Hawking-Page phase transition for pure Lovelock black holes

arXiv:2606.10647 · gr-qc, hep-th, quant-ph · Submitted 2026-06-09 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Hawking-Page phase transition for pure Lovelock black holes".

Mira: We investigate the thermodynamic properties of static, spherically symmetric Anti-de Sitter (AdS) black holes in pure Lovelock gravity to understand how higher-curvature corrections modify phase transitions and geometric universality.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So we're diving into this paper today called "Hawking-Page phase transition for pure Lovelock black holes." We're going to look at what the authors are claiming about these black holes and how their thermodynamic properties behave in these higher-order gravity theories.

Mira: That sounds like a deep dive, Kai; I'm curious to hear what the core thesis is, specifically how they connect characteristic temperatures and that Ruppeiner scalar curvature.

Lev: From my side, I'm thinking about the practical implications—how does this mathematical framework translate when we try to implement these concepts on actual quantum hardware?

Kai: Well, what this paper is really investigating is how higher-curvature corrections in pure Lovelock gravity alter the phase transitions and geometric universality that we see in simpler theories. The abstract states they are looking at the relation between the minimum temperature and the Hawking–Page transition temperature for these static, spherically symmetric Anti-de Sitter black holes.

Mira: That’s interesting because it suggests that even with these higher-order terms present, there should still be some kind of predictable relationship governing when a stable black hole solution appears in a fixed temperature ensemble.

Lev: If there's a relation, how robust is it? On real hardware, we need stability and predictable phase boundaries to design error correction codes or thermalization protocols, so I'm wondering about the reliability of these predicted relationships.

Kai: The paper claims that for the electromagnetically neutral case in Einstein gravity, the minimum temperature in (d + one) dimensions matches exactly with the HP transition temperature in d dimensions, but that this relation gets modified by a dimension- and order-dependent factor when we move to higher pure Lovelock theories <ref:2606.10647#pg0,for the electromagnetically neutral case in Einstein gravity, the minimum temperature in>.

Mira: So, it’s not as simple as it is in standard Einstein gravity; those higher-curvature terms introduce a scaling factor that depends on the specific Lovelock order, which makes sense if the new physics introduces new characteristic scales.

Lev: A dimension-dependent factor sounds like something we might need to account for when mapping dual field theories onto our physical systems; it suggests that the structure isn't just a simple rescaling of known results.

Kai: Furthermore, they point out a dual relation comparing the HP phase transition temperature THP with the minimum temperature T0, stating that T zero(d + one) = THP(d) / (d-two) one/2n, which is independent of all thermodynamic parameters.

Paper summary: Mira: That independence from thermodynamic parameters is significant because it implies this duality relation holds regardless of the specific mass or charge we put into the system, as long as we're in the neutral case.

Lev: That kind of parameter-independent scaling suggests a deeper structural property of the geometry itself, which would be very useful if we were trying to characterize emergent phenomena in a more fundamental way.

Kai: Beyond temperature relations, they also look at the normalized Ruppeiner scalar curvature at this transition point and claim that for electromagnetically neutral black holes in pure Lovelock theories, this RN is a universal constant depending only on the spacetime dimension for all d at least 2n + one <ref:2606.10647#pg0,black holes in pure Lovelock>.

Mira: That universality of the Ruppeiner scalar curvature is a strong statement; it means that even though the phase transition itself involves a change from thermal AdS to a black hole phase, the geometry at that point shares a specific intrinsic curvature scale determined only by how many dimensions we're in and the order of Lovelock gravity.

Lev: If RN is universal, it suggests that whatever microscopic interactions are governing this system are constrained by the underlying gravitational dynamics in a very tight way; it simplifies things conceptually, but I still need to know how this relates to the actual observables.

Kai: They also discuss phase transition dynamics, noting that increasing the Lovelock order n generically shifts the free-energy curves leftward, meaning G=zero occurs at a lower temperature and higher-curvature interactions enhance the preference for the black hole phase over thermal AdS.

Mira: That shift in the free-energy curves tells us exactly how higher curvature affects stability; it pushes the system toward having more black holes favored at lower temperatures, which is a key thermodynamic signature.

Lev: From an error correction viewpoint, that shift implies that our effective Hamiltonian or energy landscape will be biased toward states with larger excitations when we consider these higher-order corrections. We’d need to model how this bias affects the fidelity of any quantum state we try to maintain near the transition point.

Kai: Finally, they address charge effects by noting that in Einstein gravity, the HP transition and minimum temperature only differ by a factor determined solely by dimension, but for higher-curvature Lovelock theories, no universal mapping between temperatures survives when charge is included.

