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Long lived quasi normal modes

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For large Chern-Simons coupling, quasinormal modes of magnetically charged AdS black branes become purely real, implying long-lived current oscillations in the dual theory.

desk verdict Nice generalization, but the existence proof for real QNM frequencies has a real gap; referee it with a demand to fix or reframe. read the letter →

arxiv 2411.18039 v1 pith:KN3FVBYP submitted 2024-11-27 hep-th gr-qc

Sebastian Waeber , Amos Yarom This is my paper · ORCID
classification hep-thgr-qc PACS 11.25.Tq04.70.-s
keywords quasinormalmodesChern-SimonscouplingAdSblackbranesholographicdualitylong-livedanomalousU(1)currentSturm-Liouvilleproblemmagneticallychargedhole
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

This paper claims that in holographic theories with a sufficiently strong Chern-Simons (anomalous) coupling, magnetically charged AdS black branes whose zero-temperature geometry has vanishing entropy possess quasinormal modes with exactly real frequencies, meaning the dual currents oscillate without decaying. The result is meant to be robust: it does not depend on the specific matter content coupled to the gauge field, only on the near-horizon scaling exponents satisfying $2x+4y>1+z$ and on local stability against rotational symmetry breaking. If correct, it makes the long-lived modes found in earlier special models generic and potentially observable in condensed matter systems with an anomalous U(1) current, such as Weyl semimetals.

What carries the argument

The central object is the transformed fluctuation of the gauge field, $\delta a_3 = e^{-i\int_r^\infty \omega/f\, dx} k$, which converts the second-order equation (2.8) into the singular Sturm-Liouville equation $(f L V_F k')' - \frac{576 B^2}{L^3 V_F} k = -\omega^2 \frac{L V_F}{f} k$ with boundary conditions $k(0)=k(\infty)=0$. This machinery carries the argument: positivity of the eigenvalues follows from an integration by parts, existence follows from a finite-interval truncation and continuity in the boundary value $k_L$, and the condition $2x+4y>1+z$ ensures the magnetic-field term dominates near the horizon, making the problem genuinely Sturm-Liouville.

What would settle it

Compute numerically the lowest eigenvalue $\Omega$ of the singular Sturm-Liouville problem (3.14) with boundary conditions $k(0)=k(\infty)=0$ in an explicit model satisfying $2x+4y>1+z$ (e.g., the domain-wall model of section 4). If no such eigenvalue exists, or if the lowest eigenvalue is $\Omega=0$ without any accompanying rotational-symmetry-breaking mode, then the claim that $\omega$ is real and non-zero is falsified. Alternatively, compute the finite-temperature quasinormal spectrum and check whether the imaginary part extrapolates to zero as $T\to0$ in the same model.

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Extended reading notes

Core claim

The central claim is that for large $\lambda$, the quasinormal frequencies $\omega$ of the gauge field on a magnetically charged AdS black brane have vanishing imaginary part, provided the zero-temperature near-horizon asymptotics satisfy $2x+4y>1+z$ and the configuration is locally stable. The authors show this by reducing the fluctuation equation for $\delta a_3$ to a singular Sturm-Liouville problem whose eigenvalue $\Omega=\omega^2$ is shown to be non-negative; excluding a zero mode via local stability then forces $\omega$ to be real and non-zero. They argue that such real solutions exist by a continuity argument in the boundary value $k_L$, and they verify the scenario numerically in a domain-wall example with non-trivial scalar matter.

Load-bearing premise

The proof that real, non-zero frequencies exist relies on the claim that varying the boundary value $k_L$ can continuously force a solution to vanish at infinity, and on interpreting any zero mode of $\delta a_3$ as a rotational-symmetry-breaking instability; if either step fails, the conclusion that $\omega$ is real and non-zero does not follow.

Editorial extensions

If this is right

  • If the claim holds, then at low temperatures and large $\lambda$, the lowest quasinormal mode approaches the real frequency axis, so the dual anomalous current rings with a lifetime much longer than the quench that excites it.
  • The robustness of the result means long-lived current oscillations should appear across a wide class of holographic models, not just the Maxwell-plus-gravity theories studied earlier.
  • The estimated frequency $\omega_0 \sim C B \lambda c^{3/2} / \hbar^{1/2}$ gives a testable scale: for Weyl-semimetal parameters, $\omega_0 \sim 1$ THz at magnetic fields of order a Gauss, in the ballpark of experimental accessibility.
  • If a zero-temperature phase transition breaks rotational symmetry at some $T_c > 0$, the analysis still predicts increasingly small imaginary parts for $T$ just above $T_c$.
  • The condition $2x+4y>1+z$ ties the existence of long-lived modes to vanishing zero-temperature entropy; thus the third law of thermodynamics may select which holographic materials exhibit the effect.

