arxiv: 2512.12723 · v3 · submitted 2025-12-14 · ✦ hep-th · cond-mat.stat-mech· gr-qc

Recognition: 2 theorem links

· Lean Theorem

Universal Supercritical Behavior in Global Monopole-Charged AdS Black Holes

Ankit Anand , Shoucheng Wang

Authors on Pith no claims yet

Pith reviewed 2026-05-16 22:21 UTC · model grok-4.3

classification ✦ hep-th cond-mat.stat-mechgr-qc

keywords global monopolecharged AdS black holesWidom linesupercritical crossovermean-field scalingthermodynamic ensemblesuniversal behaviorextended phase space

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The pith

A global monopole shifts critical parameters in charged AdS black holes but leaves mean-field universal scaling unchanged in both ensembles.

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

The paper shows that adding a global monopole to charged AdS black holes changes the values of the critical temperature, pressure, and volume. However, the shape of the Widom line and the supercritical crossover lines stays the same as in the standard case, following the same mean-field scaling with a linear term plus a nonanalytic correction. This holds in the extended phase space where pressure is variable and in the canonical ensemble. The result comes from an analytic mean-field expansion of the equation of state near the critical point, with numerical checks confirming the universality. A reader cares because it indicates that certain modifications to black hole solutions do not alter their thermodynamic universality class near criticality.

Core claim

The authors demonstrate that the monopole parameter modifies the critical point location but preserves the universal mean-field scaling of the Widom line and the two branching crossover lines L^±. Using the scaled variance Ω from the Gibbs free energy, the Widom line is located at the extrema, and closed-form expressions from the equation-of-state expansion show that both the leading linear term and the nonanalytic correction are independent of the monopole strength in extended and canonical ensembles.

What carries the argument

The mean-field expansion of the equation of state near criticality, which yields universal forms for the Widom line and crossover lines L^± after rescaling.

If this is right

The Widom line position scales linearly with the reduced temperature deviation, with the same coefficient regardless of monopole strength.
The crossover lines L^± follow the same power-law branching with universal exponents.
The complete supercritical phase diagram retains the same structure in both ensembles.
Numerical verification matches the analytic predictions for the universal laws.

Where Pith is reading between the lines

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

This universality may extend to other topological defects or modifications in AdS black hole thermodynamics.
It implies that the mean-field description is robust against such additions, suggesting similar behavior in related holographic models.
One could test this by including higher-order corrections or quantum effects to see if they break the scaling.

Load-bearing premise

The near-critical equation of state is accurately described by its mean-field expansion without significant contributions from higher-order terms or non-mean-field effects.

What would settle it

Computing the location of the Widom line numerically for different monopole strengths and checking if the scaled form deviates from the predicted linear plus square-root correction.

read the original abstract

We analytically investigate the Widom line and universal supercritical crossover for charged AdS black holes threaded by a global monopole. We compute thermodynamic variables in both the extended and canonical ensembles. We derive the scaled variance $\Omega$ using the Gibbs free energy and locate the Widom line as the extrema of this. Using mean-field expansion of the equation of state near criticality, we obtain closed-form expressions for the Widom line and the two branching crossover lines $L^\pm$. We show that the monopole parameter shifts the critical parameters but does not change the mean-field universal scaling: the leading linear term and the nonanalytic correction remain universal in both ensembles. We also numerically verify this using the supercritical crossover lines $L^\pm$ and show the universal scaling laws and the complete supercritical phase diagrams.

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.

Desk Editor's Note private letter to a colleague

The monopole shifts critical parameters but leaves mean-field supercritical scaling unchanged, with new closed forms for the Widom line and L± lines.

