Pole-Skipping in Holographic Thermal Dynamics

• Pole-skipping is a phenomenon where retarded thermal Green’s functions become indeterminate (0/0) at special complex frequency-momentum pairs due to coinciding zeros and poles.
• It arises from near-horizon degeneracies in black hole geometries, linking chaos parameters like the Lyapunov exponent (λL) and butterfly velocity (vB) without directly computing OTOCs.
• The concept extends to various fields including shockwaves, entanglement-wedge reconstruction, and duality analyses in different backgrounds and higher-derivative theories.

Pole-skipping is the phenomenon in which a retarded thermal Green’s function $G^R(\omega,k)$ becomes intrinsically non-unique at special complex frequency-momentum pairs $(\omega_*,k_*)$ because a pole and a zero coincide, producing a local $0/0$ ambiguity. In holographic realizations, this boundary indeterminacy is the image of a bulk near-horizon degeneracy: the linearized field equations admit an extra ingoing solution, so the usual regularity prescription fails to fix a unique response. In the energy-density channel, the leading upper-half-plane point encodes $\lambda_L=2\pi T$ and the butterfly velocity, while more general probe sectors exhibit infinite towers at Matsubara frequencies in the lower half-plane. Recent work has further tied pole-skipping to shockwaves, entanglement-wedge geometry, and replica manifolds, enlarging its role from a diagnostic of holographic chaos to a structural feature of thermal correlators and black-hole response (Natsuume et al., 2019, Chua et al., 10 Apr 2025).

1. Definition and near-horizon mechanism

At finite temperature, a standard diagnostic is the retarded energy-density correlator

$$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$

Pole-skipping refers to special points where the correlator takes the schematic form

$$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$

because the would-be pole and the zero of the numerator coincide. In holographic language, this occurs when the near-horizon recursion relations lose a constraint, so ingoing boundary conditions no longer select a unique bulk solution (Chua et al., 10 Apr 2025, Wang et al., 2022).

For planar black holes in ingoing Eddington–Finkelstein coordinates,

$ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$

metric perturbations of the form $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$ admit a horizon expansion

$$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$

In the energy-density channel, the leading nontrivial equation is $E_{vv}$. At the first special point it decouples and reduces to the homogeneous transverse equation

$(\omega_*,k_*)$0

whose nontrivial normalizable solution fixes $(\omega_*,k_*)$1. This yields the first chaos-related point

$(\omega_*,k_*)$2

and defines the pole-skipping butterfly velocity

$(\omega_*,k_*)$3

In that setup, the pole-skipping points lie on a half-line $(\omega_*,k_*)$4, $(\omega_*,k_*)$5, and the lowest nontrivial point $(\omega_*,k_*)$6 carries the Lyapunov exponent $(\omega_*,k_*)$7 (Chua et al., 10 Apr 2025).

The same Frobenius logic applies more broadly. For generic bulk perturbations, the two indicial exponents near the horizon take the form $(\omega_*,k_*)$8 and $(\omega_*,k_*)$9, so whenever $0/0$0 is an integer one must test whether the logarithmic obstruction disappears. This produces a truncated near-horizon matrix $0/0$1, and pole-skipping occurs when

$0/0$2

This mechanism yields the standard lower-half-plane Matsubara tower for scalar, vector, and tensor channels (Natsuume et al., 2019, Wu, 2019).

2. Regularity, non-uniqueness, and local analytic structure

The defining issue is not only degeneration of the recursion matrix but also regularity of the resulting horizon solutions. In the SAdS$0/0$3 sound channel, the generic outgoing mode is singular at the future horizon in the upper-half $0/0$4-plane: a gauge-invariant curvature invariant such as the linearized Kretschmann scalar remains finite for the ingoing branch but diverges for the would-be outgoing branch. At the special point, however, the regular singular point becomes an ordinary point and both independent solutions are regular. The ingoing mode therefore ceases to be uniquely defined precisely at the pole-skipping point (Natsuume et al., 2019).

From the boundary perspective, this produces a multi-valued retarded Green’s function. Approaching $0/0$5 along different slopes in the complex $0/0$6-plane gives different limiting values. In the Rarita–Schwinger case, the local structure takes the standard form

$0/0$7

with $0/0$8 determined by the next-order horizon equations (Ceplak et al., 2021). This slope dependence is the operational form of the $0/0$9 ambiguity.

