Pith. sign in

REVIEW 2 major objections 4 minor 5 cited by

Comparative study of the butterfly velocity in holographic QCD models at finite temperature and chemical potential

T0 review · 2 major / 4 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 Four holographic QCD models share one butterfly velocity: it rises with temperature, falls with chemical potential, and saturates to the chargeless plasma value at high T.

desk verdict Useful model-by-model computation of v_B in holographic QCD, but the analytic bottom-up section contains a load-bearing typo: Eq. (4.9) is the temperature, not the butterfly velocity. read the letter →

arxiv 2505.15357 v3 pith:SS64UIFH submitted 2025-05-21 hep-th hep-lathep-phnlin.CDquant-ph

Nikesh Lilani , Dilpreet Sandhu , Subhash Mahapatra This is my paper
classification hep-thhep-lathep-phnlin.CDquant-ph
keywords butterflyvelocityholographicQCDquantumchaosEinstein-Maxwell-dilatongravitypole-skippingout-of-time-orderedcorrelatorentanglementwedgechemicalpotential
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 aims to show that the butterfly velocity—the speed at which a localized perturbation spreads through a strongly coupled plasma—obeys one universal pattern across four different holographic QCD models. Using three independent methods, entanglement wedge reconstruction, out-of-time-ordered correlator shockwave analysis, and pole-skipping, the authors obtain exactly the same value of $v_B$ in each model. In thermodynamically stable phases, $v_B$ increases with temperature and decreases with chemical potential, and in the high-temperature limit every model approaches the conformal chargeless value $v_B=\sqrt{2/3}$. If true, this tells a curious reader that the chaotic spread of information in the quark-gluon plasma is a robust feature tied to near-horizon geometry, with a density dependence that could connect to transport measurements.

What carries the argument

The load-bearing object is the general planar isotropic black hole metric $ds^2 = -h(z)g(z)dt^2 + \frac{f(z)}{g(z)}dz^2 + k(z)dx^i dx^i$ with horizon at $g(z_h)=0$. Near the horizon, three computations feed on the same data: the extremal Ryu-Takayanagi surface used in entanglement wedge reconstruction; the shock-wave shift $\bar h(x)$ obtained from the Einstein equation in the OTOC analysis; and the pole-skipping point—where pole and zero lines of the retarded energy-density Green's function intersect—located at $\omega_* = 2\pi i T$ and $q_*^2 = -(d-1)\pi T k'(z_h)\sqrt{h(z_h)f(z_h)}$, provided the perturbed matter component $\delta T^z_v$ vanishes at the horizon. All three reduce to $v_B^2 = g'_o h_o/[(d-1)k'_o]$.

What would settle it

Numerically solve the linearized Einstein-Maxwell-dilaton equations for the Section 5 background with regular horizon conditions, evaluate $\delta T^z_v$ at the horizon, and recompute the pole-skipping point from the full equation $\delta E^z_v = \delta T^z_v$; a nonzero horizon value would shift the pole-skipping $v_B$ away from $1/(6\eta'(z_h)G_0^{\mathrm{far}})$ and break the three-way equality for that model.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that for the 1RCBH and 2RCBH top-down models, a potential-reconstruction analytic bottom-up model, and a numerical bottom-up model—all built from the Einstein-Maxwell-dilaton action—the butterfly velocity computed from entanglement wedge reconstruction, OTOC shockwaves, and pole-skipping is identical. The common formula is $v_B^2 = g'_o h_o/[(d-1)k'_o]$ in terms of near-horizon metric data. The paper further claims a universal trend: in thermodynamically stable phases $v_B$ grows with $T$, decreases with $\mu$, and as $T\to\infty$ saturates to $v_B=\sqrt{2/3}$, the value for the chargeless AdS-Schwarzschild plasma.

Load-bearing premise

The pole-skipping calculation assumes that the perturbed matter fields near the horizon are regular enough that the stress-tensor component which would modify the simple horizon equations is exactly zero; the paper asserts this is straightforward for the numerical model rather than demonstrating it.

Editorial extensions

If this is right

  • In all four models the entanglement wedge, OTOC, and pole-skipping methods give exactly the same butterfly velocity, so one can compute $v_B$ from near-horizon metric data alone without building the full correlation function.
  • In thermodynamically stable phases $v_B$ increases monotonically with temperature and decreases monotonically with chemical potential, across top-down and bottom-up EMD models.
  • In the high-temperature limit each model saturates to $v_B=\sqrt{2/3}$, the chargeless AdS-Schwarzschild value, so conformal scale symmetry controls the ultraviolet chaotic speed.
  • In the 1RCBH and 2RCBH models the dilaton is secondary hair: at vanishing chemical potential both reduce to chargeless plasma and $v_B$ becomes temperature-independent, while in the potential-reconstruction model the primary dilaton makes $v_B$ depend on $T$ even at $\mu=0$.
  • The numerical EMD model shows $v_B$ varying nonmonotonically with temperature below roughly 130-140 MeV, near the inferred deconfinement scale, while in the high-temperature deconfined phase it is monotonic; the parallel with the speed of sound suggests $v_B$ carries phase-structure information.

Reading between the lines

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

  • If the paper is right, the near-horizon formula makes $v_B$ a cheap diagnostic: any new EMD background can have its chaotic speed read off from horizon data, so scanning different dilaton potentials and gauge couplings for deviations from the universal trend is a direct next step.
  • An extension the paper leaves implicit is to verify $\delta T^z_v = 0$ explicitly on the numerical backgrounds; doing so would turn the asserted three-way equality there into a demonstrated one and show whether the equality is exact or only approximate in fully numerical models.
  • Connecting to the paper's cited relation between $v_B$ and heavy-quark drag or jet quenching, the universal decrease of $v_B$ with chemical potential would predict that energy-loss observables in baryon-rich quark-gluon plasma are suppressed at larger density; heavy-ion data at finite baryon density could look for this.
  • The observed parallel between $v_B$ and the speed of sound in the numerical model suggests a possible shared infrared origin; comparing $v_B$ with $c_s$ in the analytic potential-reconstruction model, where both are available in closed form, would test whether the parallel is general.
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

2 major / 4 minor

Summary. The paper computes the butterfly velocity v_B in four holographic QCD models — the 1RCBH and 2RCBH top-down models, a potential-reconstruction analytic bottom-up model, and a numerical bottom-up model — using three different methods: entanglement wedge reconstruction, OTOC/shock-wave analysis, and pole-skipping. The central claims are that all three methods give identical results in all models, that v_B increases with temperature and decreases with chemical potential in thermodynamically stable phases, and that v_B approaches the chargeless AdS-Schwarzschild value sqrt(2/3) at high temperature. The paper also includes an analysis of the RN-AdS plasma in an appendix and reports a numerical scan of about 5×10^5 charged black hole solutions for the bottom-up model.

