Pith. sign in

REVIEW 3 major objections 6 minor 2 cited by

Probing Thermodynamic Phase Transitions of 4D R-Charged Black Holes via Lyapunov Exponent

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that the discontinuous jump in the Lyapunov exponent of unstable circular orbits acts as an order parameter for 4D R-charged black hole phase transitions, with critical exponent $\delta = 1/2$.

desk verdict Competent extension of the Lyapunov-probe program to R-charged black holes, but the written derivation of δ=1/2 is internally inconsistent. read the letter →

arxiv 2505.20800 v1 pith:F7J77MXC submitted 2025-05-27 gr-qc

Mozib Bin Awal , Prabwal Phukon This is my paper
classification gr-qc PACS 04.70.-s05.70.Fh05.45.-a
keywords LyapunovexponentR-chargedblackholesholephasetransitionsorderparametercriticalunstablecircularorbitsgaugedsupergravityAdSthermodynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that a quantity from dynamical chaos, the Lyapunov exponent of unstable circular orbits, can serve as a thermodynamic probe for four-dimensional R-charged black holes. Across three charge configurations (all equal, two equal and two unequal, all unequal), the thermal profile of the exponent is multivalued exactly in the regime where the black hole has small, intermediate, and large branches, and it becomes single-valued above the critical charge. The paper further claims that the discontinuous jump in the exponent at the phase transition behaves as an order parameter, with a critical exponent $\delta = 1/2$ for both massless and massive particles. If correct, this supports the universality of that exponent and strengthens the proposed link between chaotic dynamics and black hole thermodynamics.

What carries the argument

The load-bearing object is the Lyapunov exponent computed from the linear stability matrix of circular equatorial geodesics, $\lambda = \sqrt{-V_{\rm eff}''(r_0)/(2\dot{t}^2)}$, where $V_{\rm eff}$ is the radial effective potential and $r_0$ is the radius of an unstable circular orbit; for massless particles this reduces to Eq. (13) and for massive particles to Eq. (16). The thermodynamic side is the Hawking temperature $T(r_+)$ of the R-charged black hole, whose critical point is fixed by $\partial T/\partial r_+ = \partial^2 T/\partial r_+^2 = 0$. The critical-exponent result follows from expanding both $\lambda$ and $T$ about the critical horizon radius: the vanishing first derivative of $T$ at criticality and the quadratic term in Eq. (41) produce the square-root scaling, and combining the two expansions yields $\Delta\lambda/\lambda_c = k\sqrt{t - 1}$ with $\delta = 1/2$.

What would settle it

Evaluate $\lambda_s$ and $\lambda_l$ from the full formulas near the critical point for the equal-charge configuration and plot $\log|\Delta\lambda|$ against $\log|\tilde{T} - \tilde{T}_c|$; the paper's claim predicts a straight line of slope $1/2$, and a fitted slope clearly different from $1/2$ would refute it.

Watch

Extended reading notes

Core claim

The central claim is that the Lyapunov exponent for massless and massive particles in unstable circular orbits carries the same phase information as the Gibbs free energy of the 4D R-charged black hole. For charge $\tilde{q}$ below the critical value $\tilde{q}_c$, the exponent is a multivalued function of Hawking temperature between the two turning points; its three branches correspond to the small, intermediate, and large black hole phases. For $\tilde{q} > \tilde{q}_c$, the multivaluedness disappears, mirroring the disappearance of the phase transition. Defining the discontinuity $\Delta\lambda = \lambda_s - \lambda_l$ between the small and large branches at the transition temperature, the paper derives $\Delta\lambda/\lambda_c = k\sqrt{t - 1}$ with $t = \tilde{T}/\tilde{T}_c$ by expanding about the critical horizon radius, so the order parameter vanishes with critical exponent $\delta = 1/2$. Numerical fits to the same form for all three charge configurations and for both particle types give configuration-dependent constants $k$ but the same $\delta$, which the paper takes as evidence for universality.

Load-bearing premise

The load-bearing premise is that the Hawking temperature near the critical point can be approximated by a quadratic curve with a nonzero second derivative, even though the critical point is defined by setting that second derivative to zero; if this premise gives way, the written derivation of the critical exponent $\delta = 1/2$ collapses.

Editorial extensions

If this is right

  • The Lyapunov exponent's thermal profile can locate the small/intermediate/large black hole phase structure without building the free energy, since multivaluedness appears only for $\tilde{q} < \tilde{q}_c$ and disappears above it.
  • The discontinuity $\Delta\lambda$ behaves as an order parameter whose critical exponent $\delta = 1/2$ matches the van der Waals mean-field value, placing the chaotic probe in the same universality class as the thermodynamic one.
  • For massive particles, $\lambda$ approaches zero in the large black hole phase, so the loss of unstable-circular-orbit chaos marks that phase across all charge configurations.
  • The critical exponent is independent of the charge configuration and of whether the probe particle is massless or massive, supporting the conjectured universality of the Lyapunov-based order parameter.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • As an extension, a rigorous version of the $\delta = 1/2$ derivation would have to start from the cubic leading term of the temperature expansion, since the quadratic term used in the paper vanishes at the critical point; the exponent may still be correct, but the written proof does not deliver it.
  • As an extension, applying the same numerical fit to the two additional charge configurations mentioned but not shown (two pairs of equal charges and three equal charges) would directly test the claimed universality.
  • As an extension, the multivalued Lyapunov profile could be cross-checked against other dynamical signatures of the same phase transitions, such as quasinormal-mode frequencies or shadow radius, which the paper does not attempt.
  • As an extension, because the paper finds no clear Lyapunov signature of the zeroth-order phase transition, searching for such a signature is a natural next step.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper applies the Lyapunov-exponent probe of black hole phase transitions to four-dimensional R-charged black holes with three charge configurations. For massless and massive particles on unstable circular orbits, the authors compute Lyapunov exponents as functions of Hawking temperature and horizon radius, showing multivalued branches for q < q_c that mirror the small/intermediate/large black hole phases seen in the Gibbs free energy. They then define the discontinuity Δλ between small and large branches as an order parameter and claim a universal critical exponent δ = 1/2, supported by an analytical expansion in Section V and by numerical fits in Figs. 13–14.

