Interior structure of the holographic s + p superconductor and chaotic-stable transition near the black hole singularity

TL;DR

This work analyzes the interior structure of a holographic $s+p$ superconductor, constructed from a coupled scalar $\psi$ (s-wave) and massless vector $\rho_\mu$ (p-wave) in AdS$_4$, with a shared gauge field. The interior near the black-hole singularity exhibits Kasner-like behavior whose exponents depend on two free parameters $\beta$ and $\gamma$, leading to a generalized Kasner inversion described by $\frac{\gamma}{\gamma_a}=\frac{\beta}{\beta_a}$ and $\gamma\gamma_a+\beta\beta_a=1$. Crucially, the presence of scalar hair induces a chaotic-to-stable transition in the near-singularity structure, suppressing the chaotic Kasner dynamics typical of pure p-wave interiors and yielding a stable Kasner epoch in the coexistence region. This holographic interior phenomenon mirrors the boundary system’s secondary condensation and is interpreted within a hyperbolic billiard framework, offering insights into how multi-band order parameters influence black-hole interior geometry and potentially observable holographic signatures.

Abstract

In this work, we investigate the interior structure of a holographic multi-band superconductor with the coexistence of s-wave and p-wave order parameters. Especially, we investigate the singularity structure of this multi-band model. Different from the single p-wave case, the alternation rule is jointly determined by parameters involving both s-wave order and p-wave order. In the coexistence region, we derive the Kasner alternation laws from both analytical and numerical methods which fit each other nicely. Furthermore, we find that the occurrence of the s-wave order parameter will lead to a chaotic-stable transition for the near singularity structure which matches the expectation of cosmological billiard approach. This novel transition for the near singularity structure constitutes a holographic counterpart of the secondary condensation in boundary superconducting system, offering a complementary perspective for characterizing the properties of boundary condensed matter systems.

Figures (3)

• Figure 1: Phase structures in $p$ and $s+p$ holographic superconductor, we choose specific parameters which is $q_s=2.14$ and $q_p=2$. Left panel: Stable solution of p superconductor Right panel: Stable solutions of $s+p$ superconductor, where black dot ($T'_c/T_c$) denotes the critical temperature for the onset of scalar order parameter with $T'_c/T_c$=0.648942
• Figure 2: Kasner generalized inversion behavior for $T=0.640473 T_c$, $T=0.616679 T_c$, $T=0.592379 T_c$ and $T=0.581368 T_c$, all of those examples satisfy the three relations in Eq. (\ref{['KasLawAll']}).
• Figure 3: Kasner geometry inside the holographic superconductor with coexistence of s-wave order and p-wave order. The temperature in this case is chosen to be $T/T_c=0.648941$. In this specific temperature, there are multiple rounds of generalized Kasner inversion. Left Panel:Kasner transition behavior after first inversion Right Panel: Kasner transition behavior after second inversion.