Significance. If the central claims hold, the paper provides a useful cross-check of the equivalence between three standard holographic chaos diagnostics in phenomenologically relevant holographic QCD constructions, and it identifies a potentially universal trend for v_B as a function of T and μ. The inclusion of the 2RCBH model and the RN-AdS appendix broadens the scope of existing chaos studies. The paper is also commendable for presenting explicit analytic expressions in the first three models and for the large numerical scan in the fourth. However, two load-bearing displayed formulas — Eq. (4.9) and Eq. (2.47) — are internally inconsistent as printed, so the analytic bottom-up section and the general pole-skipping derivation need correction before the claims can be accepted. The concern that the pole-skipping condition in Sec. 5.4 is an unsupported assertion does not, in my reading, land: the EMD stress-tensor algebra in Sec. 3.4 is solution-independent and applies equally to the numerical model.

major comments (2)
  1. [4.2, Eq. (4.9)] The right-hand side of Eq. (4.9) is textually identical to the temperature expression in Eq. (4.7). Thus, as printed, the paper asserts v_B^2 = T, which is dimensionally inconsistent (v_B^2 is dimensionless while T carries energy units) and incompatible with the high-temperature expansion in Eq. (4.10), where v_B^2 tends to 2/3. Because Sections 4.3 and 4.4 claim to reproduce Eq. (4.9), and Figure 6 is presented as its consequence, the analytic bottom-up model's v_B(T, μ), the universal monotonicity trend, and the three-method equivalence for this model are not supported by the displayed formulas. The correct expression obtained from Eq. (2.13) is v_B^2 = 2πT/(3z_h + 6a/z_h) in the coordinates of this model, which is not equal to Eq. (4.7); the printed Eq. (4.9) must be corrected and the subsequent derivations and figures rechecked.
  2. [2.3, Eq. (2.47)] The pole-skipping momentum quoted in Eq. (2.47), q_*^2 = -(d-1)πT k'(z_h) sqrt(h(z_h) f(z_h)), is not consistent with the model computations that follow. Combining Eq. (2.47) with ω_* = 2πiT gives v_B^2 = 4πT / [(d-1)k' sqrt(hf)], which reduces to Eq. (2.13) only when f = 1/h; for the 1RCBH and 2RCBH metrics this condition fails. The explicitly computed pole-skipping points in Eqs. (3.33) and (3.35) instead correspond to q_*^2 = -(d-1)πT k'(z_h)/sqrt(h(z_h) f(z_h)). The general formula in Eq. (2.47) therefore appears to have the inverse ratio (k'/sqrt(hf)) and needs to be corrected; otherwise the claimed equality of the pole-skipping result with the other two methods does not follow from the printed derivation.
minor comments (4)
  1. [3.1, text after Eq. (3.7)] The text states that the small black hole branch is the 'large zh solution'; this should read 'small zh solution'.
  2. [4.3, Eq. (4.13)] The displayed near-horizon limit A(0) contains an explicit factor of v. Since z = z_h - uv, the u → 0 limit of A(u,v) in a static black hole should be independent of v, as in Eqs. (3.17) and (3.23). This suggests a typo in the OTOC computation for the bottom-up model.
  3. [4.2, text after Eq. (4.9)] The statement that the large-temperature limit corresponds to z_h → ∞ and the expansion in Eq. (4.10) should be re-derived from the corrected v_B^2 expression; with the displayed Eq. (4.9) the expansion does not follow.
  4. [Figure 6 caption] There is a typo in the caption: 'temperture' should be 'temperature'.

Circularity Check

1 steps flagged · score 6.0 of 10

As printed, Eq. (4.9) sets v_B^2 equal to the temperature expression (4.7), so the Section 4 model's butterfly velocity is the input T relabeled; the rest of the derivation is not circular.

  1. other [Section 4.2, Eq. (4.9); compared with Eq. (4.7); reproduced in Secs. 4.3 and 4.4]
    "v2B = 9a2e3a/z2h/(2πzh(e3a/z2h(3a−z2h)+z2h)) + μ2e3a/z2h( (9a2c(e(3a+c)/z2h(3a+c−z2h)+z2h))/((3a+c)2(e3a/z2h(3a−z2h)+z2h)) − cec/z2h )/(2πz3h(ec/z2h−1)2). (4.9)"

    The right-hand side of Eq. (4.9) is term-by-term identical to the temperature expression in Eq. (4.7). As printed, therefore, the claimed analytic butterfly velocity for the potential-reconstruction model is just the input temperature T, not an independent dimensionless v_B^2. This is confirmed by the paper's own high-temperature expansion, Eq. (4.10), which has v_B^2 → 2/3, whereas Eq. (4.9) would grow like T. Since Sections 4.3 and 4.4 state that the OTOC and pole-skipping methods recover exactly Eq. (4.9), and Figure 6 is evidently generated from it, the model's v_B(T, μ), the claimed three-method equivalence for this model, and the model's contribution to the universal trend all reduce, as printed, to a relabeling of the input temperature rather than a derived prediction.

