From Critical Zero Modes to Interior Dynamics in Holographic Superconductors

Abstract

We study how the critical behavior of holographic superconductors controls the interior geometry of charged black holes. Although the exterior condensate vanishes at criticality, the critical zero mode develops logarithmic oscillations near the Reissner--Nordstr\"om Cauchy horizon, rendering the interior critical limit nonuniform. We show that the transmitted critical-mode amplitude, together with local Cauchy-horizon quantities, fixes both the leading Einstein--Rosen collapse $\Delta z_{\rm ER}$ width and the phase $\Theta_J$ in the subsequent Josephson regime. If the condensate obeys $t\equiv1-T/T_c\propto\epsilon^{2r}$, then $\Delta z_{\rm ER}\propto t^{1/r}$ and $\Theta_J\propto t^{-1/r}$. This Josephson phase is transferred to the first Kasner epoch, producing oscillatory Kasner exponents and an accumulation law $t_n\sim n^{-r}$ for successive zeros of the scalar velocity. The critical zero mode fixes the leading collapse and phase coefficients, which are quantitatively confirmed by fully nonlinear solutions in both the ordinary and tricritical cases. At low temperature, the interior instead depends on the infrared completion of the zero-temperature geometry. Thus near-critical scaling is governed by critical-mode transmission, whereas the low-temperature behavior is controlled by the infrared endpoint.