Shear subdiffusion in non-relativistic holography

TL;DR

This work studies shear fluctuations of strongly coupled non-relativistic systems using Lifshitz holography in Einstein–Maxwell–dilaton gravity with torsional Newton–Cartan boundaries. It shows a universal subdiffusive shear mode with dispersion $\omega = -i D_4 k^4$ (valid for $1<z<2$) by a higher-order matched asymptotic expansion linking near-horizon and far-region solutions, and confirms the result with numerical quasinormal modes. The numerics also reveal a purely imaginary, gapped first non-hydrodynamic mode with $\omega = -i\omega_0 - i D k^2$ and indicate pole-skipping phenomena for both the hydrodynamic and first non-hydrodynamic modes. The findings establish Lifshitz holography as a robust framework for anomalous transport in strongly coupled non-relativistic quantum matter and point to intriguing future directions, including connections to fracton physics and extensions to larger Lifshitz exponents $z$ or momentum-dissipation mechanisms.

Abstract

We study shear fluctuations in non-relativistic holographic systems coupled to torsional Newton-Cartan geometry, using asymptotically Lifshitz spacetimes in Einstein-Maxwell-dilaton gravity. We identify a universal subdiffusive shear mode characterized by the quartic dispersion relation $ω=-iD_4 k^4$, in sharp contrast to the conventional hydrodynamic diffusion. We derive this result analytically through a systematic higher-order matched asymptotic expansion connecting near-horizon and far-region solutions, and we verify it with direct numerical quasinormal mode calculations. Our numerics demonstrate that the first non-hydrodynamic mode is purely imaginary and gapped, following the dispersion relation $ω=-iω_0-i D k^2$, and that both the hydrodynamic and the first non-hydrodynamic modes pass through pole-skipping points. These results highlight Lifshitz holography as an efficient framework for anomalous transport in strongly coupled non-relativistic quantum matter.

Figures (2)

• Figure 1: Imaginary parts of frequencies of the hydrodynamic modes ( blue dots) and non-hydrodynamic modes ( orange dots) for $z=3/2$. They pass through pole-skipping points ( open dots).
• Figure 2: left: The subdiffusive $D_4$ as a function of $z$. The blue dots are from \ref{['eq:D4']} while the black dots are from numerical QNM calculation. Right: The disperssion relations for hydrodynamic modes (dots) at different $z$. The lines are analytic dispersion relations \ref{['eq:dis']} with $D_4$ from \ref{['eq:D4']}.