Pole Skipping in Holography

• Pole Skipping is a phenomenon in retarded Green’s functions marked by the coincidence of a zero and a pole at special complex frequency-momentum points, leading to directional ambiguity in the correlator.
• It arises from horizon degeneracy in holographic setups, where the failure of unique ingoing boundary conditions enables extraction of chaos parameters like the Lyapunov exponent and butterfly velocity.
• Analytic and numerical methods, including matrix Frobenius analysis, reveal its impact on quantum chaos, hydrodynamic transport, and phase transitions across diverse models such as BTZ black holes and CFT₂.

Pole skipping is a particular analytic feature of retarded Green’s functions of conserved quantities, and more generally of holographic two-point functions, in which a zero and a pole coincide at a special complex frequency–momentum point $(\omega_*,k_*)$, so that the correlator is locally of type $0/0$ and its value depends on the direction of approach in $(\omega,k)$-space. In the best-studied cases this structure appears in energy-density or stress-tensor correlators, where the leading point often takes the form $\omega_*=i\lambda$, $k_*=i\lambda/v_B$, linking linear response to the same chaotic data that appear in out-of-time-order correlators. The phenomenon has been analyzed in holographic black branes, two-dimensional conformal field theories on a torus, rotating BTZ black holes, SYK-like systems, anisotropic plasmas, holographic superfluids, the AdS soliton, and cosmological horizons (Wang et al., 2022, Ramirez, 2020, Ahn et al., 21 Aug 2025).

1. Analytic structure of skipped poles

A standard schematic representation is

$G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$

Poles correspond to zeros of $D$, zeros to zeros of $N$, and pole skipping occurs when

$N(\omega_*,k_*)=0,\qquad D(\omega_*,k_*)=0.$

Near such a point, the correlator is finite but ambiguous because different ratios $\delta\omega/\delta k$ give different limits. In this sense, the would-be pole is “skipped” rather than realized as an isolated singularity (Ramirez, 2020).

In holographic settings, pole-skipping points commonly organize into towers at discrete imaginary frequencies. In many isotropic examples and in several matter channels one finds lower-half-plane points at $0/0$0, while the sound-channel point relevant for chaos lies in the upper half-plane at $0/0$1 (Sil, 2020). In diffeomorphism-invariant bulk theories with bosonic fields, the frequencies can be organized by a weight construction: if $0/0$2 is the highest weight, then candidate pole-skipping frequencies take the form $0/0$3 with $0/0$4, and the corresponding momenta are determined by a determinant condition $0/0$5 (Wang et al., 2022).

The same analytic pattern is not confined to conserved stress tensors. It also appears for scalar and Maxwell fields in black-hole backgrounds, for scalar and tensor channels in the AdS soliton, and for scalar, Maxwell, and gravitational waves in de Sitter and Schwarzschild–de Sitter geometries (Natsuume et al., 2020, Natsuume et al., 2023, Ahn et al., 21 Aug 2025).

2. Horizon degeneracy, regularity, and bulk mechanisms

The bulk signature of pole skipping is a failure of uniqueness of the ingoing boundary value problem. In ingoing Eddington–Finkelstein coordinates, generic Fourier modes are fixed at the horizon by regularity together with the radial recursion relations. At a pole-skipping point, one horizon equation degenerates, an extra ingoing solution appears, and the boundary Green’s function becomes non-unique (Natsuume et al., 2019).

A detailed regularity analysis in planar Schwarzschild–AdS$0/0$6 showed that, in the upper half $0/0$7-plane, the mode usually interpreted as outgoing is generically singular at the future horizon and produces a curvature singularity, whereas at the special point both independent solutions are regular. At that point the notion of an incoming mode is itself not unique, and the retarded energy-density Green’s function acquires the characteristic slope dependence of pole skipping (Natsuume et al., 2019).

For general diffeomorphism-invariant bosonic theories, the near-horizon equations can be organized by weight. Positive-weight background tensor components vanish at the horizon, and the action of $0/0$8 on weight-$0/0$9 components produces the factors that enforce the discrete pole-skipping frequencies. In theories without higher-spin fields, the highest-weight energy-density pole skipping occurs at $(\omega,k)$0, and the momentum $(\omega,k)$1 is determined by the highest-weight equation $(\omega,k)$2 evaluated on $(\omega,k)$3 (Wang et al., 2022).

Coupled systems need not admit a single master variable. A matrix Frobenius formalism was developed to treat directly a first-order system $(\omega,k)$4, with horizon data organized into an ingoing basis. Pole-skipping points are then identified by the vanishing of a residue matrix that controls the singular part of the near-horizon recursion. This was used in holographic superfluids precisely because a conventional single-master-field treatment is inadequate there (Natsuume et al., 16 Dec 2025).