full rationale

The paper is not globally circular: the three methods are reduced in Section 2 to the same general metric expression for v_B, and for the 1RCBH, 2RCBH, and numerical EMD models the subsequent substitutions use known analytic solutions or numerical backgrounds whose parameters were fixed against external QCD inputs (deconfinement temperature, Regge slopes, lattice thermodynamics), not against butterfly-velocity data. The self-citations to prior model constructions are therefore not load-bearing in a circular way. The one concrete reduction-by-construction is in Section 4: Eq. (4.9), presented as the analytic v_B^2 of the potential-reconstruction model, has a right-hand side identical to the temperature formula Eq. (4.7). That makes one of the paper's four model predictions literally equal to an input quantity and inconsistent with the same paper's Eq. (4.10). The numerical-model pole-skipping condition in Section 5.4 is asserted rather than verified, but it is an unverified assumption based on diffeomorphism invariance and external pole-skipping results, not a fit or self-citation; it lowers correctness confidence but does not add circularity. Because the reduction affects the central equivalence claim for one of the four models, the score is 6 rather than 0-2.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced. The paper's free parameters are inherited from the underlying holographic QCD models, fitted to QCD thermodynamics and spectra in prior work. The central computations use standard holographic chaos techniques; the main new input is the choice of models.

free parameters (5)
  • a (potential reconstruction model) = 0.15 GeV^2
    Fixed in the potential-reconstruction model to reproduce the deconfinement transition temperature around 0.270 GeV in the pure glue sector (Sec. 4.1).
  • c (potential reconstruction model) = 1.16 GeV^2
    Fixed to reproduce the linear Regge trajectory of heavy mesons (Sec. 4.1).
  • c1, c2, c4, c6 (numerical model dilaton potential) = 0.606, 0.703, -0.1, 0.0034
    Parameters of the dilaton potential in the numerical bottom-up model, fitted to lattice QCD thermodynamics (Sec. 5.1).
  • b1, b2, b3 (numerical model gauge-kinetic function) = 1.2, 0.69, 100
    Parameters of the gauge-kinetic function; b3=100 is chosen by hand to enforce f(0)=1, creating a narrow spike discussed in Sec. 5.1 and 5.4.
  • Lambda (energy scale) = 831 MeV
    Energy scale introduced in Sec. 5.1 to convert numerical black hole data to physical units; acts as a unit-setting input rather than a fit to the butterfly velocity.
assumptions (5)
  • domain assumption Gauge/gravity duality maps the strongly coupled QCD-like theory to a classical Einstein-Maxwell-dilaton gravity in AdS.
    Used throughout (Secs. 1, 3, 4, 5) to describe the boundary plasma via bulk black hole solutions. This is the foundational framework of holographic QCD.
  • domain assumption The butterfly velocity is extracted from the near-horizon exponential ansatz u(r,t) ~ e^{sigma r - 2 pi t / beta} / r^n for the RT surface (Eqs. 2.7-2.10).
    Adapted from [68,57], this assumes the operator growth front is captured by the near-horizon RT surface departure scale. If the ansatz fails, formula (2.13) for v_B would be invalid.
  • domain assumption Pole-skipping occurs at omega_* = i lambda_L = 2 pi i T and q_* = i lambda_L / v_B, and the matter sector satisfies delta T^z_v = 0 at the horizon (Eqs. 2.36, 2.44).
    Key assumption for the pole-skipping method; for the numerical model it is asserted rather than shown (Sec. 5.4). If delta T^z_v != 0, Eq. (2.47) for q_* would be modified.
  • domain assumption The three methods (entanglement wedge, OTOC, pole-skipping) compute the same butterfly velocity for metrics of the form (2.1).
    Equivalence established in [57,67]; the paper re-derives it for the general isotropic planar metric and applies it to the models.
  • domain assumption The EMD models are valid holographic descriptions of the deconfined QCD phase.
    The paper analyzes four EMD models; if any model fails to capture QCD physics, the universal trend claim is weakened.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Comparative study of the butterfly velocity in holographic QCD models at finite temperature and chemical potential." pith.science (2026). https://pith.science/paper/SS64UIFH

@misc{pith2026250515357,
  author       = {Pith},
  title        = {Pith review of: Comparative study of the butterfly velocity in holographic QCD models at finite temperature and chemical potential},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SS64UIFH}},
  note         = {Machine review of arXiv:2505.15357}
}
read the original abstract

In this work, we study quantum chaos in a variety of holographic QCD models at finite temperature and chemical potentials. This includes the 1 R-Charge black hole (1RCBH) model, the 2 R-Charge black hole (2RCBH) model, a potential reconstruction-based analytic bottom-up model, and a numerical bottom-up model. All these models are different avatars of the Einstein-Maxwell-dilaton gravity action, distinguished by their specific choices of dilaton potentials and gauge-kinetic coupling functions. We focus on computing the chaos parameter, the butterfly velocity, using three distinct methods: entanglement wedge reconstruction, out-of-time-ordered correlators (OTOCs), and pole-skipping. We show that all three methods yield identical results for the butterfly velocity across all the holographic QCD models considered, further establishing the equivalence between the three approaches. Furthermore, we analyze in detail the behavior of the butterfly velocity as a function of chemical potential and temperature. Interestingly, a universal trend emerges across all models: the butterfly velocity increases/decreases with temperature/chemical potential for thermodynamically stable phases. Additionally, in the high-temperature limit, the butterfly velocity in all models approaches that of the chargeless plasma.

Figures

Figures reproduced from arXiv: 2505.15357 by the authors.

Figure 1
Figure 1. The yellow shaded region represents the entanglement wedge and the red dot represents [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The Penrose diagram (corresponding to a two-sided black hole geometry) illustrates the [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. , we have only plotted the thermodynamically stable solution. Note that the range of values of µ1/T1 probed by the 1RCBH model is [0,π/ √ 2] and this is automatically taken care of, due to the presence of the square root (q π 2T 2 1 −2µ 2 1 ) in (3.11), as is also evident from the figures. We find that, in the allowed range of parameters µ1 and T1, v 2 B always exhibits a monotonically decreasing profile with respec… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Variation of vB with respect to µ2 and T2 for the 2RCBH model. 3.3 Calculation of vB using OTOC method The metric functions for the 1RCBH model are given in (3.8). Substituting these metric functions into (2.22) and using (2.25), we get the following expressions for A(…
Figure 5
Figure 5. Figure 5: Variation of deconfinement transition temperature [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]
Figure 6
Figure 6. Figure 6: Variation of vB with respect to µ and T for the analytic bottom-up holographic QCD model. monotonically with the chemical potential in the deconfined phase. This is true for all temperatures greater than the deconfined temperature. Similarly, for a fixed µ, v 2 B incre…
Figure 7
Figure 7. Figure 7: Pole-skipping points q∗ and ω∗ as a function of temperature and chemical potential. For completeness, we also analyze the overall structure of the pole-skipping points. This is shown in [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Variation of vB with respect to µ and T. In units of MeV. The numerical results of the butterfly velocity as a function of temperature and chemical potential are shown in [PITH_FULL_IMAGE:figures/full_fig_p031_8.png]
Figure 9
Figure 9. Figure 9: Pole-skipping points ω∗ and q∗ as a function of temperature and chemical potential. In units of MeV. It is also interesting to analyze the structure of the pole-skipping points in this model. The results are shown in [PITH_FULL_IMAGE:figures/full_fig_p033_9.png]
Figure 10
Figure 10. Figure 10: Variation of vB with respect to µ and T for RN-AdS background. The near-horizon limits are A(0) = (Q 2 −2)zh α2 , ∂u∂vB(0) = −2zh . (A.9) Putting these near-horizon values into (2.35), we get the same result as (A.5). A.4 Calculation of vB using pole-skipping method S…

