Holographic Lattices Give the Graviton a Mass

TL;DR

This paper addresses momentum dissipation in holographic CFTs by introducing a spatial lattice and analyzing how it endows the bulk graviton with an effective mass. Using a perturbative holographic lattice in Einstein–Maxwell theory with a neutral scalar, the authors show that a bulk phonon is eaten by the metric, yielding a radially varying graviton mass $M^2(r)$ and a simplified set of perturbation equations resembling massive gravity. They derive a horizon-based expression for the DC conductivity, $\sigma_{DC}$, and, consequently, a scattering rate $\Gamma \sim \frac{s}{4\pi}\frac{M^2(r_h)}{\mathcal{E}+\mathcal{P}}$, with $M^2(r_h)\sim \epsilon^2 k_L^2 \phi_0(r_h)^2$. In locally critical (AdS$_2$) IR, this leads to the key scaling $\rho \sim \epsilon^2 k_L^2\,T^{2\Delta_{k_L}}$, matching Hartnoll–Hofman field-theory results, and providing an analytic holographic derivation of the same behavior. The work thus connects holographic lattices, massive gravity, and horizon data to momentum relaxation, offering analytic insight beyond prior numerical studies.

Abstract

We discuss the DC conductivity of holographic theories with translational invariance broken by a background lattice. We show that the presence of the lattice induces an effective mass for the graviton via a gravitational version of the Higgs mechanism. This allows us to obtain, at leading order in the lattice strength, an analytic expression for the DC conductivity in terms of the size of the lattice at the horizon. In locally critical theories this leads to a power law resistivity that is in agreement with an earlier field theory analysis of Hartnoll and Hofman.