Advertisement

Pre-processing the nuclear many-body problem

Importance truncation versus tensor factorization techniques
  • A. TichaiEmail author
  • J. Ripoche
  • T. Duguet
Regular Article - Theoretical Physics
  • 43 Downloads

Abstract.

The solution of the nuclear A -body problem encounters severe limitations from the size of many-body operators that are processed to solve the stationary Schrödinger equation. These limitations are typically related to both the (iterative) storing of the associated tensors and to the computational time related to their multiple contractions in the calculation of various quantities of interest. However, not all the degrees of freedom encapsulated into these tensors equally contribute to the description of many-body observables. Identifying systematic and dominating patterns, a relevant objective is to achieve an a priori reduction to the most relevant degrees of freedom via a pre-processing of the A-body problem. The present paper is dedicated to the analysis of two different paradigms to do so. The factorization of tensors in terms of lower-rank ones, whose know-how has been recently transferred to the realm of nuclear structure, is compared to a reduction of the tensors' index size based on an importance truncation. While the objective is to eventually utilize these pre-processing tools in the context of non-perturbative many-body methods, benchmark calculations are presently performed within the frame of perturbation theory. More specifically, we employ the recently introduced Bogoliubov many-body perturbation theory that is systematically applicable to open-shell nuclei displaying strong correlations. This extended perturbation theory serves as a jumpstart for non-perturbative Bogoliubov coupled cluster and Gorkov self-consistent Green's function theories as well as to particle-number projected Bogoliubov coupled cluster theory for which the pre-processing will be implemented in the near future. Results obtained in “small” model spaces are equally encouraging for tensor factorization and importance truncation techniques. While the former requires significant numerical developments to be applied in large model spaces, the latter is presently applied in this context and demonstrates great potential to enable high-accuracy calculations at a much reduced computational cost.