A separate geometric reformulation identifies the pole-skipping mode itself with a distinguished near-horizon metric perturbation. A regularized limit of that perturbation reproduces the shock-wave solution, and the same mode also appears as the linearized replica-manifold deformation associated with a late-time entanglement wedge (Chua et al., 10 Apr 2025).

3. Relation to chaos, butterfly velocities, and its limitations

In maximally chaotic holographic systems, the leading sound-channel pole-skipping point has the form

$(\omega,k)$5

with $(\omega,k)$6. In Einstein gravity this relation is standard, and in general higher-derivative theories without higher-spin fields the same highest-weight point persists: the Lyapunov exponent saturates the chaos bound and the butterfly velocity obtained from pole skipping matches the shock-wave result (Wang et al., 2022).

In anisotropic plasma, pole skipping remains tied to chaos but the momentum is direction dependent. In the Mateos–Trancanelli background, the sound-channel point stays at $(\omega,k)$7 to order $(\omega,k)$8, while the momenta shift differently for propagation parallel and perpendicular to the anisotropy. The resulting butterfly velocities satisfy

$(\omega,k)$9

whereas $\omega_*=i\lambda$0 remains unchanged (Sil, 2020).

Away from maximal chaos, however, pole skipping need not determine the true Lyapunov exponent. In the large-$\omega_*=i\lambda$1 SYK chain, the proposal is that pole skipping is controlled by the stress-tensor contribution to chaos,

$\omega_*=i\lambda$2

rather than by the full $\omega_*=i\lambda$3. On this view, pole skipping always reflects the maximal stress-tensor contribution, while the true chaos data satisfy $\omega_*=i\lambda$4 and conjecturally $\omega_*=i\lambda$5 (Choi et al., 2020).

The same paper gives a stringent test of that proposal: the exact retarded energy-density Green’s function of the SYK chain has its diffusion pole line intersecting the skipped point at $\omega_*=i\lambda$6 in units $\omega_*=i\lambda$7, and the model obeys $\omega_*=i\lambda$8, with equality at strong coupling (Choi et al., 2020).

A broader conceptual explanation for the coincidence of different butterfly velocities was supplied later: in certain holographic theories the pole-skipping mode is the replica-manifold perturbation for the late-time entanglement wedge, and its imaginary part is the shock wave computing the OTOC. In that class of models,

$\omega_*=i\lambda$9

so pole skipping, shock-wave scattering, and entanglement-wedge growth are governed by the same near-horizon mode (Chua et al., 10 Apr 2025).

4. Exact results in two-dimensional CFT and BTZ geometries

Two-dimensional conformal field theory provides one of the few settings where pole skipping can be computed exactly from symmetry. For a CFT$k_*=i\lambda/v_B$0 on a spatial circle of circumference $k_*=i\lambda/v_B$1 at inverse temperature $k_*=i\lambda/v_B$2, the retarded energy-density correlator on the torus takes the form

$k_*=i\lambda/v_B$3

Pole skipping occurs when the zero of the numerator coincides with the light-cone denominator, giving

$k_*=i\lambda/v_B$4

with $k_*=i\lambda/v_B$5. Modular invariance then implies

$k_*=i\lambda/v_B$6

for compact, unitary CFT$k_*=i\lambda/v_B$7 with $k_*=i\lambda/v_B$8, and Hartman–Keller–Stoica sparsity implies saturation above the Hawking–Page transition in large-$k_*=i\lambda/v_B$9 theories with sparse light spectrum (Ramirez, 2020, Hartman et al., 2014).

This finite-size torus analysis resolves a tension from infinite-line results. On the line, pole skipping in the stress-tensor sector is universal at $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$0; on the circle it depends on the spectrum through $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$1 or $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$2. This suggests that the universal infinite-volume result is a Cardy high-temperature limit rather than a generic statement that every CFT$G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$3 is maximally chaotic (Ramirez, 2020).

Rotating BTZ backgrounds expose a different structure. For a minimally coupled scalar in rotating BTZ, the exact Green’s function factorizes into left- and right-moving Gamma-function ratios depending separately on $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$4 and $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$5, and pole skipping occurs only when a pole in one chiral sector coincides with a zero in the other. In the non-extremal case the skipped points therefore depend on both $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$6 and $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$7; in the extremal limit $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$8, generic pole skipping disappears because the left sector collapses to a branch structure, but a special $G^R(\omega,k)=\frac{N(\omega,k)}{D(\omega,k)}.$9 case survives with

$D$0

at right-moving Matsubara frequencies (Natsuume et al., 2020).

A broader rotating-BTZ analysis extended this to spins $D$1. For non-extremal rotating BTZ, the leading pole-skipping point obeys

$D$2

or, for the frequency alone,

$D$3

For extremal BTZ, the leading pole-skipping frequency can occur only when $D$4, in which case

$D$5

These results were obtained analytically from exact Green’s functions for fermions and vectors and checked independently by near-horizon analysis (Jeong et al., 2023).

