Dispersing quasinormal modes in 2+1 dimensional conformal field theories

TL;DR

This work uses AdS/CFT to study the finite-temperature charge response of 2+1D CFTs, focusing on current correlators and their quasinormal mode spectra as a function of momentum. By incorporating a four-derivative Weyl term and exploiting S-duality, it reveals how QNMs disperse, induce hydrodynamic-to-relativistic crossovers, and produce D-type modes that modify transport and sum rules. The analysis unifies holographic results with large-$N$ vector models, showing consistent sum rules and a deep link between bulk gauge dynamics, boundary spectral structure, and emergent excitations. The findings illuminate non-quasiparticle transport in quantum critical systems and offer potential benchmarks for experiments probing universal charge dynamics in 2+1D quantum critical states.

Abstract

We study the charge response of conformal field theories (CFTs) at non-zero temperature in 2+1 dimensions using the AdS/CFT correspondence. A central role is played by the quasinormal modes (QNMs), specifically, the poles and zeros of the current correlators. We generalize our recent study of the QNMs of the a.c. charge conductivity to include momentum dependence. This sheds light on the various excitations in the CFT. We begin by discussing the R-current correlators of the N=8 SU(N) super-Yang-Mills theory at its conformal fixed point using holography. For instance, transitions in the QNM spectrum as a function of momentum clearly identify hydrodynamic-to-relativistic crossovers. We then extend our study to include four-derivative terms in the gravitational description allowing us to study more generic charge response as well as the role of S-duality, which plays a central role in understanding the correlators. The presence of dispersing Drude-like QNMs can lead to new behavior, distinct from what occurs in the aforementioned gauge theory. We also extend previous conductivity sum rules to finite momentum and discuss their interpretation in the gravity picture. A comparison is finally made with the conformal fixed point of the vector O(N) model in the large-N limit.

Figures (15)

• Figure 1: Frequency and momentum dependence of the longitudinal and transverse R-current correlators, $C_{tt}$ and $C_{yy}$, respectively, normalized by $-\chi_0$. Note the reflection property under exchange of $w$ and $q$, a remnant from the Lorentz invariant $T=0$ form.
• Figure 2: Comparison of $\Im C_{yy}(w,q)$ with the zero temperature form $\sqrt{-w^2+q^2}$. From left to right: $q=0,2,4,6,8$. The plots of $C_{yy}$ are in units of $-\chi_0$. The plot of $-\Re C_{yy}(q,w)$ (note the arguments are interchanged) illustrates the "reflection property".
• Figure 3: Poles (crosses) and zeros (circles) in the lower-half complex frequency plane of $C_{tt}=-\chi^2 q^2/\Pi^T(w,q)$, the R-charge density correlation function of the $\mathcal{N}=8$ supersymmetric Yang-Mills CFT. The positions of the poles and zeros are interchanged for $C_{yy}=\Pi^T$.
• Figure 4: General mechanism according to which two poles detach from the imaginary axis. A double pole exists at the intermediate step. The same mechanism applies to zeros.
• Figure 5: Hydrodynamic-to-relativistic crossover. Momentum dispersion of the position of the peak of $C_{tt}(w,q)$, $w_{\rm max}(q)$, and the norm of the diffusive QNM, $w_{\rm qnm}(q)$. A quadratic hydrodynamic scaling is seen at small $q$ (as a guide, the thin blue line is $q^2$), while a linear relativistic scaling emerges at large $q$. The vertical lines at $q=0.339$ and $q=0.557$ signal momenta at which pairs of QNMs detach from the imaginary axis. The second one is where the diffusive QNM detaches.