The thermoelectric properties of inhomogeneous holographic lattices

TL;DR

This work studies thermoelectric transport in inhomogeneous holographic lattices within D=4 Einstein-Maxwell theory. It derives exact horizon-data expressions for DC conductivities, unifies them with horizon data, and confirms these results numerically by constructing monochromatic, dichromatic, and dirty lattices to compute AC transport. The results reveal Drude-like AC behavior, resonances tied to sound modes, absence of intermediate-frequency scaling, and low-temperature IR flow to AdS$_2\times\mathbb{R}^2$ with scalable DC conductivities and diverging $ZT$. The findings illuminate momentum dissipation mechanisms in strongly coupled systems and offer robust tools for analyzing transport in holographic lattices, with implications for strange metal phenomenology and thermoelectric efficiency in strongly interacting media.

Abstract

We consider inhomogeneous, periodic, holographic lattices of D=4 Einstein-Maxwell theory. We show that the DC thermoelectric conductivity matrix can be expressed analytically in terms of the horizon data of the corresponding black hole solution. We numerically construct such black hole solutions for lattices consisting of one, two and ten wave-numbers. We numerically determine the AC electric conductivity which reveals Drude physics as well as resonances associated with sound modes. No evidence for an intermediate frequency scaling regime is found. All of the monochromatic lattice black holes that we have constructed exhibit scaling behaviour at low temperatures which is consistent with the appearance of $AdS_2\times\mathbb{R}^2$ in the far IR at T=0.

Figures (12)

• Figure 1: The real (top left) and the imaginary (top right) parts of the optical conductivity $\sigma$ as a function of $\omega/\mu$ for a monochromatic lattice $\mu(x)/\mu=1+A\cos\left(k\,x\right)$, with $A=1/2$, $k/\mu=1/\sqrt{2}$, and various $T/\mu$ close to the origin. The conductivity clearly shows a Drude-like peak developing at low temperatures. The bottom figure shows the corresponding behaviour of $1+(\omega/\mu)\,\left|\sigma\right|^{\prime\prime}/\left|\sigma\right|^{\prime}$ and there is no evidence of a mid-frequency intermediate scaling with exponent $-2/3$. Note the different horizontal scale in the top and bottom figures.
• Figure 2: Plots of the DC conductivity for $\sigma$ (top left) and $\bar{\kappa}$ (bottom left), obtained from \ref{['finform']}, against temperature for four monochromatic lattices of the form $\mu(x)/\mu=1+A\,\cos\left(k\,x\right)$, all with $A=1/2$ and $k/\mu=\sqrt{2}/3$ (orange), $2\sqrt{2}/5$ (blue), $1/\sqrt{2}$ (red) and $\sqrt{2}$ (green). The red dashed lines on the right hand plots indicate the low-temperature scaling behaviour, given in \ref{['dcscal']} expected for black holes approaching $AdS_2\times\mathbb{R}^2$ in the far IR. The $k/\mu=\sqrt{2}$ case provides an example where $\bar{\kappa}$ diverges as $T\to 0$, while the other cases are examples where $\bar{\kappa}$ vanishes as $T\to 0$. In all cases $\kappa$ vanishes linearly with $T$. As $T\to \infty$ we see that $\sigma\to 1+2/A^2=9$, marked with a red dashed line in the top left figure, in agreement with \ref{['hightsc']}.
• Figure 3: Sum rules for monochromatic lattices. The top left panel plots the integrated spectral weight $S(\omega/\mu)$, defined in \ref{['sumrulefn']}, for a monochromatic lattice $\mu(x)/\mu=1+A\cos\left(k\,x\right)$, with $A=1/2$, $k/\mu=1/\sqrt{2}$ (as in figure \ref{['fig:S2']}) for three different temperatures, and we see it vanishes when $\omega/\mu\to\infty$ as expected from the first sum rule. As $T/\mu\to 0$ we see that $S(\omega/\mu)$ is developing a step-like behaviour corresponding to the appearance of a delta function with weight smaller than the $T=0$ AdS-RN black hole (which has the value $\sim0.45$). The top right panel considers monochromatic lattices with $k/\mu=1/\sqrt{2}$ and fixed $T/\mu=0.12$ and various $A$. As $A\to 0$ we see that the $S(\omega/\mu)$ is developing a step-like behaviour corresponding to the appearance of a delta function with the same weight as the AdS-RN black hole at the same temperature (which for this case has the value $\sim 0.51$). The bottom panel plots $S_d(\omega/\mu)$, defined in \ref{['sumrulefnd']}, for the same monochromatic lattices as in the top left panel and we see that the second sum rule is also satisfied.
• Figure 4: The real (left) and the imaginary (right) parts of the optical conductivity as a function of $\omega$ for various monochromatic lattices $\mu(x)/\mu=1+A\,\cos\left(k\,x\right)$. The three different cases have fixed temperature $T/\mu\approx 0.0795$ and period $k/\mu=\left(3\,\sqrt{2}\right)^{-1}$ but varying lattice strength $A$. We clearly see the appearance of a resonance associated with the sound mode frequency $\omega/\mu\sim v_sk/\mu\sim 1/6$.
• Figure 5: The real (left) and the imaginary (right) parts of the optical conductivity as a function of $\omega/\mu$ for the dichromatic lattice $\mu(x)/\mu=1+A\cos\left(k\,x\right)+B\cos\left(2k\,x\right)$, with $A=1/2$, $B=1$, $k/\mu=1/\left(3\sqrt{2}\right)$ and $T/\mu\approx 0.0796$. In this case we see two resonances associated with sound modes at $\omega/\mu\sim v_sk/\mu\sim1/6$ and also $\omega/\mu\sim v_s(2k)/\mu\sim 1/3$.