Discussion (0). Sign in to comment.

Forward citations

Cited by 5 Pith papers

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

  1. Cosmological pole-skipping, shock waves and quantum chaotic dynamics of de Sitter horizons

    hep-th 2025-08 conditional novelty 7.0 of 10

    Pole-skipping in Schwarzschild-de Sitter predicts superluminal and imaginary butterfly velocities, confirmed by shock wave analysis, hinting at nonlocal and non-Hermitian dual dynamics.

  2. Quark flavors in hot and dense holographic QCD: setup and comparison to data

    hep-ph 2025-07 conditional novelty 7.0 of 10

    A 2+1 flavor holographic QCD model fitted to lattice thermodynamics predicts a smoother nuclear-to-quark matter transition with lower latent heat than earlier V-QCD models.

  3. Diagnosing Critical Behavior in AdS Einstein-Maxwell-Scalar Theory via Holographic Entanglement Measures

    hep-th 2025-12 conditional novelty 4.0 of 10

    In the EMS holographic model, entanglement entropy, mutual information, wedge cross-section, and butterfly velocity all show critical exponent 1—twice the scalar order parameter—and MI grows faster than EWCS across th...

  4. A Dynamical Systems Framework for Reinforcement Learning Safety and Robustness Verification

    cs.AI 2025-08 unverdicted novelty 4.0 of 10

    The claimed RL safety verification framework is absent from the manuscript; the body text is an unrelated high-energy physics paper about de Sitter horizon chaos.

  5. Charged Black Hole with String Cloud Deformation: Entanglement and Chaos

    hep-th 2025-07 conditional novelty 4.0 of 10

    In a charged AdS black hole with a string cloud, entanglement entropy and entanglement wedge cross-section increase with charge and backreaction, while mutual information and butterfly velocity decrease with charge.

Reference graph

Works this paper leans on

120 extracted references · 18 canonical work pages · cited by 5 Pith papers

  1. [1]

    Srednicki, Chaos and Quantum Thermalization, Phys

    M. Srednicki, Chaos and Quantum Thermalization, Phys. Rev. E 50 (1994) [cond-mat/9403051]

    arXiv 1994

  2. [2]

    Suzuki and K.-i

    S. Suzuki and K.-i. Maeda, Chaos in Schwarzschild space-time: The motion of a spinning particle, Phys. Rev. D 55 (1997) 4848 [gr-qc/9604020]

    arXiv 1997

  3. [3]

    Natsuume and T

    M. Natsuume and T. Okamura, Holographic chaos, pole-skipping, and regularity, PTEP 2020 (2020) 013B07 [1905.12014]

    arXiv 2020

  4. [4]

    Maldacena, S.H

    J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [1503.01409]. 37

    arXiv 2016

  5. [5]

    Akutagawa, K

    T. Akutagawa, K. Hashimoto, K. Murata and T. Ota, Chaos of qcd string from holography, Phys. Rev. D 100 (2019) 046009

    2019

  6. [6]

    Giataganas, U

    D. Giataganas, U. G ¨ursoy and J. Pedraza, Strongly coupled anisotropic gauge theories and holography, Phys. Rev. Lett. 121 (2018) 121601

    2018

  7. [7]

    Maldacena, The Large N limit of superconformal field theories and supergravity, Adv

    J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200]

    arXiv 1998

  8. [8]

    Gubser, I.R

    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109]

    arXiv 1998

Show all 120 references
  1. [9]

    Witten, Anti-de Sitter space and holography, Adv

    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150]

    1998 arXiv

  2. [10]

    Casalderrey-Solana, H

    J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U. Achim Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions, Cambridge University Press (2014), 10.1017/9781009403504, [1101.0618]

    2014 arXiv

  3. [11]

    Rougemont, J

    R. Rougemont, J. Grefa, M. Hippert, J. Noronha, J. Noronha-Hostler, I. Portillo et al., Hot QCD phase diagram from holographic Einstein–Maxwell–Dilaton models, Prog. Part. Nucl. Phys. 135 (2024) 104093 [2307.03885]

    2024 arXiv

  4. [12]

    Shenker and D

    S.H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067 [1306.0622]

    2014 arXiv

  5. [13]

    Roberts, D

    D.A. Roberts, D. Stanford and L. Susskind, Localized shocks, JHEP 03 (2015) 051 [1409.8180]

    2015 arXiv

  6. [14]

    Roberts and B

    D.A. Roberts and B. Swingle, Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories, Phys. Rev. Lett. 117 (2016) 091602 [1603.09298]

    2016 arXiv

  7. [15]

    Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP 10 (2016) 069 [1602.08272]

    E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP 10 (2016) 069 [1602.08272]

    2016 arXiv

  8. [16]

    Jahnke,Recent developments in the holographic description of quantum chaos, Adv

    V . Jahnke,Recent developments in the holographic description of quantum chaos, Adv. High Energy Phys. 2019 (2019) 9632708 [1811.06949]

    2019 arXiv

  9. [17]

    Sekino and L

    Y . Sekino and L. Susskind,Fast Scramblers, JHEP 10 (2008) 065 [0808.2096]

    2008 arXiv

  10. [18]

    Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112]

    J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112]

    2003 arXiv

  11. [19]

