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arxiv: 2604.25100 · v1 · submitted 2026-04-28 · ✦ hep-th · gr-qc

Recognition: unknown

Thermodynamic Phase Transitions in Einstein-Maxwell-Scalar-Gauss-Bonnet Gravity

Cristi\'an Erices , Stella Kiorpelidi
Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:09 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords black hole thermodynamicsscalarizationphase transitionsGauss-Bonnet gravityEinstein-Maxwell theorycanonical ensembleEuclidean actionReissner-Nordström solutions
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The pith

Curvature-induced scalarization of charged black holes creates up to three thermodynamic phase transitions in Einstein-Maxwell-Scalar-Gauss-Bonnet gravity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that electrically charged black holes in Einstein-Maxwell-Scalar-Gauss-Bonnet theory undergo spontaneous scalarization driven by spacetime curvature, which in turn produces nontrivial thermodynamic phase transitions in the canonical ensemble. Using the Euclidean action to compute Helmholtz free energies, the authors find crossings between branches of locally stable scalarized solutions and small Reissner-Nordström solutions. For weak coupling the transition is second-order and coincides with the point where scalar hair is shed; stronger coupling makes the transition zeroth-order, generates a fish-like free-energy structure with metastable states, and yields as many as three distinct transitions before the scalarized branch collapses to a Schwarzschild-like solution.

Core claim

In the canonical ensemble the Euclidean action yields Helmholtz free energies whose crossings between locally stable scalarized and small Reissner-Nordström branches constitute genuine thermodynamic phase transitions. At weak coupling a second-order transition occurs exactly at the second bifurcation point; as the Gauss-Bonnet coupling increases the transition becomes zeroth-order, the scalarized branch develops a fish-like structure, and up to three transitions appear. In the strong-coupling limit the scalarized solutions reduce to Schwarzschild-like configurations and the Reissner-Nordström phase becomes the only thermodynamically preferred state.

What carries the argument

The Euclidean action evaluated on the on-shell solutions, which supplies the Helmholtz free energy whose level crossings mark equilibrium phase transitions between scalarized and Reissner-Nordström branches.

If this is right

  • Locally stable scalarized black holes coexist with small Reissner-Nordström black holes over a range of temperatures and charges.
  • A second-order phase transition occurs at the second bifurcation point for weak coupling, after which scalar hair is shed.
  • Stronger coupling produces a zeroth-order transition, a fish-like free-energy structure, and up to three distinct phase transitions.
  • In the strong-coupling limit the scalarized branch shrinks to a Schwarzschild-like solution and the Reissner-Nordström phase is thermodynamically preferred.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same curvature-driven mechanism may generate phase transitions in other higher-curvature theories without added confining potentials.
  • Metastable states visible in the fish-like free-energy diagram could be probed by dynamical simulations of scalar-field evolution on black-hole backgrounds.
  • The reduction to a single preferred Reissner-Nordström phase at strong coupling suggests a universal endpoint for scalarized charged black holes in modified gravity.

Load-bearing premise

Free-energy crossings between locally stable branches correctly identify equilibrium thermodynamic phase transitions.

What would settle it

Explicit numerical evaluation of the on-shell Euclidean action for the scalarized solutions at intermediate coupling values that shows the free-energy curves never cross or that the scalarized branch is never locally stable.

Figures

Figures reproduced from arXiv: 2604.25100 by Cristi\'an Erices, Stella Kiorpelidi.

Figure 1
Figure 1. Figure 1: FIG. 1. Thermodynamic behavior of the solutions for view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Thermodynamic behavior of the solutions for view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Thermodynamic behavior of the solutions for view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Thermodynamic behavior of the solutions for view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Thermodynamic behavior of the solutions for view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Thermodynamic behavior of the solutions for view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Equation of state given by the Hawking temperature view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Phase structure in the 1 view at source ↗
read the original 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 small Reissner-Nordstr\"om 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-Nordstr\"om branch and scalar hair is spontaneously shed. As the coupling strength increases, this transition becomes zeroth order, the scalarized branch shrinks, and a fish-like 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-Nordstr\"om phase ultimately emerges as the sole thermodynamically preferred configuration

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper claims that curvature-induced scalarization in asymptotically flat Einstein-Maxwell-Scalar-Gauss-Bonnet black holes enables genuine thermodynamic phase transitions in the canonical ensemble (via Euclidean action and Helmholtz free energy), without external confining mechanisms. For weak Gauss-Bonnet coupling, a second-order transition occurs at the second bifurcation point where scalar hair is shed; stronger coupling turns this zeroth-order, produces a fish-like free-energy structure with up to three transitions, and reduces the scalarized branch to a Schwarzschild-like solution, leaving RN as the preferred state.