References

  1. 1.
    J. Langhammer, R. Roth, C. Stumpf, Phys. Rev. C 86, 054315 (2012)CrossRefGoogle Scholar
  2. 2.
    B. Hu, F. Xu, Z. Sun, J.P. Vary, T. Li, Phys. Rev. C 94, 014303 (2016)CrossRefGoogle Scholar
  3. 3.
    A. Tichai, E. Gebrerufael, K. Vobig, R. Roth, Phys. Lett. B 786, 448 (2018)CrossRefGoogle Scholar
  4. 4.
    A. Tichai, P. Arthuis, T. Duguet, H. Hergert, V. Somá, R. Roth, Phys. Lett. B 786, 195 (2018)CrossRefGoogle Scholar
  5. 5.
    P. Arthuis, T. Duguet, A. Tichai, R.D. Lasseri, J.P. Ebran, arXiv:1809.01187 (2018)Google Scholar
  6. 6.
    B.S. Hu, T. Li, F.R. Xu, arXiv:1810.08804 (2018)Google Scholar
  7. 7.
    W.H. Dickhoff, C. Barbieri, Prog. Part. Nucl. Phys. 52, 377 (2004)CrossRefGoogle Scholar
  8. 8.
    V. Somà, T. Duguet, C. Barbieri, Phys. Rev. C 84, 064317 (2011)CrossRefGoogle Scholar
  9. 9.
    V. Somà, A. Cipollone, C. Barbieri, P. Navrátil, T. Duguet, Phys. Rev. C 89, 061301 (2014)CrossRefGoogle Scholar
  10. 10.
    A. Carbone, A. Cipollone, C. Barbieri, A. Rios, A. Polls, Phys. Rev. C 88, 054326 (2013)CrossRefGoogle Scholar
  11. 11.
    V. Lapoux, V. Somà, C. Barbieri, H. Hergert, J.D. Holt, S. Stroberg, Phys. Rev. Lett. 117, 052501 (2016)CrossRefGoogle Scholar
  12. 12.
    T. Duguet, V. Somà, S. Lecluse, C. Barbieri, P. Navrátil, Phys. Rev. C 95, 034319 (2017)CrossRefGoogle Scholar
  13. 13.
    F. Raimondi, C. Barbieri, Phys. Rev. C 97, 054308 (2018)CrossRefGoogle Scholar
  14. 14.
    F. Raimondi, C. Barbieri, arXiv:1811.07163 (2018)Google Scholar
  15. 15.
    G. Hagen, T. Papenbrock, M. Hjorth-Jensen, D.J. Dean, Rep. Prog. Phys. 77, 096302 (2014)CrossRefGoogle Scholar
  16. 16.
    A. Signoracci, T. Duguet, G. Hagen, G. Jansen, Phys. Rev. C 91, 064320 (2015)CrossRefGoogle Scholar
  17. 17.
    T.D. Morris, J. Simonis, S.R. Stroberg, C. Stumpf, G. Hagen, J.D. Holt, G.R. Jansen, T. Papenbrock, R. Roth, A. Schwenk, Phys. Rev. Lett. 120, 152503 (2018)CrossRefGoogle Scholar
  18. 18.
    H. Hergert, S.K. Bogner, T.D. Morris, A. Schwenk, K. Tsukiyama, Phys. Rep. 621, 165 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    H. Hergert, S.K. Bogner, J.G. Lietz, T.D. Morris, S. Novario, N.M. Parzuchowski, F. Yuan, Lect. Notes Phys. 936, 477 (2017)CrossRefGoogle Scholar
  20. 20.
    N.M. Parzuchowski, S.R. Stroberg, P. Navrátil, H. Hergert, S.K. Bogner, Phys. Rev. C 96, 034324 (2017)CrossRefGoogle Scholar
  21. 21.
    T. Duguet, J. Phys. G 42, 025107 (2015)CrossRefGoogle Scholar
  22. 22.
    T. Duguet, A. Signoracci, J. Phys. G 44, 015103 (2017)CrossRefGoogle Scholar
  23. 23.
    Y. Qiu, T.M. Henderson, T. Duguet, G.E. Scuseria, J. Chem. Phys. 147, 064111 (2017)CrossRefGoogle Scholar
  24. 24.
    F. Verstraete, V. Murg, J.I. Cirac, Adv. Phys. 57, 143 (2008)CrossRefGoogle Scholar
  25. 25.
    U. Schollwöck, Ann. Phys. 326, 96 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    R. Orus, Ann. Phys. 349, 117 (2014)MathSciNetCrossRefGoogle Scholar
  27. 27.
    R.J. Furnstahl, G. Hagen, T. Papenbrock, Phys. Rev. C 86, 031301 (2012)CrossRefGoogle Scholar
  28. 28.
    R.J. Furnstahl, S.N. More, T. Papenbrock, Phys. Rev. C 89, 044301 (2014)CrossRefGoogle Scholar
  29. 29.
    K.A. Wendt, C. Forssén, T. Papenbrock, D. Sääf, Phys. Rev. C 91, 061301 (2015)CrossRefGoogle Scholar
  30. 30.
    S.E. Koonin, D.J. Dean, K. Langanke, Phys. Rep. 278, 1 (1997)CrossRefGoogle Scholar
  31. 31.
    T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu, Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319 (2001)CrossRefGoogle Scholar
  32. 32.
    A. Tichai, R. Schutski, G.E. Scuseria, T. Duguet, arXiv:1810.08419 (2018)Google Scholar
  33. 33.
    T.M. Henderson, J. Dukelsky, G.E. Scuseria, A. Signoracci, T. Duguet, Phys. Rev. C 89, 054305 (2014)CrossRefGoogle Scholar
  34. 34.
    R. Roth, P. Navratil, Phys. Rev. Lett. 99, 092501 (2007)CrossRefGoogle Scholar
  35. 35.
    R. Roth, Phys. Rev. C 79, 064324 (2009)CrossRefGoogle Scholar
  36. 36.
    R. Roth, J.R. Gour, P. Piecuch, Phys. Rev. C 79, 054325 (2009)CrossRefGoogle Scholar
  37. 37.
    R.J. Buenker, S.D. Peyerimhoff, Theor. Chim. Acta 35, 33 (1974)CrossRefGoogle Scholar
  38. 38.
    R.J. Buenker, S.D. Peyerimhoff, Theor. Chim. Acta 39, 217 (1975)CrossRefGoogle Scholar
  39. 39.
    F. Illas, J. Rubio, J.M. Ricart, P.S. Bagus, J. Chem. Phys. 95, 1877 (1991)CrossRefGoogle Scholar
  40. 40.
    E. Giner, A. Scemama, M. Caffarel, Can. J. Chem. 91, 879 (2013)CrossRefGoogle Scholar
