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New (iterative) methods for solving the nuclear eigenvalue problem Pisa 05

New (iterative) methods for solving the nuclear eigenvalue problem Pisa 05. An importance sampling algorithm for large scale shell model calculations F. Andreozzi N. Lo Iudice A. Porrino. Currently adopted methods.

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New (iterative) methods for solving the nuclear eigenvalue problem Pisa 05

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  1. New (iterative) methods for solving the nuclear eigenvalue problem Pisa 05

  2. An importance sampling algorithm for large scale shell model calculations F. Andreozzi N. Lo Iudice A. Porrino.

  3. Currently adopted methods • Stochastic methodology: Monte Carlo (C.W. Johnson et al. PRL 92), suitable for ground state. Minus sign problem. • Direct Diagonalization: Lanczos (see G.H. Golub and C.F. Van Loan Matrix Computations 96). Critical point: Sizes of the matrix. • In between: Quantum MC (M. Honma et al. PRL 95). MC to select the relevant basis states. Problems: Redundancy, symmetries broken by the stochastic procedure.

  4. Diagonalizationalgorithm • (A. Andreozzi, A. Porrino, and N.Lo IudiceJ. Phys. A02)Iterative generation of an eigenspace • A  Symmetric matrix representing a self-adjoint operator in an orthonormal basis {| 1 > , | 2 > , … , | N> } • A  { aij} = { < i | Â | j > } • Lowest eigenvalue and eigenvector

  5. a11 a12 a13 a14 …….. a1N a21 a22 a23 a24 ……..a2Na31 a32 a33a34 …….. a3Na41 a42 a43 a44 …….. a4N………………………..aN1 ………………….. aNN

  6. a11 a21 a12 a22 λ1 = a11 | φ1 > = |1> basis{ |1>, |2> } Diagonalize the matrix λ2 , | φ2 > = k1(2) |1> + k2(2) |2>

  7. λ2 b3 b3 a33 Updated basis { | φ2 >, | 3 > } Compute b3 = < φ2| Â | 3 > = k1(2)a13 + k2(2)a23 Diagonalize the matrix λ3 , | φ3 > = Σki(3) | i > i = 1, 3

  8. λN-1 bN bN aNN Updated basis { | φN-1 >, | N >} ComputebN= < φN-1| Â | N > Diagonalize the matrix λN  E(1) , |φN > = |(1)> =  ki(N) | i> i = 1, N End first iteration loop

  9. λ1(2) b1 1 < φ1(2)|1 > b1 a11 < φ1(2)|1 > 1 First step of the second iteration Def.|φ 1(2)> = | ψ(1) > λ1(2) = E(1) Compute b1 = < φ1(2)| Â | 1 > the states {| φ1(2)>, |1>} are not linearly independent Generalized eigenvalue problem det ( - λ ) = 0

  10. E(1) , (1) E(2) , (2)…… THEOREM If the sequence E(i) converges , then E(i) E (eigenvalue of the matrix A) (i) (eigenvector of the matrix A)

  11. Λh Bh λj-1 bj bj ajj BhT Ã Simultaneous determination of v eigensolutions The structure of the algorithm unchanged

  12. λ10 …. 0 b11 ...….b1j0 λ2 ….0b21 …… b2j………. ……. 0 0 ….. λvbv1 ……. bvjb11 ...….bv1a11 …….. a1jb12 …… bv2 a21 …….. a2j……. ……b1j ……. bvjaj1 ..….. ajj

  13. Easy implementation • Variational foundation • Robust Convergence to the extremal eigenvalues Numerically stable and ghost-free solutions Orthogonality of the computed eigenvectors • Fast : O( N2) operations c

  14. IMPORTANCESAMPLING| Ψ > = Σ ci | i > i = 1,Nlocalization property only m ( « N ) ci important •diagonalization algorithm gives quite accurate solutions already in the first approximation loop

  15. Sampling procedure (F. Andreozzi, N. L. A. Porrino, J. Phys. G 03)givenε {aij}  Λ v = diag (λi) (i,j = 1, …, v)for j = v+1 , N• diagonalize Aj =bj = {b1j , … , bvj} Λv bj bjT ajj

  16. select the v lowest eigenvalues λ1’ , … , λv’• if Σ i = 1,v | λi’ - λi | > ε accept the j –th state end loop requires  N  ( v + 1)3 operations

  17. Importance sampling reduces by a factor N / Nsampled the number of operations • The effectiveness of the reduction depends on the localization properties of the wave function • Increase of the localization through the use of a correlated basis  model space partitioning

  18. Numerical Applications • Semimagic nuclei: 108Sn • N=Z 48Cr • N> Z 133Xe

  19. 108Sn 1h11/2 3s1/2 2d3/2 1g7/2 2d5/2 Realistic effective interaction deduced from Bonn A potential . Jπ = 2+ N = 17467

  20. scaling with n(number of sampled states) λ (n) = a + b (N/n) exp(-c N/n) ε (n) = (d/n2) exp(-c N/n)• itallows for high precision extrapolation n  N

