Effective holographic theories of momentum relaxation and violation of conductivity bound

TL;DR

This work extends holographic models of momentum relaxation by introducing higher-derivative couplings between translation-symmetry breaking axions and the charge sector, treated within an effective-field-theory mindset. The authors derive analytic charged black-brane solutions and compute the DC conductivity from horizon data for two explicit couplings, $Tr[XF^2]$ and $Tr[X]F^2$, revealing that the previously proposed bound on conductivity can be violated when these couplings are active. They establish parameter ranges ensuring positive conductivity and analyze zero-density Schrödinger potentials to assess stability, finding that the lower bound on $\sigma_{DC}$ is not universal in the presence of these higher-derivative interactions. The results illuminate how momentum-relaxation and charge transport interact in holography and suggest directions for refining physical constraints and exploring metal-insulator behavior in strongly coupled systems.

Abstract

We generalize current holographic models with homogeneous breaking of translation symmetry by incorporating higher derivative couplings, in the spirit of effective field theories. Focusing on charge transport, we specialize to two simple couplings between the charge and translation symmetry breaking sectors. We obtain analytical charged black brane solutions and compute their DC conductivity in terms of horizon data. We constrain the allowed values of the couplings and note that the DC conductivity can vanish at zero temperature for strong translation symmetry breaking, thus showing that in general there is no lower bound on the conductivity.

Figures (7)

• Figure 1: Plots of the DC conductivity with the $\mathcal{J}$ coupling turned on. Left: We fix $T=0$ and vary $k/\mu$. Right: We fix $k=\mu$ and vary $T/\mu$. For $\mathcal{J}>2/3$, the DC conductivity vanishes at $T=0$ for $k=k_\textrm{max}$.
• Figure 2: Plot of the parameter range where the DC conductivity \ref{['26']} is positive.
• Figure 3: Plots of the DC conductivity with the $\mathcal{K}$ coupling turned on, for both positive and negative values. Left, we fix $T=0$ and vary $k/\sqrt\rho$. Center, we fix $T=\sqrt{\rho}/3$ and vary $k/\sqrt\rho$. Right, we fix $k=\sqrt{\rho}$ and vary $T/\sqrt{\rho}$.
• Figure 4: Plot of the DC conductivity \ref{['sigmaDC2rho0']} with the $\mathcal{J}$ coupling turned on at zero density $\rho=0$, for positive (left) and negative (right) values of $\mathcal{J}$.
• Figure 5: Plots of the Schrödinger potential \ref{['Vschrcase2']} versus the radial coordinate $0<u<1$ in units of temperature, with $\tilde{k}=k/4\pi T$. Top row, for positive values of $\mathcal{J}$: left for $0<\mathcal{J}\leq2/3$, showing the presence of a negative well in the potential at the boundary, and a maximum close to the horizon; right, for $\mathcal{J}>2/3$, showing that for large enough $k$, an infinite negative well develops. Bottom row left, for $\mathcal{J}=-1$, a positive but finite potential barrier appears at the boundary with a negative well closer to the horizon. Bottom row, right, we fix $\tilde{k}=1$ and show the dependence of the height/depth of positive/negative potential barrier on the value of $\mathcal{J}$.