An exact microscopic multiphonon approach to nuclear
spectroscopyN. Lo Iudice
Università di Napoli Federico II
Naples(Andreozzi, Lo Iudice, Porrino) Prague (Knapp, Kvasil)
Collaboration
Tokyo 07
Eigenvalue problem in a multiphonon space
H | Ψ ν > = Eν | Ψ ν >
| Ψ ν > H = Σn Hn ( n= 0,1.....N )where
Hn ∋ |n, α > ~ | ν1 ν2 …..νn >
| νi > = Σph c ph(νi ) a
†p ah |0>
First goal: Generate the basis states |n, α >
We do it by constructing a set of equations of motion and solving them iteratively
EOM: Construction of the Equations • Crucial ingredient < n; β | [H, a†
p ah] | n-1; α>
• Preliminary step: Derive< n; β | [H, a†
p ah] | n-1; α> = ( Eβ
(n) - Eα
(n-1)) < n; β | a
†
p ah| n-1; α >
(LHS) (RHS)
The RHS comes from
* property < n; β | a†
pa
h | n’; γ > = δ
n’,n-1 < n; β | a†
pa
h | n-1; γ >
** request
< n; β | H | n; α > = E α
(n) δαβ
Commutator expansion
< n; β | [H, a†p ah] | n-1; α > =
(εp- εh) < n; β | a†p ah| n-1; α >. +
1/2 Σijp’
Vhjpk < n; β | a†p’ ah a
†i aj | n-1; α > + …
Linearization
< n; β | [H, a†p ah] | n-1; α > =
Σp’h’γ [(εp- εh) δγβ
+ 1/2 Σij
Vhjp’k’ < n-1; γ | a†i aj | n-1; α > +
………] < n; β | a†p’ ah’ | n-1; γ >
= Σp’h’γ
Aαγ(n)(ph;p’h’) < n; β | a†
p’ ah’ | n-1; γ >
Î = Σ γ |n-1; γ >< n-1; γ|
Equations of Motion : LHS
LHS=RHS AX = EX
jγ Aαγ(n) ( ij) Xγβ
(n)(j) = (Eβ
(n) - Eα(n1)) Xαβ
(n)(i) where
Xαβ(n) (i )
= < n; β | a†
p ah| n-1; α >
Aαγ(n) ( ij) = [εp–εh] δij
(n-1) δαβ (n-1)
+ [VPHρH + VHPρP + VPPρP + VHHρH ]αiβj
ρH ≡ {< n,γ|a†
hah’|n,α>} ρP ≡ {< n,γ|a†
pap’|n,α>} n =1 (ρP = 0 ρH = δhh’)
A(1) Xα(1) = (Eα
(1) - E0 (0) ) Xα
(1) Tamm-Dancoff
A(1) (ij)= δij[εp–εh] + V(p’hh’p)
Structure of multiphonon states: Overcompletness
The eigenvalue equations yield states of the structure
|n; β> = Σ α ph cα
ph a†
p ah | n-1; α >
• Problem: The multiphonon states are not fully antysymmetrized !!!
a†p ah | n-1; α > ≡
p h p h
The multiphonon states form an overcomplete set
Solution of the redundancy Reminder
Insert |n; β> = Σα ph Cαph a†
p ah | n-1; α >
Xαβ
(n)(ph)
= < n; β | a†
p ah | n-1; α >
X = DC (AD)C = H C = E DC
where
Dij = < n-1; α’| ah’ a†
p’ a†
p ah | n-1; α> overlap or metric matrix
Problem due to redundancy Det D = 0
γ Aαγ(n) Xγβ(n)= (Eβ
(n) - Eα(n-1 ) )Xαβ(n)
A X = E X
Solution of the redundancy *Removal of redundancy: Choleski decomposition(no
diagonalization)
D Ď ** Matrix inversion
Exact eigenvectors
|n; β> = Σ αph Cαph a
†
p ah | n-1; α > H n (phys)
● Now compute in Hn
i. X(n)
ii. ρ(n) recursive formula
X= D C
ρ(n) = C X (n) + C ρ (n-1) X (n)
HC = (Ď-1AD)C = E C
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 !!!
H: Spectral decomposition, diagonalization
H = Σ nα
E α
(n) |n; α><n;α| + (diagonal)
+ Σ nα β
|n; α><n;α| H |n’;β><n’;β| (off-diagonal)
n’ = n ±1, n±2
Off-diagonal terms: Recursive formulas
< n; α | H| n-1; β > = Σphγ
ϑαγ
(n-1)(ph) Xγβ
(n)(ph)
< n; α | H| n-2; β > = Σ V pp’hh’
Xγβ
(n) (ph)
Xγβ
(n-1) (p’h’)
Outcome of diagonalization H |Ψν> = Eν |Ψν>
|Ψν> = Σnα Cα(ν) (n) | n;α> |n;α> = Σγ Cγ
(n) | n-1;γ>
Numerical test: A = 16 ∙ Calculation up to 3-phonons and 3ħω
Hamiltonian
H = H0 + V = Σi hNils
(i)+ Gbare
( VBonnA
⇨ Gbare)
• CM motion (F. Palumbo Nucl. Phys. 99 (1967))
H H + Hg
Hg = g [ P2/(2Am) + (½) mA ω2 R2 ] • Consistent choice of ph space: It must includes all ph
configurations up to 3ħω
Ground state
|Ψ0> = C(0)
0 |0>
+ Σλ C
λ
(0) |λ, 0>
+ Σ λ1λ2 Cλ1λ2
(0) |λ 1 λ
2, 0
>
|λ, 0> |λ 1 λ
2, 0
>
1 = < Ψ0|Ψ
0> = P
0 + P
1 + P
20
10
20
30
40
50
60
70
80
0ph 2ph 4ph 6ph
EMNocoreHJ
0
10
20
30
40
50
60
70
80
0ph 1ph 2ph
P(n) %noCMcorr
16O negative parity spectrum • Up to three phonons
IVGDR
|1->IV ~ |1(p-h) (1ħω)>
ISGDR |1->IS ~ |1(p-h) (3 ħω)> + |2(p-h) (1ħω + 2ħω)> + |3(p-h) (1ħω )>
Toroidal
Octupole modes |3->IS ~ |1(p-h) (3 ħω)> + |2(p-h) (1ħω + 2ħω)> + |3(p-h) (1ħω )>
Low-lying
Effect of CM motion
Effect of the CM motion
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).
