Pole-Skipping Points in Holography

• Pole-skipping points are defined as special locations in the complex frequency–momentum space where both the numerator and denominator of holographic correlators vanish, leading to directional ambiguity.
• They arise from the degeneracy of near-horizon boundary conditions in black brane geometries and are classified into distinct types based on analytic structure and approach paths.
• Pole-skipping reveals connections to quantum chaos by linking special frequencies to parameters like the Lyapunov exponent and butterfly velocity, influencing holographic transport and stability.

Searching arXiv for recent and foundational papers on pole-skipping points to ground the article. Pole-skipping points are special complex frequency–momentum locations $(\omega_\star,k_\star)$ of retarded Green’s functions at which the correlator is not uniquely defined because a pole and a zero coincide. In the standard holographic formulation, if $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$, pole-skipping occurs when $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$, so the limiting value of $G_R$ depends on the direction of approach in complex momentum space. The bulk counterpart is a degeneracy of the horizon value problem: instead of a unique regular ingoing solution, the linearized equations admit two such solutions. Pole-skipping appears both in the lower half $\omega$-plane for generic probe operators and, in special stress-tensor or energy-density channels, at an upper-half-plane point associated with chaos data such as $\lambda_L$ and $v_B$ (Ahn et al., 2020, Wu, 2019, Jeong et al., 2021).

1. Definition and local analytic structure

The defining feature of a pole-skipping point is the simultaneous vanishing of the numerator and denominator of the retarded Green’s function. In local form, the correlator then has slope-dependent behavior such as

$$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$

or, more generally, dependence on the approach direction $dq/d\omega$. This nonuniqueness is equivalent to the statement that a line of poles intersects a line of zeroes at the same complex point, so the residue of the would-be pole vanishes there (Natsuume et al., 2020).

A more refined classification distinguishes several local behaviors. Type-I pole-skipping points exhibit nontrivial dependence already along linear approaches in $(\delta\omega,\delta k)$. Type-II points require quadratic or higher-order paths to reveal the ambiguity; the Green’s function may look trivial along linear approaches but remain path-dependent along curved ones. Type-III points are more subtle: they can arise at non-integer $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$0 values because of a specific UV condition rather than non-uniqueness of the near-horizon ingoing boundary condition. This classification clarifies that not every $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$1 point is governed by the same mechanism, and that near-horizon analysis alone may not always capture the full pole-skipping structure (Ahn et al., 2020).

In exact examples, the analytic origin of the phenomenon is often transparent. For scalar or spinning fields in BTZ, the Green’s function is expressed through products and ratios of $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$2-functions, and pole-skipping is realized when poles from one factor coincide with zeroes from another. In hyperbolic-space correlators, the same structure can be traced to exact thermal Green’s functions on $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$3, where one can explicitly follow the intersecting pole and zero curves and verify the local path dependence (Natsuume et al., 2020, Ahn et al., 2020).

2. Near-horizon mechanism and recursion criteria

The standard bulk diagnosis uses ingoing Eddington–Finkelstein coordinates, in which regularity at the future horizon implements the retarded prescription. One expands the master field near the horizon as a Frobenius or Taylor series,

$G_R(\omega,k)=b(\omega,k)/a(\omega,k)$4

and substitutes it into the radial equation. For generic $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$5, the recursion relations determine all higher coefficients in terms of one free ingoing datum. Pole-skipping occurs when the recursion degenerates so that an extra coefficient remains free; equivalently, the horizon no longer fixes a unique ingoing solution (Wu, 2019).

For a broad class of second-order equations, the $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$6-th pole-skipping point is obtained by a pair of algebraic conditions: one quantizes the frequency, and the other fixes the momentum. In matrix language, if the near-horizon recursion is encoded in a truncated system $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$7, then the conditions take the form

$G_R(\omega,k)=b(\omega,k)/a(\omega,k)$8

The first condition yields the special Matsubara-like frequencies, while the determinant condition selects the corresponding $G_R(\omega,k)=b(\omega,k)/a(\omega,k)$9 or $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$0 (Wu, 2019).

This logic extends to coupled systems. In holographic axion theories at finite $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$1 and $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$2, the metric, gauge, and axion perturbations are organized into gauge-invariant variables in spin-0, spin-1, and spin-2 channels. The near-horizon expansion then produces a block-lower-triangular matrix equation $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$3. Regular pole-skipping points satisfy

$a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$4

whereas singular pole-skipping points arise when the coefficients of the gauge-invariant equations themselves diverge and one must impose additional singular constraints before counting free horizon data (Ahn et al., 2024).

