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Factorizing Defects from Generalized Pinning Fields

Fedor K. Popov1 and Yifan Wang2

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Phys. Rev. Lett. 135, 201601 – Published 12 November, 2025

DOI: https://doi.org/10.1103/rh7j-yg11

Abstract

We introduce generalized pinning fields in conformal field theory that model a large class of critical impurities at large distance, enriching the familiar universality classes. We provide a rigorous definition of such defects as certain unbounded operators on the Hilbert space and prove that when inserted on codimension-one surfaces they factorize the spacetime into two halves. The factorization channels are further constrained by symmetries in the bulk. As a corollary, we solve such critical impurities in the 2D minimal models and establish the factorization phenomena previously observed for localized mass deformations in the 3D O(N) model.

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References (77)

  1. I. Affleck and A. W. W. Ludwig, Phys. Rev. Lett. 67, 161 (1991).
  2. D. Friedan and A. Konechny, Phys. Rev. Lett. 93, 030402 (2004).
  3. K. Jensen and A. O’Bannon, Phys. Rev. Lett. 116, 091601 (2016).
  4. H. Casini, I. Salazar Landea, and G. Torroba, J. High Energy Phys. 04 (2019) 166.
  5. N. Kobayashi, T. Nishioka, Y. Sato, and K. Watanabe, J. High Energy Phys. 01 (2019) 039.
  6. Y. Wang, J. High Energy Phys. 11 (2021) 122.
  7. Y. Wang, J. High Energy Phys. 02 (2022) 061.
  8. G. Cuomo, Z. Komargodski, and A. Raviv-Moshe, Phys. Rev. Lett. 128, 021603 (2022).
  9. T. Shachar, R. Sinha, and M. Smolkin, SciPost Phys. 15, 240 (2023).
  10. H. Casini, I. Salazar Landea, and G. Torroba, J. High Energy Phys. 07 (2023) 004.
  11. A. Konechny, J. Phys. A 53, 155401 (2020).
  12. R. Thorngren and Y. Wang, J. High Energy Phys. 09 (2021) 017.
  13. O. Aharony, G. Cuomo, Z. Komargodski, M. Mezei, and A. Raviv-Moshe, Phys. Rev. Lett. 130, 151601 (2023).
  14. Y. Choi, B. C. Rayhaun, Y. Sanghavi, and S.-H. Shao, Phys. Rev. D 108, 125005 (2023).
  15. O. Aharony, G. Cuomo, Z. Komargodski, M. Mezei, and A. Raviv-Moshe, J. High Energy Phys. 12 (2023) 183.
  16. L. Bhardwaj, C. Copetti, D. Pajer, and S. Schafer-Nameki, SciPost Phys. 19, 061 (2025).
  17. Y. Choi, B. C. Rayhaun, and Y. Zheng, arXiv:2409.02159.
  18. A. Antinucci, C. Copetti, G. Galati, and G. Rizi, Phys. Rev. D 111, 065025 (2025).
  19. O. Diatlyk, H. Khanchandani, F. K. Popov, and Y. Wang, J. High Energy Phys. 09 (2024) 006.
  20. O. Diatlyk, H. Khanchandani, F. K. Popov, and Y. Wang, Phys. Rev. Lett. 133, 261601 (2024).
  21. P. Kravchuk, A. Radcliffe, and R. Sinha, arXiv:2406.04561.
  22. G. Cuomo, Y.-C. He, and Z. Komargodski, J. High Energy Phys. 11 (2024) 061.

  23. For general topological defects, the defect entropy is non-negative (for p odd) and the defect conformal anomaly always vanishes (for p even).

  24. A. Krishnan and M. A. Metlitski, SciPost Phys. 15, 090 (2023).
  25. M. Trépanier, J. High Energy Phys. 09 (2023) 074.
  26. A. Raviv-Moshe and S. Zhong, J. High Energy Phys. 08 (2023) 143.
  27. S. Giombi and B. Liu, J. High Energy Phys. 12 (2023) 004.
  28. O. Diatlyk, Z. Sun, and Y. Wang, J. High Energy Phys. 06 (2025) 131.
  29. M. A. Metlitski, SciPost Phys. 12, 131 (2022).
  30. W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974).
  31. T. Banks and A. Zaks, Nucl. Phys. B196, 189 (1982).
  32. A. Karch and L. Randall, J. High Energy Phys. 05 (2001) 008.
  33. A. Karch and L. Randall, J. High Energy Phys. 06 (2001) 063.
  34. O. DeWolfe, D. Z. Freedman, and H. Ooguri, Phys. Rev. D 66, 025009 (2002).
  35. T. Takayanagi, Phys. Rev. Lett. 107, 101602 (2011).
  36. M. Fujita, T. Takayanagi, and E. Tonni, J. High Energy Phys. 11 (2011) 043.
  37. Y. Nakayama and H. Ooguri, J. High Energy Phys. 10 (2015) 114.
  38. J. L. Cardy, Nucl. Phys. B324, 581 (1989).

  39. For instance, in the case of nonlocal theories (e.g., generalized free fields) this assumption fails and factorization does not hold [40].

