an exact microscopic multiphonon approach
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An exact microscopic multiphonon approach. Naples F. Andreozzi N. Lo Iudice A. Porrino Prague F. Knapp - PowerPoint PPT PresentationTRANSCRIPT
An exact microscopic multiphonon approach
Naples F. Andreozzi N. Lo Iudice A. Porrino
Prague F. Knapp J. Kvasil
From mean field to multiphonon approaches
is responsible for collective modes
ij ijkl
kljiji aaaaklVijaajTiH ||4
1||
iresiii Vaa
resV
standard approach to collective modes: RPA (TDA)RPA (TDA) – harmonic approximations:
OOH ],[where
RPA:
TDA:
ph
phphhpph aaYaaXO ][ )()(
ph
hpph aaXO )(
for anharmonic effects multiphonon approaches are needed
OOOOOOO
problems:
1. )()( hphp aaOaaOOO lack of antisymmetry overcomplete set
2. brute approximations and/or formidable calculations are needed
sophisticated methods (with minimal approximation) should be developed for investigation of anharmonic effects
Fermion-Boson mapping A. Klein and E. R. Marshalek, Rev. Mod. Phys. 63, 375 (1991)
operator mapping
(S.T.Belyaev, V.G. Zelevinsky, Nucl.Phys. 39, 582 (1962)
ph
hpph aaCb
i ijkkjiijkiiB BBBXBXb ),3(),1()(
ijjiijB BByya ),2(),0()(
where
constraint: the exact fermionic anticommutator ijji aa },{ should be
used in the calculation of ],[ bb
ijji BB ],[
state vector mapping (T. Marumori et al., Progr.Theor.Phys. 31, 1009 (1964) 0|| bbbn )0|)| kji BBBn
constrant: )||(|| mOnmOn BF
problems in practical calculations: - in general slow convergence of the boson expansion - involved calculations
ij
jijkir aaCCa )()(
Practical examples of fermion – boson (FB) mapping
IBM – phenomenological FB mapping
ij
JjiijJ aaCA 2,02,0 )( ),( ds bosons
- originally s,d bosons were introduced from algebraic group relations and corresponding Hamiltonian was phenomenologically parametrized
- Marumori mapping of IBM was proved only within one single j shell and for only pairing plus quadrupole Hamiltonian (see T. Otsuka, A. Arima, F. Iachello, Nucl.Phys. A309, 1 (1978))- IBM successful in low-energy spectroscopy but purely phenomenological
quasi-particle + phonon model (QPM) (inspired by FB mapping)V.G.Soloviev, Theory of Atomic Nuclei: Quasiparticles and Phonons, Ins. Ph. Bristol, 1992
),,( ijjihp aaaaaaH ),( ijjiH ),( OOH BCS RPA
ijijijjiij YXO ][ )()(
RPAOOCOCn |}{,2| ),2(),1(
limitations: - Pauli principle valids only partly - valid (used) only for separable interactions - correlations not explicitely included in the ground state
QPM is microscopic and successful at low and high energies
Further attempts for multiphonon approachesMultistep Shell Model (MSM) R.L.Liotta, C.Pomar, Nucl.Phys. A382, 1 (1982)
ph
hpi
phi aaXO ),1(
ijjiij OOXn 0|,2| ),2(
iii nOXn ,2|,3| ),3(
iE ,1
,2E
,3E
They expand 0|]],,[[|, ji OOHn and for 2n (2 phonon) space theykeep only linear terms in two-phonon operators (linearization) and they geteigenvalue equation in two-phonon space
ji OO
),2(),2(,1,1,2 )( ijijji XAXEEE
where is expressed in terms of TDA eigenvalues and eigenstates.A- eigenvalue equations generate an overcomplete two-phonon set of states but redundancy and Pauli principle violation are cured by graphical method in the combination with the diagonalization of the metric matrix
Multiphonon Model M. Grinberg, R. Piepenbring et al., Nucl.Phys. A597, 355 (1996)along the same line (instead of a graphical method they give the complex recurentformulas between and ),| n ,1| n
Both MSM and MPM look involved and of problematic applicability (indeed, they have not been widely adopted).
