- 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

Imprints of UA(1) chiral anomaly and disorder in the Dirac eigenspectrum of QCD at finite temperature

Ravi Shanker*, Harshit Pandey, and Sayantan Sharma

  • *Contact author: rshanker@imsc.res.in
PDF

Phys. Rev. D 114, 014519 – Published 27 July, 2026

DOI: https://doi.org/10.1103/571b-lm3m

Abstract

We perform a comprehensive study of the properties of the Dirac eigenvalue spectrum in QCD as a function of temperature on the lattice. In addition to effects due to the interplay between interactions and disorder inherently present in a many-body system, the Dirac spectrum also contains crucial information about the effective restoration of different subgroups of almost exact two-flavor chiral symmetry in QCD. We calculate the infrared eigenvalues of the overlap Dirac operator on 2+1 flavor QCD ensembles generated using domain wall fermion discretization, on a large volume lattice. From the normalized level spacing ratios, we identify those eigenvalues that have intermediate level statistics, distinctly different from the majority in the bulk spectrum that follow universal level fluctuations similar to a random matrix of Gaussian unitary type. We provide an explanation of these intermediate level ratios in terms of a specific random matrix model and quantify the correlation between these eigenstates and disorder in the gauge fields manifested in the renormalized Polyakov loop values. Whereas the existence of intermediate eigenmodes is intimately connected to the effective restoration of different subgroups of chiral symmetry close to the chiral crossover transition, their origin can be traced to random uncorrelated disorder at higher temperatures when the UA(1) is effectively restored. We also, for the first time, calculate the Thouless conductance for the Dirac spectrum that quantifies the structural rigidity of the eigenvectors, and use it as a diagnostic tool to understand the restoration of the anomalous UA(1) subgroup of chiral symmetry and localization driven by disorder.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (96)