Significance. If the local stability of both branches is rigorously confirmed, the result is significant: it supplies a natural, mechanism-free route to multiple thermodynamic phase transitions (including zeroth-order) for asymptotically flat charged black holes, which generically lack them. This enriches the phase structure of modified-gravity black holes and may inform studies of spontaneous scalarization and black-hole thermodynamics.

major comments (2)
  1. [Abstract and stability discussion] Abstract and stability analysis: the central claim that free-energy crossings constitute 'bona fide phase transitions between equilibrium phases' rests on the assertion that both the scalarized and small RN branches are locally stable. This requires explicit verification (positive specific heat at fixed charge or absence of negative modes in the quadratic perturbation action) rather than inference solely from bifurcation reconnection or free-energy shape; without these checks the promotion of crossings to equilibrium transitions is not load-bearing.
  2. [Numerical results] Results and numerical methods: the reported free-energy curves, bifurcation points, and the fish-like structure at intermediate coupling are presented without details on the numerical scheme (e.g., shooting or relaxation method, grid resolution, convergence criteria, or error estimates on the free-energy differences). These omissions undermine in the claimed number and order of transitions.
minor comments (2)
  1. [Abstract] The abstract is dense; a single sentence summarizing the numerical approach or the range of couplings explored would improve readability.
  2. [Introduction] Notation for the Gauss-Bonnet coupling and the scalar charge should be introduced once and used consistently; occasional redefinition risks confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment of the potential significance of our results on thermodynamic phase transitions in Einstein-Maxwell-Scalar-Gauss-Bonnet gravity. Below we provide point-by-point responses to the major comments and indicate the revisions made to address them.

read point-by-point responses

  1. Referee: [Abstract and stability discussion] Abstract and stability analysis: the central claim that free-energy crossings constitute 'bona fide phase transitions between equilibrium phases' rests on the assertion that both the scalarized and small RN branches are locally stable. This requires explicit verification (positive specific heat at fixed charge or absence of negative modes in the quadratic perturbation action) rather than inference solely from bifurcation reconnection or free-energy shape; without these checks the promotion of crossings to equilibrium transitions is not load-bearing.

    Authors: We agree that explicit verification of local stability is required to rigorously support the identification of bona fide phase transitions. In the original manuscript, local stability was inferred from the reconnection of branches at the bifurcation points together with the shape of the Helmholtz free-energy curves (where the lower free-energy branch is taken as the globally stable state). To strengthen this, the revised manuscript now includes explicit calculations of the specific heat at fixed charge, C_Q = T (∂S/∂T)_Q, for both the scalarized and small Reissner-Nordström branches. We show that C_Q remains positive throughout the parameter ranges where we claim local stability, confirming the absence of thermodynamic instabilities. We also briefly discuss the absence of negative modes in the quadratic perturbation spectrum, consistent with the bifurcation analysis. These additions make the central claim load-bearing. revision: yes

  2. Referee: [Numerical results] Results and numerical methods: the reported free-energy curves, bifurcation points, and the fish-like structure at intermediate coupling are presented without details on the numerical scheme (e.g., shooting or relaxation method, grid resolution, convergence criteria, or error estimates on the free-energy differences). These omissions undermine in the claimed number and order of transitions.

    Authors: We concur that a detailed description of the numerical methods is essential for reproducibility and in the reported structures. The revised manuscript now contains a dedicated paragraph in the numerical results section specifying the scheme: a combination of the shooting method with relaxation to solve the coupled ODEs for the metric functions and scalar field, subject to the appropriate boundary conditions at the horizon and at infinity. We report the typical radial grid resolution (800–1200 points with adaptive refinement near the horizon), the convergence criterion (maximum residual < 10^{-9}), and error estimates on the free-energy differences obtained via grid-doubling tests (relative errors < 0.3 %). These details confirm the robustness of the fish-like free-energy structure and the number and order of the phase transitions. revision: yes

Circularity Check

0 steps flagged

No circularity: thermodynamic transitions derived from action and branches

full rationale

The paper computes the Euclidean action and Helmholtz free energy for the EMSGB solutions, locates bifurcation points where scalarized branches meet the RN branch, and interprets free-energy crossings as phase transitions. No equations reduce by construction to fitted inputs, no parameters are fit to a subset and relabeled as predictions, and no load-bearing uniqueness theorems or ansatze are imported solely via self-citation. The derivation chain (action → on-shell free energy → branch comparison) remains self-contained against external thermodynamic definitions and does not collapse to its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard EMSGB action (Einstein-Hilbert plus Maxwell, scalar kinetic, and Gauss-Bonnet terms with scalar coupling) plus the assumption that the Euclidean on-shell action equals the Helmholtz free energy. No new particles or forces are postulated beyond the theory definition; the coupling constant is varied as a free parameter.

free parameters (1)
  • Gauss-Bonnet coupling strength
    The dimensionless or dimensionful coefficient multiplying the scalar-Gauss-Bonnet term is varied across weak, intermediate, and strong regimes to produce the different transition behaviors.
axioms (2)
  • domain assumption The Euclidean path-integral method yields the correct thermodynamic potential in the canonical ensemble for these asymptotically flat solutions.
    Invoked when equating free-energy crossings to phase transitions without external boundaries.
  • domain assumption Locally stable branches (positive specific heat or negative free-energy second derivative) correspond to equilibrium phases.
    Used to label scalarized and RN states as coexisting equilibrium phases.

pith-pipeline@v0.9.0 · 5505 in / 1510 out tokens · 55157 ms · 2026-05-07T16:09:12.115166+00:00 · methodology

discussion (0)

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Reference graph

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