Significance. If the central exponent claim were established, the paper would be a useful extension of the Lyapunov-probe program to R-charged black holes and would strengthen the case for universality of δ = 1/2 across black hole families. The paper is systematic in covering three charge configurations and both particle types, and it includes explicit formulas for λ and detailed plots. However, the written derivation of the exponent is internally inconsistent, and the numerical verification does not independently determine the scaling law; these issues must be repaired before the main claim is reliable.

major comments (3)
  1. [Section V, Eqs. (26) and (41)] The Taylor expansion of the Hawking temperature in Eq. (41) retains the quadratic term (1/2) r_c^2 T''_c Δ^2, but Eq. (26) defines the critical point precisely by T'_c = T''_c = 0. The quadratic coefficient therefore vanishes at the expansion point, and the leading correction is cubic, T - T_c ~ (1/6) r_c^3 T'''_c Δ^3. Since Eq. (40) gives Δλ/λ_c ∝ Δ, the written derivation yields δ = 1/3 rather than δ = 1/2. This is a load-bearing inconsistency in the central claim of the paper; if the intended result is δ = 1/2, a different expansion (for example along the coexistence curve with the charge q as control parameter) must be supplied.
  2. [Section V, Eq. (43)] The prefactor k in Eq. (43) does not follow from Eqs. (40)–(42). Equation (40) contains (∂λ/∂r_+)_c, while Eq. (43) contains (∂Δλ/∂r_+)_c, and the factor of 2 arising from Δ_s - Δ_l = 2√(T_c/a)√(t-1) is missing. The expression for k should be re-derived and corrected.
  3. [Section V.1, Figs. 13–14] The numerical verification fits the data to the assumed form k√(t-1) and tabulates k; it does not test the scaling exponent against alternatives. The displayed range t-1 ≲ 10^{-3} is too narrow for the eye to distinguish a square-root law from a cube-root law with a different prefactor, and no log-log plot or free-exponent fit is provided. A fit with δ as a free parameter, or a log-log slope plot, is required to support the claimed exponent.
minor comments (6)
  1. [Section V, final paragraph] The sentence 'the critical exponent δ related to the order parameter Δλ near the critical point in 1/2' is incomplete; it should read 'is 1/2'.
  2. [Section IV.3] The text 'q1 =, q' contains a typographical error; it should read 'q1 = q'.
  3. [Section IV.3.2] The heading 'Massive paritcles' contains a typo for 'particles'.
  4. [Section VI] The statement that two additional charge configurations were studied but not shown cannot be independently checked; the consistency claim for those configurations is therefore unverifiable in this manuscript.
  5. [Section V, Eq. (37)] Equation (37) defines δ through Δλ ~ |T - T_c|^δ; in standard statistical mechanics this is the order-parameter exponent usually denoted β, whereas δ conventionally refers to the critical-isotherm exponent. The notation should be defined or justified.
  6. [Section IV.1.2 and later massive-particle sections] For massive particles the angular momentum is fixed to L = 20; a brief statement on whether the critical exponent is independent of L would clarify the universality claim.

Circularity Check

2 steps flagged · score 6.0 of 10

The δ=1/2 central claim is built on a quadratic Taylor term that the paper's own critical-point condition (26) sets to zero; the numerical verification then fits the same square-root form to self-generated data.

  1. self definitional [Section V, Eq. (41) combined with Eq. (26)]
    "The location of the critical point can be identified by satisfying the condition for an inflection point, ∂˜T/∂˜r+ = ∂²˜T/∂²˜r+ = 0, ... In a similar way, we also Taylor expand the Hawking temperature about the critical point ˜rc and obtain ˜T = ˜Tc + ˜r²c/2 [∂²˜T/∂˜r²+] ∆² where we have omitted the higher order terms and used [∂˜T/∂˜r+]c → 0."

    Equation (41) retains the quadratic coefficient (˜rc²/2)(∂²˜T/∂˜r²+)_c as the source of the √(t−1) scaling, but equation (26) — the paper's own definition of the critical point — sets this coefficient exactly to zero. The leading correction at the critical point is therefore cubic (T − Tc ∝ ∆³), so combining with Eq. (40), where ∆λ ∝ ∆, gives ∆λ ∝ (t−1)^{1/3}, not (t−1)^{1/2}. The square-root exponent is thus not derived; it is an artifact of keeping a term that the paper's own equations eliminate. This is a self-definitional reduction: the term the derivation depends on is defined away by Eq. (26).

  2. fitted input called prediction [Section V.1, Figs. 13-14 and Table I]
    "We find that the expression ∆λ/λc = k√t − 1 fits our numerically calculated data remarkably well across all considered charge configurations. The value of the constant k varies depending on the specific charge configuration but the critical exponent δ remains fixed i.e. 1/2."

    The 'data' being fitted are produced by the paper's own formulas for λ and T rather than by an independent observable, and the fitting function already contains the square-root form with a free prefactor k. The subsequent statement 'the critical exponent δ remains fixed i.e. 1/2' reads off the exponent that was put into the fit curve, not one extracted freely from the data. Over the displayed range t−1 ≲ 10⁻³, square-root and cube-root curves are nearly indistinguishable, so the fit cannot confirm δ=1/2 against the δ=1/3 implied by the corrected expansion. The numerical section therefore validates consistency with the assumed form rather than providing independent evidence for it.

full rationale

The thermodynamic background and the Lyapunov-exponent formulas are re-derived in the paper from the metric and standard geodesic equations, so the bulk of the analysis is self-contained and does not rest on load-bearing self-citations; the citations to the authors' own earlier papers (e.g., [30,37,74,82,84,85]) concern charge configurations, topology, or related probes and do not by themselves force the headline claim. The central difficulty is Section V: the written derivation of δ=1/2 depends on Eq. (41), which retains the quadratic term (1/2)˜rc² T''_c ∆², while the paper's own critical-point condition, Eq. (26), sets T''_c = 0. With the leading correction cubic, the written derivation implies δ=1/3, not δ=1/2. The numerical 'verification' then fits the assumed k√(t−1) form to data generated from the same equations, so the reported agreement and the constant δ in Table I are consistency checks of the assumed functional form, not independent measurements. Thus the headline exponent is effectively assumed and re-fit rather than derived, warranting a partial-circularity score of 6.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new particles, forces, or dimensions are introduced. The free parameters are the fitted scaling prefactors k and the chosen angular momentum L=20 for massive particles. The main unproved input is the imported conjecture that Lyapunov exponent branches correspond to thermodynamic phases, plus an invalid Taylor expansion step.