Paper summary: Mira: That lack of a universal mapping in the charged case suggests that introducing electric charges in these higher-order theories breaks the specific duality structure that holds in Einstein gravity.

Lev: So if we're looking at implementing this on hardware, the presence of charge seems to introduce new complexity, requiring us to account for both the Lovelock order and the electrostatic sector simultaneously, which complicates error correction significantly.

Kai: The paper itself concludes by summarizing that these findings are important because pure Lovelock gravity allows us to isolate a single Lovelock order, giving us genuine higher-curvature effects without interference from other terms.

Mira: It really highlights how pure Lovelock gravity serves as a controlled arena to test if those dimensional thermodynamic duality relations we see in Einstein and Gauss-Bonnet black holes hold up when you introduce more complex curvature structures.

Lev: If the results are correct, it means that the geometric universality we see in these specific black hole solutions isn't just an artifact of Einstein gravity but a feature tied specifically to the structure of pure Lovelock gravity.

Kai: That means for experimentalists, this paper gives us a clearer picture of what kind of universal geometric signatures we should be looking for when probing more complex gravitational models in the future.

Mira: Exactly; it provides a precise thermodynamic fingerprint—the behavior of RN—that should be measurable if we can access those dual field theory observables you mentioned earlier.

Lev: I just hope that the theoretical structure they've laid out provides enough constraints so that when we eventually build something, we don't end up with an intractable problem where every parameter needs to be tuned perfectly.

Kai: So to wrap up, the paper "Hawking-Page phase transition for pure Lovelock black holes" shows how higher-curvature terms modify temperature relations and maintain a specific geometric universality at the HP transition point, even when charge is involved in some cases.

Mira: It’s about finding those subtle, dimension-dependent factors that govern the relationship between different thermodynamic limits, which is vital for understanding phase transitions in gravity.

Lev: For experimentalists and error correction folks, it provides a clearer idea of how complexity from higher curvature translates into changes in the stability landscape and phase boundaries.

Conclusion: Kai: So, we've been looking at this paper titled "Hawking-Page phase transition for pure Lovelock black holes," and now it’s time to talk about what that actually means in simpler terms.

Mira: From a condensed matter standpoint, the authors are essentially exploring how adding more complex curvature terms into gravity changes the fundamental rules governing when a system switches from one state to another.

Lev: And from my side, I'm wondering if this kind of geometric structure has any direct bearing on the stability of physical systems we might be trying to model with quantum error correction codes.

Kai: Exactly; the authors are showing that even in these higher-order gravity scenarios, there are still predictable temperature relationships and geometric constants that hold true for neutral black holes.

Mira: It seems they’re demonstrating that the core thermodynamic behavior, like the Hawking–Page transition, isn't completely lost when you move beyond Einstein gravity into pure Lovelock theories.

Lev: That suggests a level of structural robustness in these gravitational solutions that could be relevant for designing more stable quantum states under extreme conditions.

Kai: It really points to the fact that these higher-curvature interactions don't just introduce noise; they actually define new, consistent geometric constraints on the phase transitions themselves.

Mira: The implications are that we can use this framework to understand how microscopic interactions manifest in macroscopic thermodynamic behavior across different dimensions and gravity models.

Lev: If we can map these geometric constants to physical observables, it opens up entirely new avenues for testing theories of quantum gravity through observable thermodynamic quantities.

Indian Institute of Astrophysics · Pondicherry University

gr-qc, hep-th, quant-ph

Submitted: 2026-06-09

Updated: 2026-10-01

Comments: 2 figures, 25 pages, Accepted in PRD

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 80/100

The gist: We investigate the thermodynamic properties of static, spherically symmetric Anti-de Sitter (AdS) black holes in pure Lovelock gravity to understand how higher-curvature corrections modify phase

Key concepts

Hawking–Page Transition Temperature (THP): This temperature marks the point where a thermal AdS space becomes thermodynamically unstable and transitions to a large black hole phase. In pure Lovelock gravity, this temperature is related to the minimum possible temperature by a factor that depends on the spacetime dimension and the order of the Lovelock theory.
Normalized Ruppeiner Scalar Curvature (RN): This geometric measure is calculated at the Hawking-Page transition point. For electromagnetically neutral black holes in pure Lovelock theories, this value is a universal constant determined solely by the spacetime dimension, indicating a unique geometric feature at the phase transition.
Temperature Duality Relation: A novel relationship found comparing the minimum temperature (T0) and the HP transition temperature (THP) in higher-order Lovelock theories. This relation shows that black hole existence onset is dual to dominance in a lower dimension, independent of specific thermodynamic parameters.
Gibbs Free Energy (G): The Gibbs free energy, defined as G = M - T S for neutral cases, determines the globally preferred thermodynamic phase. The Hawking-Page transition occurs precisely when this free energy reaches zero (G=0), signifying the boundary between thermal AdS and black hole dominance.

