The quasi-normal modes of quantum criticality

TL;DR

The work addresses charge transport at quantum critical points described by $2+1$-D CFTs using a holographic AdS$_4$ framework with a four-derivative coupling, revealing that the conductivity $\sigma( u)$ is governed by a spectrum of quasi-normal modes with complex frequencies. It demonstrates two exact sum rules for $ ext{Re}\,\sigma( u)$ and $ ext{Re}\,1/\sigma( u)$, with holography uniquely satisfying both via poles and S-dual zeros, and shows how a Drude-like single-pole feature emerges at low frequency while a full meromorphic structure captures the entire frequency dependence. The paper also compares holographic results with large-$N$ O($N$) and fermionic CFTs, showing that Drude behavior is a common low-frequency feature while the holographic approach provides the complete analytic structure via QNMs and S-duality. Overall, the quasi-normal mode perspective offers a powerful, general framework for understanding and analytically continuing quantum-critical transport in strongly interacting systems.

Abstract

We study charge transport of quantum critical points described by conformal field theories in 2+1 spacetime dimensions. The transport is described by an effective field theory on an asymptotically anti-de Sitter spacetime, expanded to fourth order in spatial and temporal gradients. The presence of a horizon at non-zero temperatures implies that this theory has quasi-normal modes with complex frequencies. The quasi-normal modes determine the poles and zeros of the conductivity in the complex frequency plane, and so fully determine its behavior on the real frequency axis, at frequencies both smaller and larger than the absolute temperature. We describe the role of particle-vortex or S-duality on the conductivity, specifically how it maps poles to zeros and vice versa. These analyses motivate two sum rules obeyed by the quantum critical conductivity: the holographic computations are the first to satisfy both sum rules, while earlier Boltzmann-theory computations satisfy only one of them. Finally, we compare our results with the analytic structure of the O(N) model in the large-N limit, and other CFTs.

Figures (12)

• Figure 1: (a) Perspective on approaches to the charge transport properties of strongly interacting CFTs in 2+1 dimension. The quantum Boltzmann approach applies to the $1/N$ expansion of the $O(N)$ model: its starting point assumes the existence of weakly interacting quasiparticles, whose collisions control the transport properties. In the present paper we start from the "nearly perfect" quantum liquid obtained in the $N_c \rightarrow \infty$ limit of a SU($N_c$) super Yang-Mills theory, which has no quasiparticle description. Holographic methods then allow expansion away from this liquid ($\lambda$ is the 't Hooft coupling of the gauge theory). (b) Structure of the charge conductivity in the quantum Boltzmann approach. The dashed line is the $N=\infty$ result: it has a delta function at zero frequency, and a gap below a threshold frequency. The full line shows the changes from $1/N$ corrections. (c) Structure of the charge conductivity in the holographic approach. The $N_c = \infty$ result is the dashed line, and this is frequency independent. The full line is the conductivity obtained by including four-derivative terms in the effective holographic theory for $\gamma > 0$.
• Figure 2: AdS spacetime with a planar black brane. The current ($J_\mu$) correlators of the CFT are related to those of the U(1) gauge field ($A_\mu$) in the AdS (bulk) spacetime. The temperature of the horizon of the black brane is equal to the temperature of the CFT. The horizon acts as a "leaky" boundary to the bulk $A_\mu$ normal modes, which consequently become quasi-normal modes with complex frequencies. These quasi-normal modes specify the finite temperature dynamic properties of the CFT.
• Figure 3: (a) Poles (crosses) and zeros (circles) of the holographic conductivity at $\gamma = 1/12$. (b) Real and imaginary parts of the holographic conductivity on the real frequency axis. (c) Poles and zeros of the $O(N)$ model at $N=\infty$; the zeros coincide with branch points, and the associated branch cuts have been chosen suggestively, indicating that the branch cuts transform into lines of poles and zeros after collisions have been included. (d) Conductivity of the $O(N)$ model at $N=\infty$; note the delta function in the real part at $\omega = 0$, and the co-incident zero in both the real and imaginary parts at $\omega = 2 \Delta$. In these figures $\Delta/T = 2 \ln ((\sqrt{5}+1)/2)$, and the $O(N)$ computation is reviewed in Appendix \ref{['app:ana']}.
• Figure 4: Conductivity $\sigma$ and its S-dual $\hat{\sigma}=1/\sigma$ in the LHP, $w"=\Im w \leq 0$, for $|\gamma|=1/12$. The zeros of $\sigma(w;\gamma)$ are the poles of $\hat{\sigma}(w;\gamma)$. We further note the qualitative correspondence between the poles of $\sigma(w;\gamma)$ and the zeros of $\hat{\sigma}(w;-\gamma)$.
• Figure 5: Quasi-normal modes (bright spots) of the transverse gauge mode for $\gamma=|1/12|$ in the complex frequency plane, $w=w'+iw"$. The QNMs correspond to the poles of the conductivity (a & c). EM duality yields the QNMs of the dual gauge mode, and these correspond to the poles of the dual conductivity, $\hat{\sigma}(w)=1/\sigma(w)$, i.e. the zeros of $\sigma(w)$, see panels b & d.