  1. O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett. 52, 1 (1984).
  2. M. V. Berry and M. Tabor, Proc. R. Soc. A 356, 375 (1977).
  3. J. M. Deutsch, Rept. Prog. Phys. 81, 082001 (2018).
  4. O. Bohigas, S. Tomsovic, and D. Ullmo, Phys. Rep. 223, 43 (1993).
  5. M. V. Berry and M. Robnik, J. Phys. A 17, 2413 (1984).
  6. T. Prosen and M. Robnik, J. Phys. A 27, 8059 (1994).
  7. S. H. Tekur, S. Kumar, and M. S. Santhanam, Phys. Rev. E 97, 062212 (2018).
  8. R. Nandkishore and D. A. Huse, Annu. Rev. Condens. Matter Phys. 6, 15 (2015).
  9. M. Serbyn and J. E. Moore, Phys. Rev. B 93, 041424 (2016).
  10. M. Amini, E. Cuevas, I. M. Khaymovich, and V. E. Kravtsov, New J. Phys. 17, 122002 (2015).
  11. S. Moudgalya and O. I. Motrunich, Phys. Rev. X 12, 011050 (2022).
  12. S. Moudgalya, B. A. Bernevig, and N. Regnault, Rep. Prog. Phys. 85, 086501 (2022).
  13. D. Adler, D. Wei, M. Will, K. Srakaew, S. Agrawal, P. Weckesser, R. Moessner, F. Pollmann, I. Bloch, and J. Zeiher, Nature (London) 636, 80 (2024).
  14. B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianides, and H. B. Shore, Phys. Rev. B 47, 11487 (1993).
  15. P. W. Anderson, Phys. Rev. 109, 1492 (1958).
  16. D. Das, L. Ebner, S. V. Kadam, I. Raychowdhury, A. Schäfer, and X. Yao, Phys. Rev. D 113, 074514 (2026).
  17. S. Sharma, Proc. Sci. CPOD2017 (2018) 086.
  18. M. P. Lombardo and A. Trunin, Int. J. Mod. Phys. A 35, 2030010 (2020).
  19. A. V. Smilga and J. Stern, Phys. Lett. B 318, 531 (1993).
  20. A. Bazavov et al. (HotQCD Collaboration), Phys. Lett. B 795, 15 (2019).
  21. F. Burger, E.-M. Ilgenfritz, M. P. Lombardo, and A. Trunin, Phys. Rev. D 98, 094501 (2018).
  22. S. Borsanyi, Z. Fodor, J. N. Guenther, R. Kara, S. D. Katz, P. Parotto, A. Pasztor, C. Ratti, and K. K. Szabo, Phys. Rev. Lett. 125, 052001 (2020).
  23. J. J. M. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994).
  24. J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 70, 3852 (1993).
  25. J. J. M. Verbaarschot, Nucl. Phys. B, Proc. Suppl. 53, 88 (1997).
  26. T. Kanazawa and M. Kieburg, J. Phys. A 51, 345202 (2018).
  27. P. Petreczky, H.-P. Schadler, and S. Sharma, Phys. Lett. B 762, 498 (2016).
  28. R. V. Gavai, M. E. Jaensch, O. Kaczmarek, F. Karsch, M. Sarkar, R. Shanker, S. Sharma, S. Sharma, and T. Ueding, Phys. Rev. D 111, 034507 (2025).
  29. V. Dick, F. Karsch, E. Laermann, S. Mukherjee, and S. Sharma, Phys. Rev. D 91, 094504 (2015).
  30. S. Sharma, V. Dick, F. Karsch, E. Laermann, and S. Mukherjee, Nucl. Phys. A956, 793 (2016).
  31. L. Holicki, E.-M. Ilgenfritz, and L. von Smekal, Proc. Sci. LATTICE2018 (2018) 180 [arXiv:1810.01130].
  32. A. Tomiya, G. Cossu, S. Aoki, H. Fukaya, S. Hashimoto, T. Kaneko, and J. Noaki, Phys. Rev. D 96, 034509 (2017); 96, 079902(E) (2017).
  33. S. Aoki, Y. Aoki, G. Cossu, H. Fukaya, S. Hashimoto, T. Kaneko, C. Rohrhofer, and K. Suzuki (JLQCD Collaboration), Phys. Rev. D 103, 074506 (2021).
  34. S. Aoki, Y. Aoki, H. Fukaya, S. Hashimoto, C. Rohrhofer, and K. Suzuki (JLQCD Collaboration), Prog. Theor. Exp. Phys. 2022, 023B05 (2022).
  35. T.-W. Chiu, Proc. Sci. LATTICE 2013 (2014) 165.
  36. M. I. Buchoff et al., Phys. Rev. D 89, 054514 (2014).
  37. O. Kaczmarek, R. Shanker, and S. Sharma, Phys. Rev. D 108, 094501 (2023).
  38. A. Alexandru and I. Horváth, Phys. Rev. Lett. 127, 052303 (2021).
  39. A. Alexandru, I. Horváth, and N. Bhattacharyya, Phys. Rev. D 109, 014501 (2024).
  40. A. Alexandru, C. Bonanno, M. D’Elia, and I. Horváth, Phys. Rev. D 110, 074515 (2024).
  41. H. Pandey, R. Shanker, and S. Sharma, Nucl. Phys. B1025, 117401 (2026).
  42. A. Alexandru and I. Horváth, Phys. Rev. D 92, 045038 (2015).
  43. A. Alexandru and I. Horváth, Phys. Rev. D 100, 094507 (2019).
  44. R. Bignell, G. Aarts, C. Allton, B. Jäger, S. Kim, J.-I. Skullerud, and A. Smecca, in Proceedings of The 42nd International Symposium on Lattice Field Theory (2026), arXiv:2601.21178.
  45. T.-W. Chiu and T.-H. Hsieh, arXiv:2602.14811.
  46. H. T. Ding, S. T. Li, S. Mukherjee, A. Tomiya, X. D. Wang, and Y. Zhang, Phys. Rev. Lett. 126, 082001 (2021).
  47. T. G. Kovacs, Phys. Rev. Lett. 132, 131902 (2024).
  48. S. Borsanyi et al., Nature (London) 539, 69 (2016).
