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arxiv: 2606.22204 · v1 · pith:B54DNIRZnew · submitted 2026-06-20 · ✦ hep-th

Thermodynamic and Topological Phase Transitions of AdS Black Holes with Nonminimal F^(αβ)F^(γλ)R_(αγ)R_(βλ) Coupling

Mehdi Sadeghi , Faramarz Rahmani This is my paper

Pith reviewed 2026-06-26 11:27 UTC · model grok-4.3

classification ✦ hep-th
keywords AdS black holesnonminimal couplingtopological phase transitionsvan der Waals transitionsHawking-Page transitionsthermodynamic topology
0
0 comments X

The pith

A non-minimal coupling term changes the topological class of four-dimensional AdS black holes from W^{1+} to W^{0-}.

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

The paper examines how a non-minimal coupling between the electromagnetic field and curvature affects both standard thermodynamic and topological phase transitions in asymptotically AdS black holes. In the absence of the coupling, the system matches the Reissner-Nordström-AdS black hole in the W^{1+} class. Introducing the coupling parameter ε to first order in perturbation shifts the topological invariant to W^{0-}, creating a hybrid system where van der Waals-like first-order transitions occur within a broader Hawking-Page structure. This shows that ε functions as a parameter capable of deforming the topological classification without eliminating the conventional phase behavior.

Core claim

Switching on the non-minimal coupling fundamentally transforms the topology to the W^{0-} class (W = 0). This transition occurs while the van der Waals-type first-order phase transition survives in the intermediate region, embedded within the overall Hawking--Page pattern, rendering the system a hybrid black hole thermodynamic system. The coupling ε thus acts as a topological deformation parameter that alters the universal classification of the system, despite the perturbative nature of the solution.

What carries the argument

The coupling constant ε in the term F^{αβ}F^{γλ}R_{αγ}R_{βλ}, which deforms the topological class of the black hole thermodynamics.

If this is right

  • The first law of thermodynamics and Smarr relation continue to hold to first order in ε.
  • The van der Waals-type first-order phase transition persists in the intermediate region.
  • The overall thermodynamic structure combines Hawking-Page and van der Waals features into a hybrid pattern.
  • The topological class switches from W^{1+} (W = +1) to W^{0-} (W = 0) when the coupling is activated.

Where Pith is reading between the lines

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

  • Comparable non-minimal couplings in higher dimensions or with different matter fields may produce similar shifts in topological class.
  • If topological class influences any observable thermodynamic quantity, measurements could distinguish coupled from uncoupled black holes.
  • Going beyond first-order perturbation in ε would test whether the topology change survives in the full nonlinear regime.

Load-bearing premise

The first-order perturbative expansion in the coupling ε is sufficient to reliably determine the change in the topological invariant W.

What would settle it

An exact non-perturbative computation of the topological number W that yields a value other than zero for nonzero ε would falsify the topology change.

Figures

Figures reproduced from arXiv: 2606.22204 by Faramarz Rahmani, Mehdi Sadeghi.

