Horizon constraints on holographic Green's functions

TL;DR

This work shows that near-horizon perturbations in holographic black-brane geometries enforce a universal pole-skipping structure in thermal Green's functions at negative imaginary Matsubara frequencies. By analyzing a minimally coupled scalar and then conserved currents and the energy–momentum tensor, the authors derive a tower of pole-skipping points (ω_n,k_n) determined by horizon data, with lines of poles and zeros crossing these points that constrain boundary dispersion relations at ω ~ T. They verify the predictions in BTZ and higher-dimensional AdS–Schwarzschild spacetimes and discuss ramifications for hydrodynamics, transport, and potential connections to chaos, highlighting both universal horizon-driven constraints and model-dependent details. The results provide a horizon-based, non-perturbative mechanism for linking near-horizon physics to the short-distance structure of holographic correlators and their transport properties.

Abstract

We explore a new class of general properties of thermal holographic Green's functions that can be deduced from the near-horizon behaviour of classical perturbations in asymptotically anti-de Sitter spacetimes. We show that at negative imaginary Matsubara frequencies and appropriate complex values of the wavenumber the retarded Green's functions of generic operators are not uniquely defined, due to the lack of a unique ingoing solution for the bulk perturbations. From a boundary perspective these `pole-skipping' points correspond to locations in the complex frequency and momentum planes at which a line of poles of the retarded Green's function intersects with a line of zeroes. As a consequence the dispersion relations of collective modes in the boundary theory at energy scales $ω\sim T$ are directly constrained by the bulk dynamics near the black-brane horizon. For the case of conserved $U(1)$ current and energy-momentum tensor operators we give examples where the dispersion relations of hydrodynamic modes pass through a succession of pole-skipping points as real wavenumber is increased. We discuss implications of our results for transport, hydrodynamics and quantum chaos in holographic systems.

Figures (3)

• Figure 1: The left hand plot shows the locations \ref{['locations']} where our study of near horizon perturbations predicts pole-skipping for a field in the BTZ background with $\Delta = 2.5$ and for $n=1,2,3,4$. The right hand plots shows the dispersion relations \ref{['disppoles']} and \ref{['dispzeroes']} of the lines of poles (dashed) and zeroes (solid) in the $\Delta = 2.5$ Green's function \ref{['btzgreensfunction']}. These lines can be seen to intersect precisely at the pole-skipping locations \ref{['locations']}, as expected from our analysis in Section \ref{['sec:higherpoleskipping']}.
• Figure 2: Blue dots denoting the location of poles of the boundary retarded Green's function for a massless scalar field in AdS$_6$-Schwarzschild. The locations were determined numerically (using the procedure described in Section 4.2 of Denef:2009yy) for four values of $k/r_0$: $0$ (top left), $3.0i$ (top right), $3.1i$ (bottom left) and $3.16i$ (bottom right). The pole locations are consistent with our near-horizon analysis, which indicates that $k_1/r_0=k_2/r_0=\sqrt{10}i\sim 3.16i$ is a pole-skipping wavenumber for the first two pole-skipping frequencies $\omega_1$ and $\omega_2$.
• Figure 3: Dispersion relation of the charge diffusion mode from AdS$_5$-Schwarzschild (left) and of the momentum diffusion mode from AdS$_4$-Schwarzschild (right). Black dots show the exact dispersion relation determined by numerical integration of the appropriate perturbation equations in $(t,r)$ coordinates, solid blue lines show the diffusive hydrodynamic dispersion relations (\ref{['chargehydro']} and \ref{['viscdisp']} respectively), and the intersections of the black dashed lines correspond to the (real $k$) pole-skipping points (\ref{['eq:ads5gaugepoints']} and \ref{['eq:poleskippinglocstransversemom']} respectively). The short-distance corrections to diffusive hydrodynamics are such that the pole passes through a succession of pole-skipping points.