A recurring structural feature is that the special frequencies are more rigid than the special momenta. Finite-coupling and higher-curvature analyses show that the Matsubara frequencies $\lambda_L=2\pi T$0 are not explicitly shifted by Gauss–Bonnet, $\lambda_L=2\pi T$1, or $\lambda_L=2\pi T$2 corrections, while the momenta $\lambda_L=2\pi T$3 or $\lambda_L=2\pi T$4 do shift. In some sectors, special points can even disappear at critical higher-derivative couplings; in the AdS$\lambda_L=2\pi T$5 tensor channel, this happens at $\lambda_L=2\pi T$6 (Natsuume et al., 2019, Wu, 2019). This separation between protected $\lambda_L=2\pi T$7 and theory-dependent $\lambda_L=2\pi T$8 has become a central feature of the subject.

3. Relation to chaos, shockwaves, and entanglement

The original prominence of pole-skipping came from the energy-density correlator, where the leading upper-half-plane point is directly tied to many-body chaos: $\lambda_L=2\pi T$9 This identifies the Lyapunov exponent and butterfly velocity without computing an out-of-time-order correlator explicitly (Natsuume et al., 2019).

A deeper unification is furnished by the relation among pole-skipping, OTOC shockwaves, and entanglement-wedge reconstruction. In the late-time, large-region regime, the extremal HRT surface lies infinitesimally shifted off the bifurcation surface along the horizon. In Kruskal coordinates, the limiting surface can be written as

$$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$0

and the extremal-surface equation reduces in the near-horizon limit to

$$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$1

The pole-skipping mode in Kruskal gauge has the form

$$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$2

so the $$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$3 replica geometry controlling late-time entanglement is precisely the pole-skipping mode. After regulating $$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$4,

$$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$5

and the imaginary part of the pole-skipping perturbation becomes the standard gravitational shockwave computing the OTOC. Since the shockwave profile, the limiting HRT surface, and the pole-skipping mode obey the same transverse equation, one obtains the triple equality

$$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$6

in Einstein gravity and, more generally, in higher-derivative $$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$7 theories with minimally coupled matter whenever the standard pole-skipping mode exists (Chua et al., 10 Apr 2025).

This coincidence is not confined to static isotropic branes. In the simply-spinning Myers-Perry–AdS$$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$8 plasma, there is a general proof of pole-skipping in the retarded energy-density Green’s function whenever the spatial profile of the perturbation satisfies the same shockwave equation that governs the OTOC. In the large-black-hole limit, the associated Lyapunov exponents and butterfly velocities can be extracted analytically, and the sound-mode dispersion relations numerically pass through the predicted pole-skipping locations (Amano et al., 2022).

4. Fields, symmetries, mass, and duality

Pole-skipping extends far beyond metric perturbations. For a minimally coupled Rarita–Schwinger field in AdS–Schwarzschild, the near-horizon $$G^R(\omega,k)=\langle T_{00}T_{00}\rangle^{\mathrm{ret}}_\beta(\omega,k).$$9 subsystem becomes degenerate at

$$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$0

At that point, two linearly independent regular ingoing solutions exist. The same analysis suggests a spin-dependent pattern

$$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$1

For $$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$2, the positive imaginary frequency is not a genuine instability, because $$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$3 is also complex and the ingoing prescription is still imposed (Ceplak et al., 2021).

For massless $$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$4 $$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$5-forms in thermal AdS/CFT, the first-order pole-skipping points occur at

$$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$6

with the upper sign corresponding to the longitudinal mode and the lower sign to the transverse mode. The dependence on $$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$7 leads to a trans-mode equivalence under electromagnetic duality: the longitudinal sector of a $$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$8-form is mapped to the transverse sector of its $$G^R(\omega,k)\sim \frac{N(\omega,k)}{D(\omega,k)}\to \frac{0}{0},$$9-form dual, and the pole-skipping loci agree (Wang et al., 2022).

Mass terms enlarge the pattern. In massive $ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$0-form theories, extra first-order points appear, the massless limit separates into the massless longitudinal $ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$1-form branch and the massless transverse $ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$2-form branch, and there is an additional zeroth-order point at

$ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$3

A complementary Stueckelberg analysis shows why the spectrum doubles: breaking gauge invariance promotes the would-be gauge sector to dynamical status, the number of skipped poles doubles, and the extra pole-skipping points are associated with the Stueckelberg fields. As the mass varies, some skipped-pole wave numbers move from a non-physical region with complex $ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$4 to a physical region with real $ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$5 (Wang et al., 2022, Pan et al., 2024).

These examples establish that pole-skipping is sensitive both to the kinematics of horizon regularity and to the operator content of the bulk theory. The frequencies are often universal, but the momenta can depend on spin, mass, duality frame, and symmetry-breaking data.

5. Beyond static isotropic black branes

The phenomenon is not restricted to standard AdS black branes. In acoustic black holes, the analogue of pole-skipping appears for scalar perturbations at

$ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$6

with the first point at $ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$7. In that setting, the lower-half-plane pole-skipping frequencies track the imaginary parts of quasinormal modes and encode dissipation and instability timescales of the fluid perturbations (Yuan et al., 2021).