,|, nn
method proposed here: Equation of Motion Method (EM)
eigenvalue problem || EH
is solved in a multiphonon space
N
nn
0
HH
where nn 21|,|nH
phhpphi aaC i 0||
Tamm-Dancoff
phonon
eigenvalue problem is solved in two steps:
- generating the multiphonon basis
- construction and diagonalization of the total Hamiltonian matrix in the whole space
nn 21|,| )(nE
H
ij ijkl
kljiijkljiij aaaaVaatH4
1
multiphonon basis1-st step: generating the multiphonon basis ,| n
We require: )(,||, nEnHn ),|,|( )( nEnH nor
for basis states with following orthogonality properties,| n)( hpph aab
)(,||,,||,
)(,||,,||,
)(,1||,,||,
,|,
)(,,
)(,,
)(1,1,
hhnaannaan
ppnaannaan
phXnbnnbn
nn
nnnhhnnhh
nnnppnnpp
nnnphnnph
nn
with closure relation:
n
nnI |,,|
Using we have),|,|( )( nEnH n
)()(,1|],[|, )()1()( phXEEnbHn nnnph (A)
From other side the closure relation and orthogonality properties above give
)(
1|)||()||(|
1|],[|
nHnnbbnnHn
nbHn
nphph
n
ph
substituting for and taking into account that only terns can contributeH b
multiphonon basis
1 1
11
21
21
21
21
21
})()(
]))((2
1[
])(2
1[{
1)1(
1)1(
21)1(
21)1(
h p
nphhp
nphhp
hhn
hhhhhhhhpp
pp
npppp
hppphh
ppVhhV
hhV
ppV
(B)
where
The comparison of (A) with (B) gives the eigenvalue equation for )()( phX n
)()( hpX n
hp
nnnnnnnn XEXAphXEhpXhpphA
)()()()()()()()( )()(),(
with
1
1
1
1
21
2121
21
21
)()(
]))((2
1[
])(2
1[),(
1)1(
1)1(
21)1(
21)1()1()(
p
nphhp
h
nphhp
hhhh
nhhhhhhpp
pp
npppppphh
npphh
n
ppVhhV
hhV
ppVEhpphA
(C)
1
11h
phhppppp Vt 1
11h
hhhhhhhh Vt
multiphonon basisIn the Hartree-Fock basis we have:
hhh
hhhhhhhhhhhh
hphphpppppppp
hhVt
ppVt
)()(
0)()(
1
11
1
11
)0(
)0(
and thenphphhppphh VhpphA )(),()1(
and (C) is the standard Tamm-Dancoff equation:for TDA states:
)()(),( )1(0
)1()1(0
)1(0 phXEhpXhpphA
hp
ph
phphXn |)(,1| )1(0
For the first sight one can expect that solutions of (C), , represent coefficients inthe expansion of the state in terms of states(because )
)()( phX n
,| n
,1| naa hp
1||)()( naanphX hpn
However, states represents the redundant (overcomplete) basis: ,1| naa hp
hhppphhp nbbn ,1||,1
multiphonon states are not fully antisymmetrizes !! ,1| naa hp
We should extract physical nonredundant basis fromthe redundant basis of states
,1| naa hp
multiphonon basis
In order to extract physical basis let us expand the exact eigenstatesin the redundant basis
,| n ,1| nbph
hpphhp
nph
n
ihp
N
i
n
nbbnhpCnbnphX
nbCnii
r
i
,1||,1)(,1||,)(
,1|,|
)()(
1
)(
),()1( hpphd n
metric matrix:
)()()(
,1||,1,1||,1
,1||,1,1||,1),(
)1()1()1(
)1(
hhpphh
naaaannaan
naaaannbbnhpphd
nnnpp
hhpphhpp
hpphphhpn
so, in matrix form: )()1()( nnn CDX
(D)
eigenvalue equation (C) can be rewritten:
)()1()()()1()()()()()( nnnnnnnnnn CDECDAXEXA
However, metric matrix is singular (det{D}=0) it is not possible to invert it
multiphonon basisusual solving of the singular metric matrix problem – diagonalization of D(Rowe J., Math.Phys. 10, 1774 (1969)
UDUDD 1
0..00000
........
0.00.000
0.00.00
........
0.0.0.0
.0......
