Optical Conductivity with Holographic Lattices

TL;DR

The paper addresses the zero-frequency delta function and infinite DC conductivity that arise from translational invariance in holographic models of finite-density matter. It introduces a gravitational lattice by sourcing a neutral scalar with a periodic boundary condition and solves the full Einstein-Maxwell-scalar system using the DeTurck method to obtain a backreacted lattice geometry. By perturbing this background, it computes the optical conductivity and finds a low-frequency Drude peak, a mid-frequency power-law fall-off with $|oldsymbol{\sigma}( ilde{ m omega})|\, o B/ ilde{ m omega}^{2/3}+C$, and a high-frequency approach to the conformal value, with the DC resistivity $ ho$ showing lattice-spacing–dependent scaling consistent with locally critical behavior near the horizon. The results reproduce qualitative features observed in cuprates, illustrating how holographic lattices can capture dissipative transport and nontrivial frequency dependence in strongly coupled systems, and highlighting paths for further analytic understanding and model extensions.

Abstract

We add a gravitational background lattice to the simplest holographic model of matter at finite density and calculate the optical conductivity. With the lattice, the zero frequency delta function found in previous calculations (resulting from translation invariance) is broadened and the DC conductivity is finite. The optical conductivity exhibits a Drude peak with a cross-over to power-law behavior at higher frequencies. Surprisingly, these results bear a strong resemblance to the properties of some of the cuprates.

Figures (4)

• Figure 1: $\Delta_N$ as a function of the number of grid points $N$. The vertical scale is logarithmic, and the data is well fit by an exponential decay: $\log(\Delta_N) = -17.5-0.22\,N$.
• Figure 2: On the left we show $Q_{xz}$ and on the right $\phi$, for $k_0=2$, $A_0 =1$, $\mu = 1.4$ and $T/\mu = 0.1$. Note that $Q_{xz}$ has an effective wavenumber of $2k_0$.
• Figure 3: On the left we show the charge density $\tilde{\rho}(x)$ and on the right the absolute value of the non-zero coefficients of its Fourier series.
• Figure 4: The real and imaginary parts of the perturbation of the scalar field are shown for $\mu = 1.4$, $T/\mu = .115$, $k_0 = 2$, $A_0 = 1.5$. The two figures on the top have $\omega/T = 0.06$, whereas the two figures on the bottom have $\omega/T = 0.6$.