https://doi.org/10.1140/epjc/s10052-026-15407-7
Regular Article - Theoretical Physics
Scalar-hairy AdS black hole in the Einstein–Maxwell-scalar theory: first-order phase transition with a critical point
1
Particle Theory and Cosmology Group, Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), 34126, Daejeon, Republic of Korea
2
College of Physics and Materials Science, Tianjin Normal University, 300387, Tianjin, China
3
Center for Quantum Spacetime, Sogang University, 04107, Seoul, Republic of Korea
a
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Received:
11
January
2026
Accepted:
3
February
2026
Published online:
28
February
2026
Abstract
In asymptotically anti-de Sitter (AdS) spacetime, we consider a real massive scalar field in the Einstein–Maxwell-Scalar (EMS) model and examine both scalar-hairy black hole solutions induced by the nonminimal coupling to the Maxwell field and tachyonic-hairy solutions driven by the scalar potential. When the scalar potential vanishes, scalar-hairy black holes emerge with profiles and properties similar to those observed in flat spacetime. The presence of the scalar potential additionally induces tachyonic-hairy solutions, leading to the coexistence of these two distinct hairy phases in different regions of the parameter space. The phase diagram reveals a first-order phase transition line between the tachyonic-hairy and scalar-hairy phases, originating at a critical point in the extreme temperature and chemical potential regime. Our detailed analysis shows that this phase transition is directly associated with the self-overlap region of the scalar-hairy phase and its start point. Moreover, increasing the coupling strength 
shifts the critical point to higher temperature and chemical potential.
© The Author(s) 2026
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