Pole-skipping also occurs without a horizon. In the AdS$ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$8 soliton, regularity at the tip replaces the ingoing horizon condition. For scalar, Maxwell, and gravitational perturbations with compact-circle momentum $ds^2=-f(r)\,dv^2+2\,dv\,dr+r^2\,dx_i\,dx^i,$9, the pole-skipping values are

$$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$0

and at each such value one would-be normal mode is absent. The resulting “missing states” resolve otherwise puzzling features of the soliton spectrum and show that pole-skipping is not exclusively a black-hole phenomenon (Natsuume et al., 2023).

In holographic axion theories, the use of gauge-invariant variables reveals two distinct classes: regular pole-skipping points, where the near-horizon matrix remains finite and $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$1, and singular pole-skipping points, where some entries diverge and a singular expansion is required. The lower-half-plane points are regular, whereas the leading spin-0 chaos point is singular. The same framework also produces a finite-$$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$2 singular shear point at $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$3, and shows that if the pole-skipping momentum is purely real or purely imaginary at $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$4, it retains that character for $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$5 and $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$6 (Ahn et al., 2024).

Spatial anisotropy and rotation deform, but do not erase, the structure. In anisotropic plasma, the sound-channel upper-half-plane point remains at $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$7, still yields $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$8, and produces anisotropy-dependent corrections to $$\delta g_{MN}(v,r,x)=e^{-i\omega v+ikx^1}\delta g_{MN}(r)$$9; in the shear channel, the analytically continued momentum-diffusion branch passes through the first three pole-skipping points (Sil, 2020). For rotating BTZ black holes, exact Green’s functions for spin-$$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$0 and spin-1 fields give a full tower of pole-skipping points, and in the non-extremal case the leading frequency is

$$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$1

In the extremal case, a leading point survives only when $$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$2, with

$$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$3

(Jeong et al., 2023).

Pole-skipping has also been used as a probe of phase structure. In the 1RCBH model, the butterfly velocity probes critical behavior, pole-skipping occurs at the chaos point, and one finds $$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$4 throughout the parameter space studied for both 1RCBH and AdS–RN (Amrahi et al., 2023). In the Einstein–Maxwell–Chern–Simons model with a quantum critical point, the quantities

$$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$5

vanish in the disordered state and become positive in the ordered state, providing order parameters built directly from pole-skipping data (Abbasi et al., 2023).

6. Algebraic structures and spectral interpretations

A recent development is the recognition that pole-skipping points possess a hidden algebraic organization. For Klein–Gordon-type equations $$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$6, the determinant condition at level $$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$7 is a degree-$$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$8 polynomial in $$\delta g_{MN}(r)=\sum_{n=0}^{\infty}\delta g_{MN}^{(n)}(r-r_0)^n.$$9, but only a subset of the $E_{vv}$0 is independent: the rest satisfy universal homogeneous polynomial identities. In this framework, the full static planar black-hole metric can be reconstructed analytically from the infinite set of pole-skipping data $E_{vv}$1 by solving linear equations for the near-horizon coefficients, and even the near-horizon vacuum Einstein equations can be rewritten entirely in terms of the symmetric polynomials of the pole-skipping momenta (Lu et al., 17 Jul 2025).

This algebraic perspective clarifies a common misconception: pole-skipping is not merely a statement about poles. Zeros are equally essential. That point is especially clear in Kerr perturbation theory near algebraically special frequencies. There, anomalous quasinormal-mode behavior is explained by tracking poles and zeros of the Teukolsky Green-function building blocks across different Riemann sheets. The apparent disappearance of a Kerr quasinormal mode is due to pole-skipping: a quasinormal-mode pole is cancelled by a Matsubara-mode zero, producing a $E_{vv}$2 structure. The accompanying bifurcation is resolved as an avoided crossing with resonant excitation rather than as a literal splitting of a single mode (Kubota et al., 18 May 2026).

Related channel structures arise already in four-dimensional massive black holes with maximally symmetric horizons. There, pole-skipping points separate into algebraically special points and a second set common to the even and odd gravitational channels, with the Darboux transformation providing the relation between the two sectors. In that classification, the $E_{vv}$3 even-channel solution is the chaos point with $E_{vv}$4 (Grozdanov et al., 2023).

Across these developments, a consistent picture has emerged. Pole-skipping is determined by IR regularity data at the horizon or its analogue, survives substantial deformations of the bulk dynamics, constrains the analytic continuation of hydrodynamic and nonhydrodynamic response, and in the energy-density channel furnishes a direct bridge between retarded two-point functions, shockwaves, and entanglement structures (Wang et al., 2022, Chua et al., 10 Apr 2025).