# Dispersion relations in non-relativistic two-dimensional materials from quasinormal modes in Ho\v{r}ava Gravity

## TL;DR

This paper computes and analyzes the dispersion relations of modes in a non-relativistic strongly coupled 2D quantum field theory using quasinormal modes of a black brane in Hořava Gravity, revealing new analytic and numerical insights.

## Contribution

It introduces a method to compute QNMs in Hořava Gravity, including an analytic expression for the decoupled khronon mode and an extension of sound mode dispersion relations.

## Key findings

- Reproduces known momentum diffusion mode numerically.
- Extends analytic expression for sound modes across khronon couplings.
- Demonstrates perturbative stability over various parameters.

## Abstract

We compute dispersion relations of non-hydrodynamic and hydrodynamic modes in a non-relativistic strongly coupled two-dimensional quantum field theory. This is achieved by numerically computing quasinormal modes (QNMs) of a particular analytically known black brane solution to 3+1-dimensional Ho\v{r}ava Gravity. Ho\v{r}ava Gravity is distinguished from Einstein Gravity by the presence of a scalar field, termed the khronon, defining a preferred time-foliation. Surprisingly, for this black brane solution, the khronon fluctuation numerically decouples from all others, having its own set of purely imaginary eigenfrequencies, for which we provide an analytic expression. All other Ho\v{r}ava Gravity QNMs are expressed analytically in terms of QNMs of Einstein Gravity, in units involving the khronon coupling constants and various horizons. Our numerical computation reproduces the analytically known momentum diffusion mode, and extends the analytic expression for the sound modes to a wide range of khronon coupling values. In the eikonal limit (large momentum limit), the analytically known dispersion of QNM frequencies with the momentum is reproduced by our numerics. We provide a parametrization of all QNM frequencies to fourth order in the momentum. We demonstrate perturbative stability in a wide range of coupling constants and momenta.