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Pole skipping and chaos in anisotropic plasma: a holographic study

Karunava Sil

TL;DR

This work analyzes pole skipping as a diagnostic of chaos in a spatially anisotropic holographic plasma. By performing near-horizon analyses of scalar, axion, and metric perturbations and employing gauge-invariant master variables, it derives how the anisotropy shifts pole-skipping points and extracts chaos parameters. The Lyapunov exponent remains at the chaos bound, while the butterfly velocity and momentum-diffusion characteristics acquire anisotropic corrections; moreover, the transverse diffusion dispersion relation is shown to pass through the first three pole-skipping points. The results establish a coherent holographic framework for connecting chaos diagnostics to anisotropic near-horizon data and provide analytic expressions for the lowest-order pole-skipping points across several channels, supported by targeted numerical checks.

Abstract

Recently, a direct signature of chaos in many body system has been realized from the energy density retarded Green's function using the phenomenon of `pole skipping'. Moreover, special locations in the complex frequency and momentum plane are found, known as the pole skipping points such that the retarded Green's function can not be defined uniquely there. In this paper, we compute the correction/shift to the pole skipping points due to a spatial anisotropy in a holographic system by performing near horizon analysis of EOMs involving different bulk field perturbations, namely the scalar, the axion and the metric field. For vector and scalar modes of metric perturbations we construct the gauge invariant variable in order to obtain the master equation. Two separate cases for every bulk field EOMs is considered with the fluctuation propagating parallel and perpendicular to the direction of anisotropy. We compute the dispersion relation for momentum diffusion along the transverse direction in the shear channel and show that it passes through the first three successive pole skipping points. The pole skipping phenomenon in the sound channel is found to occur in the upper half plane such that the parameters Lyapunov exponent $λ_{L}$ and the butterfly velocity $v_{B}$ are explicitly obtained thus establishing the connection with many body chaos.

Pole skipping and chaos in anisotropic plasma: a holographic study

TL;DR

This work analyzes pole skipping as a diagnostic of chaos in a spatially anisotropic holographic plasma. By performing near-horizon analyses of scalar, axion, and metric perturbations and employing gauge-invariant master variables, it derives how the anisotropy shifts pole-skipping points and extracts chaos parameters. The Lyapunov exponent remains at the chaos bound, while the butterfly velocity and momentum-diffusion characteristics acquire anisotropic corrections; moreover, the transverse diffusion dispersion relation is shown to pass through the first three pole-skipping points. The results establish a coherent holographic framework for connecting chaos diagnostics to anisotropic near-horizon data and provide analytic expressions for the lowest-order pole-skipping points across several channels, supported by targeted numerical checks.

Abstract

Recently, a direct signature of chaos in many body system has been realized from the energy density retarded Green's function using the phenomenon of `pole skipping'. Moreover, special locations in the complex frequency and momentum plane are found, known as the pole skipping points such that the retarded Green's function can not be defined uniquely there. In this paper, we compute the correction/shift to the pole skipping points due to a spatial anisotropy in a holographic system by performing near horizon analysis of EOMs involving different bulk field perturbations, namely the scalar, the axion and the metric field. For vector and scalar modes of metric perturbations we construct the gauge invariant variable in order to obtain the master equation. Two separate cases for every bulk field EOMs is considered with the fluctuation propagating parallel and perpendicular to the direction of anisotropy. We compute the dispersion relation for momentum diffusion along the transverse direction in the shear channel and show that it passes through the first three successive pole skipping points. The pole skipping phenomenon in the sound channel is found to occur in the upper half plane such that the parameters Lyapunov exponent and the butterfly velocity are explicitly obtained thus establishing the connection with many body chaos.

Paper Structure

This paper contains 23 sections, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Details of the anisotropic background
  3. Pole skipping in energy density Green's function
  4. Perturbation parallel to the direction of anisotropy
  5. Perturbation perpendicular to the direction of anisotropy
  6. Gauge invariant variable and pole skipping phenomenon in anisotropic plasma
  7. Scalar field fluctuation
  8. Axion field fluctuation
  9. Metric Perturbation
  10. Shear channel
  11. Transverse momentum diffusion in shear channel
  12. Sound channel
  13. Conclusion
  14. Construction of the gauge invariant variable
  15. Scalar modes of metric perturbation
  16. ...and 8 more sections

Figures (3)

  • Figure 1: a) Locations of the pole skipping points for the scalar field obtained by means of numerical calculations with the dimensionless parameter $b=0.1$, b) Plot showing the pole skipping points obtained numerical calculations for the axion field with $b=0.15$.
  • Figure 2: Plot showing the movement of the poles of retarded Green's function as the value of the momentum is increased from zero to finite values.
  • Figure 3: Plot showing the dispersion relation for momentum diffusion which passes through the first three successive pole skipping points. The calculation for the dispersion relation is done using numerical methods with $b=.5$