Solid Holography and Massive Gravity

TL;DR

This paper establishes a comprehensive framework for holographic massive gravity (HMG) as a model of momentum relaxation in strongly coupled systems. By classifying phases according to preserved diffeomorphisms, it distinguishes solids and fluids through elastic response and the presence of transverse phonons, and it demonstrates that a broad class of fully consistent EFTs can realize these phases beyond the usual dRGT setup. A two-field holographic model captures solids and fluids, yielding finite DC conductivity determined by a horizon function $M^2(r)$ and a rigidity modulus sourced by the tensor mass $m_2(r)$, while the AC response and phonon dynamics reveal metallic/insulating behavior and phonon-related resonances. The analysis also addresses nonlinear consistency and stability, showing BD ghost–free constructions and detailing when additional helicity-0 modes emerge, thereby providing a solid foundation for applying HMG to condensed-matter phenomena and transport in strongly correlated media.

Abstract

Momentum dissipation is an important ingredient in condensed matter physics that requires a translation breaking sector. In the bottom-up gauge/gravity duality, this implies that the gravity dual is massive. We start here a systematic analysis of holographic massive gravity (HMG) theories, which admit field theory dual interpretations and which, therefore, might store interesting condensed matter applications. We show that there are many phases of HMG that are fully consistent effective field theories and which have been left overlooked in the literature. The most important distinction between the different HMG phases is that they can be clearly separated into solids and fluids. This can be done both at the level of the unbroken spacetime symmetries as well as concerning the elastic properties of the dual materials. We extract the modulus of rigidity of the solid HMG black brane solutions and show how it relates to the graviton mass term. We also consider the implications of the different HMGs on the electric response. We show that the types of response that can be consistently described within this framework is much wider than what is captured by the narrow class of models mostly considered so far.

Figures (4)

• Figure 1: The electric DC conductivity (\ref{['dc_cond']}) as a function of temperature for $\mu=1$ and $m_1=1$ for different values of the parameter $\nu$.
• Figure 2: The DC resistivity $\rho=1/\sigma_{DC}$ as a function of temperature for parameter values $\nu = -3$, $\mu=1$, $m_1=1$. The dashed line is a linear fit $\rho\propto T$.
• Figure 3: Real part of the optical conductivity for different choices of $\nu$ and parameters $\mu=1.5$, $m_1=1$. We fix $r_h=1$ to have the same DC value for different $\nu$.
• Figure 4: Modulus of rigidity $G$ for the mass parameter $m_2^2(r)=r^{-2}$ and chemical potential $\mu=1$: Left: mass dependence; Center: temperature dependence with $m_2=1$; Right: Temperature dependence with $m_2=1$ normalizing the quantity by the T dependent energy density $\epsilon$.