Figure 1
Figure 1. Figure 1: (a) Isothermal P–rh diagrams for fixed Q = 0.7 at temperatures above, at, and below the critical temperature Tc = 0.1234. At small horizon radii, the behavior resembles that of a Hawking–Page system, while below Tc, a characteristic van der Waals oscillation appears in the intermediate region. (b) Pressure as a function of horizon radius for different values of the electric charge Q at fixed temperature. F… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Temperature as a function of horizon radius for fixed [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Helmholtz free energy F as a function of temperature T for different values of Q and P. Rows correspond to (top) Q = 0.7, (middle) Q = 1, and (bottom) Q = 0.4. Columns correspond to (left) P = Pc, (middle) P < Pc, and (right) P > Pc. The swallowtail structure in the middle column indicates a first-order phase transition. The local stability of the system can be examined via its heat capacity. Since the pre… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Heat capacity CP as a function of horizon radius rh for Q = 0.7 and P = 0.01 < Pc = 0.0269. Three divergences separate four regions with alternating stability: unstable, stable, unstable, stable, reflecting the hybrid Hawking–Page and van der Waals nature. The first divergence at small rh is characteristic of the HP transition. (b) Heat capacity at the critical pressure P = Pc, where the two intermedia… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Unit vector field for P = 0.01, Q = 0.7, and τ = 10.5. The direction of the vector field at the boundaries places the system in the W 0− topological class. (b) rh–τ diagram for Q = 0.7 and P = 0.01 < Pc, illustrating the global thermodynamic behavior. The colored regions indicate different phases, with the total winding number W = 0 in each region. To verify the winding numbers associated with each def… view at source ↗
Figure 6
Figure 6. Figure 6: (a) Deflection angle Ω(ϑ) for a closed loop enclosing four defects at ZP1 = 0.389, ZP2 = 0.531, ZP3 = 0.834, and ZP4 = 3.694, for Q = 0.7, P = 0.01, and τ = 10.5. The total deflection after one full circuit yields the winding numbers w = −1, +1, −1, +1, respectively. (b) Contour mapping in the (ϕ rh , ϕΘ) plane for the same parameter values. Each closed curve corresponds to a contour encircling a defect in… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Deflection angle Ω(ϑ) for a closed loop enclosing four defects located at ZP1 = 0.4639, ZP2 = 0.7901, ZP3 = 1.070, and ZP4 = 6.899, for Q = 1, P = 0.004, and τ = 15. The total deflection after one full circuit yields the winding numbers w = −1, +1, −1, +1, respectively. (b) Contour mapping in the (ϕ rh , ϕΘ) plane for the same parameter values. Each closed curve corresponds to a contour encircling a de… view at source ↗
Figure 8
Figure 8. Figure 8: Panels (a), (b), (c), and (d) correspond to the following parameter sets, respectively: [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Unit vector field for Q = 1, P = 0.004 < Pc = 0.013, and τ = 15, showing four topological defects. (b) Unit vector field for Q = 0.4, P = 0.04 < Pc = 0.079, and τ = 6; the defects are closely spaced, and the conventional loops encircling each zero point are omitted for clarity. (c) Unit vector field for Q = 0.1, P = 0.2 < Pc = 0.509, and τ = 2.5, exhibiting four distinct defects. (d) Unit vector field … view at source ↗
read the original abstract

The conventional and topological phase transitions of a four-dimensional asymptotically AdS black hole with a non-minimal coupling term $F^{\alpha\beta}F^{\gamma\lambda}R_{\alpha\gamma}R_{\beta\lambda}$ are investigated. Using a perturbative approach to first order in the coupling $\epsilon$, the thermodynamic quantities are derived and the first law and Smarr relation are verified. Intriguingly, while the $\epsilon = 0$ limit yields the standard Reissner--Nordstr\"om--AdS black hole belonging to the $W^{1+}$ topological class ($W = +1$), switching on the non-minimal coupling fundamentally transforms the topology to the $W^{0-}$ class ($W = 0$). This transition occurs while the van der Waals-type first-order phase transition survives in the intermediate region, embedded within the overall Hawking--Page pattern, rendering the system a hybrid black hole thermodynamic system. The coupling $\epsilon$ thus acts as a topological deformation parameter that alters the universal classification of the system, despite the perturbative nature of the solution.

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

1 major / 1 minor

Summary. The manuscript studies four-dimensional asymptotically AdS black holes with a non-minimal coupling ε F^{αβ} F^{γλ} R_{αγ} R_{βλ}. Working perturbatively to first order in ε, the authors derive the metric and thermodynamic quantities, verify the first law and Smarr relation, and compute the topological invariant W. They report that the ε=0 limit recovers the RN-AdS black hole in the W^{1+} class (W=+1), while any nonzero ε shifts the system to the W^{0-} class (W=0). The van der Waals-type first-order phase transition persists in an intermediate regime, embedded in the Hawking–Page structure, producing a hybrid thermodynamic system in which ε acts as a topological deformation parameter.

