Hydrodynamic diffusion and its breakdown near AdS$_2$ quantum critical points

TL;DR

The paper investigates how diffusive hydrodynamics breaks down in quantum critical states with emergent AdS$_2$-like infrared dynamics, using holographic models and SYK-chain analogues. It identifies a universal mechanism: the diffusive pole collides with infrared AdS$_2$ poles, yielding local equilibration scales $ω_{eq}=2πΔT$ and $k_{eq}^2=ω_{eq}/D$, controlled by the IR scaling dimension Δ. The authors demonstrate this across neutral and charged holographic states and corroborate it in an SYK chain at strong coupling, linking transport, equilibration, and scrambling in non-quasiparticle quantum matter. The results propose a unified framework for understanding diffusivity near quantum critical points and offer a concrete route to quantify transport in such regimes.

Abstract

Hydrodynamics provides a universal description of interacting quantum field theories at sufficiently long times and wavelengths, but breaks down at scales dependent on microscopic details of the theory. In the vicinity of a quantum critical point, it is expected that some aspects of the dynamics are universal and dictated by properties of the critical point. We use gauge-gravity duality to investigate the breakdown of diffusive hydrodynamics in two low temperature states dual to black holes with AdS$_2$ horizons, which exhibit quantum critical dynamics with an emergent scaling symmetry in time. We find that the breakdown is characterized by a collision between the diffusive pole of the retarded Green's function with a pole associated to the AdS$_2$ region of the geometry, such that the local equilibration time is set by infra-red properties of the theory. The absolute values of the frequency and wavevector at the collision ($ω_{eq}$ and $k_{eq}$) provide a natural characterization of all the low temperature diffusivities $D$ of the states via $D=ω_{eq}/k_{eq}^2$ where $ω_{eq}=2πΔT$ is set by the temperature $T$ and the scaling dimension $Δ$ of an operator of the infra-red quantum critical theory. We confirm that these relations are also satisfied in an SYK chain model in the limit of strong interactions. Our work paves the way towards a deeper understanding of transport in quantum critical phases.

Figures (12)

• Figure 1: Frequencies of the hydrodynamic and longest lived infra-red modes at $T/m=10^{-3}$. Black circles are numerical results and red lines are the analytic expressions \ref{['eq:scalingdispersion']} and \ref{['eq:IRpolelocations']}. For real $k$, all poles displayed have purely imaginary frequencies.
• Figure 2: Frequencies of the hydrodynamic and longest lived infra-red modes at $T/m=10^{-3}$, zooming in on the region near $\omega_0$. The black dots are the numerical results (the bottom dots are the hydrodynamic mode, the top ones the longest-lived non-hydrodynamic mode), the red lines show the dispersion relations extracted analytically from \ref{['eq:Gnearcollision']} for real values of $k$. The pole collision is not visible on this figure as it happens at a complex value of $k$.
• Figure 3: Motion of the hydrodynamic (starting in the bottom left of the plot) and longest-lived infra red mode (starting in the top left of the plot) in the complex $\omega$ plane as $\left|k\right|/T$ is increased (from approximately $101.09$ to approximately $101.15$) at fixed $T/m=10^{-3}$ and fixed phase of the wavenumber $\phi_k=7.095\times 10^{-4}$. There is a collision for $\left|k\right|\simeq 101.12T$. Equation \ref{['eq:Gnearcollision']} predicts a collision at $\left|k\right|\simeq 101.125T$ and $\phi_k=7.374\times 10^{-4}$. In Figure \ref{['fig:axionequilibrationdata']} in Appendix \ref{['sec:AxionAppendix']}, we show that the discrepancy between the numerical and analytical values from equation \ref{['eq:Gnearcollision']} decreases with temperature.
• Figure 4: Numerical results for the frequencies of the hydrodynamic and longest lived infra-red modes of the neutral, translation-symmetry breaking model at $T/m=10^{-3}$ (circles). Away from $\omega_{n\ge0}$, the analytic dispersion relation \ref{['eq:scalingdispersion']} (solid red line) provides an excellent approximation to the exact location of a pole.
• Figure 5: Numerically obtained local equilibration data for diffusive hydrodynamics in $G_{\varepsilon\varepsilon}$ (black circles) and $G_{\Pi\Pi}$ (red squares) of the charged state.