0.0...01
nN
i
D
ijd iji
rnihp NNjinbiii
,,,,1,,,1|
CECA
CUDECUDUAUCUDUECUDUACDECDA
111
CUCUAUA
,1
effectively we obtain linearlyindependent (physical) eigenvectors
nn NN
0 inNifor
problems: - a bruto forced calculation (diagonalization) of very lengthy and difficult for , practically impossible for - diagonalization of changes the structure of the multiphonon states (now the vector - see (D) – contains instead of )
D2n 2n
D,| n C
Cin our EM approach the redundancy of the overcomplete basis, , is removedby Choleski decomposition: - no diaginalization of - much faster and more effective
,1| nbph
D J.H.Wilkinson, The Algebraic Eigenvalue, Clarendon Oxford, 1965
n
multiphonon basisremoving of the redundancy of the overcomplete basis by Choleski decomposition:
any real non negative definnite symmetric matrix can be rewritte as
TLLD where is the lower triangular matrixwith defined by recursive formulas:
}{ ijlL
ijl
1
1
1
1
22
1111
11211
,,2
i
kjkikjijiii
i
kikiiii
jj
lldll
ldl
njdll
dl
22222
211
2 }det{}det{ nnii llllLD
decomposition of using recursiveformulas goes until a diagonal element - in this moment we know that -th basis vector is a linear combinationof vectors -th vector is discarded and we dropped -th row and -th column from and decomposition continues with -thbasis vector. This procedure continuesuntil the whole redundant basis is exhausted.
D
0iil0}det{
1
2
n
jjjlD
i1,,2,1 i
ii i D
)1( i
During decomposition we can rearrange thebasis in the decreasing way according towhich gives us the maximum of(maximum overlap)
iil}det{D
Choleski decomposition – from vectors of the overcomplete basis we extract linearly independent vectors ( )
nn nn
+
multiphonon basisCholeski method we obtain ( ) linearly independentbasis states, , for the subspace with nonsingularmatrix
;1| nbph nH},1||,1{)1(
nbbnD phhp
n
nN nn NN
nn NN ( - type matrix )
we can solve the eigen-value problem
)(nH we obtain eigen solutions (physical, nonredundant) in the subspace :nN
n
jjji
N
j
nijhpjj
ni EnbhpCn
1
)()( ,1|)(,|
nH
Now we can go from n – phonon subspace(we know ))()1()()()( ,, nnnnn CDXEC
nH (n+1) – phonon subspaceand solve: )1()1()1()1( nnnn CECH
to 1nH
where for the creation of and we need and :)( )1()()1( nnn ADH )()( 21)(
21)( hhpp nn
})({)()(
,||,)(
2)1(
11
)()(
21)(
2
21
pphpXhpC
naanpp
jn
pp
N
jj
njj
n
ppn
j
n
with a similar expression for )( 21)( hhn
recursive
formula
nNnN
(E)
)()()()1()(1)1()()1()()()1()( )( nnnnnnnnnnnn CECDADCDECDA
multiphonon basisIterative generating of phonon basis
starting point 0|
)1()1( , X
)2()2( , X
)1()1( , nnX
)()()(2
)(1
)()()()(
|,| ni
in
ii
nnnn
Ein
CECH
)2()(2
)(1
)2()2()2()2(
|,2| iii Ein
CECH
)1()(1
)1()1()1()1(
|,1| ii Ein
CECH
)()( , nnX
multiphonon basis is generated
full eigenvalue problem
Full eigenvalue problem
Once the multiphonon basis has been generated the total Hamiltonian matrix can begenerated and digonalized.
n
nn
n nnHnnnnEH |,,||,,||,,|)(
The second term involves only nondiagonal matrix lementsgiven by recursive formulas:
)2,1( nnn
)()(4
1
,2||,4
1,2||,
11)1()(
22
121
1
11
11
2
hpXphXV
naaaanVnHnH
nn
hphphhpp
ijklnkljinijklnn
nnnn
n
nn
with
21 21
2111112111211})({)( 21
)1(21
)1()(
pp hhhh
nhhph
nhppp
ph
nnnnnnnhhVppV -1n
V
1
111
1
)(2
1
,1||,4
1,1||,
)()(
11
n
nnnn
nn
ph
nph
ijklnkljinijklnn
phX
naaaanVnHnH
V
full eigenvalue problemBy the diagonalization of the total Hamiltonian matrix:
nnnnnn
nnnnnn
nnnnnn
HHH
HHH
HHH
HHH
HHHH
HHHHH
HHHHH
HHHH
HHH
H
12
11112
21222
554535
45443424
3534332313
2423221202
13121101
020100
0000000
000000
00000
0
00000
00000
0000
0000
00000
000000
where
},3||,3{
},2||,2{
},2||0,0{},1||,1{
},1||0,0{0,0||0,0
3333
2222
2021111