Terminology

Anti-de Sitter — 9× in this paper
Phase Transitions — 9× in this paper · explained in 2 episodes
Area Law — 2× in this paper · explained in 1 episode
Core — 2× in this paper · explained in 2 episodes
First-Order Transition — 2× in this paper · explained in 2 episodes
Generalization — 2× in this paper · explained in 13 episodes
Robustness — 2× in this paper · explained in 5 episodes
Universality Class — 2× in this paper · explained in 2 episodes
Consistency — 1× in this paper · explained in 3 episodes
Degeneracy — 1× in this paper · explained in 3 episodes
Effective Hamiltonian — 1× in this paper · explained in 2 episodes
Fidelity — 1× in this paper · explained in 2 episodes
Quintessence — 1× in this paper · explained in 2 episodes
Scaling Laws — 1× in this paper · explained in 4 episodes

Summary

We investigate the thermodynamic properties of static, spherically symmetric Anti-de Sitter (AdS) black holes in pure Lovelock gravity to understand how higher-curvature corrections modify phase transitions and geometric universality. The gist: In pure Lovelock theories, the Hawking–Page transition temperature is related to the minimum temperature by a dimension- and order-dependent factor, while the normalized Ruppeiner scalar curvature at this transition is a universal constant depending only on the spacetime dimension for electromagnetically neutral black holes.

Thermodynamic Framework and Black Hole Solutions

The study focuses on static, spherically symmetric AdS black hole solutions in pure Lovelock gravity within the grand canonical ensemble, which treats families of black holes as parametrized by the outer horizon radius and electrostatic potential. The metric function is given by a complex expression involving parameters such as mass parameter M and charge Q, where the sign choice reflects different branches of solutions. The thermodynamic properties are encoded in the behavior of this metric function at the event horizon, specifically near-horizon quantities evaluated at r = r+. The Hawking temperature T(r+) is derived from the regularity of the Euclidean section or surface gravity at the outer horizon and exhibits a non-monotonic structure, leading to a minimum temperature Tmin below which no equilibrium AdS black hole solution exists in the fixed temperature ensemble.

The Novel Temperature Duality Relation

A central finding involves the relation between characteristic temperatures:

For electromagnetically neutral cases in Einstein gravity, the minimum temperature in (d + 1) dimensions coincides exactly with the Hawking–Page transition temperature in d dimensions.
In higher pure Lovelock theories, this relation is modified by a dimension- and order-dependent factor, which reduces to the Einstein result in appropriate limits.
For charged AdS black holes in general relativity, the two temperatures differ by a simple dimension-dependent factor, whereas no universal relation persists in higher curvature pure Lovelock theories.
A remarkable dual relation emerges comparing the HP phase transition temperature THP with the minimum temperature T0: T0(d + 1) = THP(d) / (d−2)(1/2n). This suggests that the onset of black hole existence in a given dimension is dual to the onset of black hole dominance in a lower dimension, and this relation is independent of all thermodynamic parameters.

Universality of Ruppeiner Scalar Curvature

The analysis extends to thermodynamic geometry by examining the normalized Ruppeiner scalar curvature (RN) at the Hawking-Page transition point.

For electromagnetically neutral black holes in pure Lovelock theories, RN is a universal constant depending only on the spacetime dimension for all d ≥ 2n + 1.
This constancy indicates that despite the discontinuous change in the dominant thermodynamic phase (thermal AdS to large black hole), the transition corresponds to a geometrically distinguished hypersurface characterized by an intrinsic curvature scale determined solely by the underlying gravitational dynamics.
For general relativity (n = 1), RN at the HP transition is given by RN = −(d − 3)(d − 1)2.
For higher-order pure Lovelock theories (n > 1), RN generally depends on pressure and electrostatic potential, but asymptotically approaches a constant value in the limits of large pressure or simultaneous large potential and large pressure.

Phase Transition Dynamics and Microstructure

The Gibbs free energy G = M − T S (for neutral case) determines the globally preferred phase, with the HP transition occurring at G = 0.

