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 mappingA. 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

8. 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

11. 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 (det{D}=0) 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 ( )

14. 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 ) (n+1) – phonon subspace and solve: to where for the creation of and we need and : recursive formula (E) with a similar expression for

15. multiphonon basis Iterative generating of phonon basis starting point multiphonon basis is generated

16. 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:

18. 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

20. 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.

21. 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 Palumbo’s procedure for the elimination of the CM spurious mode also for large configuration spaces

22. 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 0+1-phonon and 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)

25. 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)

29. 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 Palumbo’s 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