Scalar-hairy AdS Black Hole in the Einstein-Maxwell-Scalar Theory: first-order phase transition with a critical point

TL;DR

This work analyzes the Einstein–Maxwell–scalar theory in AdS with a real massive scalar, focusing on two hairy black hole branches: scalar-hairy solutions arising from nonminimal coupling and tachyonic-hairy solutions driven by the scalar potential. By solving the coupled equations for $\psi$, $A_t$, and the metric with $f(\psi)=e^{-\lambda \psi^2}$ and $V(\psi)=(1/2)m^2\psi^2$, the authors map out existence domains and demonstrate a rich phase structure, including a first-order phase transition between the tachyonic-hairy and scalar-hairy phases that originates at a critical point on the overlap of these branches. The transition point shifts to higher temperature and chemical potential as the coupling $\lambda$ increases, and the transition is intimately linked to the self-overlap region of the scalar-hairy phase rather than merely the coexistence region. The results connect with previous holographic models (EMD) and offer insight into how hair and coupling strength can mimic QCD-like phase transitions in strongly coupled boundary theories.

Abstract

In asymptotically anti-de Sitter (AdS) spacetime, we consider a real massiver 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 $λ$ shifts the critical point to higher temperature and chemical potential.

Figures (11)

• Figure 1: Profiles of the field functions for $\psi_0=1$ and $\lambda=1$, shown for different scalar charges $\Psi_0=0.15,0.2,0.3,0.4,0.5,0.6$.
• Figure 2: Left: Hawking temperature of the scalar-hairy black hole as a function of the electric charge $Q$ for different scalar charges $\Psi_0$; Right: Scalar hair as a function of $T$ for different coupling value $\lambda$ with $\Psi_0=1$ and $\Psi_0=2$, respectively.
• Figure 3: Existence domain of the scalar-hairy black holes in the parameter space of $(\Psi_0, Q)$.
• Figure 4: Example profiles of field functions for tachyonic-hairy black hole solutions (left) and scalar-hairy black hole solutions (right).
• Figure 5: Scalar hair (left) and induced entropy density (right) as functions of temperature for several electric charges. Dashed segments denote scalar-hairy solutions, while solid segments denote tachyonic-hairy solutions, the black dots mark the phase boundaries between them.