- 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

Comparative study of the butterfly velocity in holographic QCD models at finite temperature and chemical potential

Nikesh Lilani*, Dilpreet Sandhu†, and Subhash Mahapatra‡

  • *Contact author: imnikeshlilani@gmail.com
  • †Contact author: dilpreetsandhu2255@gmail.com
  • ‡Contact author: mahapatrasub@nitrkl.ac.in
PDF

Phys. Rev. D 112, 046012 – Published 15 August, 2025

DOI: https://doi.org/10.1103/mvs2-45rn

Abstract

In this work, we study quantum chaos in a variety of holographic QCD models at finite temperature and chemical potentials. This includes the 1 and 2 R-charge black hole models, a potential reconstruction-based analytic bottom-up model, and a numerical bottom-up model. All these models are different avatars of the Einstein-Maxwell-dilaton gravity action, distinguished by their specific choices of dilaton potentials and gauge-kinetic coupling functions. We focus on computing the chaos parameter, the butterfly velocity, using three distinct methods: entanglement wedge reconstruction, out-of-time-ordered correlators, and pole skipping. We show that all three methods yield identical results for the butterfly velocity across all the holographic QCD models considered, further establishing the equivalence between the three approaches. Furthermore, we analyze in detail the behavior of the butterfly velocity as a function of chemical potential and temperature. Interestingly, a universal trend emerges across all models: the butterfly velocity increases/decreases with temperature/chemical potential for thermodynamically stable phases. Additionally, in the high-temperature limit, the butterfly velocity in all models approaches that of the chargeless plasma.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (120)