free parameters (2)
  • k (scaling prefactor) = Table I: 3.97061, 3.73783, 3.93412, 3.93947, 4.02495, 4.02990
    Fitted as the coefficient in Delta-lambda/lambda_c = k sqrt(t-1) for each charge configuration and particle type; no error bars reported.
  • massive particle angular momentum L = 20 (chosen; no scan)
    Massive-particle Lyapunov exponents depend on L; the paper fixes L=20 in all massive-particle plots and does not show that the phase-transition probe or critical exponent is independent of L.
assumptions (4)
  • standard math Standard geodesic Lagrangian and effective potential formalism (Section II)
    The Lyapunov exponent formulas (13) and (16) are quoted as known results; no formal proof is given.
  • domain assumption The conjecture that Lyapunov exponent multivaluedness signals black hole phase transitions (ref [69])
    The paper's central interpretive step, that the folded structure of lambda versus T is a probe of thermodynamic phases, is imported from earlier work, not derived here.
  • ad hoc to paper Taylor expansion of T about the critical point keeps a nonzero second-order term (Eq. 41)
    This premise is inconsistent with Eq. (26), which sets both first and second derivatives to zero at the critical point; the expansion is therefore not valid as written.
  • domain assumption Massive-particle results at L=20 are representative
    The massive Lyapunov exponent and the fitted k values depend on the chosen angular momentum; no L-scan is provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Probing Thermodynamic Phase Transitions of 4D R-Charged Black Holes via Lyapunov Exponent." pith.science (2026). https://pith.science/paper/F7J77MXC

@misc{pith2026250520800,
  author       = {Pith},
  title        = {Pith review of: Probing Thermodynamic Phase Transitions of 4D R-Charged Black Holes via Lyapunov Exponent},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F7J77MXC}},
  note         = {Machine review of arXiv:2505.20800}
}
abstract

We investigate the thermodynamic phase structure of four-dimensional \textit{R}-charged black holes--characterized by four independent $U(1)$ charges--through the lens of Lyapunov exponents associated with unstable circular orbits of both massless and massive particles. Considering three distinct charge configurations (equal, partially unequal, and fully unequal), we show that the thermal profile of the Lyapunov exponent can effectively probe the black hole phase transitions. Furthermore, we demonstrate that the discontinuous jump in the Lyapunov exponent acts as an order parameter, with the associated critical exponent $\delta = 1/2$ near the critical point--consistent across all charge configurations studied. This supports the proposed universality of the exponent and the broader connection between dynamical chaos and thermodynamic phase transitions.

Figures

Figures reproduced from arXiv: 2505.20800 by the authors.

Figure 1
Figure 1. FIG. 1: Hawking temperature as a function of horizon radius for different values of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Gibbs free energy as a function of temperature [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. a where we use the same colour coding as used in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Hawking temperature as a function of horizon radius for different values of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Gibbs free energy as a function of temperature [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Hawking temperature as a function of horizon radius for different values of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Gibbs free energy as a function of temperature [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13: Rescaled discontinuity in Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14: Rescaled discontinuity in Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Unifying topological, geometric, and complex classifications of black hole thermodynamics

    gr-qc 2026-04 unverdicted novelty 7.0 of 10

    Three classification schemes for black hole thermodynamics are equivalent, with the count of temperature extrema determining the class in each framework.

  2. Gaussian curvature and Lyapunov exponent as probes of black hole phase transitions

    gr-qc 2025-09 conditional novelty 4.0 of 10

    Gaussian curvature at the light ring inherits the swallowtail multivaluedness of free energy in first-order black hole phase transitions, following directly from K = -λ².

Reference graph

Works this paper leans on

100 extracted references · 28 canonical work pages · cited by 2 Pith papers

  1. [82]

    On The Phase Structure and Thermodynamic Geometry of R-Charged Black Holes

    A. Sahay, T. Sarkar and G. Sengupta, “On The Phase Structure and Thermodynamic Geometry of R-Charged Black Holes,” JHEP 11, 125 (2010) doi:10.1007/JHEP11(2010)125 [arXiv:1009.2236 [hep-th]]

    work page Pith review arXiv 2010

  2. [1]

    Gravitational radiation from colliding black holes,

    S. W. Hawking, “Gravitational radiation from colliding black holes,” Phys. Rev. Lett. 26, 1344-1346 (1971) doi:10.1103/PhysRevLett.26.1344

    doi:10.1103/physrevlett.26.1344 1971

  3. [2]

    Bekenstein-Hawking entropy,

    J. Bekenstein, “Bekenstein-Hawking entropy,” Scholarpedia 3, no.10, 7375 (2008) doi:10.4249/scholarpedia.7375

    doi:10.4249/scholarpedia.7375 2008

  4. [3]

    Black hole explosions,

    S. W. Hawking, “Black hole explosions,” Nature 248, 30-31 (1974) doi:10.1038/248030a0

    doi:10.1038/248030a0 1974

  5. [4]

    ˜r+c = 0.215966, ˜qc = 0.167433, ˜Tc = 0.227425 (32) The variation of the Hawking temperature with horizon radius for different values of ˜q is shown in Fig

    Hence, the entropy and temperature given in equations (20) and (22) becomes, S = √ 2π p (q + r+) 2 (q + 2r+) (q + 4r+) (28) T = 8l2r2 + − (q + r+) (q + 3r+) q2 − 4qr+ − 8r2 + 8 √ 2πl2r+ p (q + r+) 2 (q + 2r+) (q + 4r+) (29) The expression of mass simplifies to M = 1 4 (q + 2r+) (q + 4r+) (q + r+) 2 l2r+ + 11q + 8r+ (30) With these quantities, the Gibbs fr...

  6. [5]

    In these cases, the temperature profile exhibits two turning points, indicating the presence of three distinct black hole branches

    The yellow and blue curves correspond to values of ˜q less than the critical value ˜qc. In these cases, the temperature profile exhibits two turning points, indicating the presence of three distinct black hole branches. The red curve represents the critical case ˜q = ˜qc, while the green curve, corresponding to ˜q >˜qc, displays a single smooth branch wit...

  7. [6]

    The expression is too large and not quite intuitive so we just present its variation with the Hawking temperature

    In this charge configuration the entropy and the temperature becomes S = π p (q + r+) (q + 2r+) (q + 4r+) (q + 6r+)√ 3 (33) T = √ 3 r2 + l2 + 7q2 6 − q4 48 + 23qr3 + 6 + 3r4 + πl2r+ p (q + r+) (q + 2r+) (q + 4r+) (q + 6r+) (34) The mass expression simplifies to M = r+ 8r+ 6l2 + 7q2 + 46l2q + 13q3 + 92qr2 + + 48r3 + + q4 24l2r+ (35) The free energy is calc...