Reading between the lines

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

  • The continuity-in-$k_L$ argument may be fragile: $k_L$ is a scale-invariant normalization, not a continuous physical tuning parameter, so the claim that some eigenvalue has $k(\infty)=0$ could require a separate existence proof that does not rely on varying the boundary value.
  • The exclusion of a zero mode identifies a zero-mode solution of $\delta a_3$ with rotational symmetry breaking, yet the paper's local stability condition only forbids modes that break rotation or translation invariance; if a zero mode exists without such breaking, the conclusion that $\omega$ is non-zero would need an additional argument.
  • A natural test beyond the paper is to compute the low-temperature imaginary part of the lowest quasinormal frequency in a concrete model and verify that it decreases as a power of $T$ with an exponent determined by the near-horizon scaling, rather than merely approaching zero.
  • The result suggests that anomalously long-lived oscillations could be searched for in driven Weyl semimetals by looking for a resonance peak in the current response whose width shrinks as temperature is lowered, a prediction that does not require a precise holographic dual.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper argues that magnetically charged AdS5 black branes with a sufficiently large Chern-Simons coupling λ have long-lived quasi-normal modes: for large λ, the imaginary part of the gauge-field quasi-normal frequencies becomes small, and in the zero-temperature limit the frequencies become real, provided the near-horizon asymptotics satisfy 2x+4y>1+z and the background is locally stable. The setup is a general action (2.1) in which the gauge field couples to unspecified matter only through VF and the Chern-Simons term. The fluctuation equation for δa3 is reduced to a Sturm-Liouville problem (3.14), and the paper proves non-negativity of ω² conditional on existence. Existence is then claimed via a truncated-interval Sturm-Liouville argument with a continuity step in the parameter kL. The paper also constructs an explicit zero-temperature domain-wall example satisfying the condition and presents finite-temperature numerical quasi-normal-mode data in Appendix B.

Significance. If the central claim holds, the paper would establish a robust, matter-agnostic holographic mechanism for anomalously long-lived current resonances, extending earlier special-case results [25-32] and giving a concrete experimental target in Weyl semimetals. The main strengths are the clean reduction to a singular Sturm-Liouville problem, the positivity argument leading to Ω≥0, the explicit domain-wall realization satisfying 2x+4y>1+z, and the numerical trends in which quasi-normal modes approach the real axis at low temperature. However, the existence proof for the required eigenmodes is not rigorous as written, and the exclusion of Ω=0 is not fully tied to the paper's own stability definition. The significance is therefore conditional on repairing these two load-bearing steps.

major comments (3)
  1. [Section 3, Eqs. (3.14)-(3.21)] The existence argument is not valid. The condition k(L)=k_L in Eq. (3.17b) is a normalization condition, not a homogeneous boundary condition. For the linear second-order equation (3.14a) with the Robin condition (3.17a) at r=ε, any solution can be rescaled to satisfy k(L)=k_L for any nonzero k_L, so the problem on [ε,L] does not define a standard Sturm-Liouville eigenvalue problem with a discrete spectrum Ω_0<Ω_1<... . Consequently the 'continuity in k_L' step leading to Eq. (3.21) has no basis: rescaling a solution changes k_L without changing Ω, and there is no continuous family of boundary-value problems connecting the boundary value +k_L to -k_L except through the genuine homogeneous condition k(L)=0. Without this step, the paper has established only the conditional statement that any mode satisfying k(0)=k(∞)=0 has Ω>0, not that such a mode exists.
  2. [Section 3, Eqs. (3.15)-(3.16)] The exclusion of Ω=0 is not established. The paper states that Ω=0 would imply a zero mode and hence instability to spontaneous rotational symmetry breaking, but the definition of local stability in Section 2 only forbids zero modes that break rotational or translational invariance. A homogeneous δa3 perturbation is invariant under translations and under rotations about the magnetic-field axis, so it is not shown that the Ω=0 solution of (3.14) corresponds to a mode excluded by the stated stability criterion. This matters because the conclusion that ω is nonzero real requires Ω>0 rather than merely Ω≥0.
  3. [Appendix B and Section 4] The numerical evidence is suggestive but not a substitute for the missing existence proof. The computation in Appendix B is performed at finite λ and finite T, for a multiply charged SO(3)-symmetric deformation, and the text does not specify the fluctuation equations solved, the numerical method, the boundary conditions, or convergence checks. It therefore does not directly test the singular Sturm-Liouville problem (3.14) with k(0)=k(∞)=0, and it cannot compensate for the invalid continuity argument in Section 3.
minor comments (4)
  1. [Section 3, below Eq. (3.3b)] The text says a is a positive integer, but the subsequent discussion uses a>-2 and later requires a>0; please clarify the intended range of a.
  2. [Appendix A, near Eq. (A.4)] There is a typo: 'neccessarily' should be 'necessarily'.
  3. [Figures 2 and 3 captions] The parameter α is used in the captions but not defined in the text; please define it explicitly.
  4. [Section 5, Eq. (5.2)] The order-of-magnitude estimate in Eq. (5.2) depends on the unstated choices C≈2 and λ=N/8π²; please state these assumptions explicitly and indicate the uncertainty in the numerical prefactor.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central derivation is self-contained; self-citations are interpretive, not load-bearing.