    Akutagawa, K

    T. Akutagawa, K. Hashimoto, T. Miyazaki and T. Ota, Phase diagram of QCD chaos in linear sigma models and holography, PTEP 2018 (2018) 063B01 [1804.01737]

    2018 arXiv

  12. [20]

    Akutagawa, K

    T. Akutagawa, K. Hashimoto, K. Murata and T. Ota, Chaos of QCD string from holography, Phys. Rev. D 100 (2019) 046009 [1903.04718]

    2019 arXiv

  13. [21]

    Hashimoto, K

    K. Hashimoto, K. Murata and K. Yoshida, Chaos in chiral condensates in gauge theories, Phys. Rev. Lett. 117 (2016) 231602 [1605.08124]. 38

    2016 arXiv

  14. [22]

    Hashimoto, K

    K. Hashimoto, K. Murata and N. Tanahashi, Chaos of Wilson Loop from String Motion near Black Hole Horizon, Phys. Rev. D 98 (2018) 086007 [1803.06756]

    2018 arXiv

  15. [23]

    Shukla, D

    B. Shukla, D. Dudal and S. Mahapatra, Anisotropic and frame dependent chaos of suspended strings from a dynamical holographic QCD model with magnetic field, JHEP 06 (2023) 178 [2303.15716]

    2023 arXiv

  16. [24]

    Shukla, J

    B. Shukla, J. Nongmaithem, D. Dudal and S. Mahapatra, Interplay of magnetic field and chemical potential induced anisotropy and frame dependent chaos of a QQ¯ pair in holographic QCD, Phys. Rev. D 111 (2025) 106002 [2411.17279]

    2025 arXiv

  17. [25]

    Colangelo, F

    P. Colangelo, F. Giannuzzi and N. Losacco, Chaotic dynamics of a suspended string in a gravitational background with magnetic field, Phys. Lett. B 827 (2022) 136949 [2111.09441]

    2022 arXiv

  18. [26]

    Colangelo, F

    P. Colangelo, F. De Fazio and N. Losacco, Chaos in a Q ¯Q system at finite temperature and baryon density, Phys. Rev. D 102 (2020) 074016 [2007.06980]

    2020 arXiv

  19. [27]

    Li, Y .-p

    S.-w. Li, Y .-p. Zhang and H.-q. Li,Out-of-time-order correlator as a detector of baryonic phase structure in holographic QCD with instanton, Phys. Lett. B 857 (2024) 138972 [2403.07335]

    2024 arXiv

  20. [28]

    Anegawa, N

    T. Anegawa, N. Iizuka and M. Nishida, Krylov complexity as an order parameter for deconfinement phase transitions at large N, JHEP 04 (2024) 119 [2401.04383]

    2024 arXiv

  21. [29]

    Baishya, S

    B. Baishya, S. Chakrabarti, D. Maity and K. Nayek, Pole-skipping and chaos in D3-D7 brane systems, Phys. Rev. D 110 (2024) 086003 [2312.01829]

    2024 arXiv

  22. [30]

    Shukla, O

    B. Shukla, O. Riyaz and S. Mahapatra, Classical and quantum chaos of closed strings on a charged confining holographic background, Phys. Rev. D 111 (2025) 066019 [2411.12536]

    2025 arXiv

  23. [31]

    Pando Zayas and C.A

    L.A. Pando Zayas and C.A. Terrero-Escalante, Chaos in the Gauge / Gravity Correspondence, JHEP 09 (2010) 094 [1007.0277]

    2010 arXiv

  24. [32]

    P. Basu, D. Das, A. Ghosh and L.A. Pando Zayas, Chaos around Holographic Regge Trajectories, JHEP 05 (2012) 077 [1201.5634]

    2012 arXiv

  25. [33]

    Ishii, K

    T. Ishii, K. Murata and K. Yoshida, Fate of chaotic strings in a confining geometry, Phys. Rev. D 95 (2017) 066019 [1610.05833]

    2017 arXiv

  26. [34]

    P. Basu, P. Chaturvedi and P. Samantray,Chaotic dynamics of strings in charged black hole backgrounds, Phys. Rev. D 95 (2017) 066014 [1607.04466]

    2017 arXiv

  27. [35]

    Basu and L.A

    P. Basu and L.A. Pando Zayas, Chaos rules out integrability of strings on AdS5 × T 1,1, Phys. Lett. B 700 (2011) 243 [1103.4107]

    2011 arXiv

  28. [36]

    Banerjee, A

    A. Banerjee, A. Kundu and R.R. Poojary, Strings, branes, Schwarzian action and maximal chaos, Phys. Lett. B 838 (2023) 137632 [1809.02090]

    2023 arXiv

  29. [37]

    J. Xie, Y . Wang and B. Tang,Chaotic dynamics of strings around the Bardeen-AdS black hole surrounded by quintessence dark energy, Phys. Dark Univ. 40 (2023) 101184 [2210.08641]

    2023 arXiv

  30. [38]

    D.-Z. Ma, F. Xia, D. Zhang, G.-Y . Fu and J.-P. Wu,Chaotic dynamics of string around the conformal black hole, Eur. Phys. J. C 82 (2022) 372 [2205.00226]. 39

    2022 arXiv

  31. [39]

    Pen ´ın and K.C

    J.M. Pen ´ın and K.C. Rigatos, Evidence for a four-dimensional N=1 integrable quiver in massive type IIA supergravity, Phys. Rev. D 109 (2024) 126007 [2412.12257]

    2024 arXiv

  32. [40]

    Rigatos, Nonintegrability of La,b,c quiver gauge theories, Phys

    K.S. Rigatos, Nonintegrability of La,b,c quiver gauge theories, Phys. Rev. D 102 (2020) 106022 [2009.11878]

    2020 arXiv

  33. [41]

    P. Basu, D. Das and A. Ghosh, Integrability Lost, Phys. Lett. B 699 (2011) 388 [1103.4101]

    2011 arXiv

  34. [42]

    Swingle, G

    B. Swingle, G. Bentsen, M. Schleier-Smith and P. Hayden, Measuring the scrambling of quantum information, Phys. Rev. A 94 (2016) 040302 [1602.06271]

    2016 arXiv

  35. [43]