  40. D. Ge and Y. Nakayama, arXiv:2505.15018.
  41. S. Rychkov, EPFL Lectures on Conformal Field Theory in D>=3 Dimensions, SpringerBriefs in Physics (2016), arXiv:1601.05000.
  42. I. M. Gel’fand and N. Y. Vilenkin, Generalized Functions: Applications of Harmonic Analysis (Academic Press, New York, 2014), Vol. 4.
  43. I. Gel’fand and G. E. Šilov, Generalized Functions: Theory of Differential Equations (Academic Press, New York, 1967).
  44. See SM A at http://link.aps.org/supplemental/10.1103/rh7j-yg11 for analysis of conforma line defects in the other minimal models. Supplemental Material B for the proof of the existence of self-adjoint operator O^Δ. See SM E for an example where this limit does not exist. See SM C for the proof of this result. One can show under the weak limit that w−limb→∞Ub|ψ⟩=0 for any |ψ⟩∈H (see Sec. D for details).

  45. Note that the matrix element as defined in B.1 [YW: fix] is IR finite only when Δ>[(d−1)/2], a condition stemming from the use of the planar setup for the defect. However these divergences are naturally regularized in the spherical conformal frame, which is related to the planar frame by a conformal transformation. For the sake of conciseness, we choose to work in the planar frame with this understanding.

  46. C. Bachas and I. Brunner, J. High Energy Phys. 02 (2008) 085.
  47. C. Bachas, I. Brunner, and D. Roggenkamp, J. Stat. Mech. (2013) P08008.
  48. A. Söderberg, J. High Energy Phys. 04 (2021) 087.
  49. A. Söderberg Rousu, J. High Energy Phys. 10 (2023) 183.
  50. D. G. Delmastro, J. Gomis, P.-S. Hsin, and Z. Komargodski, SciPost Phys. 15, 079 (2023).
  51. T. D. Brennan, C. Cordova, and T. T. Dumitrescu, arXiv:2206.15401.
  52. T. Bartsch, M. Bullimore, A. E. V. Ferrari, and J. Pearson, SciPost Phys. 17, 015 (2024).
  53. T. Bartsch, M. Bullimore, A. E. V. Ferrari, and J. Pearson, SciPost Phys. 17, 067 (2024).
  54. L. Bhardwaj and S. Schafer-Nameki, SciPost Phys. 16, 093 (2024).
  55. L. Bhardwaj and S. Schafer-Nameki, SciPost Phys. 19, 098 (2025).
  56. C. Copetti, arXiv:2408.01490.
  57. C.-M. Chang, Y.-H. Lin, S.-H. Shao, Y. Wang, and X. Yin, J. High Energy Phys. 01 (2019) 026.

  58. We emphasize that the general factorization property proved in the Factorization section does not rely on the additional assumptions introduced here, which are included purely for an intuitive connection to bulk RG flows, as illustrated in Fig. 1.

  59. Z. Komargodski, K. Ohmori, K. Roumpedakis, and S. Seifnashri, J. High Energy Phys. 03 (2021) 103.
  60. T.-C. Huang, Y.-H. Lin, and S. Seifnashri, J. High Energy Phys. 12 (2021) 028.
  61. A. Konechny, J. Phys. A 50, 145403 (2017).
  62. J. Cardy, SciPost Phys. 3, 011 (2017).
  63. M. Lencses, J. Viti, and G. Takacs, J. High Energy Phys. 01 (2019) 177.
  64. A. Konechny, J. High Energy Phys. 11 (2023) 004.
  65. P. Etingof, D. Nikshych, and V. Ostrik, Quantum Topol. 1, 209 (2010).
  66. C. L. Douglas, C. Schommer-Pries, and N. Snyder, Kyoto J. Math. 59, 167 (2019).
  67. R. Thorngren and Y. Wang, J. High Energy Phys. 04 (2024) 132.
  68. M. Oshikawa and I. Affleck, Nucl. Phys. B495, 533 (1997).
  69. M. Lassig, G. Mussardo, and J. L. Cardy, Nucl. Phys. B348, 591 (1991).

  70. For the pinning flow with O=σ, to completely fix the answer, we have additionally assumed that the unique factorization channel is via a stable boundary state. This also agrees with the findings of [62, 63, 64].

  71. H. W. Diehl, Int. J. Mod. Phys. B 11, 3503 (1997).
  72. F. P. Toldin, Phys. Rev. B 108, L020404 (2023).
  73. Z. Zhou and Y. Zou, SciPost Phys. 18, 031 (2025).
  74. M. Dedushenko, arXiv:2407.15948.
  75. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Courier Corporation, Moscow, 1957), Vol. 1.

  76. If O^Δ also respects additional symmetries, these eigenstates are further diagonalized with respect to the corresponding charges.

  77. N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Rev. Mod. Phys. 84, 1419 (2012).

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Phys. Rev. Lett. 135, 201601– Published 12 November, 2025
  • Received 23 April 2025
  • Revised 4 September 2025
  • Accepted 14 October 2025
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