  8. [7]

    Charged black holes and phase transitions,

    P. Hut,“Charged black holes and phase transitions,” Monthly Notices of the Royal Astronomical Society, vol. 180, pp. 379–389, 10 1977

    1977

Show all 100 references
  1. [8]

    Particle Creation by Black Holes,

    S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43, 199-220 (1975) [erratum: Com- mun. Math. Phys. 46, 206 (1976)] doi:10.1007/BF02345020

    1975 doi

  2. [9]

    The Four laws of black hole mechanics,

    J. M. Bardeen, B. Carter and S. W. Hawking, “The Four laws of black hole mechanics,” Commun. Math. Phys. 31, 161-170 (1973) doi:10.1007/BF01645742

    1973 doi

  3. [10]

    Thermodynamics of Black Holes,

    P. C. W. Davies, “Thermodynamics of Black Holes,” Proc. Roy. Soc. Lond. A 353, 499-521 (1977) doi:10.1098/rspa.1977.0047

    1977

  4. [11]

    P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space,

    R. G. Cai, L. M. Cao, L. Li and R. Q. Yang, “P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space,” JHEP 09, 005 (2013) doi:10.1007/JHEP09(2013)005 [arXiv:1306.6233 [gr-qc]]

    2013 arXiv

  5. [12]

    The Large N limit of superconformal field theories and supergravity,

    J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2, 231-252 (1998) doi:10.4310/ATMP.1998.v2.n2.a1 [arXiv:hep-th/9711200 [hep-th]]

    1998 arXiv

  6. [13]

    Here the dots are the data points that we have calculated numerically and the solid coloured lines represent the fitted curves

    In this figure, the plot is shown only near the critical value. Here the dots are the data points that we have calculated numerically and the solid coloured lines represent the fitted curves. We find that the expression ∆λ/λc = k√t − 1 fits our numerically calculated data rema...

  7. [14]

    P-V criticality of charged AdS black holes,

    D. Kubiznak and R. B. Mann, “P-V criticality of charged AdS black holes,” JHEP 07, 033 (2012) doi:10.1007/JHEP07(2012)033 [arXiv:1205.0559 [hep-th]]

    2012 arXiv

  8. [15]

    Thermodynamics of Black Holes in anti-De Sitter Space,

    S. W. Hawking and D. N. Page, “Thermodynamics of Black Holes in anti-De Sitter Space,” Commun. Math. Phys. 87, 577 (1983) doi:10.1007/BF01208266

    1983 doi

  9. [16]

    Enthalpy and the Mechanics of AdS Black Holes,

    D. Kastor, S. Ray and J. Traschen, “Enthalpy and the Mechanics of AdS Black Holes,” Class. Quant. Grav. 26, 195011 (2009) doi:10.1088/0264-9381/26/19/195011 [arXiv:0904.2765 [hep-th]]

    2009 arXiv

  10. [17]

    The cosmological constant and the black hole equation of state,

    B. P. Dolan, “The cosmological constant and the black hole equation of state,” Class. Quant. Grav. 28, 125020 (2011) doi:10.1088/0264-9381/28/12/125020 [arXiv:1008.5023 [gr-qc]]

    2011 arXiv

  11. [18]

    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, 22 235017 (2011) doi:10.1088/0264-9381/28/23/235017 [arXiv:1106.6260 [gr-qc]]

    2011 arXiv

  12. [19]

    Compressibility of rotating black holes,

    B. P. Dolan, “Compressibility of rotating black holes,” Phys. Rev. D 84, 127503 (2011) doi:10.1103/PhysRevD.84.127503 [arXiv:1109.0198 [gr-qc]]

    2011 arXiv

  13. [20]

    Black hole chemistry: thermodynamics with Lambda,

    D. Kubiznak, R. B. Mann and M. Teo, “Black hole chemistry: thermodynamics with Lambda,” Class. Quant. Grav. 34, no.6, 063001 (2017) doi:10.1088/1361-6382/aa5c69 [arXiv:1608.06147 [hep-th]]

    2017 arXiv

  14. [21]

    Gauss-Bonnet coupling constant as a free thermodynamical variable and the as- sociated criticality,

    W. Xu, H. Xu and L. Zhao, “Gauss-Bonnet coupling constant as a free thermodynamical variable and the as- sociated criticality,” Eur. Phys. J. C 74, 2970 (2014) doi:10.1140/epjc/s10052-014-2970-8 [arXiv:1311.3053 [gr-qc]]

    2014 arXiv

  15. [22]

    Critical phenomena of static charged AdS black holes in conformal gravity,

    W. Xu and L. Zhao, “Critical phenomena of static charged AdS black holes in conformal gravity,” Phys. Lett. B 736, 214-220 (2014) doi:10.1016/j.physletb.2014.07.019 [arXiv:1405.7665 [gr-qc]]

    2014 arXiv

  16. [23]

    Reentrant phase transitions and triple points of topological AdS black holes in Born-Infeld-massive gravity,

    M. Zhang, D. C. Zou and R. H. Yue, “Reentrant phase transitions and triple points of topological AdS black holes in Born-Infeld-massive gravity,” Adv. High Energy Phys. 2017, 3819246 (2017) doi:10.1155/2017/3819246 [arXiv:1707.04101 [hep-th]]

    2017 arXiv

  17. [24]

    Ruppeiner, Thermodynamic curvature: pure fluids to black holes, J

    G. Ruppeiner, Thermodynamic curvature: pure fluids to black holes, J. Phys. Conf. Ser. 410, 012138 (2013) doi:10.1088/1742-6596/410/1/012138 [arXiv:1210.2011 [gr-qc]]

    2013 arXiv

  18. [25]

    Y . G. Miao and Z. M. Xu, Microscopic structures and thermal stability of black holes conformally coupled to scalar fields in five dimensions, Nucl. Phys. B 942, 205-220 (2019) doi:10.1016/j.nuclphysb.2019.03.015 [arXiv:1711.01757 [hep-th]]

    2019 arXiv

  19. [26]

    X. Y . Guo, H. F. Li, L. C. Zhang and R. Zhao, Microstructure and continuous phase transition of a Reissner- Nordstrom-AdS black hole, Phys. Rev. D 100, no.6, 064036 (2019) doi:10.1103/PhysRevD.100.064036 [arXiv:1901.04703 [gr-qc]]

    2019 arXiv

  20. [27]

    S. W. Wei, Y . X. Liu and R. B. Mann, Ruppeiner Geometry, Phase Transitions, and the Microstructure of Charged AdS Black Holes, Phys. Rev. D 100, no.12, 124033 (2019) doi:10.1103/PhysRevD.100.124033 [arXiv:1909.03887 [gr-qc]]

