An exact microscopic multiphonon approach

1
An exact microscopic multiphonon approach

• Naples F. Andreozzi
• N. Lo
  Iudice
• A.
  Porrino
• Prague F. Knapp
• J.
  Kvasil

2
From mean field to multiphonon approaches
is responsible for collective modes
standard approach to collective modes RPA
(TDA) RPA (TDA) harmonic approximations
where
RPA
TDA
3
for anharmonic effects multiphonon approaches are
needed
problems
1.
lack of antisymmetry overcomplete
set

• brute approximations and/or formidable
  calculations
• are needed

sophisticated methods (with minimal
approximation) should be developed for
investigation of anharmonic effects
4
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)
where
constraint the exact fermionic anticommutator
should be
used in the calculation of
state vector mapping
(T. Marumori et al., Progr.Theor.Phys. 31, 1009
(1964)
constrant
problems in practical calculations - in general
slow convergence of the boson expansion
-
involved calculations
5
Practical examples of fermion boson (FB)
mapping
IBM phenomenological FB mapping
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
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
6
Further attempts for multiphonon approaches
Multistep Shell Model (MSM)
R.L.Liotta, C.Pomar, Nucl.Phys. A382, 1 (1982)
They expand
and for
(2 phonon) space they
keep only linear terms in two-phonon operators
(linearization) and they
get eigenvalue equation in two-phonon space
where is expressed in terms of TDA
eigenvalues and eigenstates.

• 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 recurent formulas
between and )
Both MSM and MPM look involved and of problematic
applicability (indeed, they have not been widely
adopted).
7
method proposed here Equation of Motion Method
(EM)
eigenvalue problem
is solved in a multiphonon space
Tamm-Dancoff phonon
where
eigenvalue problem is solved in two steps

• generating the multiphonon basis
• construction and diagonalization of the total
  Hamiltonian matrix in
• the whole space

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1-st step generating the multiphonon basis
multiphonon basis
We require
or
for basis states with following
orthogonality properties
with closure relation
Using
we have
(A)
From other side the closure relation and
orthogonality properties above give
substituting for and taking into account
that only terns can contribute
9
multiphonon basis
(B)
where
The comparison of (A) with (B) gives the
eigenvalue equation for
(C)
with
10
multiphonon basis
In the Hartree-Fock basis we have
and then
and (C) is the standard Tamm-Dancoff
equation for TDA states
For the first sight one can expect that solutions
of (C), , represent
coefficients in the expansion of the state
in terms of states (because
)
However, states
represents the redundant (overcomplete) basis
multiphonon states are not fully antisymmetrizes
!!
We should extract physical nonredundant basis
from the redundant basis of states
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multiphonon basis
In order to extract physical basis let us expand
the exact eigenstates in the redundant basis
(D)
metric matrix
so, in matrix form
eigenvalue equation (C) can be rewritten
However, metric matrix is singular (detD0)
it is not possible to invert it
12
multiphonon basis
usual solving of the singular metric matrix
problem diagonalization of D
(Rowe J., Math.Phys. 10, 1774 (1969)
effectively we obtain
linearly independent (physical) eigenvectors
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 )
in our EM approach the redundancy of the
overcomplete basis, , is
removed by Choleski decomposition - no
diaginalization of
- much faster and more effective
J.H.Wilkinson, The Algebraic Eigenvalue,
Clarendon Oxford, 1965
13
removing of the redundancy of the overcomplete
basis by Choleski decomposition
multiphonon basis
any real non negative definnite symmetric matrix
can be rewritte as
decomposition of using recursive formulas
goes until a diagonal element - in
this moment we know that -th
basis vector is a linear combination of vectors
-th vector is discarded and we dropped -th
row and -th column from and
decomposition continues with -th basis
vector. This procedure continues until the
whole redundant basis is exhausted.
where is the lower triangular
matrix with defined by recursive formulas