A recent algebraic reformulation goes further: for static planar black holes and Klein–Gordon-type equations $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$5, the near-horizon metric coefficients can be reconstructed recursively from pole-skipping data. At each order $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$6, only two linear equations are needed to determine the two new horizon coefficients, while the remaining equations become homogeneous polynomial identities among the pole-skipping momenta. In that formulation, only a subset of pole-skipping points is independent (Lu et al., 17 Jul 2025).

3. Matsubara towers, channels, and the chaos point

A recurring pattern is the lower-half-plane tower

$a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$7

which appears for generic scalar, vector, and metric perturbations in many black-brane backgrounds. Higher-curvature corrections, including $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$8 and $a(\omega_\star,k_\star)=b(\omega_\star,k_\star)=0$9 terms, do not explicitly alter these lower-half-plane frequencies, although they do shift the corresponding special momenta $G_R$0. This supports the view that the frequency quantization is controlled by near-horizon thermal structure, whereas the momenta are channel- and theory-dependent observables (Wu, 2019).

The upper-half-plane point is conceptually different. In the sound or energy-density channel, the special location

$G_R$1

encodes the Lyapunov exponent and butterfly velocity. In Einstein-axion models with explicit or spontaneous translation breaking, and also in magnetically charged black holes, the near-horizon Einstein equation continues to yield the universal chaos form

$G_R$2

independently of the symmetry-breaking pattern. What changes is the hydrodynamic mode that intersects this point: energy diffusion for explicit breaking, crystal diffusion for spontaneous breaking, and the magnetohydrodynamic diffusive branch in the magnetic case (Jeong et al., 2021).

Anisotropic plasma provides another example of this split between robust frequency data and theory-dependent momentum data. In the Mateos–Trancanelli geometry, anisotropy does not shift the special frequencies in scalar, axion, shear, or sound sectors, but it shifts the momenta differently for propagation parallel and perpendicular to the anisotropy. In the sound channel, the upper-half-plane point still gives $G_R$3, while the butterfly velocity becomes direction-dependent, $G_R$4 (Sil, 2020).

Gauge-invariant analyses at finite chemical potential and momentum relaxation sharpen this picture further. In the five-dimensional linear-axion model, lower-half-plane pole-skipping points are regular, while the spin-0 chaos point is singular in the gauge-invariant formalism. The leading points are summarized by

$G_R$5

with the spin-0 point obeying the standard relation to $G_R$6 and $G_R$7 (Ahn et al., 2024).

4. Rotation, left–right thermal structure, and extremality

Rotating BTZ black holes make the left/right structure of pole-skipping explicit. Because the dual theory is a $G_R$8-dimensional CFT with independent left- and right-moving temperatures,

$G_R$9

the pole-skipping frequencies in rotating BTZ are not governed by a single temperature. For a minimally coupled scalar, exact Green’s functions show that pole-skipping occurs only when a left pole coincides with a right zero, or a right pole coincides with a left zero, leading to

$\omega$0

or

$\omega$1

In the static limit $\omega$2, the $\omega$3-dependence collapses to the familiar negative imaginary bosonic Matsubara values $\omega$4 (Natsuume et al., 2020).

For fermions and vectors in non-extremal rotating BTZ, the leading pole-skipping point takes a universal spin-dependent form,

$\omega$5

with $\omega$6 the Hawking temperature, $\omega$7 the angular velocity, and $\omega$8. Exact towers were derived for $\omega$9 and $\lambda_L$0, and near-horizon analysis reproduces the leading result across $\lambda_L$1 (Jeong et al., 2023).

Extremality is qualitatively subtler because the horizon becomes an irregular singular point and the non-extremal pole-skipping analysis does not commute with the zero-temperature limit. For the rotating BTZ scalar with noninteger $\lambda_L$2, taking $\lambda_L$3 turns the left-moving $\lambda_L$4-function ratio into a power law in $\lambda_L$5. The left sector then contributes a branch point rather than isolated poles or zeroes, so the two thermal towers no longer intersect in the way required for pole-skipping. The special exception is $\lambda_L$6, equivalently $\lambda_L$7, where pole-skipping survives at

$\lambda_L$8

that is, at right-moving Matsubara frequencies even in the extremal limit (Natsuume et al., 2020).

For more general spins in extremal rotating BTZ, the surviving pattern is even more restrictive. The leading extremal pole-skipping point occurs only when

$\lambda_L$9

and then

$v_B$0

This cannot be obtained by naively taking the non-extremal limit $v_B$1, $v_B$2 of the non-extremal formula, reflecting a genuinely different analytic structure at extremality (Jeong et al., 2023).

5. Classification, gauge symmetry, and massive fields

Gauge symmetry and its breaking reorganize pole-skipping in a sharp way. For massless $v_B$3-gauged $v_B$4-forms in asymptotically $v_B$5 black branes, the first-order pole-skipping points occur at

$v_B$6

with

$v_B$7

plus a longitudinal zeroth-order point $v_B$8. The dependence on $v_B$9 produces a simple form-number pattern, and the paper identifies a trans-mode equivalence between dual $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$0- and $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$1-form fields consistent with electromagnetic duality (Wang et al., 2022).