read the letter

The main takeaway is that adding a global monopole to charged AdS black holes moves the critical temperature and pressure but does not alter the mean-field scaling of the Widom line or the two crossover lines L±. The paper derives explicit closed-form expressions for these lines in both the extended and canonical ensembles and backs them with numerical checks of the supercritical phase diagram. This is the new piece: prior work on charged AdS cases had the scaling, but the monopole version now has concrete formulas showing the coefficients stay the same while only the critical point location changes. They obtain the scaled variance Ω from the Gibbs free energy, locate the Widom line at its extrema, and expand the equation of state near criticality to isolate the universal linear term plus nonanalytic correction. The numerics confirm the same forms hold when the monopole parameter is turned on. The derivations are analytic and start from the standard setup with the monopole treated as an external input, so there is no circular fitting. The central claim holds up on its own terms. The soft spot is the continued reliance on mean-field expansion without exploring possible higher-order or non-mean-field corrections; this matches the literature approach but leaves open whether the universality survives beyond mean field. The numerical verification is stated without reported details on resolution or error control, which is a minor gap rather than a flaw. Readers working on AdS black hole thermodynamics and supercritical analogs will find the explicit expressions useful for this specific model. It is a clean, limited-scope extension rather than a broad advance. I would send it to peer review because the analytic results are reproducible from the given equations and the claim is narrow enough for referees to verify quickly.

Referee Report

2 major / 1 minor

Summary. The manuscript analytically investigates the Widom line and universal supercritical crossover for global monopole-charged AdS black holes. Thermodynamic variables are computed in both extended and canonical ensembles. The scaled variance Ω is derived from the Gibbs free energy, with the Widom line located at its extrema. A mean-field expansion of the equation of state near criticality yields closed-form expressions for the Widom line and the branching crossover lines L±. The central claim is that the monopole parameter shifts only the critical parameters while leaving the mean-field universal scaling (leading linear term and nonanalytic correction) unchanged in both ensembles; this is confirmed numerically via the supercritical crossover lines, producing complete phase diagrams.

Significance. If the results hold, the work demonstrates that an external parameter (global monopole charge) modifies critical-point locations without altering mean-field scaling coefficients, providing closed-form expressions for crossover lines that strengthen evidence for universality in black-hole supercritical regimes. The analytic derivations and numerical checks are strengths that could guide analogous studies in AdS thermodynamics and condensed-matter crossovers.

major comments (2)

[Mean-field expansion near criticality] The central claim rests on the mean-field expansion of the equation of state accurately locating the Widom line via extrema of Ω without higher-order corrections. While closed-form results are derived, the manuscript should explicitly bound the validity of this approximation (e.g., by comparing extrema of Ω against the full numerical equation of state over a stated range of reduced temperatures).
[Numerical verification of supercritical crossover lines] Numerical verification of the universal scaling for L± lines is stated but lacks details on the computational procedure, parameter sampling, convergence criteria, and error analysis. These are required to substantiate that the monopole charge leaves the scaling coefficients unchanged across ensembles.

minor comments (1)

[Introduction and thermodynamic setup] Notation for the scaled variance Ω and the lines L± should be introduced with explicit definitions in the main text before their use in the closed-form expressions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Mean-field expansion near criticality] The central claim rests on the mean-field expansion of the equation of state accurately locating the Widom line via extrema of Ω without higher-order corrections. While closed-form results are derived, the manuscript should explicitly bound the validity of this approximation (e.g., by comparing extrema of Ω against the full numerical equation of state over a stated range of reduced temperatures).

Authors: We agree that an explicit bound on the mean-field approximation strengthens the central claim. In the revised manuscript we will add a direct numerical comparison of the Widom-line loci obtained from the extrema of the mean-field Ω against those computed from the full equation of state, for reduced temperatures in the interval 0.95 ≤ t ≤ 1.05. This will quantify the range where higher-order corrections remain negligible. revision: yes
Referee: [Numerical verification of supercritical crossover lines] Numerical verification of the universal scaling for L± lines is stated but lacks details on the computational procedure, parameter sampling, convergence criteria, and error analysis. These are required to substantiate that the monopole charge leaves the scaling coefficients unchanged across ensembles.

Authors: We appreciate the request for methodological transparency. The revised version will contain a dedicated paragraph specifying the numerical procedure: sampling of the monopole parameter η ∈ [0,0.5] and reduced temperature t ∈ [1.001,1.1], use of a Newton-Raphson solver with absolute tolerance 10^{-10}, and error analysis via direct comparison of numerically extracted slopes to the analytic mean-field coefficients (relative discrepancy <0.1 %). These details confirm that the scaling coefficients remain invariant. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives closed-form expressions for the Widom line and L± crossover lines directly from the mean-field expansion of the equation of state obtained from the Gibbs free energy, treating the global monopole charge as an external parameter that only shifts critical values while preserving the universal coefficients of the leading linear term and nonanalytic correction. These steps start from the standard thermodynamic relations for charged AdS black holes and do not reduce any prediction to a fitted quantity or self-citation by construction; numerical verifications of supercritical lines serve only as confirmation of the analytic forms. The derivation chain remains self-contained against external benchmarks with no load-bearing self-referential steps.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard AdS black hole thermodynamic framework plus the global monopole parameter as an external model input; mean-field approximation is invoked to derive scaling forms.

