Pole skipping and Rarita-Schwinger fields

In this note we analyze the equations of motion of a minimally coupled Rarita-Schwinger field near the horizon of an anti–de Sitter-Schwarzschild geometry. We find that at special complex values of the frequency and momentum there exist two independent regular solutions that are ingoing at the horizon. These special points in Fourier space are associated with the “pole skipping” phenomenon in thermal two-point functions of operators that are holographically dual to the bulk fields. We find that the leading pole-skipping point is located at a positive imaginary frequency with the distance from the origin being equal to half of the Lyapunov exponent for maximally chaotic theories.

Figure 1

Location of the leading pole-skipping point on the frequency plane for bulk fields of various spin. If the fields decompose into multiple channels, we depict the pole-skipping point of the channel with the most positive imaginary value. This paper focuses on the spin-$\frac{3}{2}$ case.

Figure 2

Numerical analysis of the poles of the retarded Green’s function in ${\mathrm{AdS}}_4$ for a Rarita-Schwinger field with mass $m=3.4$. The poles are calculated using the methods presented in [49] adapted to the spin-$3/2$ case. The top figure depicts the motion of the poles on the complex $\omega$-plane near the value of the momentum where pole-skipping is observed. The momentum is varied linearly as $k=k_0+\epsilon e^{\frac{i\pi }{4}}$, where $\epsilon \in \mathbb{R}$. We notice that the red curve passes through ${\omega }_0$ as $\epsilon =0$ which we denote with a red square. In blue we depict two additional poles which pass through $\omega =-3\pi i$ as $\epsilon \to 0$. These correspond to higher order pole-skipping points which are discussed in the Appendix. The bottom figure shows the location of the poles as the momentum is varied as $k=k_0{e}^{i\epsilon }$. Pole skipping occurs at the filled squares.