Turning on a mass produces an abrupt change. For massive $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$2-forms and dRGT massive gravity, pole-skipping points are computed by near-horizon methods and are generically doubled relative to the massless case. In the simplest massive vector longitudinal channel, the single massless first-order point splits into two branches,

$$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$3

both at $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$4. The discontinuity at $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$5 is explained by the Stueckelberg formalism: the extra pole-skipping points are associated with the Stueckelberg fields that restore gauge invariance and account for the extra degrees of freedom activated when the mass term breaks the gauge symmetry (Pan et al., 2024).

This interpretation extends to massive $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$6-forms and to dRGT massive gravity. In both cases, the additional skipped poles of the massive theory do not simply disappear in the $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$7 limit; rather, they become the skipped poles of the decoupled Stueckelberg sector. The paper also notes that as the mass varies, some special wave numbers can move from a non-physical region with complex $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$8 to a physical region with real $$G_R \sim \frac{\delta\omega+\delta q}{\delta\omega-\delta q},$$9, which suggests a nontrivial interplay between pole-skipping and the kinematic accessibility of the corresponding modes (Pan et al., 2024).

The classification into type-I, type-II, and type-III fits naturally into this broader gauge-theoretic picture. Type-I and type-II are tied to non-unique near-horizon boundary conditions, while type-III can arise from UV conditions even when the incoming bulk solution remains unique. This means that the full pole-skipping structure of holographic correlators need not be exhausted by the standard horizon argument, particularly in channels with gauge constraints or special asymptotic prefactors (Ahn et al., 2020).

6. Beyond standard horizons: non-black-hole geometries, gapped modes, and algebraic reconstruction

Pole-skipping is not restricted to black-hole horizons. In the AdS soliton, obtained by double Wick rotation from the AdS black hole,

$dq/d\omega$0

the universal structure moves from frequency to momentum along the compact circle: $dq/d\omega$1 The relevant IR condition is regularity at the smooth cap, with

$dq/d\omega$2

for $dq/d\omega$3. In this horizonless geometry, hydrodynamic-type points such as $dq/d\omega$4 and a chaotic-type gravitational scalar point $dq/d\omega$5 occur in the physical region. The paper emphasizes, however, that in the AdS soliton these do not diagnose black-hole-like chaos; instead, the “chaotic” point is interpreted as a missing normal mode or missing state in the confining spectrum (Natsuume et al., 2023).

Related horizon-based structures also appear in Lifshitz, hyperscaling-violating, AdS$dq/d\omega$6, and Rindler geometries. In Lifshitz backgrounds, the lower-half-plane frequencies remain Matsubara-like, $dq/d\omega$7, while the momenta depend on the scaling exponents and the perturbation sector; the analytically continued diffusive hydrodynamic curves pass through these pole-skipping points. In AdS$dq/d\omega$8, the locations are independent of the choice between standard and alternative quantization. In Rindler, a full holographic Green’s function is not available, but the near-horizon equations still admit “special points” with two incoming solutions, which are physically analogous to pole-skipping (Yuan et al., 2020).

Pole-skipping also constrains non-hydrodynamic, gapped spectra. For a massive scalar in AdS$dq/d\omega$9 Schwarzschild, the special frequencies are

$(\delta\omega,\delta k)$0

and the corresponding pole-skipping points lie on the dispersion relations of the gapped quasinormal modes continued to imaginary momentum. The paper finds a hierarchy in which the $(\delta\omega,\delta k)$1 gapped QNM is constrained by pole-skipping levels $(\delta\omega,\delta k)$2, and studies derivative expansions

$(\delta\omega,\delta k)$3

about gapped poles. The radii of convergence of these expansions are numerically bounded from above by the nearest pole-skipping points, and a transition between two classes of critical points occurs at a particular conformal dimension $(\delta\omega,\delta k)$4 for the scalar case (Abbasi et al., 2020).

Finally, the algebraic approach developed for static planar black holes shows that pole-skipping data can be used to reconstruct both the exterior and interior geometry. For master equations of the form $(\delta\omega,\delta k)$5, the near-horizon coefficients $(\delta\omega,\delta k)$6 and $(\delta\omega,\delta k)$7 are solved recursively from pole-skipping symmetric polynomials using only linear equations, while the remaining relations become universal homogeneous polynomial identities among the $(\delta\omega,\delta k)$8. In that framework, vacuum Einstein equations themselves can be rewritten directly as algebraic constraints on pole-skipping data. This suggests that pole-skipping points encode substantially more geometric information than the original “skipped pole” terminology would imply (Lu et al., 17 Jul 2025).