Hawking-Page Transitions Overview

• Hawking-Page transitions are first-order thermodynamic phase transitions between black hole and thermal configurations in spacetimes with finite cosmological constant.
• They are analyzed via Euclidean path integrals and off-shell free energy landscapes to determine critical temperatures and phase stability in holographic duals.
• Recent studies extend these transitions to modified gravity, higher-curvature corrections, and topological classifications, offering insights into gravitational microstructure and quantum information.

A Hawking-Page transition is a first-order thermodynamic phase transition between a black hole phase and a thermal (horizonless) background in spacetimes with finite negative or positive cosmological constant or in thermal ensembles with appropriate infrared regulators. In the canonical AdS context, it delineates the confinement/deconfinement transition of the dual gauge theory via holographic correspondence. The underlying mechanism involves a competition between semiclassical gravitational saddles, with the favored phase identified via the global minimum of the (Euclidean) free energy. This transition structure persists and exhibits deep modifications—topological, kinetic, information-theoretic, and field-theoretic—across a wide range of generalizations.

1. Thermodynamic Foundations and On-shell Free Energies

The simplest realization utilizes the Euclidean path integral approach in Einstein gravity with fixed boundary conditions. For Schwarzschild–AdS, the Euclidean action consists of the bulk term, Gibbons–Hawking surface integral, and counterterms: $I_{\rm tot} = I_g + I_{\rm GH} + I_{\rm ct}$ with the metric

$$ds^2 = -f(r)dt^2 + f(r)^{-1} dr^2 + r^2 d\Omega_{d-2}^2,\qquad f(r) = 1 - \frac{2M}{r^{d-3}} + \frac{r^2}{L^2}$$

and $L$ the AdS scale. The Euclidean action yields the Gibbs or Helmholtz free energy %%%%1%%%% as a function of temperature $T$ and other ensemble parameters. Setting $F(T_c) = 0$ identifies the Hawking–Page temperature separating the thermal (AdS) and large black-hole phases (Eune et al., 2013).

The transition is first-order: the entropy and energy exhibit discontinuities, corresponding to the latent heat of the phase transition. In dimensions $d>3$, the canonical transition temperature is

$T_c = \frac{d-2}{2\pi L}$

for Schwarzschild–AdS (Wei et al., 2020). In lower (BTZ, $d=3$) or more elaborate setups, the analysis of on-shell actions for black holes, solitons, or global AdS must account for additional features such as soliton mass gaps (Eune et al., 2013, Sun, 2021).

2. Off-shell Free Energy Landscapes and Stochastic Kinetics

Going beyond equilibrium, one can embed the system in an off-shell Gibbs free energy landscape $G(r_h, T)$, where the horizon radius $r_h$ serves as an order parameter and “off-shell” geometries interpolate between phases. Minima correspond to global or local thermodynamic stability, maxima to barriers separating phases.

The stochastic dynamics of transitions between free-energy minima, governed by a Langevin equation, can be formulated as a Fokker–Planck process. Additive noise results in the standard Gibbs distribution, while multiplicative (horizon-size dependent) noise yields an effective free energy: $$G_{\rm eff}(r, T) = \int_{r_0}^r \frac{G_0'(u, T)}{g(u)^2} du + kT \ln g(r)$$ where $g(r)$ is a noise profile. Large multiplicative noise generically suppresses large black-hole nucleation and can shift or eliminate the Hawking–Page transition altogether unless extrema in $g(r)$ and $G_0(r,T)$ coincide (Ho, 29 Sep 2025).

For transition kinetics, the barrier crossing rate is sensitive to non-Markovian effects. Memory-dependent (non-Markovian) friction kernels in the generalized Langevin equation accelerate crossing compared to classical Kramers transitions. Exponential memory promotes tunneling, while oscillatory friction can show non-monotonic behavior (Li et al., 2022).

3. Topological and Geometrical Classification

Recent developments exploit the Bragg–Williams construction for the off-shell free energy to define vector fields in order-parameter space, allowing a topological classification via winding numbers. Specifically:

• $\psi$-mapping: Zeros correspond to global coexistence points (e.g., Hawking–Page points, with $\psi$-charge $+1$).
• $\xi$-mapping: Identifies local (temperature-dependent) phase transitions (e.g., Van der Waals criticality).
• $\eta$-mapping: Zeros classify equilibrium phases; stable (unstable) phases carry index $+1$ ($-1$).

The topological class of the Hawking–Page transition (total $\psi$-charge $+1$, $\eta$-charge $0$) is distinct from that of Van der Waals–type transitions (total $\psi$-charge $0$) (Barzi et al., 2023, Yerra et al., 2023). Identical integer-valued winding numbers characterize bulk and boundary transitions, indicating a deep correspondence across statistical ensembles (e.g., Rényi–Bragg–Williams in flat space and Boltzmann–Gibbs in AdS).

4. Extensions: Modified Gravity, Topology, and Higher Curvature

Massive Gravity, Cavity, and Boundary Conditions

In models with momentum dissipation (massive gravity), the transition temperature decreases with graviton mass $m$ and can be driven to zero at strong dissipation: $$T_c(m) = \frac{1}{\pi L}\sqrt{1 - \frac{1}{2}(mL)^2 }$$ (Adams et al., 2014). For spatially noncompact AdS boundaries, the graviton mass provides the required IR scale for the HP transition: $$T_c(m,L,n) = \frac{n}{2\pi} m\sqrt{c_2} + \frac{c_1 m^2 L}{4\pi}$$ (Braga et al., 16 Sep 2025).