10100
nHnH
nHnH
nHnHnHnH
nHnHnHnH
we obtain exact nuclear eigenvalues and eigenvectors:
n n
nnn
ni
n
n cnc
1)( |,||
1
1E
(F)
transition amplitudesTransition amplitudes
Let us consider the transition amplitudes ififM ||M
of some single-particle transition operator kl
lkkl aaMM
Substitution of (F) gives:
klnn
nnfikl
nlkn n
nkl
klfi
if
nn
nnnn
n n
nn
klcc
naanccM
)(
,||,
)()()(
)()(
M
M
transition amplitude involves density matrix elementswhich are given by recursive formula (E)
)()( klnn
nn
elimination of spuriosityElimination of the center of mass spuriosity
Following the method: F. Palumbo, Nucl.Phys. 99, 100 (1967) we add to the startingHamiltonian
a center of mass (CM) oscillator Hamiltonian multiplied by a constant :
ij ijkl
kljiji aaaaklVijaajTiH ||4
1||
g
)2
3
2
1
2( 22
2
RmA
mA
PgH g
A
i ji
CMij
CMi gUh
1
)(
2
3
where
A
i
A
ijiji
CMijii
iiCM
i
A
i
rrmppmA
gUprmRMP
rmm
p
A
ghrmRM
1 1
2)(
22)(
1
)1
(
)2
1
2(
full space Schr. equation:
oscosc
oscosc
oscoscosc
nn
oscnn
nnn
g
EEE
EH
HHH
)(
for physical states:
)2
3( )(
000 gE oscnn oscosc
for spurious states:)
2
3( )(
111 ggE oscnn oscosc
for big spurious statesare shifted to high energies
by the shift:
spurious states can be easily tagged and eliminated
gEE oscosc )(0
)(1
g
effec
tive o
nly in
J = 1-
channel
elimination of spuriosity
reminder :
The Hamiltonian is effective only in ph- channel. gH 1J
For exact factorization of the wave function and tagging the spurious mode is easy. In our approach the ph (1- phonon) states represent building blocks of all n- phonon states we are able to identify and eliminate exactly spurious states also for
1n
1n
1n
There is a necessary condition for the tagging and elimination of spurious modes by the method given above: all shells up to given are to be involved in the space
The fulfilment of this condition is possible in our approach (see futher)but is not easy in the standard shell model calculations.
n
evaluation of the method
Pluses and minuses of the method
Pluses Minuses
simple structure of the eigenvalueequation in any n- phonon subspace
only density matrix elements , and have to becomputed
)()( ppn )()( hhn )()( phX n
relatively simple recursive formulashold for , andfor all other quantities
)()( ppn )()( hhn
overcompletness of multiphonon basisstates (in our approach eliminated byelegant Choleski method)
number of density matrix elements to be computed increaseswith the number of phonons
for higher numerical process may become slow
)()( ijn
n
our method enables to use Palumbo’sprocedure for the elimination of theCM spurious mode also for largeconfiguration spaces
numerical resultsNumerical test: 16Ocalculation up to 3- phonon states ( ) with unperturbed energies up to3,2,1,0n 3
Hamiltonian :
ij ijkl
kljiji aaaaklVijaajTiH ||4
1||
iresiii Vaa
ibare
iNilss Gh )(
bareABonn GV
Elimination of the CM spurious mode : F. Palumbo, Nucl. Phys. 99, 100 (1967)needed condition: construction of the subspaces from n- phonon ( np-nh configuration) states with unperturbed energies up to 3
)3,2,1( nnH
snl
dnl
gnlN
33,0
22,2
11,44
pnl
fnlN
22,1
11,33
snl
dnlN
22,0
11,22
1)6(11,11 pnlN
4)30(
3)20(
2)12(
0)2(11,00 snlN
3,4,133
)1)(1,2()1)(1,2(2
)1)(1,2(1
0|0
1
11
1
toupNforstatesphononnon
pdspdsonlyn
pfpn
n
111
11
1
)1)(1,2()1)(1,2()1)(1,2(3
)1)(1,2()1)(1,2(2
)1)(1,2,3(1
1
pdspdspdsonlyn
pfppdsn
pgdsn
,2,1,0
,2,1
1)1(2
l
n
nN
4N
0
0,5
1
1,5
2
2,5
3
3,5
4
1ph 2ph 3ph
Nmx
dependence of the maximum major number N ofshell one has to include in order to have all and onlyconfigurations up to 3
numerical resultspositive parity
0 ph 1 ph 2 ph 3 ph 4 ph0
10
20
30
40
50
60
70
80
n ph- component
Pn [
% ]
our BG HJ B