  41. 41.
    J.E. Deustua, J. Shen, P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)CrossRefGoogle Scholar
  42. 42.
    G.H. Booth, A.J.W. Thom, A. Alavi, J. Chem. Phys. 131, 054106 (2009)CrossRefGoogle Scholar
  43. 43.
    D. Cleland, G.H. Booth, A. Alavi, J. Chem. Phys. 132, 041103 (2010)CrossRefGoogle Scholar
  44. 44.
    A.J.W. Thom, Phys. Rev. Lett. 105, 263004 (2010)CrossRefGoogle Scholar
  45. 45.
    J.S. Spencer, A.J.W. Thom, J. Chem. Phys. 144, 084108 (2016)CrossRefGoogle Scholar
  46. 46.
    C.J.C. Scott, A.J.W. Thom, J. Chem. Phys. 147, 124105 (2017)CrossRefGoogle Scholar
  47. 47.
    A. Signoracci, T. Duguet, G. Hagen, G.R. Jansen, Phys. Rev. C 91, 064320 (2015)CrossRefGoogle Scholar
  48. 48.
    P. Ring, P. Schuck, The Nuclear Many-Body Problem (Springer Verlag, New York, 1980)Google Scholar
  49. 49.
    R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer, P. Navrátil, Phys. Rev. Lett. 109, 052501 (2012)CrossRefGoogle Scholar
  50. 50.
    E. Gebrerufael, A. Calci, R. Roth, Phys. Rev. C 93, 031301(R) (2016)CrossRefGoogle Scholar
  51. 51.
    J. Ripoche, A. Tichai, T. Duguet (2019) unpublishedGoogle Scholar
  52. 52.
    A. Tichai, P. Arthuis, J. Ripoche, T. Duguet (2019) unpublishedGoogle Scholar
  53. 53.
    D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific Publishing Company, 1988)Google Scholar
  54. 54.
    S. Weinberg, Phys. Lett. B 251, 288 (1990)CrossRefGoogle Scholar
  55. 55.
    S. Weinberg, Nucl. Phys. B 363, 3 (1991)CrossRefGoogle Scholar
  56. 56.
    E. Epelbaum, Lectures given at the 2009 Joliot-Curie School, Lacanau, France, 27 September - 3 October 2009, arXiv:1001.3229 [nucl-th] (2009)Google Scholar
  57. 57.
    D.R. Entem, R. Machleidt, Phys. Rev. C 68, 041001(R) (2003)CrossRefGoogle Scholar
  58. 58.
    P. Navrátil, Few Body Syst. 41, 117 (2007)CrossRefGoogle Scholar
  59. 59.
    R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer, P. Navratil, Phys. Rev. Lett. 109, 052501 (2012)CrossRefGoogle Scholar
  60. 60.
    S.K. Bogner, R.J. Furnstahl, R.J. Perry, Phys. Rev. C 75, 061001(R) (2007)CrossRefGoogle Scholar
  61. 61.
    H. Hergert, R. Roth, Phys. Rev. C 75, 051001(R) (2007)CrossRefGoogle Scholar
  62. 62.
    R. Roth, S. Reinhardt, H. Hergert, Phys. Rev. C 77, 064003 (2008)CrossRefGoogle Scholar
  63. 63.
    R. Roth, J. Langhammer, A. Calci, S. Binder, P. Navrátil, Phys. Rev. Lett. 107, 072501 (2011)CrossRefGoogle Scholar
  64. 64.
    E.D. Jurgenson, P. Maris, R.J. Furnstahl, P. Navrátil, W.E. Ormand, J.P. Vary, Phys. Rev. C 87, 054312 (2013)CrossRefGoogle Scholar
  65. 65.
    A. Tichai, J. Langhammer, S. Binder, R. Roth, Phys. Lett. B 756, 283 (2016)CrossRefGoogle Scholar
  66. 66.
    B.S. Hu, F.R. Xu, Z.H. Sun, J.P. Vary, T. Li, Phys. Rev. C 94, 014303 (2016)CrossRefGoogle Scholar
  67. 67.
    E.G. Hohenstein, R.M. Parrish, T.J. Martínez, J. Chem. Phys. 137, 044103 (2012)CrossRefGoogle Scholar
  68. 68.
    E.G. Hohenstein, R.M. Parrish, C.D. Sherrill, T.J. Martínez, J. Chem. Phys. 137, 221101 (2012)CrossRefGoogle Scholar
  69. 69.
    R. Schutski, J. Zhao, T.M. Henderson, G.E. Scuseria, J. Chem. Phys. 147, 184113 (2017)CrossRefGoogle Scholar
  70. 70.
    D. Braess, W. Hackbusch, IMA J. Numer. Anal. 25, 685 (2005)MathSciNetCrossRefGoogle Scholar
  71. 71.
    P. Piecuch, M. Wloch, J. Chem. Phys. 123, 224105 (2005)CrossRefGoogle Scholar
  72. 72.
    P. Piecuch, M. Wloch, J.R. Gour, A. Kinal, Chem. Phys. Lett. 418, 467 (2006)CrossRefGoogle Scholar
  73. 73.
    J. Shen, P. Piecuch, Chem. Phys. 401, 180 (2012)CrossRefGoogle Scholar
  74. 74.
    J. Shen, P. Piecuch, J. Chem. Phys. 136, 144104 (2012)CrossRefGoogle Scholar
  75. 75.
    J. Shen, P. Piecuch, J. Chem. Theory Comput. 8, 4968 (2012)CrossRefGoogle Scholar
  76. 76.
    N.P. Bauman, J. Shen, P. Piecuch, Mol. Phys. 115, 2860 (2017)CrossRefGoogle Scholar
  77. 77.
    C. Barbieri, T. Duguet, P. Navrátil, F. Raimondi, V. Somà (2018) unpublishedGoogle Scholar
  78. 78.
    J. Ripoche, R. Wirth, T. Duguet, A. Tichai (2019) unpublishedGoogle Scholar

Copyright information

© Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.ESNT, CEA-Saclay, DRF, IRFU, Département de Physique NucléaireUniversité de Paris SaclayGif-sur-YvetteFrance
  2. 2.CEA, DAMDIFArpajonFrance
  3. 3.IRFU, CEAUniversité Paris-SaclayGif-sur-YvetteFrance
  4. 4.KU Leuven, Instituut voor Kern- en StralingsfysicaLeuvenBelgium

Personalised recommendations