  1. Mark Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  2. Shingo Suzuki and Kei-ichi Maeda, Chaos in Schwarzschild space-time: The motion of a spinning particle, Phys. Rev. D 55, 4848 (1997).
  3. Makoto Natsuume and Takashi Okamura, Holographic chaos, pole-skipping, and regularity, Prog. Theor. Exp. Phys. 2020, 013B07 (2020).
  4. Juan Maldacena, Stephen H. Shenker, and Douglas Stanford, A bound on chaos, J. High Energy Phys. 08 (2016) 106.
  5. T. Akutagawa, K. Hashimoto, K. Murata, and T. Ota, Chaos of QCD string from holography, Phys. Rev. D 100, 046009 (2019).
  6. D. Giataganas, U. Gürsoy, and J. F. Pedraza, Strongly coupled anisotropic gauge theories and holography, Phys. Rev. Lett. 121, 121601 (2018).
  7. Juan Martin Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2, 231 (1998).
  8. S. S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428, 105 (1998).
  9. Edward Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2, 253 (1998).
  10. Jorge Casalderrey-Solana, Hong Liu, David Mateos, Krishna Rajagopal, and Urs Achim Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions (Cambridge University Press, Cambridge, England, 2014).
  11. Romulo Rougemont, Joaquin Grefa, Mauricio Hippert, Jorge Noronha, Jacquelyn Noronha-Hostler, Israel Portillo, and Claudia Ratti, Hot QCD phase diagram from holographic Einstein–Maxwell–Dilaton models, Prog. Part. Nucl. Phys. 135, 104093 (2024).
  12. Stephen H. Shenker and Douglas Stanford, Black holes and the butterfly effect, J. High Energy Phys. 03 (2014) 067.
  13. Daniel A. Roberts, Douglas Stanford, and Leonard Susskind, Localized shocks, J. High Energy Phys. 03 (2015) 051.
  14. Daniel A. Roberts and Brian Swingle, Lieb-Robinson bound and the butterfly effect in quantum field theories, Phys. Rev. Lett. 117, 091602 (2016).
  15. Eric Perlmutter, Bounding the space of holographic CFTs with chaos, J. High Energy Phys. 10 (2016) 069.
  16. Viktor Jahnke, Recent developments in the holographic description of quantum chaos, Adv. High Energy Phys. 2019, 9632708 (2019).
  17. Yasuhiro Sekino and Leonard Susskind, Fast scramblers, J. High Energy Phys. 10 (2008) 065.
  18. Juan Martin Maldacena, Eternal black holes in anti-de Sitter, J. High Energy Phys. 04 (2003) 021.
  19. Tetsuya Akutagawa, Koji Hashimoto, Takeshi Miyazaki, and Toshihiro Ota, Phase diagram of QCD chaos in linear sigma models and holography, Prog. Theor. Exp. Phys. 2018, 063B01 (2018).
  20. Tetsuya Akutagawa, Koji Hashimoto, Keiju Murata, and Toshihiro Ota, Chaos of QCD string from holography, Phys. Rev. D 100, 046009 (2019).
  21. Koji Hashimoto, Keiju Murata, and Kentaroh Yoshida, Chaos in chiral condensates in gauge theories, Phys. Rev. Lett. 117, 231602 (2016).
  22. Koji Hashimoto, Keiju Murata, and Norihiro Tanahashi, Chaos of Wilson loop from string motion near black hole horizon, Phys. Rev. D 98, 086007 (2018).
  23. Bhaskar Shukla, David Dudal, and Subhash Mahapatra, Anisotropic and frame dependent chaos of suspended strings from a dynamical holographic QCD model with magnetic field, J. High Energy Phys. 06 (2023) 178.
  24. Bhaskar Shukla, Jasper Nongmaithem, David Dudal, and Subhash Mahapatra, Interplay of magnetic field and chemical potential induced anisotropy and frame dependent chaos of a QQ− pair in holographic QCD, Phys. Rev. D 111, 106002 (2025).
  25. Pietro Colangelo, Floriana Giannuzzi, and Nicola Losacco, Chaotic dynamics of a suspended string in a gravitational background with magnetic field, Phys. Lett. B 827, 136949 (2022).
  26. P. Colangelo, F. De Fazio, and N. Losacco, Chaos in a QQ¯ system at finite temperature and baryon density, Phys. Rev. D 102, 074016 (2020).
  27. Si-wen Li, Yi-peng Zhang, and Hao-qian Li, Out-of-time-order correlator as a detector of baryonic phase structure in holographic QCD with instanton, Phys. Lett. B 857, 138972 (2024).
