- 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

Thermodynamic phase transitions in Einstein-Maxwell-scalar-Gauss-Bonnet gravity

Cristián Erices* and Stella Kiorpelidi†

  • *Contact author: cristian.erices@ucentral.cl
  • †Contact author: stellak.postdoc@ucentral.cl
PDF

Phys. Rev. D 114, 084053 – Published 19 October, 2026

DOI: https://doi.org/10.1103/1xt9-w373

Abstract

Although asymptotically flat black holes generically lack thermodynamic phase transitions, we show that curvature-induced scalarization of electrically charged black holes in Einstein-Maxwell-scalar-Gauss-Bonnet theory provides a natural setting for nontrivial thermodynamic behavior, without invoking external confining mechanisms or an extended thermodynamic formalism. Working within the canonical ensemble and employing the Euclidean approach, we identify the coexistence of locally stable scalarized and Reissner-Nordström thermal states, which promotes free-energy crossings to bona fide phase transitions between equilibrium phases. For weak coupling, a second-order phase transition coincides with the second bifurcation point, at which the scalarized branch reconnects with the Reissner-Nordström branch and scalar hair is spontaneously shed. As the coupling strength increases, this transition becomes zeroth order, the scalarized branch shrinks, and a fishlike structure develops in its Helmholtz free energy, rendering locally stable thermal states partially metastable, and yielding up to three phase transitions. In the strong-coupling limit, the scalarized branch reduces to a Schwarzschild-like solution, and the Reissner-Nordström phase ultimately emerges as the sole thermodynamically preferred configuration.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (69)