    G. Zhu, M. Hafezi and T. Grover, Measurement of many-body chaos using a quantum clock, Phys. Rev. A 94 (2016) 062329 [1607.00079]

    2016 arXiv

  36. [44]

    J. Li, R. Fan, H. Wang, B. Ye, B. Zeng, H. Zhai et al., Measuring Out-of-Time-Order Correlators on a Nuclear Magnetic Resonance Quantum Simulator, Phys. Rev. X 7 (2017) 031011 [1609.01246]

    2017 arXiv

  37. [45]

    N.Y . Yao, F. Grusdt, B. Swingle, M.D. Lukin, D.M. Stamper-Kurn, J.E. Moore et al., Interferometric Approach to Probing Fast Scrambling, 1607.01801

    arXiv

  38. [46]

    Blake, Universal charge diffusion and the butterfly effect in holographic theories, Phys

    M. Blake, Universal charge diffusion and the butterfly effect in holographic theories, Phys. Rev. Lett. 117 (2016) 091601

    2016

  39. [47]

    Grozdanov, On the connection between hydrodynamics and quantum chaos in holographic theories with stringy corrections, JHEP 01 (2019) 048

    S. Grozdanov, On the connection between hydrodynamics and quantum chaos in holographic theories with stringy corrections, JHEP 01 (2019) 048

    2019

  40. [48]

    Y . Ling, P. Liu and J. Wu,Holographic butterfly effect at quantum critical points, JHEP 10 (2017) 025

    2017

  41. [49]

    G ¨ursoy, M

    U. G ¨ursoy, M. J¨arvinen, G. Nijs and J. Pedraza, On the interplay between magnetic field and anisotropy in holographic qcd, JHEP 03 (2021) 180

    2021

  42. [50]

    Ageev, Chaotic nature of holographic QCD, Phys

    D.S. Ageev, Chaotic nature of holographic QCD, Phys. Rev. D 104 (2021) 126013 [2105.04589]

    2021 arXiv

  43. [51]

    DeWolfe, S.S

    O. DeWolfe, S.S. Gubser and C. Rosen, Dynamic critical phenomena at a holographic critical point, Phys. Rev. D 84 (2011) 126014 [1108.2029]

    2011 arXiv

  44. [52]

    DeWolfe, S.S

    O. DeWolfe, S.S. Gubser and C. Rosen, Fermi surfaces in N=4 Super-Yang-Mills theory, Phys. Rev. D 86 (2012) 106002 [1207.3352]

    2012 arXiv

  45. [53]

    Dudal and S

    D. Dudal and S. Mahapatra, Thermal entropy of a quark-antiquark pair above and below deconfinement from a dynamical holographic QCD model, Phys. Rev. D 96 (2017) 126010 [1708.06995]

    2017 arXiv

  46. [54]

    Bohra, D

    H. Bohra, D. Dudal, A. Hajilou and S. Mahapatra, Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model, Phys. Lett. B 801 (2020) 135184 [1907.01852]

    2020 arXiv

  47. [55]

    DeWolfe, S.S

    O. DeWolfe, S.S. Gubser and C. Rosen, A holographic critical point, Phys. Rev. D 83 (2011) 086005 [1012.1864]. 40

    2011 arXiv

  48. [56]

    Rougemont, A

    R. Rougemont, A. Ficnar, S. Finazzo and J. Noronha, Energy loss, equilibration, and thermodynamics of a baryon rich strongly coupled quark-gluon plasma, JHEP 04 (2016) 102 [1507.06556]

    2016 arXiv

  49. [57]

    X. Dong, D. Wang, W.W. Weng and C.-H. Wu,A tale of two butterflies: an exact equivalence in higher-derivative gravity, JHEP 10 (2022) 009 [2203.06189]

    2022 arXiv

  50. [58]

    Baishya, A

    B. Baishya, A. Chakraborty and N. Padhi, Entanglement wedge method, out-of-time-ordered correlators, and pole skipping, Phys. Rev. D 111 (2025) 106013 [2406.18319]

    2025 arXiv

  51. [59]

    Lilani, Chaos in hyperscaling violating Lifshitz theories, 2411.09667

    N. Lilani, Chaos in hyperscaling violating Lifshitz theories, 2411.09667

    arXiv

  52. [60]

    D. Basu, A. Chandra and Q. Wen, Butterfly effect and TT-deformation, 2505.14331

    arXiv

  53. [61]

    DiNunno, N

    B.S. DiNunno, N. Jokela, J.F. Pedraza and A. P ¨onni, Quantum information probes of charge fractionalization in large-N gauge theories, JHEP 05 (2021) 149 [2101.11636]

    2021 arXiv

  54. [62]

    Fischler, V

    W. Fischler, V . Jahnke and J.F. Pedraza,Chaos and entanglement spreading in a non-commutative gauge theory, JHEP 11 (2018) 072 [1808.10050]

    2018 arXiv

  55. [63]

    Eccles, W

    S. Eccles, W. Fischler, T. Guglielmo, J.F. Pedraza and S. Racz, Speeding up the spread of quantum information in chaotic systems, JHEP 12 (2021) 019 [2108.12688]

    2021 arXiv

  56. [64]

    Saha and S

    A. Saha and S. Gangopadhyay, Quantum chaos in the presence of nonconformality, Phys. Rev. D 110 (2024) 026025 [2401.05814]

    2024 arXiv

  57. [65]

    Chakrabortty, H

    S. Chakrabortty, H. Hoshino, S. Pant and K. Sil, A holographic study of the characteristics of chaos and correlation in the presence of backreaction, Phys. Lett. B 838 (2023) 137749 [2206.12555]

    2023 arXiv

  58. [66]

    Karan and S

    D. Karan and S. Pant, Entanglement and Chaos near critical point in strongly coupled Gauge theory, Eur. Phys. J. C 84 (2024) 113 [2308.00018]

    2024 arXiv

  59. [67]

    W.Z. Chua, T. Hartman and W.W. Weng,Replica manifolds, pole skipping, and the butterfly effect, 2504.08139

    arXiv

  60. [68]

    Mezei and D

    M. Mezei and D. Stanford, On entanglement spreading in chaotic systems, JHEP 05 (2017) 065 [1608.05101]

    2017 arXiv

  61. [69]