  28. Takanori Anegawa, Norihiro Iizuka, and Mitsuhiro Nishida, Krylov complexity as an order parameter for deconfinement phase transitions at large N, J. High Energy Phys. 04 (2024) 119.
  29. Banashree Baishya, Sayan Chakrabarti, Debaprasad Maity, and Kuntal Nayek, Pole-skipping and chaos in D3-D7 brane systems, Phys. Rev. D 110, 086003 (2024).
  30. Bhaskar Shukla, Owais Riyaz, and Subhash Mahapatra, Classical and quantum chaos of closed strings on a charged confining holographic background, Phys. Rev. D 111, 066019 (2025).
  31. Leopoldo A. Pando Zayas and Cesar A. Terrero-Escalante, Chaos in the gauge / gravity correspondence, J. High Energy Phys. 09 (2010) 094.
  32. Pallab Basu, Diptarka Das, Archisman Ghosh, and Leopoldo A. Pando Zayas, Chaos around holographic Regge trajectories, J. High Energy Phys. 05 (2012) 077.
  33. Takaaki Ishii, Keiju Murata, and Kentaroh Yoshida, Fate of chaotic strings in a confining geometry, Phys. Rev. D 95, 066019 (2017).
  34. Pallab Basu, Pankaj Chaturvedi, and Prasant Samantray, Chaotic dynamics of strings in charged black hole backgrounds, Phys. Rev. D 95, 066014 (2017).
  35. Pallab Basu and Leopoldo A. Pando Zayas, Chaos rules out integrability of strings on AdS5×T1,1, Phys. Lett. B 700, 243 (2011).
  36. Avik Banerjee, Arnab Kundu, and Rohan R. Poojary, Strings, branes, Schwarzian action and maximal chaos, Phys. Lett. B 838, 137632 (2023).
  37. Jiayu Xie, Yaxuan Wang, and Bing Tang, Chaotic dynamics of strings around the Bardeen-AdS black hole surrounded by quintessence dark energy, Phys. Dark Universe 40, 101184 (2023).
  38. Da-Zhu Ma, Fang Xia, Dan Zhang, Guo-Yang Fu, and Jian-Pin Wu, Chaotic dynamics of string around the conformal black hole, Eur. Phys. J. C 82, 372 (2022).
  39. José Manuel Penín and Konstantinos C. Rigatos, Evidence for a four-dimensional N=1 integrable quiver in massive type IIA supergravity, Phys. Rev. D 109, 126007 (2024).
  40. Konstantinos S. Rigatos, Nonintegrability of La,b,c quiver gauge theories, Phys. Rev. D 102, 106022 (2020).
  41. Pallab Basu, Diptarka Das, and Archisman Ghosh, Integrability lost, Phys. Lett. B 699, 388 (2011).
  42. Brian Swingle, Gregory Bentsen, Monika Schleier-Smith, and Patrick Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94, 040302 (2016).
  43. Guanyu Zhu, Mohammad Hafezi, and Tarun Grover, Measurement of many-body chaos using a quantum clock, Phys. Rev. A 94, 062329 (2016).
  44. Jun Li, Ruihua Fan, Hengyan Wang, Bingtian Ye, Bei Zeng, Hui Zhai, Xinhua Peng, and Jiangfeng Du, Measuring out-of-time-order correlators on a nuclear magnetic resonance quantum simulator, Phys. Rev. X 7, 031011 (2017).
  45. Norman Y. Yao, Fabian Grusdt, Brian Swingle, Mikhail D. Lukin, Dan M. Stamper-Kurn, Joel E. Moore, and Eugene A. Demler, Interferometric approach to probing fast scrambling, arXiv:1607.01801.
  46. M. Blake, Universal charge diffusion and the butterfly effect in holographic theories, Phys. Rev. Lett. 117, 091601 (2016).
  47. S. Grozdanov, On the connection between hydrodynamics and quantum chaos in holographic theories with stringy corrections, J. High Energy Phys. 01 (2019) 048.
  48. Y. Ling, P. Liu, and J. P. Wu, Holographic butterfly effect at quantum critical points, J. High Energy Phys. 10 (2017) 025.
  49. U. Gürsoy, M. Järvinen, G. Nijs, and J. F. Pedraza, On the interplay between magnetic field and anisotropy in holographic QCD, J. High Energy Phys. 03 (2021) 180.
  50. Dmitry S. Ageev, Chaotic nature of holographic QCD, Phys. Rev. D 104, 126013 (2021).
  51. Oliver DeWolfe, Steven S. Gubser, and Christopher Rosen, Dynamic critical phenomena at a holographic critical point, Phys. Rev. D 84, 126014 (2011).
  52. Oliver DeWolfe, Steven S. Gubser, and Christopher Rosen, Fermi surfaces in N=4 Super-Yang-Mills theory, Phys. Rev. D 86, 106002 (2012).