  1. S. Carlip, Black hole thermodynamics, Int. J. Mod. Phys. D 23, 1430023 (2014).
  2. J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973).
  3. S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43, 199 (1975); 46, 206(E) (1976).
  4. J. M. Bardeen, B. Carter, and S. W. Hawking, The four laws of black hole mechanics, Commun. Math. Phys. 31, 161 (1973).
  5. G. W. Gibbons and S. W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15, 2752 (1977).
  6. S. W. Hawking, Black holes and thermodynamics, Phys. Rev. D 13, 191 (1976).
  7. R. Ruffini and J. A. Wheeler, Introducing the black hole, Phys. Today 24, No. 1, 30 (1971).
  8. S. W. Hawking, Black hole explosions, Nature (London) 248, 30 (1974).
  9. P. C. W. Davies, Thermodynamics of black holes, Rep. Prog. Phys. 41, 1313 (1978).
  10. S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun. Math. Phys. 87, 577 (1983).
  11. D. Kubiznak and R. B. Mann, Black hole chemistry, Can. J. Phys. 93, 999 (2015).
  12. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60, 064018 (1999).
  13. D. Kubiznak, R. B. Mann, and M. Teo, Black hole chemistry: Thermodynamics with lambda, Classical Quantum Gravity 34, 063001 (2017).
  14. N. Altamirano, D. Kubiznak, and R. B. Mann, Reentrant phase transitions in rotating anti–de Sitter black holes, Phys. Rev. D 88, 101502 (2013).
  15. N. Altamirano, D. Kubizňák, R. B. Mann, and Z. Sherkatghanad, Kerr-AdS analogue of triple point and solid/liquid/gas phase transition, Classical Quantum Gravity 31, 042001 (2014).
  16. S.-W. Wei and Y.-X. Liu, Triple points and phase diagrams in the extended phase space of charged Gauss-Bonnet black holes in AdS space, Phys. Rev. D 90, 044057 (2014).
  17. R. B. Mann, Black hole chemistry: The first 15 years, Int. J. Mod. Phys. D 34, 2542001 (2025).
  18. J. W. York, Jr., Black hole thermodynamics and the Euclidean Einstein action, Phys. Rev. D 33, 2092 (1986).
  19. A. P. Lundgren, Charged black hole in a canonical ensemble, Phys. Rev. D 77, 044014 (2008).
  20. D. Astefanesei, R. B. Mann, and R. Rojas, Hairy black hole chemistry, J. High Energy Phys. 11 (2019) 043.
  21. L. Hui and A. Nicolis, No-hair theorem for the Galileon, Phys. Rev. Lett. 110, 241104 (2013).
  22. C. A. R. Herdeiro and E. Radu, Asymptotically flat black holes with scalar hair: A review, Int. J. Mod. Phys. D 24, 1542014 (2015).
  23. S. S. Yazadjiev and D. D. Doneva, No-hair theorems in general relativity and scalar–tensor theories, Int. J. Mod. Phys. D 34, 2530004 (2025).
  24. C. P. Burgess, Introduction to effective field theory, Annu. Rev. Nucl. Part. Sci. 57, 329 (2007).
  25. M. Grana, Flux compactifications in string theory: A comprehensive review, Phys. Rep. 423, 91 (2006).
  26. J. Polchinski, String Theory. Vol. 1: An Introduction to the Bosonic String, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2007), 12.
  27. M. Bravo-Gaete, L. Guajardo, and J. Oliva, Nonlinear charged planar black holes in four-dimensional scalar-Gauss-Bonnet theories, Phys. Rev. D 106, 024017 (2022).
  28. N. Caceres, C. Corral, F. Diaz, and R. Olea, Holographic renormalization of Horndeski gravity, J. High Energy Phys. 05 (2024) 125.
  29. C. Erices, L. Guajardo, and K. Lara, Reverse stealth construction and its thermodynamic imprints, J. Cosmol. Astropart. Phys. 03 (2025) 051.
  30. D. Astefanesei, J. L. Blázquez-Salcedo, C. Herdeiro, E. Radu, and N. Sanchis-Gual, Dynamically and thermodynamically stable black holes in Einstein-Maxwell-dilaton gravity, J. High Energy Phys. 07 (2020) 063.
  31. D. Astefanesei, R. B. Mann, M. J. Rodriguez, and C. Stelea, Quasilocal formalism and thermodynamics of asymptotically flat black objects, Classical Quantum Gravity 27, 165004 (2010).
  32. D. Astefanesei, D. Choque, F. Gómez, and R. Rojas, Thermodynamically stable asymptotically flat hairy black holes with a dilaton potential, J. High Energy Phys. 03 (2019) 205.
  33. D. Astefanesei, J. L. Blázquez-Salcedo, F. Gómez, and R. Rojas, Thermodynamically stable asymptotically flat hairy black holes with a dilaton potential: The general case, J. High Energy Phys. 02 (2021) 233.
  34. D. Astefanesei, R. Ballesteros, P. Cabrera, G. Casanova, and R. Rojas, Existence of thermodynamically stable asymptotically flat black holes, Phys. Rev. D 110, 024045 (2024).
  35. C. Erices and M. Fathi, Thermodynamic and observational constraints on black holes with primary hair in beyond Horndeski gravity: Stability and shadows, J. Cosmol. Astropart. Phys. 01 (2025) 016.
  36. A. Bakopoulos, N. Chatzifotis, and T. Karakasis, Thermodynamics of black holes featuring primary scalar hair, Phys. Rev. D 110, L101502 (2024).
  37. K. S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16, 953 (1977).
  38. S. Mignemi and N. R. Stewart, Charged black holes in effective string theory, Phys. Rev. D 47, 5259 (1993).
  39. P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis, and E. Winstanley, Dilatonic black holes in higher curvature string gravity, Phys. Rev. D 54, 5049 (1996).