    2019 arXiv

  21. [28]

    P. Wang, H. Wu and H. Yang, Thermodynamic Geometry of AdS Black Holes and Black Holes in a Cavity, Eur. Phys. J. C 80, no.3, 216 (2020) doi:10.1140/epjc/s10052-020-7776-2 [arXiv:1910.07874 [gr-qc]]

    2020 arXiv

  22. [29]

    P. K. Yerra and C. Bhamidipati, Ruppeiner Geometry, Phase Transitions and Microstructures of Black Holes in Massive Gravity, Int. J. Mod. Phys. A 35, no.22, 2050120 (2020) doi:10.1142/S0217751X20501201 [arXiv:2006.07775 [hep-th]]

    2020 arXiv

  23. [30]

    P. K. Yerra and C. Bhamidipati, Novel relations in massive gravity at Hawking-Page transition, Phys. Rev. D 104, no.10, 104049 (2021) doi:10.1103/PhysRevD.104.104049 [arXiv:2107.04504 [gr-qc]]

    2021 arXiv

  24. [31]

    Wu, Topological classes of rotating black holes, Phys

    D. Wu, Topological classes of rotating black holes, Phys. Rev. D 107, no.2, 024024 (2023) doi:10.1103/PhysRevD.107.024024 [arXiv:2211.15151 [gr-qc]]

    2023 arXiv

  25. [32]

    Liu and J

    C. Liu and J. Wang, Topological natures of the Gauss-Bonnet black hole in AdS space, Phys. Rev. D 107, no.6, 064023 (2023) doi:10.1103/PhysRevD.107.064023 [arXiv:2211.05524 [gr-qc]]

    2023 arXiv

  26. [33]

    Z. Y . Fan, Topological interpretation for phase transitions of black holes, Phys. Rev. D107, no.4, 044026 (2023) doi:10.1103/PhysRevD.107.044026 [arXiv:2211.12957 [gr-qc]]

    2023 arXiv

  27. [34]

    N. J. Gogoi and P. Phukon, Thermodynamic topology of 4D dyonic AdS black holes in different ensembles, Phys. Rev. D 108, no.6, 066016 (2023) doi:10.1103/PhysRevD.108.066016 [arXiv:2304.05695 [hep-th]]

    2023 arXiv

  28. [35]

    M. S. Ali, H. El Moumni, J. Khalloufi and K. Masmar, Topology of Born-Infeld-AdS Black Hole Phase Transi- tion, [arXiv:2306.11212 [hep-th]]

    arXiv

  29. [36]

    M. A. Saleem and A. Taani, The chaotic behavior of black holes: Investigating a topological retraction in anti-de Sitter spaces, New Astron. 107, 102149 (2024) doi:10.1016/j.newast.2023.102149

    2024

  30. [37]

    M. U. Shahzad, A. Mehmood, S. Sharif and A. ¨Ovg¨un, Criticality and topological classes of neutral Gauss–Bonnet AdS black holes in 5D, Annals Phys. 458, no.3, 169486 (2023) doi:10.1016/j.aop.2023.169486

    2023

  31. [38]

    Z. Q. Chen and S. W. Wei, Thermodynamics, Ruppeiner geometry, and topology of Born-Infeld black hole in asymptotic flat spacetime, Nucl. Phys. B 996, 116369 (2023) doi:10.1016/j.nuclphysb.2023.116369

    2023

  32. [39]

    N. C. Bai, L. Li and J. Tao, Topology of black hole thermodynamics in Lovelock gravity, Phys. Rev. D 107, no.6, 064015 (2023) doi:10.1103/PhysRevD.107.064015 [arXiv:2208.10177 [gr-qc]]

    2023 arXiv

  33. [40]

    P. K. Yerra and C. Bhamidipati, Topology of black hole thermodynamics in Gauss-Bonnet gravity, Phys. Rev. D 105, no.10, 104053 (2022) doi:10.1103/PhysRevD.105.104053 [arXiv:2202.10288 [gr-qc]]

    2022 arXiv

  34. [41]

    Hazarika and P

    B. Hazarika and P. Phukon, Thermodynamic Topology of D = 4, 5 Horava Lifshitz Black Hole in Two Ensem- bles, [arXiv:2312.06324 [hep-th]]

    arXiv

  35. [42]

    Y . Liu, D. C. Zou and B. Wang, Signature of the Van der Waals like small-large charged AdS black hole phase 23 transition in quasinormal modes, JHEP 09, 179 (2014) doi:10.1007/JHEP09(2014)179 [arXiv:1405.2644 [hep- th]]

    2014 arXiv

  36. [43]

    D. C. Zou, Y . Liu and R. H. Yue, Behavior of quasinormal modes and Van der Waals-like phase transition of charged AdS black holes in massive gravity, Eur. Phys. J. C77, no.6, 365 (2017) doi:10.1140/epjc/s10052-017- 4937-z [arXiv:1702.08118 [gr-qc]]

    2017 arXiv

  37. [44]

    Zhang, C

    M. Zhang, C. M. Zhang, D. C. Zou and R. H. Yue, Phase transition and Quasinormal modes for Charged black holes in 4D Einstein-Gauss-Bonnet gravity, Chin. Phys. C 45, no.4, 045105 (2021) doi:10.1088/1674- 1137/abe19a [arXiv:2009.03096 [hep-th]]

    2021 arXiv

  38. [45]

    Mahapatra, Thermodynamics, Phase Transition and Quasinormal modes with Weyl corrections, JHEP 04, 142 (2016) doi:10.1007/JHEP04(2016)142 [arXiv:1602.03007 [hep-th]]

    S. Mahapatra, Thermodynamics, Phase Transition and Quasinormal modes with Weyl corrections, JHEP 04, 142 (2016) doi:10.1007/JHEP04(2016)142 [arXiv:1602.03007 [hep-th]]

    2016 arXiv

  39. [46]

    Chabab, H

    M. Chabab, H. El Moumni, S. Iraoui and K. Masmar, Behavior of quasinormal modes and high dimension RN–AdS black hole phase transition, Eur. Phys. J. C 76, no.12, 676 (2016) doi:10.1140/epjc/s10052-016-4518- 6 [arXiv:1606.08524 [hep-th]]

    2016 arXiv

  40. [47]

    S. W. Wei and Y . X. Liu, Photon orbits and thermodynamic phase transition ofd-dimensional charged AdS black holes, Phys. Rev. D 97, no.10, 104027 (2018) doi:10.1103/PhysRevD.97.104027 [arXiv:1711.01522 [gr-qc]]