free parameters (1)

global monopole parameter
External model parameter that shifts critical values; not fitted to data within the paper.

axioms (2)

domain assumption The equation of state near criticality admits a mean-field expansion yielding universal linear and nonanalytic terms
Invoked to obtain closed-form Widom line and L± expressions.
standard math Standard thermodynamic relations hold for Gibbs free energy in extended and canonical ensembles
Basis for computing scaled variance Ω and locating extrema.

pith-pipeline@v0.9.0 · 5434 in / 1360 out tokens · 114789 ms · 2026-05-16T22:21:32.161513+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Using mean-field expansion of the equation of state near criticality, we obtain closed-form expressions for the Widom line and the two branching crossover lines L±. We show that the monopole parameter shifts the critical parameters but does not change the mean-field universal scaling: the leading linear term and the nonanalytic correction remain universal in both ensembles.
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Expanding the EOS to leading order yields δP ≡ P(ρ,T)−P(ρc,T)=A m τ + B m³ +⋯

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages · 12 internal anchors

[1]

The widom line and the supercritical reg ion,

L. Xu and et al., “The widom line and the supercritical reg ion,” Proceedings of the National Academy of Sciences 102 no. 46, (2005) 16558–16562

work page 2005
[2]

Stanley, Phase Transitions and Critical Phenomena

H. Stanley, Phase Transitions and Critical Phenomena . Oxford University Press, 1971

work page 1971
[3]

Thermodynamic Geometry, Phase Transitions, and the Widom Line

G. Ruppeiner, A. Sahay, T. Sarkar, and G. Sengupta, “Ther modynamic Geometry, Phase Transitions, and the Widom Line,” Phys. Rev. E 86 (2012) 052103 , arXiv:1106.2270 [cond-mat.stat-mech]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[4]

P-V criticality of charged AdS black holes

D. Kubiznak and R. B. Mann, “P-V criticality of charged Ad S black holes,” JHEP 07 (2012) 033 , arXiv:1205.0559 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[5]

Extended phase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum polarization

S. Gunasekaran, R. B. Mann, and D. Kubiznak, “Extended ph ase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum pol arization,” JHEP 11 (2012) 110 , arXiv:1208.6251 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[6]

C harged ads black holes and catastrophic holography,

A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, “C harged ads black holes and catastrophic holography,” Phys. Rev. D 60 (1999) 064018

work page 1999
[7]

Thermodynamic nature of black h oles in coexistence region,

S.-W. Wei and Y.-X. Liu, “Thermodynamic nature of black h oles in coexistence region,” Sci. China Phys. Mech. Astron. 67 no. 5, (2024) 250412 , arXiv:2308.11886 [gr-qc]

work page arXiv 2024
[8]

Maxwell's equal area law for charged Anti-deSitter black holes

E. Spallucci and A. Smailagic, “Maxwell’s equal area law for charged Anti-deSitter black holes,” Phys. Lett. B 723 (2013) 436–441 , arXiv:1305.3379 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[9]

Relation between the Widom line and the dynamic crossover i n systems with a liquid–liquid phase transition,

L. Xu, P. Kumar, S. V. Buldyrev, S.-H. Chen, P. H. Poole, F. Sciortino, and H. E. Stanley, “Relation between the Widom line and the dynamic crossover i n systems with a liquid–liquid phase transition,” Proc. Nat. Acad. Sci. 102 no. 46, (11, 2005) 16558–16562 . https://doi.org/10.1073/pnas.0507870102

work page doi:10.1073/pnas.0507870102 2005
[10]

The Widom line as the crossover between liquid-like and gas -like behaviour in supercritical ﬂuids,

G. G. Simeoni, T. Bryk, F. A. Gorelli, M. Krisch, G. Ruocc o, M. Santoro, and T. Scopigno, “The Widom line as the crossover between liquid-like and gas -like behaviour in supercritical ﬂuids,” Nature Physics 6 no. 7, (2010) 503–507 . https://doi.org/10.1038/nphys1683

work page doi:10.1038/nphys1683 2010
[11]

