Complexified quasinormal modes and the pole-skipping in a holographic system at finite chemical potential

TL;DR

This work develops and applies a coupled-field holographic framework to AdS$_5$ RN black branes to study quasinormal modes across spin-0, -1, and -2 sectors, obtaining both nonperturbative spectra in $\mu/T$ and analytic hydrodynamic limits at small chemical potential. By analyzing complex momentum, it reveals how collisions between hydrodynamic and non-hydrodynamic poles govern the radius of convergence of derivative expansions, with spin-0 showing a rich, $\mu/T$-dependent structure including collisions between hydrodynamic modes. The authors establish a direct link between pole-skipping in energy-density correlators and quantum chaos, deriving a chaos point from shock-wave methods and identifying a $Q$-dependent boundary $Q_c$ that separates perturbative from nonperturbative regimes for chaos detection. Overall, the paper provides a constructive method for coupled bulk perturbations, clarifies the convergence landscape of holographic hydrodynamics at finite density, and connects hydrodynamic data to chaotic dynamics in strongly coupled plasmas.

Abstract

We develop a method to study coupled dynamics of gauge-invariant variables, constructed out of metric and gauge field fluctuations on the background of a AdS$_5$ Reissner-Nordström black brane. Using this method, we compute the numerical spectrum of quasinormal modes associated with fluctuations of spin 0, 1 and 2, non-perturbatively in $μ/T$. We also analytically compute the spectrum of hydrodynamic excitations in the small chemical potential limit. Then, by studying the spectral curve at complex momenta in every spin channel, we numerically find points at which hydrodynamic and non-hydrodynamic poles collide. We discuss the relation between such collision points and the convergence radius of the hydrodynamic derivative expansion. Specifically in the spin 0 channel, we find that within the range $1.1\lesssim μ/T\lesssim 2$, the radius of convergence of the hydrodynamic sound mode is set by the absolute value of the complex momentum corresponding to the point at which the sound pole collides with the hydrodynamic diffusion pole. It shows that in holographic systems at finite chemical potential, the convergence of the hydrodynamic derivative expansion in the mentioned range is fully controlled by hydrodynamic information. As the last result, we explicitly show that the relevant information about quantum chaos in our system can be extracted from the pole-skipping points of energy density response function. We find a threshold value for $μ/T$, lower than which the pole-skipping points can be computed perturbatively in a derivative expansion.

Figures (14)

• Figure 1: Left panel: Stars represent poles of the master charge density Green’s function and dots correspond with poles of the master energy density Green’s function. As ${\mathfrak{q}}$ decreases, all poles stay at a finite distance from the real axis, except for the one marked with a large star and the two marked with large dots. Large dots therefore manifest the existence of two sound modes and the large dot corresponds with the existence of a diffusive $U(1)$ charge mode in the boundary $\mathcal{N}=4$ SYM theory at finite chemical potential.
• Figure 2: Left panel: Stars are poles of the transverse master momentum density Green's function and dots correspond with the poles of the transverse master charge current Green's function. In fact, dots identify the shear poles. As ${\mathfrak{q}}$ decreases, all poles stay at a finite distance from the real axis, except for the one marked with a large dot. The latter manifests the existence of a diffusive shear mode in the boundary $\mathcal{N}=4$ SYM theory at finite chemical potential.
• Figure 3: Quasinormal modes associated with $Q=0.5$ at two different values of ${\mathfrak{q}}$, in spin 1 channel. At each value of ${\mathfrak{q}}$, we have shown five lowest quasinormal modes. As ${\mathfrak{q}}$ decreases, all poles stay at finite distances from the real axis. The latter manifests the non-existence of any spin 2 hydrodynamic mode in $\mathcal{N}=4$ SYM theory at finite chemical potential.
• Figure 4: The radius of convergence of the derivative expansion versus $Q$. As $Q$ increases, the radius of convergence of ${\mathfrak{w}}_{\text{diffusion}}$ (red curve) monotonically increases. At the same time, domain of convergence of ${\mathfrak{w}}_{\text{sound}}$ non-trivially changes. As discussed in the text, it can be studied in three different intervals: $0\le Q\le 0.386$, $0.386\le Q \le 633$ and $0.633\le Q\le 0.850$. The intersection point of blue curve with the vertical axis, related to the sound mode in the $\mathcal{N}=4$ SYM theory in the vanishing $\mu$ limit, was found in Grozdanov:2019kge.
• Figure 5: Poles of the retarded two-point function in the spin 0 channel at $Q=0.3$, in the complex ${\mathfrak{w}}-$plane, at various values of the complexified momentum ${\mathfrak{q}}^2=|{\mathfrak{q}}^2|e^{i \theta}$. Large dots and large stars correspond to the poles with purely real momentum (i.e. at $\theta=0$). As $\theta$ increases from $0$ to $2\pi$, each pole moves counter-clockwise following the trajectory whose color changes continuously from blue to red. While at $|{\mathfrak{q}}^2|=0.60$ (top left panel) all poles follow closed orbits, at $|{\mathfrak{q}}^2|=1.00$ (top middle panel), the hydrodynamic diffusion pole and the two lowest (namely nearest to the horizontal axis) star gapped poles follow open orbits. It simply means that the dispersion relation ${\mathfrak{w}}_{\text{diffusion}}({\mathfrak{q}}^2)$ has branch point singularities in the complex momentum squared plane at $(0.60)^{1/2}<|{\mathfrak{q}}_c|<(1.00)^{1/2}$. It is clear that the point $(Q=0.5, |{\mathfrak{q}}_c|)$ then lies on the red curve in Fig.\ref{['q_c_Sound_diiusion']}. At $|{\mathfrak{q}}^2|=1.50$ (top right panel) the orbits of the hydrodynamic sound pole and the two nearest dot poles are still closed. However, by further increasing $|{\mathfrak{q}}^2|$, their associated trajectories come close to each other. Finally they collide at the positions marked by black dots in the bottom row plots. The collision points are identified with critical value of momentum $|{\mathfrak{q}}_c^2|=2.32$. It is clear that the point $(Q=0.3, (2.32)^{1/2})$ lies on the blue curve in Fig.\ref{['q_c_Sound_diiusion']}. After the collision, for instance at $|{\mathfrak{q}}^2|=2.34$, the orbits of sound pole and the two nearest dot gapped poles are no longer closed: four of them exchange their positions as the phase $\theta$ increases from $0$ to $2\pi$. This is the manifestation of the $\boldsymbol{\bar{T}\bar{T}-}$crossing.