Quasinormal modes in charged fluids at complex momentum

TL;DR

We investigate the convergence of relativistic hydrodynamics in charged fluids within holography by combining nonperturbative spectral-curve analysis with high-order small-$k$ perturbations in $AdS_{d+1}$ RN black-brane backgrounds. The study shows a finite radius of convergence at zero charge for all hydrodynamic channels, while approaching extremality drives the radius to zero for shear and sound (diffusion can remain finite). Pole skipping is analyzed via Matsubara frequencies and chaos points, revealing a complex, charge-dependent structure that does not universally constrain the convergence radius. Across $d=3,4$, the perturbative and spectral-curve methods agree quantitatively, clarifying previous neutral-versus-charged results and highlighting the distinct roles of different pole-collision mechanisms.

Abstract

We investigate the convergence of relativistic hydrodynamics in charged fluids, within the framework of holography. On the one hand, we consider the analyticity properties of the dispersion relations of the hydrodynamic modes on the complex frequency and momentum plane and on the other hand, we perform a perturbative expansion of the dispersion relations in small momenta to a very high order. We see that the locations of the branch points extracted using the first approach are in good quantitative agreement with the radius of convergence extracted perturbatively. We see that for different values of the charge, different types of pole collisions set the radius of convergence. The latter turns out to be finite in the neutral case for all hydrodynamic modes, while it goes to zero at extremality for the shear and sound modes. Furthermore, we also establish the phenomenon of pole-skipping for the Reissner-Nordstrom black hole, and we find that the value of the momentum for which this phenomenon occurs need not be within the radius of convergence of hydrodynamics.

Figures (4)

• Figure 1: Radius of convergence as a function of the charge in the system for the hydrodynamic modes in our system, namely shear, sound and charge diffusion in $d=3$ and $d=4$. Each line on these plots corresponds to a different branch of pole collisions, with the solid (dashed) lines indicating where each branch is dominant (subdominant). As $\tilde{Q}=Q/Q_{max}$ is varied, different branches are responsible for setting the radius of convergence. Transition points are indicated with dashed vertical lines. The circles correspond to the results for the radius of convergence obtained using a perturbative expansion of the corresponding dispersion relations in small momenta up to a very high order, as discussed in section \ref{['sec:Pert']}. We see that the two methods are in good quantitative agreement.
• Figure 2: Quasinormal modes in the shear channel in $d=3$ at fixed $|k|^2$, illustrating the qualitatively different types of collisions. Purple diamonds indicate the QNMs at real momentum and stars indicate the collisions, with the color matching the color in figure \ref{['fig:RC2']}. The red arrows indicate the movement of the modes as the phase approaches the critical value.
• Figure 3: Spectral curve for the sound hydrodynamic mode for $d=4, \tilde{Q}=Q/Q_{max}=0.445$, at fixed $|k|^2 = |k_c|^2 = 4.90$, which corresponds to the transition point between the blue branch and the green branch. The brown line corresponds to the hydrodynamic sound mode, the green line to a diffusive non-hydrodynamic mode in the sound channel and the lilac line to the diffusion mode. The pole collisions are indicated by the blue stars. At the critical point located at $\lambda=-i$ we see a collision between 4 QNMs implying a multiplicity 4 branch point.
• Figure 4: Perturbative coefficients of the small-$k$ expansion of the dispersion relation of the shear mode for $\tilde{Q}=Q/Q_{max}=0.5, d=4$ (panel a) and the sound mode for $\tilde{Q}=Q/Q_{max}=0.2, d=3$ (panel b). By fitting appropriately these coefficients, as depicted by the solid black lines, we are able to extract the corresponding radius of convergence to be $k_c^2=-6.01$ in panel (a) and $k_c^2= 3.75\, e^{\pm\,i\pi\, 0.054}$ in panel (b).