Time-Crystalline Phase in a Single-Band Holographic Superconductor

TL;DR

Using a holographic single-band superconductor, the paper develops a nonlinear gauge-scalar coupling under external driving to produce time-crystalline dynamics via coupled plasma and Higgs modes. It derives effective boundary ODEs from a bulk reduction, performs a multi-scale analysis to predict subharmonic resonances, and validates them with bulk numerical simulations of quasinormal modes. The results offer a controlled, strongly coupled framework for studying time crystals and suggest potential connections to ultrafast control of quantum materials.

Abstract

We investigate the emergence of a time-crystalline phase in a single-band holographic superconductor, extending the AdS/CFT framework. By incorporating a nonlinear gauge-scalar coupling and external driving, we derive coupled equations of motion for the plasma and Higgs modes, analogous to those in high-Tc superconductors. Multi-scale analysis reveals a sum resonance with subharmonic growth indicating broken time-translation symmetry. We perform numerical computation of quasinormal mode and demonstrate the transition to the time-crystalline phase. The holographic model may serve as a robust tool for studying strongly coupled time crystals.

Figures (3)

• Figure 1: (Left) Contour plot of given boundary conditions for fixed $\mu$ and $\psi^{(1)}=0$ in the shooting parameters space. Solutions exist at intersecting points. For higher chemical potential (for instant, $\mu=10$), there exists multiple solutions for excited states. (Right) In our simulation, we use the ground state wavefunctions $\psi(z)$ and $\phi(z)$ for $\mu=2.5$.
• Figure 2: (Top Left) A typical Higgs mode $h$ responds to small driving current $j$ in sum channel. (Top Right) Phase diagram of $j$ v.s. $h$ shows a closed and stable trajectory. (Bottom Left) Distorted Higgs mode $h$ responds to large driving current $j$ in sum channel, where envelope amplitude is growing and will blow up at later time (not shown), indicating failure of perturbation approach. (Bottom Right) Phase diagram shows uneven and non repeating trajectory. We remark that the amplitudes of $j(t)$ and $h(t)$ have been rescaled for presentation purpose. We only show first $150$ sampling points in time domain.
• Figure 3: (Top Left) Power spectrum of Higgs mode in 2:1 channel for $\omega_d = \omega_J$. The vertical red dashed lines indicate $\omega_H$, $\omega_J$, $\omega_H+\omega_J$ from left to right. Among many resonance peaks, the strongest resonance happens at $2\omega_J$. (Top Right) Power spectrum of Higgs mode in 2:1 channel for $2\omega_d = \omega_H$. The extra vertical dashed line (most left) indicates $\omega_d$. The strongest resonance happens at $\omega_H$. (Bottom Left) Power spectrum of Higgs mode in sum channel for $\omega_d = \omega_J+\omega_H$. Resonance happens at integer multiples of $(\omega_J+\omega_H)/2$. (Bottom Right) Power spectrum of Higgs mode in sum channel, but for runaway Higgs amplitude (large driving current). Vanishing subharmonic peaks signify the system has left time-crystalline phase. We note that parameters for this simulation are $\mu=2.5$, $\omega_H=0.95$, $\omega_J=1.63$, $\gamma_H=3.35$, $\gamma_J=2.61$, $g=4.88$, $\chi=5.58$ (rescaled).