Momentum relaxation in holographic massive gravity

TL;DR

This work analyzes momentum relaxation in a holographic theory by introducing a bulk graviton mass, which breaks momentum conservation in the dual field theory. It shows that a single momentum-relaxation timescale suffices to describe the low-energy dynamics in a modified hydrodynamic framework, and identifies a wall of stability where the relaxation time vanishes. Analytic results for the zero-temperature AC conductivity reveal Drude-like behavior with calculable corrections and finite DC conductivity, while a two-region (inner AdS2 and outer) analysis clarifies the near-horizon contributions. A preliminary stability study finds no spatially modulated instabilities of the examined transverse sector, supporting the viability of this holographic toy model for momentum-dissipating systems. The results illuminate how a simple bulk mass term maps to observable transport features and invite further exploration of longitudinal modes and more comprehensive stability analyses.

Abstract

We study the effects of momentum relaxation on observables in a recently proposed holographic model in which the conservation of momentum in the field theory is broken by the presence of a bulk graviton mass. In the hydrodynamic limit, we show that these effects can be incorporated by a simple modification of the energy-momentum conservation equation to account for the dissipation of momentum over a single characteristic timescale. We compute this timescale as a function of the graviton mass terms and identify the previously known "wall of stability" as the point at which this relaxation timescale becomes negative. We also calculate analytically the zero temperature AC conductivity at low frequencies. In the limit of a small graviton mass this reduces to the simple Drude form, and we compute the corrections to this which are important for larger masses. Finally, we undertake a preliminary investigation of the stability of the zero temperature black brane solution of this model, and rule out spatially modulated instabilities of a certain kind.

Figures (3)

• Figure 1: Dependence of the imaginary part of the shear diffusion mode upon the field theory chemical potential $\mu$, for $r_0^2m_\alpha^2=r_0^2m_\beta^2=0.01$. The real part (not shown) is zero. The black dots are the results from numerically integrating the equations of motion, and the solid red line is the expression (\ref{['eq:nonzerodensitydispersionrelation']}).
• Figure 2: The real and imaginary parts of the conductivity $\sigma$ for $m_\beta^2r_0^2\approx0.44$ (top) and $m_\beta^2r_0^2\approx1.04$ (bottom). The black dots are the exact numerical results, the dashed black line is the Drude conductivity (\ref{['eq:ApproxofAcConductivitytoDrudeModel']}) and the solid red line is the result (\ref{['eq:ZeroTACConductivityResult']}) which includes corrections to the simple Drude formula. $\sigma$ is plotted in units of $L^2/\left(4\kappa_4^2r_0\mu\right)$.
• Figure 3: The potential scaling exponent (left) as described in the main text with $c=-8.9$, and phase (right), of the conductivity $\sigma$ when $m_\beta^2r_0^2\approx0.44$. Black dots show the exact numerical results, the solid red line is the analytic expression (\ref{['eq:ZeroTACConductivityResult']}) and the dashed black lines denote the exponent and phase found experimentally in experimentalpaper1. Both $c$ and $\sigma$ are given in units of $L^2/\left(4\kappa_4^2r_0\mu\right)$.