    Czech, J.L

    B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The Gravity Dual of a Density Matrix, Class. Quant. Grav. 29 (2012) 155009 [1204.1330]

    2012 arXiv

  62. [70]

    Wall, Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class

    A.C. Wall, Maximin Surfaces, and the Strong Subadditivity of the Covariant Holographic Entanglement Entropy, Class. Quant. Grav. 31 (2014) 225007 [1211.3494]

    2014 arXiv

  63. [71]

    Headrick, V .E

    M. Headrick, V .E. Hubeny, A. Lawrence and M. Rangamani,Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [1408.6300]

    2014 arXiv

  64. [72]

    X. Dong, D. Harlow and A.C. Wall, Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality, Phys. Rev. Lett. 117 (2016) 021601 [1601.05416]. 41

    2016 arXiv

  65. [73]

    Kim and M.J

    I.H. Kim and M.J. Kastoryano, Entanglement renormalization, quantum error correction, and bulk causality, JHEP 04 (2017) 040 [1701.00050]

    2017 arXiv

  66. [74]

    Ryu and T

    S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001]

    2006 arXiv

  67. [75]

    Hubeny, M

    V .E. Hubeny, M. Rangamani and T. Takayanagi,A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [0705.0016]

    2007 arXiv

  68. [76]

    Grozdanov, K

    S. Grozdanov, K. Schalm and V . Scopelliti,Black hole scrambling from hydrodynamics, Phys. Rev. Lett. 120 (2018) 231601 [1710.00921]

    2018 arXiv

  69. [77]

    Blake, H

    M. Blake, H. Lee and H. Liu, A quantum hydrodynamical description for scrambling and many-body chaos, JHEP 10 (2018) 127 [1801.00010]

    2018 arXiv

  70. [78]

    Blake, R.A

    M. Blake, R.A. Davison, S. Grozdanov and H. Liu, Many-body chaos and energy dynamics in holography, JHEP 10 (2018) 035 [1809.01169]

    2018 arXiv

  71. [79]

    Blake, R.A

    M. Blake, R.A. Davison and D. Vegh, Horizon constraints on holographic Green’s functions, JHEP 01 (2020) 077 [1904.12883]

    2020 arXiv

  72. [80]

    Wang and Z.-Y

    D. Wang and Z.-Y . Wang,Pole Skipping in Holographic Theories with Bosonic Fields, Phys. Rev. Lett. 129 (2022) 231603 [2208.01047]

    2022 arXiv

  73. [81]

    S. Ning, D. Wang and Z.-Y . Wang,Pole skipping in holographic theories with gauge and fermionic fields, JHEP 12 (2023) 084 [2308.08191]

    2023 arXiv

  74. [82]

    Jeong, K.-Y

    H.-S. Jeong, K.-Y . Kim and Y .-W. Sun,Bound of diffusion constants from pole-skipping points: spontaneous symmetry breaking and magnetic field, JHEP 07 (2021) 105 [2104.13084]

    2021 arXiv

  75. [83]

    Behrndt, M

    K. Behrndt, M. Cvetic and W.A. Sabra, Nonextreme black holes of five-dimensional N=2 AdS supergravity, Nucl. Phys. B 553 (1999) 317 [hep-th/9810227]

    1999 arXiv

  76. [84]

    Cvetic and S.S

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

    1999 arXiv

  77. [85]

    Ebrahim, M

    H. Ebrahim, M. Asadi and M. Ali-Akbari, Evolution of Holographic Complexity Near Critical Point, JHEP 09 (2019) 023 [1811.12002]

    2019 arXiv

  78. [86]

    Ebrahim and G.-M

    H. Ebrahim and G.-M. Nafisi, Holographic Mutual Information and Critical Exponents of the Strongly Coupled Plasma, Phys. Rev. D 102 (2020) 106007 [2002.09993]

    2020 arXiv

  79. [87]

    Amrahi, M

    B. Amrahi, M. Asadi and F. Taghinavaz, Chaos near to the critical point: butterfly effect and pole-skipping, Eur. Phys. J. C 84 (2024) 505 [2305.00298]

    2024

  80. [88]

    de Oliveira, W

    G. de Oliveira, W. Barreto and R. Rougemont, Universality of the entropy stairway in homogeneous isotropization, 2506.10294

  81. [89]

    de Oliveira and R

    G. de Oliveira and R. Rougemont, New purely damped pairs of quasinormal modes in a hot and dense strongly-coupled plasma, JHEP 11 (2024) 079 [2408.09498]. 42

    2024 arXiv

  82. [90]

    Finazzo, R

    S.I. Finazzo, R. Critelli, R. Rougemont and J. Noronha, Momentum transport in strongly coupled anisotropic plasmas in the presence of strong magnetic fields, Phys. Rev. D 94 (2016) 054020 [1605.06061]

    2016 arXiv

  83. [91]

    Mahapatra and P

    S. Mahapatra and P. Roy, On the time dependence of holographic complexity in a dynamical Einstein-dilaton model, JHEP 11 (2018) 138 [1808.09917]

    2018 arXiv

  84. [92]

    Aref’eva and K

    I. Aref’eva and K. Rannu, Holographic Anisotropic Background with Confinement-Deconfinement Phase Transition, JHEP 05 (2018) 206 [1802.05652]

    2018 arXiv

  85. [93]

    Aref’eva, K

    I.Y . Aref’eva, K. Rannu and P. Slepov,Holographic anisotropic model for light quarks with confinement-deconfinement phase transition, JHEP 06 (2021) 090 [2009.05562]

    2021 arXiv

  86. [94]

    Mahapatra, S

    S. Mahapatra, S. Priyadarshinee, G.N. Reddy and B. Shukla, Exact topological charged hairy black holes in AdS Space in D-dimensions, Phys. Rev. D 102 (2020) 024042 [2004.00921]

    2020 arXiv

  87. [95]

    Priyadarshinee, S

    S. Priyadarshinee, S. Mahapatra and I. Banerjee, Analytic topological hairy dyonic black holes and thermodynamics, Phys. Rev. D 104 (2021) 084023 [2108.02514]

    2021 arXiv

  88. [96]

    Priyadarshinee and S

    S. Priyadarshinee and S. Mahapatra, Analytic three-dimensional primary hair charged black holes and thermodynamics, Phys. Rev. D 108 (2023) 044017 [2305.09172]