The multi-branch structure visible in all panels indicates competing black-hole saddles (small/large branches), and the cusp/turning-point behavior corresponds to the boundary of local thermodynamic stability where heat capacity changes sign.
Increasing the Lovelock order n generically shifts the free-energy curves leftward, causing G=0 to occur at a lower temperature, indicating that higher-curvature interactions enhance the thermodynamic preference for the black-hole phase over thermal AdS.
The Ruppeiner scalar curvature encodes both nature and effective strength of microscopic interactions; its sign reveals whether effective microinteractions are dominantly attractive (RN 0).

Charge Effects and Stability Limits

The inclusion of electric charge qualitatively modifies the structure:

In Einstein gravity, the HP transition temperature and minimum temperature differ only by a multiplicative factor determined solely by the spacetime dimension, suggesting Einstein gravity retains a nontrivial remnant of neutral-sector universality even with charge.
For higher-curvature Lovelock theories, no universal mapping between temperatures survives in the presence of charge; instead, higher-curvature terms introduce additional theory-dependent scales that make the phase structure sensitive to both Lovelock order and electrostatic sector.
The HP transition is not possible above a certain large electrostatic potential for fixed parameters, as this leads to the black hole crossing an extremal limit where no horizon exists.

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed this paper, Hawking–Page phase transition for pure Lovelock black holes, focusing on its theoretical advancements in gravitational thermodynamics.

Here are the specific improvements that can be made to AI systems using this research, categorized by the capability they would gain:

)1. Enhanced Predictive Modeling of High-Curvature Phase Transitions

The paper provides explicit, dimension-dependent scaling relations (Eqs. 15, 20) and universal geometric invariants (Eq. 26) for the Hawking–Page (HP) transition in pure Lovelock gravity across different orders of curvature and dimensions.

The improved AI system can perform high-fidelity predictive modeling of phase transitions in higher-dimensional gravitational theories, specifically those governed by pure Lovelock gravity.

• It can predict the critical temperature thresholds for black hole phase transitions by inputting the spacetime dimension and the Lovelock order (n), yielding results that are exact in certain limits (e.g., Einstein limit, or large pressure/potential limits).

• It can identify whether a given physical system's transition is governed by universal geometric constraints versus theory-specific parameters. For instance, it can distinguish between the Einstein-like scaling structure and the corrections introduced by higher curvature terms in pure Lovelock models.

)2. Geometric Interpretation of Thermodynamic Stability

The paper establishes that the normalized Ruppeiner scalar curvature at the HP transition is a universal constant depending only on dimension and Lovelock order (Eq. 26). This provides a geometric fingerprint for the transition state, independent of thermodynamic variables like temperature or pressure in certain limits.

The improved AI system can interpret complex thermodynamic data by mapping it onto a universal geometric framework, allowing it to diagnose the nature of phase transitions based on intrinsic curvature invariants rather than just macroscopic quantities.

• It can analyze simulated or theoretical black hole ensembles and calculate the Ruppeiner scalar curvature at the HP transition point to classify the underlying statistical interaction as either dominantly attractive (negative RN) or repulsive (positive RN).

• It can use this geometric invariant to determine if a phase transition is truly first-order (as suggested by constant RN) versus critical, offering a more robust classification of system behavior in complex gravitational settings.

)3. Multi-Scale and Ensemble Robustness Analysis

The research explicitly addresses the interplay between different thermodynamic ensembles (canonical vs. grand canonical ensemble), electric charge, and higher-curvature terms, showing how these factors modify or preserve universality (e.g., the failure of universal relations for charged Lovelock theories).

The improved AI system can perform robust analysis across multiple thermodynamic ensembles and physical regimes to determine the stability and universality of black hole solutions.

• It can accurately predict when a simple scaling relation between temperatures (like the Einstein duality) holds, and precisely quantify the theory-dependent factors that modify this relation in pure Lovelock theories.

• It can model systems where multiple physical parameters (charge Q, pressure P, potential Φ) compete to determine the phase structure, predicting regimes where transitions become improbable (e.g., large electrostatic potential limits).

)4. Cross-Dimensional Mapping and Holographic Interpretation

The paper derives a novel dual relation linking the minimum temperature in a higher dimension to the HP transition temperature in a lower dimension (Eq. 15), suggesting a recursive structure between global phase dominance and local existence bounds across dimensions.

The improved AI system can perform cross-dimensional thermodynamic mapping, enabling it to relate physical constraints observed in one spacetime dimension to corresponding stability or dominance conditions in an adjacent dimension.

• It can use this relationship to infer the necessary boundary conditions or existence bounds for a gravitational theory by observing the phase structure in a lower-dimensional holographic dual.

• It can interpret black hole phase transitions not just as local instabilities, but as manifestations of global geometric constraints that are preserved or reorganized under dimensional reduction, aiding in understanding holographic confinement/deconfinement transitions.