Criticality and Phase Structures of Excited Holographic Superconductors in Nonlinear Electrodynamics

TL;DR

This work extends holographic superconductor models by incorporating Born-Infeld nonlinear electrodynamics within the extended phase space where the cosmological constant is treated as pressure $P$. Using a probe-limit Schwarzschild-AdS background, it analyzes ground and two lowest excited states (n=0,1,2) and uncovers a triplet phase structure governed by the small-to-large black hole transition at $P_c$: for $P> P_c$ GS and ES1 are gapped while ES2 is gapless; for $P\le P_c$ excited states condense into gapless phases. The critical temperatures decrease with increasing nonlinearity $b$ and increase with pressure, and the order of transitions changes (second-order for $P>P_c$, with ES1 becoming first-order at $P\le P_c$ as shown by free-energy swallow-tail). Conductivity calculations confirm the presence or absence of a hard gap in accordance with the phase structure, linking bulk geometry and nonlinear electrodynamics to boundary superconducting properties. These results highlight the rich bulk-boundary interplay in AdS/CFT and motivate future studies with alternative bulk fields and backreaction effects.

Abstract

We investigate the properties of excited states in a holographic superconductor model within the extended phase space framework, where the cosmological constant is identified as the thermodynamic pressure. Employing Born-Infeld nonlinear electrodynamics, we explore how the nonlinear parameter affects the condensation of the ground state and the two lowest excited states. Our numerical results demonstrate that the nonlinear parameter $b$ significantly modifies the critical temperature $T_c$ for all states. We focus on the phase structure near the critical pressure $P_c$ and discuss the ``triplet'' phenomenon of these states. The competition between nonlinear effects and geometrical deformation of the black hole induced by pressure is analyzed in detail. Specifically, we find that when the pressure $P$ exceeds the critical pressure $P_c$, both the ground state and the first excited state are superconducting (gapped), while the second excited state is a gapless superconductor. However, at pressures below or equal to $P_c$, while the ground state remains a gapped superconductor, the excited states undergo condensation into gapless phases without exhibiting superconducting gap behavior. (See introduction for terminology clarification.)

Figures (6)

• Figure 1: Temperature dependence of the condensation $\langle O\rangle$ for different states: ground state (solid), first excited state (dashed), second excited state (dotted), with $b = 0$ (blue), $b = 0.5$ (orange), $b = 1.0$ (red).
• Figure 2: Critical temperature $T_c$ as a function of pressure $P$ for the ground state (blue), first excited state (red), and second excited state (green). The point $P_c$ marks the small-to-large black hole phase transition.
• Figure 3: Critical temperature $T_c$ as a function of the nonlinear parameter $b$ at the critical pressure $P_c$.
• Figure 4: Phase structure of holographic superconducting states: (a) $O_1$ quantization and (b) $O_2$ quantization. The horizontal axis is temperature $T$, and the vertical axis is pressure $P$.
• Figure 5: Free energy difference $\Delta\Omega$ between condensed and normal phases for ground state (blue), first excited state (orange), second excited state (green) at various pressures.