On pole-skipping with gauge-invariant variables in holographic axion theories

We study the pole-skipping phenomenon within holographic axion theories, a common framework for studying strongly coupled systems with chemical potential ($\mu$) and momentum relaxation ($\beta$). Considering the backreaction characterized by $\mu$ and $\beta$, we encounter coupled equations of motion for the metric, gauge, and axion field, which are classified into spin-0, spin-1, and spin-2 channels. Employing gauge-invariant variables, we systematically address these equations and explore pole-skipping points within each sector using the near-horizon method. Our analysis reveals two classes of pole-skipping points: regular and singular pole-skipping points in which the latter is identified when standard linear differential equations exhibit singularity. Notably, pole-skipping points in the lower-half plane are regular, while those elsewhere are singular. This suggests that the pole-skipping point in the spin-0 channel, associated with quantum chaos, corresponds to a singular pole-skipping point. Additionally, we observe that the pole-skipping momentum, if purely real or imaginary for $\mu=\beta=0$, retains this characteristic for $\mu \neq0$ and $\beta \neq 0$.