Dynamical stability of the Holographic System with Two Competing Orders

TL;DR

This paper studies the dynamical stability of a holographic system with two competing orders, using both linear quasi-normal mode analysis and nonlinear evolution, revealing unique late-time behaviors and long relaxation times.

Contribution

It introduces a gauge-dependent formalism for calculating quasi-normal modes and demonstrates the nonlinear evolution of the system, highlighting exceptions to the usual late-time behavior predictions.

Findings

Dynamical stability aligns with thermodynamical stability.

Relaxation times are longer than in simpler holographic models.

An exception to the quasi-normal mode late-time behavior is identified.

Abstract

We investigate the dynamical stability of the holographic system with two order parameters, which exhibits competition and coexistence of condensations. In the linear regime, we have developed the gauge dependent formalism to calculate the quasi-normal modes by gauge fixing, which turns out be considerably convenient. Furthermore, by giving different Gaussian wave packets as perturbations at the initial time, we numerically evolve the full nonlinear system until it arrives at the final equilibrium state. Our results show that the dynamical stability is consistent with the thermodynamical stability. Interestingly, the dynamical evolution, as well as the quasi-normal modes, shows that the relaxation time of this model is generically much longer than the simplest holographic system. We also find that the late time behavior can be well captured by the lowest lying quasi-normal modes except for the non-vanishing order towards the single ordered phase. To our knowledge, this exception is the first counter example to the general belief that the late time behavior towards a final stable state can be captured by the lowest lying quasi-normal modes. In particular, a double relation is found for this exception in certain cases.