    2018 arXiv

  41. [48]

    S. W. Wei, Y . X. Liu and Y . Q. Wang, Probing the relationship between the null geodesics and ther- modynamic phase transition for rotating Kerr-AdS black holes, Phys. Rev. D 99, no.4, 044013 (2019) doi:10.1103/PhysRevD.99.044013 [arXiv:1807.03455 [gr-qc]]

    2019 arXiv

  42. [49]

    Zhang, S

    M. Zhang, S. Z. Han, J. Jiang and W. B. Liu, Phys. Rev. D 99, no.6, 065016 (2019) doi:10.1103/PhysRevD.99.065016 [arXiv:1903.08293 [hep-th]]

    2019 arXiv

  43. [50]

    Zhang and M

    M. Zhang and M. Guo, Can shadows reflect phase structures of black holes?, Eur. Phys. J. C80, no.8, 790 (2020) doi:10.1140/epjc/s10052-020-8389-5 [arXiv:1909.07033 [gr-qc]]

    2020 arXiv

  44. [51]

    Belhaj, L

    A. Belhaj, L. Chakhchi, H. El Moumni, J. Khalloufi and K. Masmar, Thermal Image and Phase Transi- tions of Charged AdS Black Holes using Shadow Analysis, Int. J. Mod. Phys. A 35, no.27, 2050170 (2020) doi:10.1142/S0217751X20501705 [arXiv:2005.05893 [gr-qc]]

    2020 arXiv

  45. [52]

    A.M Lyapunov, The general problem of the stability of motion Int. J. Control 55, 531 (1992). doi:https://doi.org/10.1080/00207179208934253

    1992 doi

  46. [53]

    Sandri, Numerical calculation of lyapunov exponents, Math

    M. Sandri, Numerical calculation of lyapunov exponents, Math. J. 6, 78 (1996)

    1996

  47. [54]

    Phase transition and chaos in charged SYK model,

    N. Sorokhaibam, “Phase transition and chaos in charged SYK model,” JHEP 07, 055 (2020) doi:10.1007/JHEP07(2020)055 [arXiv:1912.04326 [hep-th]]

    2020 arXiv

  48. [55]

    Quantum chaos and phase transition in the Yukawa–Sachdev-Ye-Kitaev model,

    A. Davis and Y . Wang, “Quantum chaos and phase transition in the Yukawa–Sachdev-Ye-Kitaev model,” Phys. Rev. B 107, no.20, 205122 (2023) doi:10.1103/PhysRevB.107.205122 [arXiv:2212.03265 [cond-mat.str-el]]

    2023 arXiv

  49. [56]

    Chaos and the quantum phase transition in the Dicke model,

    C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E 67, 066203 (2003) doi:10.1103/PhysRevE.67.066203 [arXiv:cond-mat/0301273 [cond-mat]]

    2003 arXiv

  50. [57]

    Phase Transitions and Chaos in Long-Range Models of Coupled Oscillators,

    G. Miritello, A. Pluchino and A. Rapisarda, “Phase Transitions and Chaos in Long-Range Models of Coupled Oscillators,” EPL 85, no.1, 10007 (2009) doi:10.1209/0295-5075/85/10007 [arXiv:0807.1870 [cond-mat.stat- mech]]

    2009 arXiv

  51. [58]

    Transitional regions of finite Fermi systems and quantum chaos,

    W. D. Heiss and A. L. Sannino, “Transitional regions of finite Fermi systems and quantum chaos,” Phys. Rev. A 43, 4159-4166 (1991) doi:10.1103/PhysRevA.43.4159

    1991 doi

  52. [59]

    Chaos in static axisymmetric space-times. 1: Vacuum case,

    Y . Sota, S. Suzuki and K. i. Maeda, “Chaos in static axisymmetric space-times. 1: Vacuum case,” Class. Quant. Grav. 13, 1241-1260 (1996) doi:10.1088/0264-9381/13/5/034 [arXiv:gr-qc/9505036 [gr-qc]]

    1996 arXiv

  53. [60]

    Chaos in static axisymmetric space-times. 2. Nonvacuum case,

    Y . Sota, S. Suzuki and K. i. Maeda, “Chaos in static axisymmetric space-times. 2. Nonvacuum case,” [arXiv:gr- qc/9610065 [gr-qc]]

  54. [61]

    Bound on the Lyapunov exponent in Kerr-Newman black holes via a charged particle,

    N. Kan and B. Gwak, “Bound on the Lyapunov exponent in Kerr-Newman black holes via a charged particle,” Phys. Rev. D 105, no.2, 026006 (2022) doi:10.1103/PhysRevD.105.026006 [arXiv:2109.07341 [gr-qc]]

    2022 arXiv

  55. [63]

    Chaotic motion in multi-black hole spacetimes and holographic screens,

    W. Hanan and E. Radu, “Chaotic motion in multi-black hole spacetimes and holographic screens,” Mod. Phys. Lett. A 22, 399-406 (2007) doi:10.1142/S0217732307022815 [arXiv:gr-qc/0610119 [gr-qc]]

    2007 arXiv

  56. [64]

    Minimal Length Effects on Chaotic Motion of Particles around Black Hole Horizon,

    F. Lu, J. Tao and P. Wang, “Minimal Length Effects on Chaotic Motion of Particles around Black Hole Horizon,” JCAP 12, 036 (2018) doi:10.1088/1475-7516/2018/12/036 [arXiv:1811.02140 [gr-qc]]

    2018 arXiv

  57. [65]

    Chaotic Motion around a Black Hole under Minimal Length Effects,

    X. Guo, K. Liang, B. Mu, P. Wang and M. Yang, “Chaotic Motion around a Black Hole under Minimal Length Effects,” Eur. Phys. J. C80, no.8, 745 (2020) doi:10.1140/epjc/s10052-020-8335-6 [arXiv:2002.05894 [gr-qc]]

    2020 arXiv

  58. [66]

    A bound on chaos,

    J. Maldacena, S. H. Shenker and D. Stanford, “A bound on chaos,” JHEP 08, 106 (2016) 24 doi:10.1007/JHEP08(2016)106 [arXiv:1503.01409 [hep-th]]

    2016 arXiv

  59. [67]

    Universality in Chaos of Particle Motion near Black Hole Horizon,

    K. Hashimoto and N. Tanahashi, “Universality in Chaos of Particle Motion near Black Hole Horizon,” Phys. Rev. D 95, no.2, 024007 (2017) doi:10.1103/PhysRevD.95.024007 [arXiv:1610.06070 [hep-th]]

