 response
46
Multiphonon excitations Exp. Evidence (deformed
nuclei)

• (Some of the) 0 states
• observed in large abundance in (p,t)
• S. R. Lesher et al. PR C 66, 051305(R)(2002).
• H.-F. Wirth et al. PRC 69, 044310 (2004).
• D. Bucurescu et al, PRC 73, 064309 (2006).
• D. A. Meyer et al., PRC 74, 044309 (2006).

• M1 mode built on excited states
• observed in two-step ?-cascade
• M. Krticka, F. Becvar et al. PRL 92 , 172501
  (2004).
• 
  

47
From TDA to RPA

• A(n) B(n)
  X(n)
• (E(n) -
  E(n-1))
  
• B(n) A(n)
  -Y(n)

48

• Aa?(n) (phph)dhhdppda?(n-1)epeh
• Sh1 V(ph1hp) ?a?(n-1) (h1h)
• Sp1 V(php1h) ?a?(n-1) (pp1)
• dhh Sp2p3 V(pp2pp3)?a?(n-1) (p2p3)
• Sh2h3 V(ph2ph3)?a?(n-1)(h2h3)
• dpp Sh2h3 V(hh2hh3)?a?(n-1)(h2h3)
• Sp2p3 V(hp2hp3)?a?(n-1)(p2p3)
• Ba?(n) (phph)
• Sh2 V(p ph2 h)?a?(n-1)(h2h)
• Sp2 V(hhp2p)?a?(n-1)(pp2)

49
AX EX

• where
• Aa?(n) (phph)dhhdppda?(n-1)epeh
• Sh1 V(ph1hp) ?a?(n-1)(h1h)
• Sp1 V(php1h) ?a?(n-1)(pp1)
• dhh1/2 Sp2p3 V(pp2pp3) ?a?(n-1)(p2p3)
• dpp1/2 Sh2h3 V(hh2hh3) ?a?(n-1)(h2h3)
• (?a?(n)(ij) ltn,aaiajn,agt)