Criticality and Phase Structures of Excited Holographic Superconductors in Nonlinear Electrodynamics

We investigate the critical properties and phase structure of excited states in a holographic superconductor model within the framework of Varying Central Charge Thermodynamics, where the cosmological constant serves as a fundamental parameter controlling the number of degrees of freedom in the boundary conformal field theory. Employing Born-Infeld nonlinear electrodynamics, we explore how the nonlinear parameter $b$ affects the condensation of the ground state (GS) and the two lowest excited states (ES1, ES2) in the background of a spherically symmetric Schwarzschild-AdS black hole. A state is classified as possessing a \textbf{hard gap} if its optical conductivity exhibits $\mathrm{Re}\sigma(\omega) = 0$ for $\omega < \omega_g$, indicating a hard energy gap in the excitation spectrum and the Meissner effect. In contrast, a \textbf{gapless superconductor} possesses a non-zero order parameter but lacks a hard gap, with $\mathrm{Re}\sigma(\omega) \neq 0$ as $\omega \to 0$. Our central finding reveals that the emergence of gapless phases in the excited states represents a genuine physical phenomenon arising from the competition between Born-Infeld nonlinear screening effects and the spatial curvature of the black hole geometry, not from numerical artifacts. Specifically, when the pressure $P$ exceeds the critical pressure $P_c$, both GS and ES1 are gapped superconductors with hard energy gaps while ES2 is a gapless superconductor. However, when $P \leq P_c$, only GS remains gapped while both ES1 and ES2 condense into gapless phases. This curvature-controlled switching of superconducting properties provides a novel mechanism for engineering gapless superconductivity in strongly coupled systems through variation of the boundary CFT degrees of freedom, with potential implications for understanding unconventional high-temperature superconductors.