- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Green’s functions under magnetic effects in a nontrivial topology

Emerson B. S. Corrêa1,2,*, Michelli S. R. Sarges3,†,§, and José A. Helayël-Neto4,‡,§

  • *Contact author: ecorreae@ufpa.br
  • †Contact author: michellissarges@gmail.com
  • ‡Contact author: helayel@cbpf.br
  • §These authors contributed equally to this work.
PDF

Phys. Rev. D 112, 125017 – Published 15 December, 2025

DOI: https://doi.org/10.1103/xbmn-1m8m

Abstract

We compute the propagators of Bose and Dirac fields in four-dimensional Euclidean space in the presence of an external magnetic field, using a hybrid formulation that combines the Ritus and Schwinger methods. This approach yields explicit expressions for both propagators in coordinate and momentum space. By performing a suitable gauge transformation, we eliminate the components of the propagators that break translation invariance. We further derive the Matsubara frequencies of the fields within a toroidal topology. The resulting framework provides a unified mathematical structure that incorporates thermal and finite-volume effects in Green’s functions under various boundary conditions, while simultaneously offering an analytically tractable representation of all Landau levels.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (75)

  1. J. M. Quashnock, A. Loeb, and D. N. Spergel, Astrophys. J. Lett. 344, L49 (1989).
  2. B. Cheng and A. V. Olinto, Phys. Rev. D 50, 2421 (1994).
  3. D. Grasso and H. R. Rubinstein, Phys. Rep. 348, 163 (2001).
  4. T. Kahniashvili, A. G. Tevzadze, A. Brandenburg, and A. Neronov, Phys. Rev. D 87, 083007 (2013).
  5. J. O. Andersen, W. R. Naylor, and A. Tranberg, Rev. Mod. Phys. 88, 025001 (2016).
  6. M. D’Elia, L. Maio, F. Sanfilippo, and A. Stanzione, Phys. Rev. D 105, 034511 (2022).
  7. K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D 78, 074033 (2008).
  8. Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosić, A. V. Fedorov, R. D. Zhong, J. A. Schneeloch, G. D. Gu, and T. Valla, Nat. Phys. 12, 550 (2016).
  9. D. E. Kharzeev, J. Liao, and P. Tribedy, Int. J. Mod. Phys. E 33, 2430007 (2024).
  10. D. P. Menezes, M. B. Pinto, S. S. Avancini, A. P. Martínez, and C. Providência, Phys. Rev. C 79, 035807 (2009).
  11. M. D’Elia, S. Mukherjee, and F. Sanfilippo, Phys. Rev. D 82, 051501(R) (2010).
  12. B. Chatterjee, H. Mishra, and A. Mishra, Phys. Rev. D 84, 014016 (2011).
  13. K. Hattori, T. Kojo, and N. Su, Nucl. Phys. A951, 1 (2016).
  14. J. Mei, R. Wen, S. Mao, M. Huang, and K. Xu, Phys. Rev. D 110, 034024 (2024).
  15. G. S. Bali, F. Bruckmann, G. Endrődi, Z. Fodor, S. D. Katz, S. Krieg, A. Schäfer, and K. K. Szabó, J. High Energy Phys. 02 (2012) 044.
  16. G. S. Bali, F. Bruckmann, G. Endrődi, Z. Fodor, S. D. Katz, and A. Schäfer, Phys. Rev. D 86, 071502(R) (2012).
  17. F. Bruckmann, G. Endrődi, and T. G. Kovács, J. High Energy Phys. 04 (2013) 112.
  18. G. S. Bali, F. Bruckmann, G. Endrődi, F. Gruber, and A. Schäfer, J. High Energy Phys. 04 (2013) 130.
  19. E. J. Ferrer, V. de la Incera, and X. J. Wen, Phys. Rev. D 91, 054006 (2015).
  20. N. Mueller and J. M. Pawlowski, Phys. Rev. D 91, 116010 (2015).
  21. A. Bandyopadhyay and R. L. S. Farias, Eur. Phys. J. Special Topics 230, 719 (2021).
  22. K. Hattori and X-G. Huang, Nucl. Sci. Tech. 28, 26 (2017).
  23. F. A. Barone, H. Boschi-Filho, and C. Farina, Am. J. Phys. 71, 483 (2003).
  24. W. Dittrich and H. Gies, Probing the Quantum Vacuum (Springer-Verlag, Berlin, 2000).
  25. J. Schwinger, Phys. Rev. 82, 664 (1951).
  26. V. I. Ritus, Ann. Phys. (N.Y.) 69, 555 (1972).
  27. V. I. Ritus, Sov. Phys. JETP 48, 788 (1978), http://jetp.ras.ru/cgi-bin/dn/e_048_05_0788.pdf.
  28. A. Chodos and K. Everding, Phys. Rev. D 42, 2881 (1990).
  29. V. R. Khalilov, Theor. Math. Phys. 125, 1413 (2000).
  30. I. D. Lawrie, Phys. Rev. Lett. 79, 131 (1997).
  31. E. Elizalde, E. Ferrer, and V. de la Incera, Phys. Rev. D 70, 043012 (2004).
  32. G. Murguía, A. Raya, Á. Sánchez, and E. Reyes, Am. J. Phys. 78, 700 (2010).
  33. I. A. Shovkovy, Lect. Notes Phys. 871, 13 (2013).