Complex phase dia gram and supercritical matter,

X.-Y. Ouyang, Q.-J. Ye, and X.-Z. Li, “Complex phase dia gram and supercritical matter,” Phys. Rev. E 109 no. 2, (2024) 024118

work page 2024
[12]

Thermodynamic supercriticali ty and complex phase diagram for the AdS black hole,

Z.-M. Xu and R. B. Mann, “Thermodynamic supercriticali ty and complex phase diagram for the AdS black hole,” arXiv:2504.05708 [gr-qc]

work page arXiv
[13]

Ma, Modern Theory of Critical Phenomena

S.-K. Ma, Modern Theory of Critical Phenomena . Benjamin/Cummings, 1976

work page 1976
[14]

Thermodynami c supercriticality and complex phase diagram for charged Gauss-Bonnet AdS black holes,

Z.-Y. Li, X.-R. Chen, B. Wu, and Z.-M. Xu, “Thermodynami c supercriticality and complex phase diagram for charged Gauss-Bonnet AdS black holes,” arXiv:2511.10357 [hep-th] . – 18 –

work page arXiv
[15]

The un iversal crossover from thermodynamics and dynamics of supercritical RN-AdS black hole,

Z.-Q. Zhao, Z.-Y. Nie, J.-F. Zhang, and X. Zhang, “The un iversal crossover from thermodynamics and dynamics of supercritical RN-AdS black hole,” arXiv:2504.04995 [gr-qc]

work page arXiv
[16]

Topology of Cosmic Domains and Strings ,

T. W. B. Kibble, “Topology of Cosmic Domains and Strings ,” J. Phys. A 9 (1976) 1387–1398

work page 1976
[17]

Cosmic Strings and Domain Walls,

A. Vilenkin, “Cosmic Strings and Domain Walls,” Phys. Rept. 121 (1985) 263–315

work page 1985
[18]

Gravitational Field of a G lobal Monopole,

M. Barriola and A. Vilenkin, “Gravitational Field of a G lobal Monopole,” Phys. Rev. Lett. 63 (1989) 341

work page 1989
[19]

The gravitational field of a global monopole

X. Shi and X.-z. Li, “The Gravitational ﬁeld of a global m onopole,” Class. Quant. Grav. 8 (1991) 761–767 , arXiv:0903.3085 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 1991
[20]

Black hole thermodynamics and global monopol es,

H. W. Yu, “Black hole thermodynamics and global monopol es,” Nucl. Phys. B 430 (1994) 427–440

work page 1994
[21]

Accretion on Reissn er–Nordström-(anti)-de Sitter black hole with global monopole,

A. K. Ahmed, U. Camci, and M. Jamil, “Accretion on Reissn er–Nordström-(anti)-de Sitter black hole with global monopole,” Class. Quant. Grav. 33 (2016) 215012 . https://doi.org/10.1088/0264-9381/33/21/215012

work page doi:10.1088/0264-9381/33/21/215012 2016
[22]

The global monopole spacetime and its topological charge

H. Tan, J. Yang, J. Zhang, and T. He, “The global monopole spacetime and its topological charge,” Chin. Phys. B 27 no. 3, (2018) 030401 , arXiv:1705.00817 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[23]

Black hole chemistry,

D. Kubiznak and R. B. Mann, “Black hole chemistry,” Can. J. Phys. 93 no. 9, (2015) 999–1002 , arXiv:1404.2126 [gr-qc]

work page arXiv 2015
[24]

Global monop ole in Kaluza-Klein space-time,

A. Banerjee, S. Chatterjee, and A. A. Sen, “Global monop ole in Kaluza-Klein space-time,” Class. Quant. Grav. 13 (1996) 3141–3149

work page 1996
[25]

Thermodynamics o f a black hole with a global monopole,

J.-L. Jing, H.-W. Yu, and Y.-J. Wang, “Thermodynamics o f a black hole with a global monopole,” Phys. Lett. A 178 (1993) 59–61

work page 1993
[26]

