Pre-processing the nuclear many-body problem

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


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.


  1. 1.
    J. Langhammer, R. Roth, C. Stumpf, Phys. Rev. C 86, 054315 (2012)CrossRefADSGoogle Scholar
  2. 2.
    B. Hu, F. Xu, Z. Sun, J.P. Vary, T. Li, Phys. Rev. C 94, 014303 (2016)CrossRefADSGoogle Scholar
  3. 3.
    A. Tichai, E. Gebrerufael, K. Vobig, R. Roth, Phys. Lett. B 786, 448 (2018)CrossRefADSGoogle Scholar
  4. 4.
    A. Tichai, P. Arthuis, T. Duguet, H. Hergert, V. Somá, R. Roth, Phys. Lett. B 786, 195 (2018)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  8. 8.
    V. Somà, T. Duguet, C. Barbieri, Phys. Rev. C 84, 064317 (2011)CrossRefADSGoogle Scholar
  9. 9.
    V. Somà, A. Cipollone, C. Barbieri, P. Navrátil, T. Duguet, Phys. Rev. C 89, 061301 (2014)CrossRefADSGoogle Scholar
  10. 10.
    A. Carbone, A. Cipollone, C. Barbieri, A. Rios, A. Polls, Phys. Rev. C 88, 054326 (2013)CrossRefADSGoogle Scholar
  11. 11.
    V. Lapoux, V. Somà, C. Barbieri, H. Hergert, J.D. Holt, S. Stroberg, Phys. Rev. Lett. 117, 052501 (2016)CrossRefADSGoogle Scholar
  12. 12.
    T. Duguet, V. Somà, S. Lecluse, C. Barbieri, P. Navrátil, Phys. Rev. C 95, 034319 (2017)CrossRefADSGoogle Scholar
  13. 13.
    F. Raimondi, C. Barbieri, Phys. Rev. C 97, 054308 (2018)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  16. 16.
    A. Signoracci, T. Duguet, G. Hagen, G. Jansen, Phys. Rev. C 91, 064320 (2015)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  18. 18.
    H. Hergert, S.K. Bogner, T.D. Morris, A. Schwenk, K. Tsukiyama, Phys. Rep. 621, 165 (2016)MathSciNetCrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  20. 20.
    N.M. Parzuchowski, S.R. Stroberg, P. Navrátil, H. Hergert, S.K. Bogner, Phys. Rev. C 96, 034324 (2017)CrossRefADSGoogle Scholar
  21. 21.
    T. Duguet, J. Phys. G 42, 025107 (2015)CrossRefADSGoogle Scholar
  22. 22.
    T. Duguet, A. Signoracci, J. Phys. G 44, 015103 (2017)CrossRefADSGoogle Scholar
  23. 23.
    Y. Qiu, T.M. Henderson, T. Duguet, G.E. Scuseria, J. Chem. Phys. 147, 064111 (2017)CrossRefADSGoogle Scholar
  24. 24.
    F. Verstraete, V. Murg, J.I. Cirac, Adv. Phys. 57, 143 (2008)CrossRefADSGoogle Scholar
  25. 25.
    U. Schollwöck, Ann. Phys. 326, 96 (2011)MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    R. Orus, Ann. Phys. 349, 117 (2014)MathSciNetCrossRefADSGoogle Scholar
  27. 27.
    R.J. Furnstahl, G. Hagen, T. Papenbrock, Phys. Rev. C 86, 031301 (2012)CrossRefADSGoogle Scholar
  28. 28.
    R.J. Furnstahl, S.N. More, T. Papenbrock, Phys. Rev. C 89, 044301 (2014)CrossRefADSGoogle Scholar
  29. 29.
    K.A. Wendt, C. Forssén, T. Papenbrock, D. Sääf, Phys. Rev. C 91, 061301 (2015)CrossRefADSGoogle Scholar
  30. 30.
    S.E. Koonin, D.J. Dean, K. Langanke, Phys. Rep. 278, 1 (1997)CrossRefADSGoogle Scholar
  31. 31.
    T. Otsuka, M. Honma, T. Mizusaki, N. Shimizu, Y. Utsuno, Prog. Part. Nucl. Phys. 47, 319 (2001)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  34. 34.