  21. Heuristic argument • consistency ε(n)  dλ / dn • from the sampling condition (one target state)Δλ = Σj Δλj = Σj bj2 / ( ajj - λ)

  22. In the convergence region • ajj -   ann -   n • bj2 = <j-1 |H| j>2 (j-1 = i ci |i>) small and random for j < n zero for j> n bj2  exp (- j/n)

  23. Δλ  b (N/n) exp(-c N/n) • ε(n)  dλ / dn  (d/n2) exp(-c N/n)

  24. Conclusions • The algorithm is simple, robust and has a variational foundation • Once endowed with the importance sampling, a) it keeps the extent of space truncation under strict control b) it allows for extrapolation to exact eigensolutions • It is very promising for heavy nuclei • It may be applied to systems others than nuclei (molecules, metal clusters, quantum dots etc.)

  25. Nuclear Eigenvalue problem in a microscopic multiphonon spaceIterative equation of motion method Naples(Andreozzi, Lo Iudice, Porrino) Prague (Knapp, Kvasil) collaboration

  26. Preliminary remarks • Standard Shell model is exact and complete within a given model space. • Often the model space is spanned by ΔN = 0 –ħω Thus it does not include the high-energy configurations building up the collective states. • TDA or RPA act in a more restricted but more selective space (p-h or 2qp configurations up to nħω) and therefore are in general more suitable for collective excitations. They, however, do not account for anharmonic effects. • A multiphonon space is needed for describing anharmonicity • Problem with multiphonon space: AntisymmetryRedundancy Proposed way out: Equation of motion method

  27. Eigenvalue problem: Formulation Goal: Solving H | α > = Eα| α > in a multiphonon space spanned by | 0 >, |1, {ν1} >, |2, {ν2} >, ……. |n, {νn }>… where |n, {νn }> = | ν1 ν2 …..νn > | νi > = Σph Cph(νi ) B+ph |0> = Σph Cph(νi ) a+p ah|0>

  28. <n; νn| [H, B+ph] | n-1; νn-1>= • Σν’ <n; νn|H|n; ν’><n; ν’| B+ph | n-1; νn-1>- • Σν’<n; νn| B+ph | n-1; ν’> <n-1; ν’|H |n-1; νn-1> • <n; νn|H|n; ν’> = Eνnδνnν’ • <n-1; ν’|H |n-1; νn-1> = Eνn-1δνn-1ν’ Amplitude <n; νn| [H, B+ph] | n-1; νn-1>= (Eνn - Eνn-1)<n; νn| B+ph| n-1; νn-1 >.

  29. Amplitudes The commutator yields: <n; νn| [H, B+ph] | n-1; νn-1> = (εp- εh) <n; νn| B+ph | n-1; νn-1 > + + [Σp’h’ Gp’ h h’ p<n; νn| B+ph’| n-1; νn-1 > + Σ Gp’p’’p’’’p <n; νn|B+p’hB+p’’p’’’ | n-1; νn-1 > +ΣGp’h’h’’p <n; νn| B+p’hB+h’h’’) | n-1; νn-1 > + Σ Gp’h p’’h’ <n; νn| B+ph’B+p’p’’ | n-1; νn-1 >) +ΣGhh’h’’h’’’ <n; νn|B+ph’B+h’’h’’’) | n-1; νn-1 > ]

  30. Linearization Amplitudes <n; νn|B+p’h’B+p’’p’’’ | n-1; νn-1 > <n; νn| B+p’h’ B+h’’h’’’) | n-1; νn-1 > <n; νn|B+p’h’ | n-1; ν’>∙ · < n-1; ν’|B+p’’p’’’ | n-1; νn-1 > Î = Σν’|n-1; ν’><n-1; ν’| <n; νn|B+p’h’ | n-1; ν’>∙ · < n-1; ν’|B+h’’h’’’ | n-1; νn-1 > Î = Σν’|n-1; ν’><n-1; ν’|

  31. Eigenvalue Equation М(n)X (n) = EνnX (n) Where X(n)νn-1ph = < n; {νn} | B+ph| n-1; {νn-1} > (М(n))ph,p’h’(νn-1ν’n-1) = A ph,p’h’δνn-1ν’n-1 + Hpp’(νn-1ν’n-1)δhh’ - Hhh’(νn-1ν’n-1)δpp’ Aph,p’h’=(ep – eh + Eνn-1) δph,p’h’ - Gphp’h’ Hpp’(νn-1ν’n-1) = Σh1h2Gph1p’h2 R h1h2(νn-1ν’n-1) - ½ Σp1p2Gp’p1p2p R p1p2(νn-1ν’n-1) Hhh’(νn-1ν’n-1) = Σp1p2Ghp1h’p2 R p1p2(νn-1ν’n-1) - ½ Σh1h2Gh’h1h2h R h1h2(νn-1ν’n-1) Rab(ν’n-1νn-1)= < n-1; {ν’n-1} | B+ab| n-1; {νn-1} >

