- 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

Six loop critical exponent analysis for Lee-Yang and percolation theory

J. A. Gracey

  • Theoretical Physics Division, Department of Mathematical Sciences, University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, United Kingdom

PDF

Phys. Rev. D 112, 105019 – Published 24 November, 2025

DOI: https://doi.org/10.1103/kphq-sc3j

Abstract

Using the recent six loop renormalization group functions for Lee-Yang and percolation theory constructed by Schnetz from a scalar cubic Lagrangian, we deduce the ε expansion of the critical exponents for both cases. Estimates for the exponents in three, four, and five dimensions are extracted using two-sided Padé approximants and shown to be compatible with values from other approaches.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (75)

  1. S. Laporta, Int. J. Mod. Phys. A 15, 5087 (2000).
  2. T. Ueda, B. Ruijl, and J. A. M. Vermaseren, Proc. Sci. LL2016 (2016) 070.
  3. T. Ueda, B. Ruijl, and J. A. M. Vermaseren, Comput. Phys. Commun. 253, 107198 (2020).
  4. O. Schnetz, Commun. Num. Theor. Phys. 08, 589 (2014).
  5. M. Borinsky and O. Schnetz, Commun. Num. Theor. Phys. 16, 515 (2022).
  6. P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Phys. Rev. Lett. 118, 082002 (2017).
  7. F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, J. High Energy Phys. 02 (2017) 090.
  8. T. Luthe, A. Maier, P. Marquard, and Y. Schröder, J. High Energy Phys. 03 (2017) 020.
  9. J. A. Gracey, A. Maier, P. Marquard, and Y. Schröder, Phys. Rev. D 112, 085029 (2025).
  10. M. V. Kompaniets and E. Panzer, Phys. Rev. D 96, 036016 (2017).
  11. O. Schnetz, Phys. Rev. D 97, 085018 (2018).
  12. M. Kompaniets and A. Pikelner, Phys. Lett. B 817, 136331 (2021).
  13. M. Borinsky, J. A. Gracey, M. V. Kompaniets, and O. Schnetz, Phys. Rev. D 103, 116024 (2021).
  14. O. Schnetz, Phys. Rev. D 112, 016028 (2025).
  15. O. Schnetz, hyperlogprocedures v0.8 (2025), https://www.physik.uni-hamburg.de/th2/ag-moch/personen/schnetz-oliver.html.
  16. M. E. Fisher, Phys. Rev. Lett. 40, 1610 (1978).
  17. G. Başar, Phys. Rev. C 110, 015203 (2024).
  18. A. Adam, S. Borsányi, Z. Fodor, J. N. Guenther, P. Kumar, P. Parotto, A. Pásztor, and C. H. Wong, arXiv:2507.13254.
  19. V. V. Skokov, SciPost Phys. Lect. Notes 91, 15 (2025).
  20. K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 (1972).
  21. A. J. Macfarlane and G. Woo, Nucl. Phys. B77, 91 (1974); B86, 548(E) (1975).
  22. O. F. de Alcantara Bonfim, J. E. Kirkham, and A. J. McKane, J. Phys. A 13, L247 (1980); 13, 3785(E) (1980).
  23. O. F. de Alcantara Bonfim, J. E. Kirkham, and A. J. McKane, J. Phys. A 14, 2391 (1981).
  24. J. A. Gracey, Phys. Rev. D 92, 025012 (2015).
  25. D. Friedan, Z. Qiu, and S. H. Shenker, Phys. Rev. Lett. 52, 1575 (1984).
  26. S. N. Lai and M. E. Fisher, J. Chem. Phys. 103, 8144 (1995).
  27. X. An, D. Mesterházy, and M. A. Stephanov, J. High Energy Phys. 07 (2016) 041.
  28. D. Dhar, Phys. Rev. Lett. 49, 959 (1982).
  29. D. Dhar, Phys. Rev. Lett. 51, 853 (1983).
  30. E. A. Cruz, I. R. Klebanov, G. Tarnopolsky, and Y. Xin, arXiv:2505.06369.
  31. R. Fan, J. Dong, and A. Vishwanath, arXiv:2505.06342.
  32. J. Miró and O. Delouche, J. High Energy Phys. 10 (2025) 037.
  33. P. Butera and M. Pernici, Phys. Rev. E 86, 011104 (2012).