  53. David Dudal and Subhash Mahapatra, Thermal entropy of a quark-antiquark pair above and below deconfinement from a dynamical holographic QCD model, Phys. Rev. D 96, 126010 (2017).
  54. Hardik Bohra, David Dudal, Ali Hajilou, and Subhash Mahapatra, Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model, Phys. Lett. B 801, 135184 (2020).
  55. Oliver DeWolfe, Steven S. Gubser, and Christopher Rosen, A holographic critical point, Phys. Rev. D 83, 086005 (2011).
  56. Romulo Rougemont, Andrej Ficnar, Stefano Finazzo, and Jorge Noronha, Energy loss, equilibration, and thermodynamics of a baryon rich strongly coupled quark-gluon plasma, J. High Energy Phys. 04 (2016) 102.
  57. Xi Dong, Diandian Wang, Wayne W. Weng, and Chih-Hung Wu, A tale of two butterflies: An exact equivalence in higher-derivative gravity, J. High Energy Phys. 10 (2022) 009.
  58. Banashree Baishya, Adrita Chakraborty, and Nibedita Padhi, Entanglement wedge method, out-of-time-ordered correlators, and pole skipping, Phys. Rev. D 111, 106013 (2025).
  59. Nikesh Lilani, Chaos in Hyperscaling Violating Lifshitz Theories, arXiv:2411.09667.
  60. Debarshi Basu, Ashish Chandra, and Qiang Wen, Butterfly effect and TT¯-deformation, arXiv:2505.14331.
  61. Brandon S. DiNunno, Niko Jokela, Juan F. Pedraza, and Arttu Pönni, Quantum information probes of charge fractionalization in large-N gauge theories, J. High Energy Phys. 05 (2021) 149.
  62. Willy Fischler, Viktor Jahnke, and Juan F. Pedraza, Chaos and entanglement spreading in a non-commutative gauge theory, J. High Energy Phys. 11 (2018) 072; 02 (2021) 149(E).
  63. Stefan Eccles, Willy Fischler, Tyler Guglielmo, Juan F. Pedraza, and Sarah Racz, Speeding up the spread of quantum information in chaotic systems, J. High Energy Phys. 12 (2021) 019.
  64. Ashis Saha and Sunandan Gangopadhyay, Quantum chaos in the presence of nonconformality, Phys. Rev. D 110, 026025 (2024).
  65. Shankhadeep Chakrabortty, Hironori Hoshino, Sanjay Pant, and Karunava Sil, A holographic study of the characteristics of chaos and correlation in the presence of backreaction, Phys. Lett. B 838, 137749 (2023).
  66. Debanjan Karan and Sanjay Pant, Entanglement and Chaos near critical point in strongly coupled Gauge theory, Eur. Phys. J. C 84, 113 (2024).
  67. Wan Zhen Chua, Thomas Hartman, and Wayne W. Weng, Replica manifolds, pole skipping, and the butterfly effect, arXiv:2504.08139.
  68. Márk Mezei and Douglas Stanford, On entanglement spreading in chaotic systems, J. High Energy Phys. 05 (2017) 065.
  69. Bartlomiej Czech, Joanna L. Karczmarek, Fernando Nogueira, and Mark Van Raamsdonk, The gravity dual of a density matrix, Classical Quantum Gravity 29, 155009 (2012).
  70. Aron C. Wall, Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy, Classical Quantum Gravity 31, 225007 (2014).
  71. Matthew Headrick, Veronika E. Hubeny, Albion Lawrence, and Mukund Rangamani, Causality & holographic entanglement entropy, J. High Energy Phys. 12 (2014) 162.
  72. Xi Dong, Daniel Harlow, and Aron C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett. 117, 021601 (2016).
  73. Isaac H. Kim and Michael J. Kastoryano, Entanglement renormalization, quantum error correction, and bulk causality, J. High Energy Phys. 04 (2017) 040.
  74. Shinsei Ryu and Tadashi Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, 181602 (2006).
  75. Veronika E. Hubeny, Mukund Rangamani, and Tadashi Takayanagi, A covariant holographic entanglement entropy proposal, J. High Energy Phys. 07 (2007) 062.
  76. Sašo Grozdanov, Koenraad Schalm, and Vincenzo Scopelliti, Black hole scrambling from hydrodynamics, Phys. Rev. Lett. 120, 231601 (2018).
  77. Mike Blake, Hyunseok Lee, and Hong Liu, A quantum hydrodynamical description for scrambling and many-body chaos, J. High Energy Phys. 10 (2018) 127.