Finite cavities (reflecting walls) in asymptotically flat space yield transitions closely paralleling the AdS case, albeit with differing pressure and potential bounds. Notably, for charged black holes in a cavity, the phase transition occurs only below a critical wall potential $\Phi < 1/7$ (Zhao et al., 2020).

Gauss-Bonnet and Higher-Curvature Corrections

In (Einstein-)Gauss–Bonnet gravity, the Hawking–Page temperature is lowered by the GB coupling $\alpha$, and transition lines terminate at a finite maximal pressure: $$T_{\rm HP} = \sqrt{\frac{p}{4\pi}}\frac{5 - 96\pi\alpha p + ...}{\sqrt{3 - 96\pi\alpha p + ...}}$$ and coexistence lines have terminal points, unlike in pure Einstein gravity (Su et al., 2019, Wang et al., 2020). Nontrivial global phase structures, including reentrant transitions and triple points (with multiple HP temperatures), appear for specific combinations of pressure and GB coupling (Cui et al., 2021). Reentrance is directly analogous to triple-phase transitions in condensed matter and QCD.

In Einstein-scalar gravity, continuous (second- or higher-order) HP transitions are possible in special “singular” limits. The order of transition is governed by the subleading structure of the scalar potential, with finite-order or BKT-type transitions appearing based on whether the subleading term is exponential or power-law (Gursoy, 2010).

General Topological Structure and Universality

For CFTs on product manifolds ($S^{d_1}\times S^{d_2}$), the Hawking–Page transition maps to a first-order topology-changing transition in the dual bulk, with phase coexistence tracked by the relative radii of the component spheres (Aharony et al., 2019). Universality emerges in the appearance of identical winding numbers and phase-structure invariants across vast classes of black hole backgrounds, statistical ensembles, and even dimensionality (Barzi et al., 2023, Yerra et al., 2023).

5. Holographic, Quantum Information, and Duality Aspects

In AdS/CFT, the HP transition is holographically dual to a confinement–deconfinement transition (Sun, 2021). Novel dualities exist, such as the exact equality between the HP transition temperature in $d$ dimensions and the minimal black hole temperature in $d+1$ dimensions: $T_{\rm HP}(d,P) = T_{\min}(d+1, P)$ accompanied by a universal, dimension-dependent constant value for the Ruppeiner thermodynamic curvature at the transition, conjectured to denote a threshold for black hole microstructure formation (Wei et al., 2020).

Quantum information diagnostics (entanglement entropy, mutual information, entanglement wedge cross-section EWCS) reveal strong signatures at the HP transition. Notably, EWCS cleanly captures both first- and second-order transitions and is configuration-independent, with all geometric quantities sharing critical exponents at the second-order point, e.g., $\beta=1/3$: $\delta S_E, \delta E_w \sim |T-T_c|^{1/3}$ (Yang et al., 31 Dec 2025). Loschmidt echo statistics in 1D quantum spin chains and free fermion models also reproduce the HP transition, establishing an experimentally accessible analogue model (Pérez-García et al., 2024).

6. Lower and Non-AdS Backgrounds

In three dimensions, the AdS$_3$ BTZ/soliton system exhibits a HP transition with a mass gap, requiring non-equilibrium off-shell constructions to interpolate between phases and compute tunneling rates (Eune et al., 2013).

In de Sitter (dS) spacetime with a nonlinear electromagnetic field, the phase coexistence curve is a closed loop in the $(T,P)$ plane, adding upper bounds to the HP temperature and pressure, contrasting with the unbounded AdS case (Du et al., 2021).

For holographic QCD, the HP transition maps confinement/deconfinement and exhibits a critical temperature $T_c(\mu)$ decreasing monotonically with chemical potential $\mu$, consistent with the standard QCD phase diagram (Braga et al., 2024).

7. Physical Significance, Open Directions, and Synthesis

The Hawking–Page transition not only encodes the thermodynamic and dynamical story of black hole–thermal competition, but also:

• Serves as a precise marker of large-$N$ phase transitions in holographic duals.
• Realizes universal relations and topological invariants (winding numbers, Ruppeiner curvature), bridging thermodynamic, topological, and geometric classification of phase structure.
• Provides a dynamical laboratory to study nucleation, tunneling, and transition kinetics in semiclassical gravity.
• Links gravitational and gauge-theoretic critical phenomena, including reentrance, triple points, and continuous transitions, and supports experimental quantum simulation via spin chains and other analog models (Pérez-García et al., 2024).
• Establishes new paradigms for probing gravitational microstructure and quantum-information content across black hole transitions, with universal exponents emerging in both geometric and entanglement diagnostics (Yang et al., 31 Dec 2025).

Contemporary research continues to extend the landscape of Hawking–Page transitions into modified gravity, higher-curvature corrections, arbitrary asymptotics, and statistical frameworks, revealing a rich web of universalities and cross-disciplinary connections.