full rationale

The central derivation is self-contained: the quasinormal-mode equation (2.8) follows from the explicit action (2.1), it is transformed into the Sturm-Liouville problem (3.14), and the positivity and existence arguments in Eqs. (3.15)-(3.21) are carried out in this paper without fitting any parameter to the claimed output. The condition 2x+4y>1+z is derived from the near-horizon asymptotics (3.5)-(3.11), not imposed as the desired conclusion. References [30,32] are used only for the interpretive statement that real quasinormal frequencies imply observable long-lived current oscillations and for prior concrete examples; they do not supply the eigenvalue existence, positivity argument, or the main inequality. These self-citations are therefore not load-bearing. A possible mathematical gap exists in the existence proof, because k(L)=k_L may act as a normalization rather than a homogeneous boundary condition and the continuity argument in k_L is not fully justified; however, that is a correctness/rigor concern, not a circular reduction. The conclusion is not assumed as an input, no parameter is fitted to the predicted spectrum, and the numerical appendix is presented as an independent check rather than a fit. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on structural assumptions about the background geometry and the stability of zero-temperature solutions, not on fitted numbers. No free parameters are introduced to make the derivation work, and no new particles or forces are postulated. The most fragile entries are the exclusion of the a3 zero mode and the assumed validity of the singular Sturm-Liouville existence step.

assumptions (6)
  • domain assumption The holographic dictionary maps bulk Chern-Simons terms to anomalies in the boundary current and quasinormal modes to relaxation times.
    Invoked throughout Sections 1 and 5 to connect real QNM frequencies to long-lived boundary modes.
  • domain assumption There exists a zero-temperature, rotation and translation invariant, locally stable solution to the decoupled equations (2.2) with near-horizon behavior (3.5) and near-boundary behavior (3.3).
    Section 2 assumes this existence and stability to set up the background; without it the QNM analysis has no target geometry.
  • domain assumption In the large lambda limit with a = lambda A fixed, the U(1) gauge field decouples from gravity and matter, yielding equations (2.2).
    Section 2, after Eq. (2.1); this decoupling is essential to make the fluctuation equation depend only on the background and VF.
  • ad hoc to paper A zero mode in delta a3 implies instability to spontaneous rotational symmetry breaking, so local stability rules out Omega = 0.
    Section 3, after Eq. (3.16). This identification is not derived and conflicts with the paper's earlier definition of local stability, which only excludes modes breaking rotation or translation.
  • domain assumption The near-horizon scaling inequality 2x+4y > 1+z holds for the zero-temperature backgrounds of interest.
    Equation (4.1) is the central condition for long-lived modes; the paper shows one explicit domain-wall example but does not prove it holds generically.
  • standard math Standard singular Sturm-Liouville theory guarantees a discrete, bounded-below spectrum for the transformed equation.
    Section 3 uses this to claim existence of eigenvalues; the paper sketches the argument but does not cite a theorem covering the singular endpoint at r = 0.

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Pith. "Pith review of Long lived quasi normal modes." pith.science (2026). https://pith.science/paper/KN3FVBYP

@misc{pith2026241118039,
  author       = {Pith},
  title        = {Pith review of: Long lived quasi normal modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KN3FVBYP}},
  note         = {Machine review of arXiv:2411.18039}
}
read the original abstract

We consider magnetically charged AdS black branes with vanishing entropy at zero temperature. We argue that in the presence of a large enough Chern-Simons coupling the quasi normal modes of the brane will have a diminishing imaginary part. Since our result is agnostic to the matter content of the theory it implies that, generically, the boundary theory will possess long lived modes which should be observable in a proper setting.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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  1. Nonequilibrium steady states in driven holographic Weyl semi-metals

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    A driven holographic Weyl semimetal supports a stable nonequilibrium steady state, becomes superharmonic and then chaotic at stronger driving, and exhibits strong-coupling chiral pumping in a magnetic field.

  2. Anomalous resonance in Weyl semimetals: A holographic study of non-linear effects

    hep-th 2025-11 conditional novelty 6.0 of 10

    Fully back-reacted holographic simulations show nonlinear corrections to anomalously long-lived current oscillations are tiny, so decay rates can be made arbitrarily small at low temperature.

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