  78. Mike Blake, Richard A. Davison, Sašo Grozdanov, and Hong Liu, Many-body chaos and energy dynamics in holography, J. High Energy Phys. 10 (2018) 035.
  79. Mike Blake, Richard A. Davison, and David Vegh, Horizon constraints on holographic Green’s functions, J. High Energy Phys. 01 (2020) 077.
  80. Diandian Wang and Zi-Yue Wang, Pole skipping in holographic theories with bosonic fields, Phys. Rev. Lett. 129, 231603 (2022).
  81. Sirui Ning, Diandian Wang, and Zi-Yue Wang, Pole skipping in holographic theories with gauge and fermionic fields, J. High Energy Phys. 12 (2023) 084.
  82. Hyun-Sik Jeong, Keun-Young Kim, and Ya-Wen Sun, Bound of diffusion constants from pole-skipping points: Spontaneous symmetry breaking and magnetic field, J. High Energy Phys. 07 (2021) 105.
  83. K. Behrndt, Mirjam Cvetic, and W. A. Sabra, Nonextreme black holes of five-dimensional N=2 AdS supergravity, Nucl. Phys. B553, 317 (1999).
  84. Mirjam Cvetic and Steven S. Gubser, Phases of R charged black holes, spinning branes and strongly coupled gauge theories, J. High Energy Phys. 04 (1999) 024.
  85. H. Ebrahim, M. Asadi, and M. Ali-Akbari, Evolution of holographic complexity near critical point, J. High Energy Phys. 09 (2019) 023.
  86. Hajar Ebrahim and Gol-Mohammad Nafisi, Holographic mutual information and critical exponents of the strongly coupled plasma, Phys. Rev. D 102, 106007 (2020).
  87. B. Amrahi, M. Asadi, and F. Taghinavaz, Chaos near to the critical point: Butterfly effect and pole-skipping, Eur. Phys. J. C 84, 505 (2024).
  88. Gustavo de Oliveira, Willians Barreto, and Romulo Rougemont, Universality of the entropy stairway in homogeneous isotropization, arXiv:2506.10294.
  89. Gustavo de Oliveira and Romulo Rougemont, New purely damped pairs of quasinormal modes in a hot and dense strongly-coupled plasma, J. High Energy Phys. 11 (2024) 079.
  90. Stefano Ivo Finazzo, Renato Critelli, Romulo Rougemont, and Jorge Noronha, Momentum transport in strongly coupled anisotropic plasmas in the presence of strong magnetic fields, Phys. Rev. D 94, 054020 (2016); 96, 019903(E) (2017).
  91. Subhash Mahapatra and Pratim Roy, On the time dependence of holographic complexity in a dynamical Einstein-dilaton model, J. High Energy Phys. 11 (2018) 138.
  92. Irina Aref’eva and Kristina Rannu, Holographic anisotropic background with confinement-deconfinement phase transition, J. High Energy Phys. 05 (2018) 206.
  93. Irina Ya. Aref’eva, Kristina Rannu, and Pavel Slepov, Holographic anisotropic model for light quarks with confinement-deconfinement phase transition, J. High Energy Phys. 06 (2021) 090.
  94. Subhash Mahapatra, Supragyan Priyadarshinee, Gosala Narasimha Reddy, and Bhaskar Shukla, Exact topological charged hairy black holes in AdS Space in D-dimensions, Phys. Rev. D 102, 024042 (2020).
  95. Supragyan Priyadarshinee, Subhash Mahapatra, and Indrani Banerjee, Analytic topological hairy dyonic black holes and thermodynamics, Phys. Rev. D 104, 084023 (2021).
  96. Supragyan Priyadarshinee and Subhash Mahapatra, Analytic three-dimensional primary hair charged black holes and thermodynamics, Phys. Rev. D 108, 044017 (2023).
  97. Rong-Gen Cai, Song He, and Danning Li, A hQCD model and its phase diagram in Einstein-Maxwell-Dilaton system, J. High Energy Phys. 03 (2012) 033.
  98. Ayan Daripa and Subhash Mahapatra, Analytic three-dimensional primary hair charged black holes with Coulomb-like electrodynamics and their thermodynamics, Phys. Rev. D 109, 124039 (2024).
  99. Steven S. Gubser, Curvature singularities: The good, the bad, and the naked, Adv. Theor. Math. Phys. 4, 679 (2000).
  100. Peter Breitenlohner and Daniel Z. Freedman, Positive energy in anti-de Sitter backgrounds and gauged extended supergravity, Phys. Lett. 115B, 197 (1982).