    R. Roth, P. Navratil, Phys. Rev. Lett. 99, 092501 (2007)CrossRefADSGoogle Scholar
  35. 35.
    R. Roth, Phys. Rev. C 79, 064324 (2009)CrossRefADSGoogle Scholar
  36. 36.
    R. Roth, J.R. Gour, P. Piecuch, Phys. Rev. C 79, 054325 (2009)CrossRefADSGoogle 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)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  42. 42.
    G.H. Booth, A.J.W. Thom, A. Alavi, J. Chem. Phys. 131, 054106 (2009)CrossRefADSGoogle Scholar
  43. 43.
    D. Cleland, G.H. Booth, A. Alavi, J. Chem. Phys. 132, 041103 (2010)CrossRefADSGoogle Scholar
  44. 44.
    A.J.W. Thom, Phys. Rev. Lett. 105, 263004 (2010)CrossRefADSGoogle Scholar
  45. 45.
    J.S. Spencer, A.J.W. Thom, J. Chem. Phys. 144, 084108 (2016)CrossRefADSGoogle Scholar
  46. 46.
    C.J.C. Scott, A.J.W. Thom, J. Chem. Phys. 147, 124105 (2017)CrossRefADSGoogle Scholar
  47. 47.
    A. Signoracci, T. Duguet, G. Hagen, G.R. Jansen, Phys. Rev. C 91, 064320 (2015)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  50. 50.
    E. Gebrerufael, A. Calci, R. Roth, Phys. Rev. C 93, 031301(R) (2016)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  55. 55.
    S. Weinberg, Nucl. Phys. B 363, 3 (1991)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  58. 58.
    P. Navrátil, Few Body Syst. 41, 117 (2007)CrossRefADSGoogle Scholar
  59. 59.
    R. Roth, S. Binder, K. Vobig, A. Calci, J. Langhammer, P. Navratil, Phys. Rev. Lett. 109, 052501 (2012)CrossRefADSGoogle Scholar
  60. 60.
    S.K. Bogner, R.J. Furnstahl, R.J. Perry, Phys. Rev. C 75, 061001(R) (2007)CrossRefADSGoogle Scholar
  61. 61.
    H. Hergert, R. Roth, Phys. Rev. C 75, 051001(R) (2007)CrossRefADSGoogle Scholar
  62. 62.
    R. Roth, S. Reinhardt, H. Hergert, Phys. Rev. C 77, 064003 (2008)CrossRefADSGoogle Scholar
  63. 63.
    R. Roth, J. Langhammer, A. Calci, S. Binder, P. Navrátil, Phys. Rev. Lett. 107, 072501 (2011)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  65. 65.
    A. Tichai, J. Langhammer, S. Binder, R. Roth, Phys. Lett. B 756, 283 (2016)CrossRefADSGoogle Scholar
  66. 66.
    B.S. Hu, F.R. Xu, Z.H. Sun, J.P. Vary, T. Li, Phys. Rev. C 94, 014303 (2016)CrossRefADSGoogle Scholar
  67. 67.
    E.G. Hohenstein, R.M. Parrish, T.J. Martínez, J. Chem. Phys. 137, 044103 (2012)CrossRefADSGoogle Scholar
  68. 68.
    E.G. Hohenstein, R.M. Parrish, C.D. Sherrill, T.J. Martínez, J. Chem. Phys. 137, 221101 (2012)CrossRefADSGoogle Scholar
  69. 69.
    R. Schutski, J. Zhao, T.M. Henderson, G.E. Scuseria, J. Chem. Phys. 147, 184113 (2017)CrossRefADSGoogle 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)CrossRefADSGoogle Scholar
  72. 72.
    P. Piecuch, M. Wloch, J.R. Gour, A. Kinal, Chem. Phys. Lett. 418, 467 (2006)CrossRefADSGoogle 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)CrossRefADSGoogle 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)CrossRefADSGoogle 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