- A riformulation for an efficient truncation is in progress
THANK YOU
E2 response up to 3 ħω: Running sum
Sn = Σn (En – E0(0) ) Bn(E2,0→2+
n)
Rn = Sn/SEW(E2)
• It is necessary to enlarge the space!!
E2 response up to 3 ħω
S(ω,E2) = Σn Bn(E2,0→2+n) ρΔ(ω-ωn)
ρΔ(x) = (Δ/2π) / [x2 + (Δ/2)2]
M(E2μ) = Σ(p)ep rp2 Y 2μ
Effect of CM on the E2 response
Growing evidence of multiphonon excitations
* 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 × Q3|0>, Q2×Q2×Q3|0>• Proton-neutron (F-spin) mixed-symmetry states (N. Pietralla et al. PRL 83, 1303 (1999))
[Q2(p) - Q2
(n)] (Q2(p) + Q2
(n)) N|0>,
** 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 (maybe) triple dipole giant resonances D × D |0>
From TDA to RPA
A(n)
B(n)
X(n)
= (E(n)
- E(n-1)
)
B(n)
A(n)
-Y(n)
Aαγ(n) (ph;p’h’)=δhh’δpp’δαγ(n-1)[εp–εh]
+ Σh1 V(p’h1h
’p) ραγ
(n-1) (h1h)
+ Σp1 V(p’hp
1h’) ραγ
(n-1) (pp1)
+ δhh’ [ Σp2p3 V(p’p2pp3)ραγ
(n-1) (p2p3)
+ Σh2h3 V(p’h2ph3)ραγ
(n-1)(h2h3)]
+ δpp’ [ Σh2h3 V(hh2h
’h3)ραγ
(n-1)(h2h3)
+ Σp2p3 V(hp2h
’p3)ραγ
(n-1)(p2p3)]
Bαγ
(n) (ph;p’h’)=
Σh2 V(p p’h2 h
’)ραγ
(n-1)(h2h)
+ Σp2 V(h’hp2p
’)ραγ
(n-1)(pp2)
AX = EXwhere
Aαγ
(n)
(ph;p’h’)=δhh’δpp’δαγ
(n-1)[εp–εh]
+ Σh1 V(p’h1h’p) ραγ
(n-1)(h1h)
+ Σp1 V(p’hp1h’) ραγ
(n-1)(pp1)
+ δhh’1/2 Σp2p3 V(p’p2pp3) ραγ
(n-1)(p2p3)
+ δpp’1/2 Σh2h3 V(hh2h’h3) ραγ
(n-1)(h2h3)
(ραγ
(n)(ij) = <n,α|a†
iaj|n,α>)
Choleski decomposition• Any real non negative definite
symmetric matrix can be written as
D = L LT
( L {lij} lower triangular matrix) Det{D} = (Det{L})2
= l112 l22
2 ...lii2…..
= λ21 …λ2
i… (D | λi > = λi | λi > )
Definition of L: Recursive formulas
l211 = d11
l11 lj1 = dj1 j=2,….,nl2
ii = dii – Σk=1,i-1 l2
ik
lii lji = dji – Σk=1,i-1 lik ljk
(It reminds the Schimdt orthogonalization method)
The decomposition goes on until lnn = 0
lnn = 0 → Det{L} =0→Det{D} =0 ⇨
⇨ | λn > (D | λi > = λi | λi > ) linearly dependent ⇨ to be discarded
• For numerical stability we need maximum overlap
λj ≤ λi j > i
• Sequence order
lii ≤ ljj j > i
Once lnn = 0 → lii = 0 i >n
1) We can stop at the nth step
2) We get Ď (Nn < N
r) with maximum determinant
(overlaps)
Elimination of CM spuriosity F. Palumbo Nucl. Phys. 99 (1967)
• HSM H = HSM + Hg
• Hg = g [ P2/(2Am) + (½) mA ω2 R
2 ] = g/A [ Σi hi + Σi<j vij ]
hi = pi
2/(2m) + (1/2) m(Aω)2ri
2
vij = (1/m) pi· pj + m (Aω)2ri · rj
Hg effective only in Jπ = 1
- ph channel.
H Ψ = (HSM + Hg) Ψ =E Ψ
In the full space
Ψ = ψ
int Φn
CM
For the physical states
Ψn = ψn Φ0
CM
with CM energy
E0
CM = 3/2 g ħω
For the spurious ones
Ψn
(1)
= ψn
Φ1
CM
at the very high CM energy
E1
CM
- E0
CM
= g ħω
They can therefore be tagged and eliminated
Top Related