Holography without translational symmetry

TL;DR

This work introduces a holographic model for momentum dissipation by endowing the bulk graviton with a Lorentz-violating mass, implemented via a spatial reference metric and Stückelberg fields to break translational symmetry while preserving homogeneous bulk dynamics. In charged AdS black brane backgrounds, it derives the DC/AC conductivities, uncovering a Drude peak that broadens with increasing graviton mass and, in an intermediate regime, an emergent non-Drude scaling $|\sigma(\omega)| \approx {A\over \omega^{\alpha}} + B$ with $\alpha$ related to the mass parameters. The results reproduce and extend lattice-holography insights (Horowitz–Santos–Tong), showing that momentum relaxation can be modeled without explicit inhomogeneous lattices, and point to rich transport phenomenology, including stability boundaries and possible phase structures, in strongly coupled systems. The framework offers a tractable avenue to study momentum dissipation and transport in holography with potential extensions to time-translation breaking and dynamical reference metrics, bridging toward more realistic condensed-matter applications.

Abstract

We propose massive gravity as a holographic framework for describing a class of strongly interacting quantum field theories with broken translational symmetry. Bulk gravitons are assumed to have a Lorentz-breaking mass term as a substitute for spatial inhomogeneities. This breaks momentum-conservation in the boundary field theory. At finite chemical potential, the gravity duals are charged black holes in asymptotically anti-de Sitter spacetime. The conductivity in these systems generally exhibits a Drude peak that approaches a delta function in the massless gravity limit. Furthermore, the optical conductivity shows an emergent scaling law: $|σ(ω)| \approx {A \over ω^α} + B$. This result is consistent with that found earlier by Horowitz, Santos, and Tong who introduced an explicit inhomogeneous lattice into the system.

Figures (3)

• Figure 1: Stability in parameter space. We set $r_h = L = m^2 = F = 1$ for the plot. Above the "wall of stability" $\beta = - {L \alpha \over 2 F r_h}$ the entropy density is larger than the usual value ("$S=A/4$") and numerical results indicate an instability. On the line $\beta = -{3 + F L m^2 r_h \alpha \over F^2 m^2 r_h^2}$, the maximal value of the chemical potential is zero. Between these two lines (yellow region) the system can be stable. The lines cross at $(\alpha,\beta)=(-{6\over F L m^2 r_h},{3\over F^2 m^2 r_h^2})$. Beyond this point there may still be stable points.
• Figure 2: Drude peak in the conductivity. The real and imaginary parts are drawn in blue and purple, respectively. At larger frequencies, the conductivity approaches a constant. The parameters were set to $\alpha=-1$, $\beta=0$, $\mu = 1.724$, $T=0.1$, $m=1$, $L=1$.
• Figure 3: Non-Drude optical conductivity. There is an approximate power-law: $|\sigma(\omega)| \approx {A \over \omega^{\gamma}} + B$. The mass is tuned to $L^2 m^2 \alpha = -0.75$ (and $\beta=0$) so that the exponent $\gamma\approx 2/3$ with an offset $B\approx -1.2$. The constants $\gamma$, $A$ and $B$ depend on the two parameters $\alpha$ and $\beta$. Fig. \ref{['fig:picrnsigma']}: The blue and purple lines are the real and imaginary parts of the conductivity, respectively. Fig. \ref{['fig:picrnexp']}: The plot shows ${d\left(|\sigma|-B\right) \over d\left( \log \omega \right)}$ which gives the exponent if there is indeed a power law. Fig. \ref{['fig:picrnphase']}: Phase of $\sigma(\omega)$. If $B$ were zero, then it would exactly be $60^\circ$ due to causality and time-reversal invariance 2003Natur.425..271M.