Quasinormal modes of a strongly coupled nonconformal plasma and approach to criticality

We study fluctuations around equilibrium in a class of strongly interacting nonconformal plasmas using holographic techniques. In particular, we calculate the quasinormal mode spectrum of black hole backgrounds that approach Chamblin-Reall plasmas in the IR. In a specific limit, related to the exact linear-dilaton background in string theory, we observe that the plasma approaches criticality and we obtain the quasinormal spectrum analytically. We regulate the critical limit by gluing the IR geometry that corresponds to the nonconformal plasma to a part of AdS space-time in the UV. Near criticality, the spectrum can still be computed analytically and we find two sets of quasinormal modes, related to the IR and UV parts of the geometry. In the critical limit, the quasinormal modes accumulate to form a branch cut in the correlators of the energy-momentum tensor on the real axis of the complex frequency plane.

Figure 1

The dependence of the four lowest nonhydrodynamic quasinormal modes on $\xi$ at $q=0$. Thick lines were obtained by directly solving the fluctuation equation (5) numerically for $4\le \xi \lesssim 20$, and the thin lines are based on the analytic approximation in (8).

Figure 2

The structure of the geometry for potentials asymptoting to CR behavior in the IR at small temperatures with large $\xi$.

Figure 3

The dependence of the location of quasinormal modes on the location of the horizon at $q=0$ in the setup where AdS and CR geometries were glued together. Left: The trajectories of the ten lowest QNMs on the complex $\varpi$-plane at $\xi \simeq 26.77$ as $T$ grows from $T=0.2T_c$ to $T=0.91T_c$. The dashed curves were added to guide the eye, they are at constant $T/T_c$ with values of the ratio indicated by the labels. The markers are at $T/T_c=0.2$, 0.3, … 0.9 for all curves. Right: The trajectories of the QNMs at $\xi =\infty$ on the complex $\varpi$-plane as $r_h$ is varied from $r_h=1.2r_c$ (lower end points of the curves) to $r_h=20r_c$ (upper end points). The dots are the locations of the QNMs for $r_h/r_c=\sqrt{2}$, 2, $2\sqrt{2},\dots 16$. The dashed lines connect the locations at the same $r_h$ as indicated by the labels.

Figure 4

The (logarithm of the) absolute value of the correlators of the energy-momentum tensor on the complex $\varpi$-plane in various approximations. The plots are for $\xi =\infty$ and $q=0$. The contours are at constant values of $|{\tilde{G}}_{\mathrm{reg}}|$, with orange/yellow colors (mostly top parts of the plots) indicating small values and blue/white colors (mostly bottom parts of the plots) indicating large values. Top left: the “glued” correlator of (11) at $r_h/r_c=2$. Top right: the correlator of (11) at $r_h/r_c=20$. Bottom left: the large $\varpi$ approximation of the correlator with $r_h/r_c=2$ [8]. Bottom right: the limit of large black hole (12) of the glued correlator.