  49. C. Bonati, M. D’Elia, G. Martinelli, F. Negro, F. Sanfilippo, and A. Todaro, J. High Energy Phys. 11 (2018) 170.
  50. M. Giordano, S. D. Katz, T. G. Kovacs, and F. Pittler, J. High Energy Phys. 02 (2017) 055.
  51. R. Kehr, D. Smith, and L. von Smekal, Phys. Rev. D 109, 074512 (2024).
  52. A. M. Garcia-Garcia and J. C. Osborn, Phys. Rev. D 75, 034503 (2007).
  53. T. G. Kovacs and F. Pittler, Phys. Rev. Lett. 105, 192001 (2010).
  54. M. Giordano, T. G. Kovacs, and F. Pittler, Phys. Rev. Lett. 112, 102002 (2014).
  55. L. Ujfalusi, M. Giordano, F. Pittler, T. G. Kovács, and I. Varga, Phys. Rev. D 92, 094513 (2015).
  56. G. Cossu and S. Hashimoto, J. High Energy Phys. 06 (2016) 056.
  57. M. Giordano, T. G. Kovacs, and F. Pittler, Phys. Rev. D 113, 114514 (2026).
  58. H. T. Ding et al. (HotQCD Collaboration), Phys. Rev. Lett. 123, 062002 (2019).
  59. J. T. Edwards and D. J. Thouless, J. Phys. C 5, 807 (1972).
  60. Z. Lin, The Chiral and U(1)A Symmetries of the QCD Phase Transition using Chiral Lattice Fermions, Ph.D. thesis, Columbia U., 2014.
  61. T. Blum et al. (RBC Collaboration and UKQCD Collaboration), Phys. Rev. D 93, 074505 (2016).
  62. C. Jung (RBC Collaboration and UKQCD Collaboration), Proc. Sci. LATTICE2021 (2022) 165.
  63. T. Bhattacharya et al., Phys. Rev. Lett. 113, 082001 (2014).
  64. T. Kalkreuter and H. Simma, Comput. Phys. Commun. 93, 33 (1996).
  65. M. Lüscher, J. High Energy Phys. 08 (2010) 071; 03 (2014) 092(E).
  66. A. Bazavov, N. Brambilla, H. T. Ding, P. Petreczky, H. P. Schadler, A. Vairo, and J. H. Weber, Phys. Rev. D 93, 114502 (2016).
  67. V. Oganesyan and D. A. Huse, Phys. Rev. B 75, 155111 (2007).
  68. Y. Y. Atas, E. Bogomolny, O. Giraud, and G. Roux, Phys. Rev. Lett. 110, 084101 (2013).
  69. O. Giraud, N. Macé, E. Vernier, and F. Alet, Phys. Rev. X 12, 011006 (2022).
  70. R. G. Edwards, U. M. Heller, J. E. Kiskis, and R. Narayanan, Phys. Rev. D 61, 074504 (2000).
  71. M. Giordano and T. G. Kovacs, Universe 7, 194 (2021).
  72. E. Hofstetter and M. Schreiber, Phys. Rev. B 48, 16979 (1993).
  73. F. J. Dyson, J. Math. Phys. (N.Y.) 3, 1191 (1962).
  74. C. M. Canali, Phys. Rev. B 53, 3713 (1996).
  75. V. Kravtsov and K. Muttalib, Phys. Rev. Lett. 79, 1913 (1997).
  76. H. Yan, Phys. Rev. E 111, 054213 (2025).
  77. N. Rosenzweig and C. E. Porter, Phys. Rev. 120, 1698 (1960).
  78. T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. 53, 385 (1981).
  79. G. Lenz and F. Haake, Phys. Rev. Lett. 67, 1 (1991).
  80. V. K. B. Kota and S. Sumedha, Phys. Rev. E 60, 3405 (1999).
  81. S. Schierenberg, F. Bruckmann, and T. Wettig, Phys. Rev. E 85, 061130 (2012).
  82. F. Bruckmann, T. G. Kovacs, and S. Schierenberg, Phys. Rev. D 84, 034505 (2011).
  83. M. Giordano, T. G. Kovacs, and F. Pittler, J. High Energy Phys. 04 (2015) 112.
  84. R. N. Larsen, S. Sharma, and E. Shuryak, Phys. Rev. D 105, L071501 (2022).
  85. R. N. Larsen, S. Sharma, and E. Shuryak, Phys. Lett. B 794, 14 (2019).
  86. R. N. Larsen, S. Sharma, and E. Shuryak, Phys. Rev. D 102, 034501 (2020).
  87. J.-P. Blaizot and M. A. Nowak, Phys. Rev. Lett. 101, 102001 (2008).
  88. S. Guin, H. Pandey, and S. Sharma, Phys. Lett. B 866, 139490 (2025).
  89. D. Basko, I. Aleiner, and B. Altshuler, Ann. Phys. (Amsterdam) 321, 1126 (2006).
  90. D. A. Abanin, E. Altman, I. Bloch, and M. Serbyn, Rev. Mod. Phys. 91, 021001 (2019).
  91. S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H. Bardarson, Phys. Rev. Lett. 115, 046603 (2015).
  92. A. Smith, J. Knolle, D. L. Kovrizhin, and R. Moessner, Phys. Rev. Lett. 118, 266601 (2017).
  93. J. H. Pixley, P. Goswami, and S. Das Sarma, Phys. Rev. Lett. 115, 076601 (2015).
  94. F. von Oppen, Phys. Rev. Lett. 73, 798 (1994).
  95. M. A. Clark, R. Babich, K. Barros, R. C. Brower, and C. Rebbi (QUDA Collaboration), Comput. Phys. Commun. 181, 1517 (2010).
  96. C. Lehner, M. Bruno, D. Richtmann, M. Schlemmer, R. Lehner, D. Knüttel, T. Wurm, L. Jin, S. Bürger, A. Hackl, and A. Klein, Grid python Toolkit (GPT), 2024-10, (2024) 10.5281/zenodo.14017415.

Outline

Information

Phys. Rev. D 114, 014519– Published 27 July, 2026
  • Received 17 March 2026
  • Accepted 8 July 2026
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