During decomposition we can rearrange the basis
in the decreasing way according to which gives us
the maximum of (maximum overlap)
Choleski decomposition from vectors of the
overcomplete basis we extract
linearly independent
vectors ( )
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Choleski method we obtain (
) linearly independent basis
states, ,
for the subspace with
nonsingular matrix
multiphonon basis
( - type matrix )
we can solve the eigen-value problem
we obtain eigen solutions (physical,
nonredundant) in the subspace
Now we can go from n phonon subspace (we know
)
(n1) phonon subspace and solve
to
where for the creation of
and we need
and
recursive formula
(E)
with a similar expression for
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multiphonon basis
Iterative generating of phonon basis
starting point
multiphonon basis is generated
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full eigenvalue problem
Full eigenvalue problem
Once the multiphonon basis has been generated the
total Hamiltonian matrix can be generated and
digonalized.
The second term involves only nondiagonal matrix
lements given by recursive formulas
with
17
By the diagonalization of the total Hamiltonian
matrix
full eigenvalue problem
where
(F)
we obtain exact nuclear eigenvalues and
eigenvectors
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Transition amplitudes
transition amplitudes
Let us consider the transition amplitudes
of some single-particle transition operator
Substitution of (F) gives
transition amplitude involves density matrix
elements which are given by recursive formula
(E)
19
elimination of spuriosity
Elimination of the center of mass spuriosity
Following the method F. Palumbo, Nucl.Phys. 99,
100 (1967) we add to the starting Hamiltonian
a center of mass (CM) oscillator Hamiltonian
multiplied by a constant
where
effective only in Jp 1- channel
full space Schr. equation
for physical states
for big spurious states are shifted to high
energies by the shift
for spurious states
spurious states can be easily tagged and
eliminated
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elimination of spuriosity
reminder
The Hamiltonian is effective only in
ph- channel.
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
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.
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evaluation of the method
Pluses and minuses of the method
Pluses
Minuses
simple structure of the eigenvalue equation in
any n- phonon subspace
overcompletness of multiphonon basis states (in
our approach eliminated by elegant Choleski
method)
only density matrix elements ,
and have to
be computed
number of density matrix elements
to be computed increases with the number of
phonons for higher
numerical process may become
slow
relatively simple recursive formulas hold for
, and for all
other quantities
our method enables to use Palumbos procedure for
the elimination of the CM spurious mode also for
large configuration spaces
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Numerical test 16O
numerical results
calculation up to 3- phonon states (
) with unperturbed energies up to
Hamiltonian
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
dependence of the maximum major number N of shell
one has to include in order to have all and
only configurations up to
23
Positive parity states
numerical results positive parity
Ground state
comparison with others
influence of the elimination of CM spurious mode
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
24
numerical results positive parity
E2 response (up to )
the big difference between 01-phonon and
012-phonon cases we need to enlarge
the space up to (in
order to have 4 ph components in the
ground and 2 states)
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numerical results positive parity
E2 response (up to ) effect of
the elimination of the CM motion
In spite of the fact that CM Hamiltonian
acts only in the channel it contributes
to the E2 strength because of the presence of the
phonon components
.
26
numerical results positive parity
E2 response (up to ) running sum
It is necessary to enlarge the space up to
see e.g. P.Ring, P.Schuck, The many Body Problem,
Springer-V., 1980
27
Negative parity states
numerical results negative parity
Isovector Giant Dipole Resonance (IVGDR) (up to
)
Practically no anharmonicity
28
numerical results negative parity
numerical results negative parity
Isoscalar Giant Dipole Resonance (ISGDR)
multiphonon components are important for the
ISGDR (large anharmonicity)
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Concluding remarks
Eigenvalue equations generating multiphonon bases
have a simple structure for any number of phonons
(ph confugurations). Redundancy of such basis is
removed by elegant Choleski method.
After the creation of the multiphonon basis the
total Hamiltonian matrix is constructed
and diagonalised. The spurious CM modes are
removed by Palumbos method. It needs
to construct each multiphonon space
by the consistent way involving all
unperturbed multiphonon states with the energy up
to given (procedure hardly
treated in the SM calculations).
We solved in fact exactly the full
eigenvalue problem for 16O in a space spanned
by multiphonon states up to 3 phonons and up to
unperturbed energy . We found
that such 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 lengthy because 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)
30
Choleski method reduction of the number of
redundant basis states to the number of
nonredundant (physical) basis states for 16O (
) (protons or neutrons)
we use axial symmetry basis where ang. momentum
projection M is a good quantum number