  34. F. Gliozzi, Phys. Rev. Lett. 111, 161602 (2013).
  35. F. Gliozzi and A. Rago, J. High Energy Phys. 10 (2014) 042.
  36. S. Hikami, Prog. Theor. Exp. Phys. 2018, 053I01 (2018).
  37. L. Zambelli and O. Zanusso, Phys. Rev. D 95, 085001 (2017).
  38. C. M. Fortuin and P. W. Kasteyn, Physica (Amsterdam) 57, 536 (1972).
  39. R. G. Priest and T. C. Lubensky, Phys. Rev. B 13, 4159 (1976); 13, 4159(E) (1976).
  40. D. J. Amit, J. Phys. A 9, 1441 (1976).
  41. R. M. Ziff, Phys. Rev. E 83, 020107(R) (2011).
  42. Wikipedia, Percolation critical exponents, https://en.wikipedia.org/wiki/Percolation_critical_exponents (2025).
  43. X. Xu, J. Wang, J.-P. Lv, and Y. Deng, Front. Phys. 9, 113 (2014).
  44. A. LeClair and J. Squires, J. Stat. Mech. (2018) P123105.
  45. S. Kirkpatrick, Phys. Rev. Lett. 36, 69 (1976).
  46. A. Sur, J. Lebowitz, J. Marro, M. Kalos, and S. Kirkpatrick, J. Stat. Phys. 15, 345 (1976).
  47. J. Adler, Y. Meir, A. Aharony, and A. B. Harris, J. Stat. Phys. 58, 511 (1990).
  48. J. Adler, Y. Meir, A. Aharony, and A. B. Harris, Phys. Rev. B 41, 9183 (1990).
  49. S. Fayfar, A. Bretaña, and W. Montfrooij, J. Phys. Comm. 5, 015008 (2021).
  50. H. Nakanishi and H. E. Stanley, Phys. Rev. B 22, 2466 (1980); 23, 1489(E) (1981).
  51. H. G. Ballesteros, L. A. Fernandez, V. Martín-Mayor, A. M. Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, Phys. Lett. B 400, 346 (1997).
  52. C. D. Lorenz and R. M. Ziff, Phys. Rev. E 57, 230 (1998).
  53. N. Jan, D. C. Hong, and H. E. Stanley, J. Phys. A 18, L935 (1985).
  54. N. Jan and D. Stauffer, Int. J. Mod. Phys. C 09, 341 (1998).
  55. P. N. Ballesteros, L. A. Fernández, V. Martín-Mayor, A. M. Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, Phys. Rev. B 61, 3215 (2000).
  56. G. Paul, R. M. Ziff, and H. E. Stanley, Phys. Rev. E 64, 026115 (2001).
  57. D. Tiggemann, Int. J. Mod. Phys. C 12, 871 (2001).
  58. Y. Deng and H. W. J. Blöte, Phys. Rev. E 72, 016126 (2005).
  59. S. Mertens and C. Moore, Phys. Rev. E 98, 022120 (2018).
  60. Z. Zhang, P. Hou, S. Fang, H. Hu, and Y. Deng, Physica (Amsterdam) 580A, 126124 (2021).
  61. D. S. Gaunt and H. Ruskin, J. Phys. A 11, 1369 (1978).
  62. M. E. Levenshteĭn, B. I. Shklovskiĭ, M. S. Shur, and A. L. Éfros, Sov. Phys. JETP 42, 197 (1976).
  63. K. Chistensen and N. R. Moloney, Complexity and Criticality (Imperial College Press, London, 2005), Vol. 1.
  64. J. Wang, Z. Zhou, W. Zhang, T. M. Garoni, and Y. Deng, Phys. Rev. E 87, 052107 (2013); 89, 069907(E) (2014).
  65. H. Hu, H. W. Blöte, R. M. Ziff, and Y. Deng, Phys. Rev. E 90, 042106 (2014).
  66. Z. Koza and J. Pola, J. Stat. Mech. (2016) P103206.
  67. X.-J. Tan, Y.-J. Deng, and J. L. Jacobsen, Front. Phys. 15, 41501 (2020).
  68. P. Brzeski and G. Kondrat, J. Stat. Mech. (2022) P053202.
  69. M. F. Sykes, D. S. Gaunt, and J. W. Essam, J. Phys. A 9, L43 (1976).
  70. D. Tiggemann, Int. J. Mod. Phys. C 17, 1141 (2006).
  71. Z. Xun, D. Hao, and R. M. Ziff, arXiv:2308.15719.
  72. B. Kozlov and M. Laguës, Physica (Amsterdam) 389A, 5339 (2010).
  73. J.-C. Gimel, T. Nicolai, and D. Durand, J. Phys. A 33, 7687 (2000).
  74. B. Nienhuis, J. Phys. A 15, 199 (1982).
  75. J. A. Gracey, arXiv:2510.05723.

Outline

Information

Phys. Rev. D 112, 105019– Published 24 November, 2025
  • Received 17 October 2025
  • Accepted 30 October 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