Simple holographic insulator

We present a simple holographic model of an insulator. Unlike most previous holographic insulators, the zero temperature infrared geometry is completely nonsingular. Both the low temperature DC conductivity and the optical conductivity at zero temperature satisfy power laws with the same exponent, given by the scaling dimension of an operator in the IR. Changing a parameter in the model converts it from an insulator to a conductor with a standard Drude peak.

Figure 1

A plot of our potential. The minimum is at ${\psi }_c=\sqrt{2}\ln (1+\sqrt{2})$ with minimum value $V({\psi }_c)=-8$.

Figure 2

The value of $⟨\mathcal{O}⟩$ vs $T/(-\kappa )$ with critical value $T_c/(-\kappa )\approx .616$.

Figure 3

Top: log-log plot of our numerical solution for $f(r)$ in the IR (blue dots). The red line (which goes through all the points) is the analytic planar AdS-Schwarzschild solution. The two curves correspond to $T/(-\kappa )=.037$ (top) and $T/(-\kappa )=7.06\times {10}^{-3}$ (bottom). One can see the transition from the linear Schwarzschild behavior to the quadratic ${\mathrm{AdS}}_4$ behavior. Bottom: the scalar field on the horizon as a function of temperature. At low $T$, it scales like $T^{{\mathrm{\Delta }}_{\psi }}$.

Figure 4

Top: plot of the optical conductivity at $T/(-\kappa )=7.06\times {10}^{-3}$ and $g=g_c$. The imaginary component vanishes as $\omega \to 0$ rather than diverging as ${\omega }^{-1}$ showing that there is no $\delta (\omega )$ in $\mathrm{Re}[\sigma (\omega )]$. Bottom: log-log plot of the real component of the optical conductivity above. The line of best fit gives $\sigma \sim {\omega }^{2.75}$. Our analytic solution says that it should be a power law with an exponent $2{\mathrm{\Delta }}_{\psi }\approx 2.744$.

Figure 5

Fit of optical conductivity to a Drude-type curve, $\sigma (\omega )=\frac{K\tau }{1-i\omega \tau }$. For this plot, we chose $g=10$ at a temperature $T/(-\kappa )=.037$, and found $K=540$, $\tau =.485$.

Figure 6

Left: plot of DC resistivity vs temperature for different values of our tuning parameter $-1/{\psi }_c^2<g<200$. Right: numerical data from [35] showing the DC resistivity of a Ga thin film. The resistivity increases as the thickness of the thin film is decreased (bottom to top).