5. Deformations, phase structure, and nonstandard realizations

Deformations that preserve the basic horizon structure can move pole-skipping momenta without moving the chaos frequency. Scalar–Gauss–Bonnet coupling in AdS$D$6 Schwarzschild illustrates this sharply. In the perturbative regime, the sound-channel chaos point remains

$D$7

so $D$8 and $D$9 are unchanged. By contrast, the shear pole-skipping momentum and the associated diffusion constant acquire nontrivial $N$0-dependent corrections, and the scalar-channel lower-half-plane point shifts with the scalar source $N$1 (Baishya et al., 2023).

Holographic quantum criticality can also reorganize the pole-skipping spectrum. In the Einstein–Maxwell–Chern–Simons model with a magnetic field, the longitudinal butterfly velocities split because of the anomaly. At the quantum critical point $N$2 and $N$3,

$N$4

In the tensor channel, the tower of pole-skipping points changes qualitatively across the transition, and the quantities

$N$5

serve as order parameters: $N$6 in the disordered phase and $N$7 in the ordered phase near zero temperature (Abbasi et al., 2023).

In holographic superfluids, the technical issue is not merely deformation but field coupling. A matrix formalism without a master variable was applied directly to the coupled Maxwell–scalar system. The result is that the Maxwell diffusion pole still gives a hydrodynamic pole-skipping point, but the critical order-parameter mode does not: the order-parameter Green’s function develops a hydrodynamic pole without the $N$8 structure required for pole skipping. This established that not all hydrodynamic poles are pole-skipping points (Natsuume et al., 16 Dec 2025).

Pole skipping can also occur without a horizon. In the AdS soliton, where the $N$9 circle caps off smoothly at the tip, the same phenomenon appears in normal-mode spectra. The key effect is that some would-be states are absent exactly at pole-skipping points, producing “missing states” in the discrete spectrum of scalar, Maxwell, and gravitational perturbations with $N(\omega_*,k_*)=0,\qquad D(\omega_*,k_*)=0.$0 momentum. Once these missing states are recognized, the puzzling ordering of the spectrum is resolved (Natsuume et al., 2023).

6. Cosmological extensions, higher spin, and open issues

Cosmological spacetimes exhibit an additional generalization. In empty de Sitter and Schwarzschild–de Sitter geometries, scalar, Maxwell, and gravitational perturbations all possess towers of pole-skipping points, and the gravitational sound channel yields chaos data at each horizon. For both the black-hole and cosmological horizons one finds

$N(\omega_*,k_*)=0,\qquad D(\omega_*,k_*)=0.$1

and the butterfly velocities extracted from pole skipping match an independent shock-wave computation. In Schwarzschild–de Sitter these velocities can be superluminal or imaginary; the paper interprets this as evidence that a putative dual description would split into two entangled sectors, with increasingly nonlocal dynamics for the black-hole sector and non-Hermitian behavior for the cosmological sector (Ahn et al., 21 Aug 2025).

Higher-spin bulk fields raise a different issue. In general bosonic theories, if the highest dynamical spin is $N(\omega_*,k_*)=0,\qquad D(\omega_*,k_*)=0.$2, the highest-weight pole-skipping frequency is

$N(\omega_*,k_*)=0,\qquad D(\omega_*,k_*)=0.$3

which matches the higher-spin Lyapunov exponent and lies outside the usual chaos bound. Without higher-spin fields, the highest-weight energy-density point returns to $N(\omega_*,k_*)=0,\qquad D(\omega_*,k_*)=0.$4, and the pole-skipping butterfly velocity matches the shock-wave result (Wang et al., 2022).

Several unresolved questions remain. Away from maximal chaos, the identification between the skipped point and the true Lyapunov exponent is not proven in general (Choi et al., 2020). In extremal limits, standard near-horizon Frobenius analyses can fail because the horizon becomes an irregular singular point (Natsuume et al., 2020). In finite-size CFT$N(\omega_*,k_*)=0,\qquad D(\omega_*,k_*)=0.$5, the relation between pole skipping and OTOCs is strongly constrained by modular invariance, but equality between the pole-skipping exponent and the OTOC Lyapunov exponent is still not automatic outside canonical holographic regimes (Ramirez, 2020). Coupled systems such as holographic superfluids show that even the hydrodynamic interpretation requires care (Natsuume et al., 16 Dec 2025).

Taken together, these developments place pole skipping at the intersection of horizon regularity, analytic structure of Green’s functions, hydrodynamic transport, and quantum chaos. The phenomenon is sufficiently rigid to survive across black holes, CFT$N(\omega_*,k_*)=0,\qquad D(\omega_*,k_*)=0.$6, anisotropic plasmas, superfluids, solitons, and cosmological horizons, yet sufficiently sensitive to spectrum, symmetry, coupling, and kinematics to serve as a diagnostic rather than a single universal observable (Chua et al., 10 Apr 2025).