    2017 arXiv

  60. [68]

    Presence of horizon makes particle motion chaotic,

    S. Dalui, B. R. Majhi and P. Mishra, “Presence of horizon makes particle motion chaotic,” Phys. Lett. B 788, 486-493 (2019) doi:10.1016/j.physletb.2018.11.050 [arXiv:1803.06527 [gr-qc]]

    2019 arXiv

  61. [69]

    Static Equilibria of Charged Particles Around Charged Black Holes: Chaos Bound and Its Violations,

    Q. Q. Zhao, Y . Z. Li and H. Lu, “Static Equilibria of Charged Particles Around Charged Black Holes: Chaos Bound and Its Violations,” Phys. Rev. D 98, no.12, 124001 (2018) doi:10.1103/PhysRevD.98.124001 [arXiv:1809.04616 [gr-qc]]

    2018 arXiv

  62. [70]

    Minimal Length Effects on Motion of a Particle in Rindler Space,

    X. Guo, K. Liang, B. Mu, P. Wang and M. Yang, “Minimal Length Effects on Motion of a Particle in Rindler Space,” Chin. Phys. C 45, no.2, 023115 (2021) doi:10.1088/1674-1137/abcf20 [arXiv:2007.07744 [gr-qc]]

    2021 arXiv

  63. [71]

    Violation of bound on chaos for charged probe in Kerr-Newman-AdS black hole,

    B. Gwak, N. Kan, B. H. Lee and H. Lee, “Violation of bound on chaos for charged probe in Kerr-Newman-AdS black hole,” JHEP 09, 026 (2022) doi:10.1007/JHEP09(2022)026 [arXiv:2203.07298 [gr-qc]]

    2022 arXiv

  64. [72]

    Bound on Lyapunov exponent in Kerr-Newman-de Sitter black holes by a charged parti- cle,

    J. Park and B. Gwak, “Bound on Lyapunov exponent in Kerr-Newman-de Sitter black holes by a charged parti- cle,” JHEP 04, 023 (2024) doi:10.1007/JHEP04(2024)023 [arXiv:2312.13075 [gr-qc]]

    2024 arXiv

  65. [73]

    Probing phase structure of black holes with Lyapunov exponents,

    X. Guo, Y . Lu, B. Mu and P. Wang, “Probing phase structure of black holes with Lyapunov exponents,” JHEP 08, 153 (2022) doi:10.1007/JHEP08(2022)153 [arXiv:2205.02122 [gr-qc]]

    2022 arXiv

  66. [74]

    Lyapunov exponents and phase transitions of Born-Infeld AdS black holes,

    S. Yang, J. Tao, B. Mu and A. He, “Lyapunov exponents and phase transitions of Born-Infeld AdS black holes,” JCAP 07, 045 (2023) doi:10.1088/1475-7516/2023/07/045 [arXiv:2304.01877 [gr-qc]]

    2023 arXiv

  67. [75]

    Probing the thermodynamics of charged Gauss Bonnet AdS black holes with the Lyapunov exponent,

    X. Lyu, J. Tao and P. Wang, “Probing the thermodynamics of charged Gauss Bonnet AdS black holes with the Lyapunov exponent,” Eur. Phys. J. C 84, no.9, 974 (2024) doi:10.1140/epjc/s10052-024-13354-9 [arXiv:2312.11912 [gr-qc]]

    2024 arXiv

  68. [76]

    Lyapunov exponents and phase structure of Lifshitz and hyperscaling violating black holes,

    A. N. Kumara, S. Punacha and M. S. Ali, “Lyapunov exponents and phase structure of Lifshitz and hyperscaling violating black holes,” JCAP 07, 061 (2024) doi:10.1088/1475-7516/2024/07/061 [arXiv:2401.05181 [gr-qc]]

    2024 arXiv

  69. [77]

    Phase structure and optical properties of the de Sitter Spacetime with KR field based on the Lyapunov exponent,

    Y . Z. Du, H. F. Li, Y . B. Ma and Q. Gu, “Phase structure and optical properties of the de Sitter Spacetime with KR field based on the Lyapunov exponent,” Eur. Phys. J. C 85, no.1, 78 (2025) doi:10.1140/epjc/s10052-025- 13809-7 [arXiv:2403.20083 [hep-th]]

    2025 arXiv

  70. [78]

    Lyapunov exponents and phase transition of Hayward AdS black hole,

    N. J. Gogoi, S. Acharjee and P. Phukon, “Lyapunov exponents and phase transition of Hayward AdS black hole,” Eur. Phys. J. C 84, no.11, 1144 (2024) doi:10.1140/epjc/s10052-024-13520-z [arXiv:2404.03947 [hep-th]]

    2024 arXiv

  71. [79]

    Interplay between the Lyapunov exponents and phase transi- tions of charged AdS black holes,

    B. Shukla, P. P. Das, D. Dudal and S. Mahapatra, “Interplay between the Lyapunov exponents and phase transi- tions of charged AdS black holes,” Phys. Rev. D 110, no.2, 024068 (2024) doi:10.1103/PhysRevD.110.024068 [arXiv:2404.02095 [hep-th]]

    2024 arXiv

  72. [80]

    Lyapunov exponents as probes for a phase transition of a Kerr-AdS black hole,

    D. Chen, C. Yang and Y . Liu, “Lyapunov exponents as probes for a phase transition of a Kerr-AdS black hole,” Phys. Lett. B 865, 139463 (2025) doi:10.1016/j.physletb.2025.139463 [arXiv:2501.16999 [hep-th]]

    2025

  73. [81]

    Euclidean Thermodynamics and Lyapunov Exponents of Einstein-Power-Yang-Mills AdS Black Holes,

    K. R., D. D., K. M. Ajith, K. Hegde, S. Punacha and A. N. Kumara, “Euclidean Thermodynamics and Lyapunov Exponents of Einstein-Power-Yang-Mills AdS Black Holes,” [arXiv:2504.12890 [gr-qc]]

  74. [83]

    Thermodynamics of R-charged black holes in AdS(5) from effective strings,

    S. S. Gubser and J. J. Heckman, “Thermodynamics of R-charged black holes in AdS(5) from effective strings,” JHEP 11, 052 (2004) doi:10.1088/1126-6708/2004/11/052 [arXiv:hep-th/0411001 [hep-th]]

    2004 arXiv

  75. [84]

