Low-Temperature Holographic Conductivity

We revisit holographic electrical conductivity in the regime of very low temperatures where quantum fluctuations in the throat of the near-extremal, asymptotically AdS${}_4$ dual black brane are strongly coupled. We treat the fluctuations via an effective two-dimensional action capturing the effects of the Schwarzian modes at the scale $1\mathrm{/}C$. Our main result is a non-monotonic temperature dependence: the quantum-corrected conductivity decreases as the temperature is lowered, reaches a minimum at $(C{T}_{\mathrm{m}\mathrm{i}\mathrm{n}}\simeq 0.023)$, and grows as $(1\mathrm{/}\sqrt{CT})$ for $(CT\ll 1)$. We further estimate the one-loop contribution to the conductivity directly from the four-dimensional gravitational path integral and find qualitative agreement with the effective two-dimensional description in the regime $(CT\gg 1)$. We find that quantum corrections substantially modify the low-temperature behavior of the holographic conductivity in both approaches.