A simple holographic model of momentum relaxation

TL;DR

This work introduces a minimal holographic model for momentum relaxation by coupling Einstein–Maxwell theory to $d-1$ massless scalars with linearly spatial sources, yielding homogeneous, isotropic black brane solutions and a finite DC conductivity with a closed-form expression. In general dimensions, the DC conductivity is $\sigma_{DC} = r_0^{d-3}\left(1 + (d-2)^2 \frac{\mu^2}{\alpha^2}\right)$, and in $d=3$ the background geometry and shear-mode correlators align with a sector of nonlinear massive gravity (with $\alpha^2 = 2 m_\beta^2$). The paper provides detailed holographic renormalization and thermodynamics for the $d=3$ case and analyzes shear fluctuations, revealing a decoupled set of master equations whose massive-gravity form matches the known results for current correlators in that sector. The study clarifies the field-theoretic interpretation of momentum relaxation in holography and suggests natural extensions, including additional couplings and potential string-embeddings. Overall, it offers a transparent, analytically tractable framework to explore momentum dissipation without lattice constructions, linking holographic momentum relaxation to massive gravity in a controlled setting.

Abstract

We consider a holographic model consisting of Einstein-Maxwell theory in (d+1) bulk spacetime dimensions with (d-1) massless scalar fields. Momentum relaxation is realised simply through spatially dependent sources for operators dual to the neutral scalars, which can be engineered so that the bulk stress tensor and resulting black brane geometry are homogeneous and isotropic. We analytically calculate the DC conductivity, which is finite. In the d=3 case, both the black hole geometry and shear-mode current-current correlators are those of a sector of massive gravity.