Significance. If the reported change in W is robust, the result supplies a concrete example in which a non-minimal curvature–Maxwell coupling alters the universal topological classification of black-hole thermodynamics while preserving local phase-transition phenomenology. The explicit verification of the first law and Smarr relation at O(ε) and the retention of the van der Waals feature are positive features of the calculation.

major comments (1)
  1. [Topological analysis section (following the thermodynamic quantities)] The central claim that the topological class changes from W^{1+} to W^{0-} at first order in ε rests on applying the standard winding-number construction directly to the O(ε)-truncated thermodynamic potentials. Because W is an integer sensitive to the global structure of the normalized vector field (including large-r and large-T asymptotics), it is not a priori clear that an O(ε) truncation preserves the reported degree; higher-order terms could shift critical-point locations or alter the contour behavior enough to restore W=+1. The manuscript does not provide an explicit check that the vector-field degree remains stable under the truncation or that the exact solution (if obtainable) yields the same class.
minor comments (1)
  1. The abstract states that thermodynamic quantities are derived to first order, but the main text should include the explicit O(ε) expressions for T, S, Φ, and the free energy used to construct the vector field for W, together with the contour chosen for the degree calculation.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comment on the topological analysis. We respond to the major comment below.

read point-by-point responses

  1. Referee: [Topological analysis section (following the thermodynamic quantities)] The central claim that the topological class changes from W^{1+} to W^{0-} at first order in ε rests on applying the standard winding-number construction directly to the O(ε)-truncated thermodynamic potentials. Because W is an integer sensitive to the global structure of the normalized vector field (including large-r and large-T asymptotics), it is not a priori clear that an O(ε) truncation preserves the reported degree; higher-order terms could shift critical-point locations or alter the contour behavior enough to restore W=+1. The manuscript does not provide an explicit check that the vector-field degree remains stable under the truncation or that the exact solution (if obtainable) yields the same class.

    Authors: We agree that the integer-valued nature of the topological invariant W requires justification when applied to perturbatively truncated potentials. Our calculation constructs the normalized vector field directly from the O(ε) thermodynamic quantities, which are obtained consistently from the perturbative metric. Within this framework the winding number evaluates to zero. To address the stability concern we will add a brief discussion in the topological section showing that the leading large-r and large-T asymptotics of the vector field remain unchanged at O(ε), so that the degree is preserved for sufficiently small nonzero ε. We will also include a short numerical scan confirming that no critical-point coalescence occurs in the small-ε regime that could allow the winding number to jump discontinuously. We note, however, that an exact non-perturbative solution is not available, as the modified field equations do not admit a closed-form solution beyond the perturbative expansion employed in the manuscript. revision: partial

standing simulated objections not resolved
  • Verification of the topological class using an exact (non-perturbative) solution to the field equations, which is not obtainable in closed form.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by constructing a perturbative metric solution to first order in the coupling ε, deriving thermodynamic quantities from it, and explicitly verifying the first law and Smarr relation on those quantities. The topological invariant W is then evaluated on the resulting free-energy or temperature function using the standard normalized-vector-field construction from prior external literature; the ε=0 limit is shown to recover the known RN-AdS class W=+1 without re-deriving that class inside the paper. No step reduces a claimed prediction to a fitted parameter defined by the same paper, nor does any load-bearing premise rest on a self-citation whose content is itself unverified. The central claim that nonzero ε shifts the class to W=0 is therefore an independent output of the perturbative calculation rather than a definitional or self-referential input.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard assumptions of general relativity in asymptotically AdS spacetime together with a perturbative treatment of the newly introduced coupling term; no new entities are postulated.

free parameters (1)
  • ε
    Coupling constant of the non-minimal term, expanded perturbatively to first order only.
axioms (2)
  • domain assumption The spacetime is asymptotically AdS
    Required for the thermodynamic analysis and Hawking-Page transition context.
  • ad hoc to paper Perturbative expansion to first order in ε captures the leading topological change
    The method chosen to obtain the modified solution and thermodynamic quantities.

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