  32. Redundancy • The states B+ph| n-1; {νn-1} > form an overcomplete linearly dependent set. Is there a way out? Yes

  33. Let us perform the expansion in the redundant basis |n; {νn}> = Σ νn-1phCph (νnνn-1)B+ph| n-1;{νn-1} > We obtain X (n)νn-1ph = < n; {νn} | B+ph| n-1; {νn-1} > = Σ ν’n-1p’h’Cp’h’(νnν’n-1)Dphp’h’(νn-1ν’n-1) where Dp’h’ph (νn-1ν’n-1) = < n-1; {ν’n-1} | Bp’h’ B+ph| n-1; {νn-1} >

  34. In matrix form X = D C Therefore МX = E X (МD)C = H C = E DC This Eq. is ill defined with respect to inversion (D is singular)

  35. The way out: Choleskimethod • Choleski selects a basis of linear independent states B+ph| n-1; {νn-1} > spanning the physical subspace of the right dimension Ng < N Using this basis, we compute a non singular matrix Ď and get (Ď-1МD)C = EC

  36. Eq. (Ď-1МD)C = E C yields Ng exact eigensolutions for the n-phonon subspace. We can now move to the (n+1)-phonon subspace. We only need to know X(n) and R(n). X(n) is given by X= D C

  37. R(n) is given by the recursive relations Rpp’(ν’nνn)= < n; {ν’n} | B+pp’| n; {νn} > =Σ ν’n-1hCp’h (νnν’n-1)X(ν’n)ν’n-1ph + Σ νn-1ν’n-1p1hCph (νnνn-1)X(ν’n)ν’n-1p1h Rpp’(ν’n-1νn-1) Rhh’(ν’nνn)= Σ ν’n-1pCph’ (νnν’n-1)X(ν’n)ν’n-1ph’ + Σ νn-1ν’n-1ph1Cph1(νnνn-1)X(ν’n)ν’n-1ph1 Rhh’(ν’n-1νn-1)

  38. Outcome of iteration:the Hamiltonian matrix Eν0{ H ν0 ν1 } { H ν0 ν2 }{ 0 } {0} Eν1 0 ………. . .0{ H ν1 ν2 } {H ν1 ν3 }{0} Eν’10……….0{ H ν’1 ν2 } {H ν’1 ν3 }{0} E ν’’1 0….0{ H ν’’1 ν2 } {H ν’’1 ν3}{0} ………………………………….. Eν2 0……….. 0{H ν2 ν3} {H ν2 ν4} Eν’2 0........ 0{H ν’2 ν3}{H ν’2 ν4} Eν’’2 0..0{H ν’’2 ν3}{H ν’2 ν4} ……........................ Eν3 0 ..0{H ν3 ν4} ……………

  39. The off diagonal terms are also computed by iteration < n-1; {νn-1} | H| n; {νn} > = Σ (ph)kC(νn)(ph)k [< n-1; {νn-1} | [H, B+(ph)k ] | n-1; {νn-1}k > + Σ lX(ph)l(νn-1 νn-2)< n-2; {νn-1}l | H| n-1; {νn}k > ] < n-2; {νn-2} | H| n; {νn} > = Σ (ph)kC(νn)(ph)k [< n-2; {νn-2} | [H, B+(ph)k ] | n-1; {νn-1}k > + X(ph)l(νn-2 νn-3)< n-3; {νn-3}l | H| n-1; {νn-1}k > ]

  40. The Hamiltonian matrix Eν0{ H ν0 ν1 } { H ν0 ν2 }{ 0 } {0} Eν1 0 ………. . .0{ H ν1 ν2 } {H ν1 ν3 }{0} Eν’10……….0{ H ν’1 ν2 } {H ν’1 ν3 }{0} E ν’’1 0….0{ H ν’’1 ν2 } {H ν’’1 ν3}{0} ………………………………….. Eν2 0……….. 0{H ν2 ν3} {H ν2 ν4} Eν’2 0........ 0{H ν’2 ν3}{H ν’2 ν4} Eν’’2 0..0{H ν’’2 ν3}{H ν’2 ν4} ……........................ Eν3 0 ..0{H ν3 ν4} ……………

  41. Properties of H • It is composed of central diagonal blocks • Each block corresponds to a given n-phonon subspace • A given n-block is coupled only to (n1)- and (n2)-blocks PartitioningImportance sampling Severe truncation

  42. Status of art: Program tests successfully completed A = 16 Phonon space p-configurations {d} h-configurations {s p}-1 Hamiltonian : BonnA

  43. Choleski effectJπ = 0+ T = 1 Two-phonon space 12226 Three phonon space 3142329

  44. Choleski effectJπ = 3- T = 1 Two-phonon space 25262 Three phonon space 149561438

  45. Future program Immediate applications • Coupled scheme p-h. Detailed study of • anharmonic effects on giant resonances • Peculiar collective modes: i. ISGDR (squeezed mode), which requires up to 3ħω p-h configurations ii. Twist mode (orbital M2 mode) iii. Double GDR

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