Pole Skipping

Pole skipping occurs when the near-horizon constraint that normally fixes one ingoing datum loses rank at a special complex frequency and momentum. In an Einstein black brane, the decisive coefficient is the one multiplying the leading horizon value of $\delta g_{vv}$ in the $vv$ Einstein equation. At $\omega_\star=i2\pi T$ it becomes $k^2+k_0^2$; setting $k_\star=ik_0$ makes the entire leading constraint vanish and leaves the energy-density Green function direction-dependent.

Required background. Gauge Theory and String-Theory Preparation provides stress-tensor and bulk-field conventions; Quantum Information provides retarded-response and chaos language.

Helpful background. Quantum Gravity supplies the semiclassical bulk regime, while Black-Hole Perturbation Theory provides ingoing coordinates and gauge-invariant horizon data.

The horizon constraint that loses rank

Write an isotropic black brane in ingoing Eddington–Finkelstein coordinates,

$$ds^2=-r^2f(r)dv^2+2\,dv\,dr+h(r)d\mathbf{x}^2, \qquad T=\frac{r_0^2f'(r_0)}{4\pi}.$$

Define the input normalization before expanding: $v$ agrees with boundary time, $h(r)$ multiplies each of the $d$ boundary spatial directions, and $k$ is conjugate to a boundary coordinate with that metric normalization. Rescaling $x$ rescales both $k_0$ and the numerical butterfly velocity, so only a consistently normalized comparison is meaningful. We also take the background matter stress tensor to be regular in these ingoing coordinates and use radial gauge only after identifying gauge-invariant rank loss.

Expand all perturbations as regular Taylor series at $r=r_0$ with dependence $e^{-i\omega v+ikx}$. In the Einstein–matter class for which $\delta T_{vv}(r_0)=T_{vr}(r_0)\delta g_{vv}^{(0)}$, the leading $vv$ equation reduces to

$$\left[-\frac{id}{2}\omega h'(r_0)+k^2\right]\delta g_{vv}^{(0)} -i(2\pi T+i\omega) \left[\omega\,\delta g_{x_i x_i}^{(0)} +2k\,\delta g_{vx}^{(0)}\right]=0.$$

The matter identity holds by horizon regularity in the Einstein–Maxwell–dilaton–axion models checked in Blake et al. 2018, Appendix A; it must be verified for a new matter action. At $\omega_\star=i2\pi T$, the second bracket vanishes. The remaining equation is

$$\left[d\pi T h'(r_0)+k^2\right]\delta g_{vv}^{(0)}=0.$$

Hence $k_0^2=d\pi T h'(r_0)$ and $k_\star=ik_0$ remove the final coefficient. Blake, Davison, Grozdanov, and Liu 2018, §II.B–C, Eqs. (2.8)–(2.11), pp. 9–10 give the complete derivation. The answer to “which coefficient vanishes?” is therefore precise: the coefficient of $\delta g_{vv}^{(0)}$ in the leading $vv$ horizon constraint, after the universal Matsubara factor has decoupled the other leading data.

From an extra ingoing solution to an ambiguous Green function

At generic $(\omega,k)$, the horizon equations fix one relation among the ingoing data. At $(\omega_\star,k_\star)$, one relation disappears, producing an extra regular ingoing solution. Boundary source and response are no longer uniquely related. If locally

$$G_R(\omega,k)=\frac{b(\omega,k)}{a(\omega,k)}$$

and both $a$ and $b$ vanish at the special point, the linearized ratio along $\delta\omega=s\,\delta k$ is

$$G_R\longrightarrow \frac{b_\omega s+b_k}{a_\omega s+a_k}.$$

Distinct numerator and denominator directions produce different limits. For example, $(\delta\omega+\delta k)/(\delta\omega+2\delta k)$ tends to $1/2$ along $s=0$ and $2/3$ along $s=1$, has a zero along $s=-1$, and diverges along $s=-2$. If numerator and denominator have proportional leading gradients, the linearized ratio alone does not establish ambiguity. This is why finding a formal $0/0$ is insufficient.

The slope dependence is worked out explicitly in Blake et al. 2018, §II.D and §III.B–C, pp. 12–23. Numerically, one should approach the point along at least two independent directions and verify different finite limits. Evaluating only at the special coordinates can miss the rank loss because a boundary solver may silently choose one linear combination of the extra solutions.

Relation to chaos data

The shockwave equation on the same horizon is

$$\left(\nabla_{\perp}^2-k_0^2\right)c(\mathbf{x}) \sim E\,e^{2\pi T t_w}\delta(\mathbf{x}).$$

It identifies $\lambda_L=2\pi T$ and $v_B=\lambda_L/k_0$ in the two-derivative Einstein regime. Thus the pole-skipping point can be written $\omega_\star=i\lambda_L$, $k_\star=i\lambda_L/v_B$. The agreement is an independent check because the shockwave calculation is nonlinear in a highly boosted perturbation, whereas pole skipping uses linearized energy-density response.

The implication is conditional. A pole-skipping point is a property of an analytically continued two-point function; it does not by itself prove that every operator has an exponential OTOC with the inferred data. In theories with nonmaximal chaos, higher derivatives, extra conserved sectors, or anisotropy, the relation between the special point and a single butterfly velocity can change.

Checks, scope, and failure cases

The calculation must fix the energy channel, Fourier sign, gauge, and horizon-regular variables. A rank test on the complete leading recursion matrix is safer than inspecting one gauge-dependent equation. Higher Matsubara frequencies can produce additional pole-skipping points, but only the leading energy-density point has the direct universal shockwave relation described here.

Failure cases are informative. If matter is singular at the horizon, the cancellation used above can fail. If translations or rotations break isotropy, $k_0$ becomes direction-dependent. If a zero and pole do not meet at the same order, the Green function may have an ordinary pole, a zero, or a removable singularity instead of slope-dependent ambiguity.

Exercises

Verify the rank loss at the special point and explain why approaching it along two slopes is a stronger diagnostic than evaluating the Green function exactly there.

Solution

Set $\omega=i2\pi T$. The terms multiplying $\delta g_{x_i x_i}^{(0)}$ and $\delta g_{vx}^{(0)}$ vanish because $2\pi T+i\omega=0$. Choosing $k^2=-d\pi T h'(r_0)$ also removes the coefficient of $\delta g_{vv}^{(0)}$, so the leading constraint becomes an identity and the horizon matrix loses one rank. Exactly at that point a solver may select an arbitrary extra ingoing combination. Two approach slopes expose the physical ambiguity: they yield different response-to-source ratios and demonstrate intersecting zero and pole lines.

References

• Blake, Mike, Richard A. Davison, Sašo Grozdanov, and Hong Liu. “Many-Body Chaos and Energy Dynamics in Holography.” Journal of High Energy Physics 10 (2018) 035. arXiv:1809.01169.