  40. T. Torii, H. Yajima, and K.-i. Maeda, Dilatonic black holes with Gauss-Bonnet term, Phys. Rev. D 55, 739 (1997).
  41. D. Ayzenberg and N. Yunes, Slowly rotating black holes in Einstein-dilaton-Gauss-Bonnet gravity: Quadratic order in spin solutions, Phys. Rev. D 90, 044066 (2014).
  42. B. Kleihaus, J. Kunz, and E. Radu, Rotating black holes in dilatonic Einstein-Gauss-Bonnet theory, Phys. Rev. Lett. 106, 151104 (2011).
  43. B. Kleihaus, J. Kunz, S. Mojica, and M. Zagermann, Rapidly rotating neutron stars in dilatonic Einstein-Gauss-Bonnet theory, Phys. Rev. D 93, 064077 (2016).
  44. C. Erices, S. Riquelme, and N. Zalaquett, EFT approach to black hole scalarization and its compatibility with cosmic evolution, Phys. Rev. D 106, 044046 (2022).
  45. T. Damour and G. Esposito-Farèse, Nonperturbative strong field effects in tensor-scalar theories of gravitation, Phys. Rev. Lett. 70, 2220 (1993).
  46. T. Damour and G. Esposito-Farese, Tensor–scalar gravity and binary pulsar experiments, Phys. Rev. D 54, 1474 (1996).
  47. D. D. Doneva and S. S. Yazadjiev, New Gauss-Bonnet black holes with curvature-induced scalarization in extended scalar-tensor theories, Phys. Rev. Lett. 120, 131103 (2018).
  48. H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, Spontaneous scalarization of black holes and compact stars from a Gauss-Bonnet coupling, Phys. Rev. Lett. 120, 131104 (2018).
  49. C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual, and J. A. Font, Spontaneous scalarization of charged black holes, Phys. Rev. Lett. 121, 101102 (2018).
  50. D. D. Doneva, S. Kiorpelidi, P. G. Nedkova, E. Papantonopoulos, and S. S. Yazadjiev, Charged Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories, Phys. Rev. D 98, 104056 (2018).
  51. P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu, and N. Sanchis-Gual, Spontaneous scalarisation of charged black holes: Coupling dependence and dynamical features, Classical Quantum Gravity 36, 134002 (2019).
  52. A. Dima, E. Barausse, N. Franchini, and T. P. Sotiriou, Spin-induced black hole spontaneous scalarization, Phys. Rev. Lett. 125, 231101 (2020).
  53. D. D. Doneva, F. M. Ramazanoğlu, H. O. Silva, T. P. Sotiriou, and S. S. Yazadjiev, Spontaneous scalarization, Rev. Mod. Phys. 96, 015004 (2024).
  54. K. İ. Ünlütürk, S. Tuna, O. K. Yamak, and F. M. Ramazanoğlu, Nature of phase transitions and metastability in scalar-tensor theories, Phys. Rev. Lett. 135, 061401 (2025).
  55. K. V. Staykov, F. M. Ramazanoğlu, D. D. Doneva, and S. S. Yazadjiev, Phase transition structure of scalarized neutron stars: The effect of rotation and linear coupling, Phys. Rev. D 114, 024082 (2026).
  56. C. Herdeiro, H. Huang, J. Kunz, M.-Y. Lai, E. Radu, and D.-C. Zou, Phase structure of scalarized black holes in Einstein-Scalar-Gauss-Bonnet gravity, Phys. Rev. D 113, 124046 (2026).
  57. J. L. Blázquez-Salcedo, D. D. Doneva, J. Kunz, and S. S. Yazadjiev, Radial perturbations of scalar-Gauss-Bonnet black holes beyond spontaneous scalarization, Phys. Rev. D 105, 124005 (2022).
  58. R. Ballesteros, M. Cárdenas, and E. Lescano, Nonclosed scalar charge in four-dimensional Einstein-scalar-Gauss-Bonnet black hole thermodynamics, Phys. Rev. D 113, 084039 (2026).
  59. R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, R3427 (1993).
  60. R. M. Wald, The thermodynamics of black holes, Living Rev. Relativity 4, 6 (2001).
  61. N. Chatzifotis, P. Dorlis, N. E. Mavromatos, and E. Papantonopoulos, Thermal stability of hairy black holes, Phys. Rev. D 107, 084053 (2023).
  62. L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, Course of Theoretical Physics Vol. 5 (Butterworth-Heinemann, Oxford, 1980).
  63. A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Holography, thermodynamics and fluctuations of charged AdS black holes, Phys. Rev. D 60, 104026 (1999).
  64. A. Eichhorn, P. G. S. Fernandes, and L. Marino, Strong breaking of black-hole uniqueness from coexisting scalarization mechanisms, Phys. Rev. D 113, 124071 (2026).
  65. N. M. Santos, C. A. R. Herdeiro, and E. Radu, Thermodynamic stability of quasibald asymptotically flat black holes, Phys. Rev. D 106, 124005 (2022).
  66. Z. Belkhadria and S. Mignemi, New model of spontaneous scalarization of black holes induced by curvature and matter, Phys. Rev. D 112, 044015 (2025).
  67. J. L. Blázquez-Salcedo, D. D. Doneva, J. Kunz, and S. S. Yazadjiev, Radial perturbations of the scalarized Einstein-Gauss-Bonnet black holes, Phys. Rev. D 98, 084011 (2018).
  68. J. L. Blázquez-Salcedo, D. D. Doneva, S. Kahlen, J. Kunz, P. Nedkova, and S. S. Yazadjiev, Axial perturbations of the scalarized Einstein-Gauss-Bonnet black holes, Phys. Rev. D 101, 104006 (2020).
  69. J. L. Blázquez-Salcedo, D. D. Doneva, S. Kahlen, J. Kunz, P. Nedkova, and S. S. Yazadjiev, Polar quasinormal modes of the scalarized Einstein-Gauss-Bonnet black holes, Phys. Rev. D 102, 024086 (2020).

Outline

Information

Phys. Rev. D 114, 084053– Published 19 October, 2026
  • Received 4 May 2026
  • Accepted 28 September 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