Holographic Lattice in Einstein-Maxwell-Dilaton Gravity

TL;DR

This work extends holographic lattice physics to Einstein-Maxwell-Dilaton gravity by constructing an ionic lattice background and computing the boundary optical conductivity. Without lattice, the charged dilatonic black brane preserves translation symmetry, yielding a divergent $\mathrm{Im}\,\sigma$ at $\omega=0$ and a delta peak in $\mathrm{Re}\,\sigma$ due to momentum conservation; with the lattice, momentum is dissipated and the low-frequency response follows a Drude form, $\sigma(\omega) = \frac{K\tau}{1-i\omega\tau}$. In an intermediate frequency range, the modulus of the conductivity exhibits a universal power-law $|\sigma(\omega)| = \frac{B}{\omega^{\gamma}} + C$ with $\gamma \approx 2/3$, independent of model parameters, echoing prior Horowitz–Santos–Tong results and supporting a degree of universality for holographic lattice effects. The study also reveals resonances at large lattice amplitude or low temperature, linked to quasinormal modes, and outlines avenues for analytic understanding and extensions to superconducting phases and zero-temperature limits.

Abstract

We construct an ionic lattice background in the framework of Einstein-Maxwell-dilaton theory in four dimensional space time. The optical conductivity of the dual field theory on the boundary is investigated. Due to the lattice effects, we find the imaginary part of the conductivity is manifestly suppressed in the zero frequency limit, while the DC conductivity approaches a finite value such that the previous delta function reflecting the translation symmetry is absent. Such a behavior can be exactly fit by the Drude law at low frequency. Moreover, we find that the modulus of the optical conductivity exhibits a power-law behavior at intermediate frequency regime. Our results provides further support for the universality of such power-law behavior recently disclosed in Einstein-Maxwell theory by Horowitz, Santos and Tong.

Figures (11)

• Figure 1: The real (left) and imaginary (right) parts of the conductivity for different $Q$ (blue for $Q=0.01$, red for $Q=0.1$, green for $Q=0.25$ and black for $Q=0.5$).
• Figure 2: We show $H_2$, $F$, $\psi$ and $\phi$ for $k_0=2$, $A_0 =0.4$, $Q = 0.1$ and $|T/\mu| = 0.457$.
• Figure 3: On the left side we show the charge density $\rho$ as a function of $x$ on the boundary, which can be read off by expanding $\psi=\mu+(\mu-\rho)z+\mathcal{O}(z^2)$. On the right we show the charge discrepancy $\Delta_N$ as a function of the number of grid points $N$, where the boundary charge is evaluated in a period as $Q=\int_0^{2\pi/k_0}dx \rho$. The vertical scale is logarithmic, and the data is well fit by an exponential decay: $\log(\Delta_N) = -6.56-0.197\,N$.
• Figure 4: The real and imaginary parts of $h_{tz}$ and $b_x$ are shown for $A_0=0.4$, $k_0=2$, $|T/\mu|=0.457$ and $\omega=0.6$.
• Figure 5: The real and imaginary parts of the conductivity, both without the lattice (dashed line) and with the lattice (solid line and data points) for $A_0=0.4$, $k_0=2$ and $Q=0.5$. Only the low frequency regime of the conductivity is changed by the lattice.