Near-extremal holographic charge correlators

We analytically compute the low-temperature charge correlators in near-extremal black holes with a planar horizon and an infrared ${\mathrm{A}\mathrm{d}\mathrm{S}}_2\times {\mathbb{R}}^2$ extremal geometry, finding excellent agreement with numerical calculations. The analytical result consistently describes the crossover between the hydrodynamic diffusive regime at low frequencies and wavenumbers, and the quantum, zero-temperature regime at high frequencies and wavenumbers. We analytically resolve the successive collisions between the diffusive pole and the non-hydrodynamic poles sourced by the infrared ${\mathrm{A}\mathrm{d}\mathrm{S}}_2\times {\mathbb{R}}^2$ geometry. We demonstrate that in the $T=0$ limit, a pair of gapless poles survive with a dispersion relation ${\omega }_{\pm }=-i{d}_2{k}^2-i{d}_4{k}^4-i{\tilde{d}}_4{k}^4(\pm i\pi +\log k^2)$. The nonanalytic contributions arise from the interplay with the branch cut formed by the condensation of the non-hydrodynamic poles. The real part is caused by the `snatching’ of one of the non-hydrodynamic poles by the hydrodynamic diffusive pole.