    Notes on R-charged black holes near criticality and gauge theory,

    S. Jain, S. Mukherji and S. Mukhopadhyay, “Notes on R-charged black holes near criticality and gauge theory,” JHEP 11, 051 (2009) doi:10.1088/1126-6708/2009/11/051 [arXiv:0906.5134 [hep-th]]

    2009 arXiv

  76. [85]

    Thermodynamic geometry of black holes enclosed by a cavity in extended phase space,

    P. Wang and F. Yao, “Thermodynamic geometry of black holes enclosed by a cavity in extended phase space,” Nucl. Phys. B 976, 115715 (2022) doi:10.1016/j.nuclphysb.2022.115715 [arXiv:2107.14640 [gr-qc]]

    2022

  77. [86]

    Topology of thermodynamics in R-charged black holes,

    N. J. Gogoi and P. Phukon, “Topology of thermodynamics in R-charged black holes,” Phys. Rev. D 107, no.10, 106009 (2023) doi:10.1103/PhysRevD.107.106009

    2023 doi

  78. [87]

    Phases of R charged black holes, spinning branes and strongly coupled gauge theories,

    M. Cvetic and S. S. Gubser, “Phases of R charged black holes, spinning branes and strongly coupled gauge theories,” JHEP 04, 024 (1999) doi:10.1088/1126-6708/1999/04/024 [arXiv:hep-th/9902195 [hep-th]]

    1999 arXiv

  79. [88]

    R-Charged Black Holes and Holographic Optics,

    P. Phukon and T. Sarkar, “R-Charged Black Holes and Holographic Optics,” JHEP 09, 102 (2013) doi:10.1007/JHEP09(2013)102 [arXiv:1305.2745 [hep-th]]

    2013 arXiv

  80. [89]

    Generalized Superconductors and Holographic Optics,

    S. Mahapatra, P. Phukon and T. Sarkar, “Generalized Superconductors and Holographic Optics,” JHEP 01, 135 (2014) doi:10.1007/JHEP01(2014)135 [arXiv:1305.6273 [hep-th]]

    2014 arXiv

  81. [90]

    Generalized superconductors and holographic optics. Part II,

    S. Mahapatra, “Generalized superconductors and holographic optics. Part II,” JHEP 01, 148 (2015) doi:10.1007/JHEP01(2015)148 [arXiv:1411.6405 [hep-th]]

    2015 arXiv

  82. [91]

    Hydrodynamics of r-charged black holes,

    D. T. Son and A. O. Starinets, “Hydrodynamics of r-charged black holes,” JHEP 03, 052 (2006) 25 doi:10.1088/1126-6708/2006/03/052 [arXiv:hep-th/0601157 [hep-th]]

    2006 arXiv

  83. [92]

    Shear viscosity from R-charged AdS black holes,

    J. Mas, “Shear viscosity from R-charged AdS black holes,” JHEP 03, 016 (2006) doi:10.1088/1126- 6708/2006/03/016 [arXiv:hep-th/0601144 [hep-th]]

    2006 arXiv

  84. [93]

    Anti-de Sitter black holes in gauged N = 8 supergravity,

    M. J. Duff and J. T. Liu, “Anti-de Sitter black holes in gauged N = 8 supergravity,” Nucl. Phys. B 554, 237-253 (1999) doi:10.1016/S0550-3213(99)00299-0 [arXiv:hep-th/9901149 [hep-th]]

    1999 arXiv

  85. [94]

    Zeroth-Order Phase Transitions

    V .P. Maslov “Zeroth-Order Phase Transitions” Mathematical Notes 76, 697–710 (2004) https://doi.org/10.1023/B:MATN.0000049669.32515.f0

    2004

  86. [95]

    Extended phase space thermodynamics for charged and ro- tating black holes and Born-Infeld vacuum polarization,

    S. Gunasekaran, R. B. Mann and D. Kubiznak, “Extended phase space thermodynamics for charged and ro- tating black holes and Born-Infeld vacuum polarization,” JHEP 11, 110 (2012) doi:10.1007/JHEP11(2012)110 [arXiv:1208.6251 [hep-th]]

    2012 arXiv

  87. [96]

    Reentrant phase transitions in rotating anti–de Sitter black holes,

    N. Altamirano, D. Kubiznak and R. B. Mann, “Reentrant phase transitions in rotating anti–de Sitter black holes,” Phys. Rev. D 88, no.10, 101502 (2013) doi:10.1103/PhysRevD.88.101502 [arXiv:1306.5756 [hep-th]]

    2013 arXiv

  88. [97]

    Reentrant phase transitions and van der Waals behaviour for hairy black holes,

    R. A. Hennigar and R. B. Mann, “Reentrant phase transitions and van der Waals behaviour for hairy black holes,” Entropy 17, no.12, 8056-8072 (2015) doi:10.3390/e17127862 [arXiv:1509.06798 [hep-th]]

    2015 arXiv

  89. [98]

    Novel phase transition in charged dilaton black holes,

    A. Dehyadegari, A. Sheykhi and A. Montakhab, “Novel phase transition in charged dilaton black holes,” Phys. Rev. D 96, no.8, 084012 (2017) doi:10.1103/PhysRevD.96.084012 [arXiv:1707.05307 [hep-th]]

    2017 arXiv

  90. [99]

    Critical behavior and phase transition of dilaton black holes with nonlinear electrodynamics,

    Z. Dayyani, A. Sheykhi, M. H. Dehghani and S. Hajkhalili, “Critical behavior and phase transition of dilaton black holes with nonlinear electrodynamics,” Eur. Phys. J. C78, no.2, 152 (2018) doi:10.1140/epjc/s10052-018- 5623-5 [arXiv:1709.06875 [gr-qc]]

    2018 arXiv

  91. [100]

    Microscopic explanation for black hole phase transitions via Ruppeiner geom- etry: Two competing factors–the temperature and repulsive interaction among BH molecules,

    Y . Chen, H. Li and S. J. Zhang, “Microscopic explanation for black hole phase transitions via Ruppeiner geom- etry: Two competing factors–the temperature and repulsive interaction among BH molecules,” Nucl. Phys. B 948, 114752 (2019) doi:10.1016/j.nuclphysb.2019.114752 [arXiv...

    2019

  92. [101]

    Critical behavior of Born Infeld AdS black holes in higher dimensions,

    R. Banerjee and D. Roychowdhury, “Critical behavior of Born Infeld AdS black holes in higher dimensions,” Phys. Rev. D 85, 104043 (2012) doi:10.1103/PhysRevD.85.104043 [arXiv:1203.0118 [gr-qc]]

    2012 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.