  34. E. B. S. Corrêa and J. E. Oliveira, Rev. Bras. Ens. Fís. 37, 3302 (2015).
  35. E. B. S. Corrêa, C. A. Bahia, and J. A. Lourenço, Can. J. Phys. 100, 1 (2022).
  36. E. B. S. Corrêa, M. G. Lima, and S. C. Q. Arruda, Contemp. Math. 3, 343 (2022).
  37. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 5th ed. (Oxford University Press, New York, 2021).
  38. J. Zinn-Justin, arXiv:hep-ph/0005272v1.
  39. H. T. C. Stoof, K. B. Gubbels, and D. B. M. Dickerscheid, Ultracold Quantum Fields (Springer, Bristol, 2009).
  40. F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, Thermal Quantum Field Theory: Algebraic Aspects and Applications (World Scientific, Singapore, 2009).
  41. T. Matsubara, Prog. Theor. Phys. 14, 351 (1955).
  42. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).
  43. P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342 (1959).
  44. L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974).
  45. N. D. Birrell and L. H. Ford, Phys. Rev. D 22, 330 (1980).
  46. F. C. Khanna, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, Phys. Rep. 539, 135 (2014).
  47. T. Inagaki, Y. Matsuo, and H. Shimoji, Prog. Theor. Exp. Phys. 013B09, 1 (2022).
  48. T. Misumi and T. Kanazawa, J. High Energy Phys. 06 (2014) 181.
  49. E. Cavalcanti, C. A. Linhares, and A. P. C. Malbouisson, Int. J. Mod. Phys. A, 33, 1850008 (2018).
  50. E. B. S. Corrêa and M. S. R. Sarges, Nucl. Phys. A1040, 122749 (2023).
  51. E. B. S. Corrêa and M. S. R. Sarges, Eur. Phys. J. Plus 139, 1097 (2024).
  52. V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. Lett. 73, 3499 (1994); 76, 1005(E) (1996).
  53. V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Nucl. Phys. B462, 249 (1996).
  54. Strong Interacting Matter in Magnetic Fields, edited by D. Kharzeev, K. Landsteiner, A. Schmitt, and H. U. Yee, Lecture Notes in Physics Vol. 871 (Springer-Verlag, Berlin-Heidelbeg, 2013).
  55. V. A. Miransky and I. A. Shovkovy, Phys. Rep. 576, 1 (2015).
  56. J. O. Andersen, Eur. Phys. J. A 57, 189 (2021).
  57. E. B. S. Corrêa, C. A. Linhares, and A. P. C. Malbouisson, Phys. Lett. A 377, 1984 (2013).
  58. R. Zhang, Wj. Fu, and Yx. Liu, Eur. Phys. J. C 76, 307 (2016).
  59. E. B. S. Corrêa, Transições de Fase em Sistemas Sujeitos a Campos Magnéticos e definidos em Espaços com Topologia Toroidal, PhD thesis, Centro Brasileiro de Pesquisas Físicas, Rio de Janeiro, 2017.
  60. E. B. S. Corrêa, C. A. Linhares, A. P. C. Malbouisson, J. M. C. Malbouisson, and A. E. Santana, Eur. Phys. J. C 77, 261 (2017).
  61. S. S. Avancini, R. L. S. Farias, and W. R. Tavares, Phys. Rev. D 99, 056009 (2019).
  62. E. B. S. Corrêa, C. A. Linhares, and A. P. C. Malbouisson, Int. J. Mod. Phys. B 32, 1850091 (2018).
  63. L. M. Abreu, E. B. S. Corrêa, C. A. Linhares, and A. P. C. Malbouisson, Phys. Rev. D 99, 076001 (2019).
  64. S. Ghosh, A. Mukherjee, N. Chaudhuri, P. Roy, and S. Sarkar, Phys. Rev. D 101, 056023 (2020).
  65. L. M. Abreu, E. B. S. Corrêa, and E. S. Nery, Physica A (Amstedam) 572, 125885 (2021).
  66. L. M. Abreu, E. B. S. Corrêa, and E. S. Nery, Phys. Rev. D 105, 056010 (2022).
  67. L. M. Abreu, E. S. Nery, and E. B. S. Corrêa, Eur. Phys. J. A 59, 157 (2023).
  68. E. B. S. Corrêa, Phys. Rev. D 108, 076002 (2023).
  69. E. B. S. Corrêa, C. A. Bahia, and M. S. R. Sarges, Rev. Bras. Ens. Fís. 46, e20230340 (2024).
  70. A. Wünsche, J. Phys. A 31, 8267 (1998).
  71. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, edited by A. Jeffrey and D. Zwillinger, 7th ed. (Academic Press, New York, 2007).
  72. J. Schwinger, Particles, Sources, and Fields (Perseus Books Publishing, L.L.C., Massachusetts, 1998), Vol. 2.
  73. E. J. Ferrer and V. de la Incera, Phys. Rev. D 76, 045011 (2007).
  74. E. J. Ferrer, V. de la Incera, D. Manreza Paret, A. Pérez Martínez, and A. Sanchez, Phys. Rev. D 91, 085041 (2015).
  75. E. J. Ferrer, V. P. Gusynin, and V. de la Incera, Phys. Lett. B 455, 217 (1999).

Outline

Information

Phys. Rev. D 112, 125017– Published 15 December, 2025
  • Received 4 September 2025
  • Accepted 26 November 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation