Holographic Q-lattices

TL;DR

We address breaking translation invariance in holographic CFTs by introducing holographic Q-lattices constructed in $D=4$ Einstein-Maxwell theory with a complex scalar, yielding ODE-based black hole solutions. The metallic phase flows to $AdS_2\times\mathbb{R}^2$ in the IR and exhibits a Drude peak with DC resistivity scaling $\rho \sim (T/\mu)^{2\Delta(k)-2}$; the insulating phase avoids this IR fixed point and shows spectral weight transfer and a mid-frequency hump in the optical conductivity. No evidence for the intermediate scaling reported in earlier lattice models is found for the parameter ranges studied, though a neutral $AdS_3\times\mathbb{R}$ fixed point is identified and possible domain-wall interpolations are discussed. The work provides a controlled, back-reacted holographic realization of metal-insulator transitions and clarifies how lattice parameters control IR physics and transport.

Abstract

We introduce a new framework for constructing black hole solutions that are holographically dual to strongly coupled field theories with explicitly broken translation invariance. Using a classical gravitational theory with a continuous global symmetry leads to constructions that involve solving ODEs. We study in detail $D=4$ Einstein-Maxwell theory coupled to a complex scalar field with a simple mass term. We construct black holes dual to metallic phases which exhibit a Drude-type peak in the optical conductivity, but there is no evidence of an intermediate scaling that has been reported in other holographic lattice constructions. We also construct black holes dual to insulating phases which exhibit a suppression of spectral weight at low frequencies. We show that the model also admits a novel $AdS_3\times\mathbb{R}$ solution.

Figures (2)

• Figure 1: Black holes in the metallic phase for lattice parameters $\lambda/\mu=1/2$ and $k/\mu=1/\sqrt{2}$. Panels (a) and (b) shows the real and imaginary parts of the optical conductivity, $Re(\sigma)$ and $Im(\sigma)$, respectively, for four different temperatures. As the the temperature is lowered, the Drude peak becomes more pronounced. Panel (c) shows the behaviour of the DC resistivity, $\rho$, as a function of $T/\mu$. The blue line is the data and the red dashed line is the scaling expected from \ref{['dares']}. Panel (d) shows a plot of $1+\omega|\sigma|"/|\sigma|'$ versus frequency; there is no evidence for an intermediate scaling of the form \ref{['scal']}, which corresponds to the red dashed line.
• Figure 2: Black holes in the insulating phase for lattice parameters $\lambda/\mu^{3-\Delta}=2$ and $k/\mu=1/2^{3/2}$. Panel (a) shows the behaviour of the DC resistivity, $\rho$, as a function of $T/\mu$. Panels (b) and (c) show the real and imaginary parts of the optical conductivity, $Re(\sigma)$ and $Im(\sigma)$, respectively, for four different temperatures. For very low temperatures we see in panel (b) the suppression of spectral weight for small $\omega$ and the development of a mid-frequency hump.