Classifying pole-skipping points

TL;DR

This work provides a comprehensive classification of pole-skipping points for scalar and vector fields in hyperbolic space, identifying three distinct types: Type-I (linear path dependence), Type-II (nonlinear path dependence or merging), and Type-III (non-integer $i\omega$ arising from UV conditions). By combining near-horizon holographic analysis with exact field-theory results on Rindler-AdS, the authors derive analytic Green's functions and a determinant framework $\det\mathcal{M}^{(n)}$ that locates pole-skipping points via $\omega_n=-in$ and corresponding $k_{\{n\}}$. They show how Type-I/II points can be read off from the near-horizon data, while highlighting subtleties such as point merging and prefactor effects that can shift the apparent type, and they introduce Type-III as a UV-driven exception. The results clarify the IR/UV interplay in pole-skipping and provide a robust, general method applicable beyond flat-space holography, with potential implications for chaos diagnostics and the structure of boundary correlators. Overall, the paper advances a unified, analytic understanding of pole-skipping towers in holography and their precise classification.

Abstract

We clarify general mathematical and physical properties of pole-skipping points. For this purpose, we analyse scalar and vector fields in hyperbolic space. This setup is chosen because it is simple enough to allow us to obtain analytical expressions for the Green's function and check everything explicitly, while it contains all the essential features of pole-skipping points. We classify pole-skipping points in three types (type-I, II, III). Type-I and Type-II are distinguished by the (limiting) behavior of the Green's function near the pole-skipping points. Type-III can arise at non-integer $iω$ values, which is due to a specific UV condition, contrary to the types I and II, which are related to a non-unique near-horizon boundary condition. We also clarify the relation between the pole-skipping structure of the Green's function and the near-horizon analysis. We point out that there are subtle cases where the near-horizon analysis alone may not be able to capture the existence and properties of the pole-skipping points.

Figures (9)

• Figure 1: The black dots represent the pole-skipping points at $d=4$ (obtained from \ref{['scalleading']}) of a scalar field propagating in a Rindler-AdS geometry. Here $\delta:=\Delta-d/2$.
• Figure 2: $\log |\mathcal{G}^\Delta(\mathrm{Im}[\omega],\mathrm{Im}[k])|$ at $d=4$. The blue lines and red lines represent zeros and poles of the Green's function, respectively. The white circles and squares are type I and type II pole-skipping points, respectively. The meaning of the type I and II classifications will be explained in section \ref{['section3']}.
• Figure 3: Locations of the pole-skipping points of longitudinal channels.
• Figure 4: $\log |\mathcal{G}_V^{\Delta=d-1}(\mathrm{Im}[\omega],\mathrm{Im}[k])|$ at $d=4,5,6$. The blue lines and red lines represent zeros and poles of the Green's function respectively. The white circles and squares are type I and type II pole-skipping points, respectively. The meaning of type I and II will be explained in section \ref{['section3']}.
• Figure 5: Zoom-in of Fig. \ref{['fig:regscalar4']} including a type-I point (B) and a type-II point (A). The dotted lines represent curves on which the Green's function has a constant value. Different colors represent different values: $\{\mathrm{black},\mathrm{orange}, \mathrm{yellow}, \mathrm{green}, \mathrm{purple}\} = \{0, 1.05, 3, -1.05, -3\}$.