  101. Bing Chen, Xun Chen, Xiaohua Li, Zhou-Run Zhu, and Kai Zhou, Exploring transport properties of quark-gluon plasma in flavor-dependent systems with a holographic model, Phys. Rev. D 111, 086033 (2025).
  102. Siddhi Swarupa Jena, Arpan Bhattacharjee, David Dudal, and Subhash Mahapatra, Probing quarkonium diffusion in a magnetized quark-gluon plasma, arXiv:2507.00746.
  103. Siddhi Swarupa Jena, Jyotirmoy Barman, Bruno Toniato, David Dudal, and Subhash Mahapatra, A dynamical Einstein-Born-Infeld-dilaton model and holographic quarkonium melting in a magnetic field, J. High Energy Phys. 12 (2024) 096.
  104. Siddhi Swarupa Jena, Bhaskar Shukla, David Dudal, and Subhash Mahapatra, Entropic force and real-time dynamics of holographic quarkonium in a magnetic field, Phys. Rev. D 105, 086011 (2022).
  105. Hardik Bohra, David Dudal, Ali Hajilou, and Subhash Mahapatra, Chiral transition in the probe approximation from an Einstein-Maxwell-dilaton gravity model, Phys. Rev. D 103, 086021 (2021).
  106. Renato Critelli, Jorge Noronha, Jacquelyn Noronha-Hostler, Israel Portillo, Claudia Ratti, and Romulo Rougemont, Critical point in the phase diagram of primordial quark-gluon matter from black hole physics, Phys. Rev. D 96, 096026 (2017).
  107. J. Knaute and B. Kämpfer, Holographic entanglement entropy in the QCD phase diagram with a critical point, Phys. Rev. D 96, 106003 (2017).
  108. Rong-Gen Cai, Song He, Li Li, and Yuan-Xu Wang, Probing QCD critical point and induced gravitational wave by black hole physics, Phys. Rev. D 106, L121902 (2022).
  109. Niko Jokela, Matti Järvinen, and Aleksi Piispa, Refining holographic models of the quark-gluon plasma, Phys. Rev. D 110, 126013 (2024).
  110. Maximilian Attems, Jorge Casalderrey-Solana, David Mateos, Ioannis Papadimitriou, Daniel Santos-Oliván, Carlos F. Sopuerta, Miquel Triana, and Miguel Zilhão, Thermodynamics, transport and relaxation in non-conformal theories, J. High Energy Phys. 10 (2016) 155.
  111. Shao-Jun Zhang, Complexity and phase transitions in a holographic QCD model, Nucl. Phys. B929, 243 (2018).
  112. Romulo Rougemont, Renato Critelli, and Jorge Noronha, Holographic calculation of the QCD crossover temperature in a magnetic field, Phys. Rev. D 93, 045013 (2016).
  113. Rong-Gen Cai, Song He, Li Li, and Hong-An Zeng, Neural ordinary differential equations for mapping the magnetic QCD phase diagram via holography, arXiv:2406.12772.
  114. Song He, Li Li, Zhibin Li, and Shao-Jiang Wang, Gravitational waves and primordial black hole productions from gluodynamics by holography, Sci. China Phys. Mech. Astron. 67, 240411 (2024).
  115. Sayak Guin, Harshit Pandey, and Sayantan Sharma, Understanding thermalization in a non-Abelian gauge theory in terms of its soft modes, Phys. Lett. B 866, 139490 (2025).
  116. Wung-Hong Huang, Butterfly velocity in quadratic gravity, Classical Quantum Gravity 35, 195004 (2018).
  117. Yanqi Wang, Peng-Ju Hu, and Yi Pang, A holographic model of magnetohydrodynamics with fortuitous SO(3) symmetry, J. High Energy Phys. 10 (2024) 035.
  118. Matteo Baggioli, Bikash Padhi, Philip W. Phillips, and Chandan Setty, Conjecture on the butterfly velocity across a quantum phase transition, J. High Energy Phys. 07 (2018) 049.
  119. Bhaskar Shukla, Pranaya Pratik Das, David Dudal, and Subhash Mahapatra, Interplay between the Lyapunov exponents and phase transitions of charged AdS black holes, Phys. Rev. D 110, 024068 (2024).
  120. Xiaobo Guo, Yuhang Lu, Benrong Mu, and Peng Wang, Probing phase structure of black holes with Lyapunov exponents, J. High Energy Phys. 08 (2022) 153.

Outline

Information

Phys. Rev. D 112, 046012– Published 15 August, 2025
  • Received 27 May 2025
  • Accepted 24 July 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