0 ph 1 ph 2 ph0
10
20
30
40
50
60
70
80
n ph- component
Pn [
% ]
w ithout CM w ith CM
comparison with others influence of the elimination of CM spurious mode
Positive parity statesGround state
21
21 21)0()0()0(
00 ||0|| ccc
2
21
1
1
)2(,21
1
)1()(,
||
0||
N
jjhp
N
jhphp
jjj
jjjj
bC
bC
1| 21000 PPP
21
21
2)0(2
2)0(1
2)0(00
||
||
||
cP
cP
cP
BG G.E.Brown, A.M.Green, Nucl.Phys. 75, 401 (1966)HJ W.C.Haxton, C.Johnson, Phys.Rev.Lett. 65, 1325 (1990)B B.R.Barret et al., private communication
numerical resultspositive parityE2 response (up to )3
2|||)2||2|)2;2( grEgrEB M(
i
ii YreE 22)2M(
)()2;2();2( EEgrEBEES
22 )2
()(
1
2)(
EEEE
the big difference between 0+1-phononand 0+1+2-phonon cases
we need to enlarge the space up to (in order to have 4 ph components in the ground and 2+ states)
4
numerical resultspositive parityE2 response (up to ) – effect of the elimination of the CM motion3
21
21 21)2()2()2(
02||0||
ccc
2
21
1
1
)2(,21
1
)1()(,
||
0||
N
jjhp
N
jhphp
jjj
jjjj
bC
bC
3,4,133
)1)(1,2()1)(1,2(2
)1)(1,2(1
0|0
1
11
1
toupNforstatesphononnon
pdspdsonlyn
pfpn
n
.gHIn spite of the fact that CMHamiltonian acts only inthe channel it contributes tothe E2 strength because of thepresence of the phononcomponents:
1
11 )1)(1,2()1)(1,2( pdspds
2n
gH
numerical resultspositive parityE2 response (up to ) – running sum3
S(E
) / S
0 10 20 30 40 50 60 70
1.0
0.0
0.2
0.4
0.6
0.8
E [ M e V ]
)()2;2();2( EEgrEBEES
It is necessary to enlarge the space up to 4
22
)(1
0
1
4
50
2)2(
);2()(
rZm
ES
EESEEdES
EWSR
E
see e.g. P.Ring, P.Schuck, The many Body Problem, Springer-V., 1980
numerical resultsnegative parity
Negative parity states
Isovector Giant Dipole Resonance (IVGDR)(up to )3
Practically no anharmonicity
i
iii YreE
grEgrEB
EEgrEBEES
3)(
13
233
33
)1,1(
|||)1,1(||1|)1;11(
)()1;11(),11(
M
M
numerical resultsnegative paritynumerical resultsnegative parity
Isoscalar Giant Dipole Resonance(ISGDR)
i
iii YreE
grE
grEB
EEgrEB
EES
)(1
33
23
3
3
3
)0,1(
|||)0,1(||1|
)1;01(
)()1;01(
),01(
M
M
multiphonon components areimportant for the ISGDR(large anharmonicity)
Concluding remarks
Eigenvalue equations generating multiphonon bases have a simple structure for any numberof phonons (ph confugurations). Redundancy of such basis is removed by elegant Choleskimethod.
After the creation of the multiphonon basis the total Hamiltonian matrix is constructed anddiagonalised. The spurious CM modes are removed by Palumbo’s method. It needs toconstruct each multiphonon space by the consistent way involving all unperturbedmultiphonon states with the energy up to given (procedure hardly treated in theSM calculations).
nHn
We solved in fact exactly the full eigenvalue problem for 16O in a space spanned bymultiphonon states up to 3 phonons and up to unperturbed energy . We found thatsuch a space is not sufficient (e.g. for the fulfilling the EWSR rule).On the other hand, in an enlarged space (up to 4 phonons) the calculation becomes lengthybecause of a large number of one-body density matrix elements. Truncation of the space is needed in this case. Fortunately, effective sampling method allows this truncation keeping the necessary accuracy: F.Andreozzi, N.Lo Iudice, A.Porrino, J. Phys. G 29, 2319 (2003)
3
Choleski method reduction of the number of redundant basis states to the number of nonredundant (physical) basis states for 16O ( ) (protons or neutrons)nn NN
13 n
67099400
62593781
50978642
36058383
21438084
10521575
4110426
104177
11328
0309
0410
33
33
33
33
33
33
33
33
33
33
33
NNM
NNM
NNM
NNM
NNM
NNM
NNM
NNM
NNM
NNM
NNM
33 ,,1,,1,3| NorNin i
we use axial symmetry basis where ang. momentum projection M is a goodquantum number