    2023 arXiv

  89. [97]

    R.-G. Cai, S. He and D. Li, A hQCD model and its phase diagram in Einstein-Maxwell-Dilaton system, JHEP 03 (2012) 033 [1201.0820]

    2012 arXiv

  90. [98]

    Daripa and S

    A. Daripa and S. Mahapatra, Analytic three-dimensional primary hair charged black holes with Coulomb-like electrodynamics and their thermodynamics, Phys. Rev. D 109 (2024) 124039 [2401.04561]

    2024 arXiv

  91. [99]

    Gubser, Curvature singularities: The Good, the bad, and the naked, Adv

    S.S. Gubser, Curvature singularities: The Good, the bad, and the naked, Adv. Theor. Math. Phys. 4 (2000) 679 [hep-th/0002160]

    2000 arXiv

  92. [100]

    Breitenlohner and D.Z

    P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-De Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197

    1982

  93. [101]

    B. Chen, X. Chen, X. Li, Z.-R. Zhu and K. Zhou, Exploring transport properties of quark-gluon plasma in flavor-dependent systems with a holographic model, Phys. Rev. D 111 (2025) 086033 [2404.18217]

    2025 arXiv

  94. [102]

    S.S. Jena, A. Bhattacharjee, D. Dudal and S. Mahapatra, Probing Quarkonium Diffusion in a Magnetized Quark-Gluon Plasma, 2507.00746

  95. [103]

    S.S. Jena, J. Barman, B. Toniato, D. Dudal and S. Mahapatra, A dynamical Einstein-Born-Infeld-dilaton model and holographic quarkonium melting in a magnetic field, JHEP 12 (2024) 096 [2408.14813]

    2024 arXiv

  96. [104]

    S.S. Jena, B. Shukla, D. Dudal and S. Mahapatra, Entropic force and real-time dynamics of holographic quarkonium in a magnetic field, Phys. Rev. D 105 (2022) 086011 [2202.01486]

    2022 arXiv

  97. [105]

    Bohra, D

    H. Bohra, D. Dudal, A. Hajilou and S. Mahapatra, Chiral transition in the probe approximation from an Einstein-Maxwell-dilaton gravity model, Phys. Rev. D 103 (2021) 086021 [2010.04578]. 43

    2021 arXiv

  98. [106]

    Critelli, J

    R. Critelli, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti and R. Rougemont, Critical point in the phase diagram of primordial quark-gluon matter from black hole physics, Phys. Rev. D 96 (2017) 096026 [1706.00455]

    2017 arXiv

  99. [107]

    Knaute and B

    J. Knaute and B. K ¨ampfer, Holographic Entanglement Entropy in the QCD Phase Diagram with a Critical Point, Phys. Rev. D 96 (2017) 106003 [1706.02647]

    2017 arXiv

  100. [108]

    R.-G. Cai, S. He, L. Li and Y .-X. Wang,Probing QCD critical point and induced gravitational wave by black hole physics, Phys. Rev. D 106 (2022) L121902 [2201.02004]

    2022 arXiv

  101. [109]

    Jokela, M

    N. Jokela, M. J ¨arvinen and A. Piispa, Refining holographic models of the quark-gluon plasma, Phys. Rev. D 110 (2024) 126013 [2405.02394]

    2024 arXiv

  102. [110]

    Attems, J

    M. Attems, J. Casalderrey-Solana, D. Mateos, I. Papadimitriou, D. Santos-Oliv ´an, C.F. Sopuerta et al., Thermodynamics, transport and relaxation in non-conformal theories, JHEP 10 (2016) 155 [1603.01254]

    2016 arXiv

  103. [111]

    Zhang, Complexity and phase transitions in a holographic QCD model, Nucl

    S.-J. Zhang, Complexity and phase transitions in a holographic QCD model, Nucl. Phys. B 929 (2018) 243 [1712.07583]

    2018 arXiv

  104. [112]

    Rougemont, R

    R. Rougemont, R. Critelli and J. Noronha, Holographic calculation of the QCD crossover temperature in a magnetic field, Phys. Rev. D 93 (2016) 045013 [1505.07894]

    2016 arXiv

  105. [113]

    R.-G. Cai, S. He, L. Li and H.-A. Zeng, Neural Ordinary Differential Equations for Mapping the Magnetic QCD Phase Diagram via Holography, 2406.12772

  106. [114]

    S. He, L. Li, Z. Li and S.-J. Wang, Gravitational waves and primordial black hole productions from gluodynamics by holography, Sci. China Phys. Mech. Astron. 67 (2024) 240411 [2210.14094]

    2024 arXiv

  107. [115]

    S. Guin, H. Pandey and S. Sharma, Understanding thermalization in a non-Abelian gauge theory in terms of its soft modes, Phys. Lett. B 866 (2025) 139490 [2501.04397]

    2025 arXiv

  108. [116]

    Huang, Butterfly Velocity in Quadratic Gravity, Class

    W.-H. Huang, Butterfly Velocity in Quadratic Gravity, Class. Quant. Grav. 35 (2018) 195004 [1804.05527]

    2018 arXiv

  109. [117]

    Wang, P.-J

    Y . Wang, P.-J. Hu and Y . Pang,A holographic model of magnetohydrodynamics with fortuitous SO(3) symmetry, JHEP 10 (2024) 035 [2408.04791]

    2024 arXiv

  110. [118]

    Baggioli, B

    M. Baggioli, B. Padhi, P.W. Phillips and C. Setty, Conjecture on the Butterfly Velocity across a Quantum Phase Transition, JHEP 07 (2018) 049 [1805.01470]

    2018 arXiv

  111. [119]

    Shukla, P.P

    B. Shukla, P.P. Das, D. Dudal and S. Mahapatra, Interplay between the Lyapunov exponents and phase transitions of charged AdS black holes, Phys. Rev. D 110 (2024) 024068 [2404.02095]

    2024 arXiv

  112. [120]

    X. Guo, Y . Lu, B. Mu and P. Wang,Probing phase structure of black holes with Lyapunov exponents, JHEP 08 (2022) 153 [2205.02122]. 44

    2022 arXiv

Pith tools

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