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