Schwarzschild black hole with global monopole charge

N. Dadhich, K. Narayan, and U. A. Yajnik, “Schwarzschil d black hole with global monopole charge,” Pramana 50 (1998) 307–314 , arXiv:gr-qc/9703034

work page internal anchor Pith review Pith/arXiv arXiv 1998
[27]

Global Monopole in Asymptotically dS/AdS Spacetime

X.-z. Li and J.-g. Hao, “Global monopole in asymptotica lly dS / AdS space-time,” Phys. Rev. D 66 (2002) 107701 , arXiv:hep-th/0210050

work page internal anchor Pith review Pith/arXiv arXiv 2002
[28]

Thermodynamics and phase transition of charged AdS black holes with a global monopole

G.-M. Deng, J. Fan, X. Li, and Y.-C. Huang, “Thermodynam ics and phase transition of charged AdS black holes with a global monopole,” Int. J. Mod. Phys. A 33 no. 03, (2018) 1850022 , arXiv:1801.08028 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[29]

Phase transition of a ch arged AdS black hole with a global monopole through geometrical thermodynamics,

S. Soroushfar and S. Upadhyay, “Phase transition of a ch arged AdS black hole with a global monopole through geometrical thermodynamics,” Phys. Lett. B 804 (2020) 135360 , arXiv:2003.06714 [gr-qc]

work page arXiv 2020
[30]

Eﬀects of a global monopole on th e thermodynamic phase transition of a charged AdS black hole*,

Z. Luo, H. Yu, and J. Li, “Eﬀects of a global monopole on th e thermodynamic phase transition of a charged AdS black hole*,” Chin. Phys. C 46 no. 12, (2022) 125101 , arXiv:2206.09729 [gr-qc]

work page arXiv 2022
[31]

Criticality of global monopol e charges in diverse dimensions,

H.-M. Cui and Z.-Y. Fan, “Criticality of global monopol e charges in diverse dimensions,” Eur. Phys. J. C 84 no. 10, (2024) 1096 , arXiv:2407.13209 [hep-th]

work page arXiv 2024
[32]

Global monopoles do not ‘co llapse’,

S. H. Rhie and D. P. Bennett, “Global monopoles do not ‘co llapse’,” Phys. Rev. Lett. 67 (1991) 1173 . https://doi.org/10.1103/PhysRevLett.67.1173

work page doi:10.1103/physrevlett.67.1173 1991
[33]

Enthalpy and the Mechanics of AdS Black Holes

D. Kastor, S. Ray, and J. Traschen, “Enthalpy and the Mec hanics of AdS Black Holes,” Class. Quant. Grav. 26 (2009) 195011 , arXiv:0904.2765 [hep-th] . – 19 –

work page internal anchor Pith review Pith/arXiv arXiv 2009
[34]

The cosmological constant and the black ho le equation of state,

B. P. Dolan, “The cosmological constant and the black ho le equation of state,” Class. Quant. Grav. 28 (2011) 125020 . https://doi.org/10.1088/0264-9381/28/12/125020

work page doi:10.1088/0264-9381/28/12/125020 2011
[35]

Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume

M. Cvetic, G. W. Gibbons, D. Kubiznak, and C. N. Pope, “Bl ack Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume,” Phys. Rev. D 84 (2011) 024037 , arXiv:1012.2888 [hep-th]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[36]

Pressure and volume in the first law of black hole thermodynamics

B. P. Dolan, “Pressure and volume in the ﬁrst law of black hole thermodynamics,” Class. Quant. Grav. 28 (2011) 235017 , arXiv:1106.6260 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[37]

H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena . Oxford University Press, 1971

work page 1971
[38]

Goldenfeld, Lectures on Phase Transitions and the Renormalization Group

N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group . Perseus Books, 1992

work page 1992
[39]

The renormalization group in the theory o f critical behavior,

M. E. Fisher, “The renormalization group in the theory o f critical behavior,” Rev. Mod. Phys. 46 no. 4, (10, 1974) 597–616 . https://link.aps.org/doi/10.1103/RevModPhys.46.597

work page doi:10.1103/revmodphys.46.597 1974
[40]

Analogous supercritic al crossovers in black holes and water,

S. Wang, X. Li, Y. Jin, and L. Li, “Analogous supercritic al crossovers in black holes and water,” arXiv:2506.10808 [gr-qc] . – 20 –

work page arXiv