Constraints on quasinormal modes and bounds for critical points from pole-skipping

TL;DR

This work investigates how pole-skipping points constrain the spectrum of gapped quasinormal modes in a holographic thermal state and how these constraints relate to critical points and derivative expansions. By analyzing massive scalar perturbations in AdS$_5$ Schwarzschild, it shows that pole-skipping data lie on QNM dispersion curves at imaginary momenta and that the nth QNM is progressively less constrained as one moves up the tower; it also establishes that the radius of convergence of the derivative expansion around a gapped mode is bounded above by pole-skipping points, with a transition at ${\Delta}_{\text{t}} \approx 5.23$ between two collision regimes. The results generalize to vector and tensor operators, while in the BTZ limit ($AdS_3$) the convergence radius becomes infinite, underscoring dimension-dependent behavior. These findings connect discrete pole-skipping information to the analytic structure of QNMs and perturbative expansions, with potential implications for quantum chaos and analytic approaches to critical points.

Abstract

We consider a holographic thermal state and perturb it by a scalar operator whose associated real-time Green's function has only gapped poles. These gapped poles correspond to the non-hydrodynamic quasinormal modes of a massive scalar perturbation around a Schwarzschild black brane. Relations between pole-skipping points, critical points and quasinormal modes in general emerge when the mass of the scalar and hence the dual operator dimension is varied. First, this novel analysis reveals a relation between the location of a mode in the infinite tower of quasinormal modes and the number of pole-skipping points constraining its dispersion relation at imaginary momenta. Second, for the first time, we consider the radii of convergence of the derivative expansions about the gapped quasinormal modes. These convergence radii turn out to be bounded from above by the set of all pole-skipping points. Furthermore, a transition between two distinct classes of critical points occurs at a particular value for the conformal dimension, implying close relations between critical points and pole-skipping points in one of those two classes. We show numerically that all of our results are also true for gapped modes of vector and tensor operators.

Figures (10)

• Figure 1: Left panel: Massive scalar dispersion relations for the lowest four QNMs evaluated at purely imaginary momenta, i.e. $\text{Re}\mathfrak{q}=0$. The stars lying on the vertical axis denote the imaginary part of the QNMs at ${\mathfrak{q}}=0$. Colorful dots show the pole-skipping points at frequencies $\text{Im}\,{\mathfrak{w}}=-1, -2, -3, -4$. Right panel: spectrum of the lowest four QNMs at ${\mathfrak{q}}=0$. These poles are exactly the poles indicated by stars in the left panel. We have checked that this same figure arises for all values of $\Delta$ as well as for massive vector and tensor fields which will be discussed in section \ref{['sec:vectorTensor']}.
• Figure 2: This figure shows the same QNMs as figure 1, however, now in the complex frequency plane. Along the trajectories, $|\mathfrak{q}^2|$ changes from 1 (dots) to 2 (circles). This range is chosen in order to resolve the apparent collision between the red and green pole in figure 1 and the apparent branch point of the red curve in that same figure, both occuring near $\text{Im}\,{\mathfrak{w}}\approx -3.5$. All modes are shown for the fixed phase $\theta=\pi$.
• Figure 3: The radius of convergence, $|\mathfrak{q}_c|$, of the derivative expansion versus the scaling dimension of the boundary operator, within the range $2 \le \Delta \le 8$. The transition between the regime of level-crossing and the regime of lowest-level-degeneracy occurs at $\Delta_\text{t}\approx 5.23$.
• Figure 4: Lowest-level-degeneracy. Poles of the retarded two-point function of the scalar operator, in the complex ${\mathfrak{w}}-$plane, at various values of the complexified momentum ${\mathfrak{q}}^2=|{\mathfrak{q}}^2|e^{i \theta}$. Blue dots correspond to the location of poles for real ${\mathfrak{q}}^2$, namely for $\theta=0$. As $\theta$ increases from $0$ to $2\pi$, each pole moves counter-clockwise following the trajectory whose color changes continuously from blue to red. The two top panels show the situation slightly before and after the collision point marked by black dots. At the critical value $|{\mathfrak{q}}_c^2|\approx 1.25$, the trajectories of the lowest-lying modes collide. After the collision, the orbits of these modes are no longer closed: the two of them exchange their positions as the phase $\theta$ increases from $0$ to $2\pi$. The second collision occurs at a higher momentum. The two bottom panels show the situation slightly before and after the second collision, marked by black crosses. At the critical value $|{\mathfrak{q}}_c^2|\approx 1.54$, the trajectories of the second QNMs collide with the common trajectory of the lowest-lying modes. After the collision, the two lowest-lying modes together with the second QNMs exchange their positions cyclically as the phase $\theta$ increase from $0$ to $2\pi$.
• Figure 5: Level-crossing. Poles of the retarded two-point function of the scalar operator, in the complex ${\mathfrak{w}}-$plane, at various values of the complexified momentum ${\mathfrak{q}}^2=|{\mathfrak{q}}^2|e^{i \theta}$. Blue dots correspond to the location of poles for real ${\mathfrak{q}}^2$, namely for $\theta=0$. As $\theta$ increases from $0$ to $2\pi$, each pole moves counter-clockwise following the trajectory whose color changes continuously from blue to red. The two panels show the situation slightly before and after the collision point marked by brown dot-crosses. Before the collision the orbits of all QNMs are closed. At the critical value $|{\mathfrak{q}}_c^2|\approx2.18$, the trajectories of the lowest-lying modes collide with those of the second QNMs. After the collision, orbits of mentioned modes are no longer closed: in both left and right sides of the plane, the lowest-lying and the second QNMs exchange their positions as the phase $\theta$ increases from $0$ to $2\pi$.