In equilibrium physics, spontaneous symmetry breaking and elementary excitation are two concepts closely related with each other: the symmetry and its spontaneous breaking not only control the dynamics and spectrum of elementary excitations, but also determine their underlying structures. In this Letter, based on an exactly solvable model, we propose a phase ramping protocol to study an excitationlike behavior of a nonequilibrium quantum matter: a discrete time crystal phase with spontaneous temporal translational symmetry breaking. It is shown that slow ramping could induce a dynamical transition between two $Z_2$ symmetry breaking time crystal phases in time domain, which can be considered as a temporal analog of the soliton excitation spatially sandwiched by two degenerate charge density wave states in polyacetylene. By tuning the ramping rate, we observe a critical value at which point the transition duration diverges, resembling the critical slowing down phenomenon in nonequilibrium statistic physics. We also discuss the effect of stochastic sequences of such phase ramping processes and its implication to the stability of the discrete time crystal phase against noisy perturbations.

中文翻译：

---

离散时间晶相之间的动态转变和临界行为

在平衡物理学中，自发对称性断裂和基本激发是两个紧密相关的概念：对称性及其自发断裂不仅控制基本激发的动力学和谱，而且还决定了其基本结构。在这封信中，基于一个完全可解的模型，我们提出了一种相变协议来研究非平衡量子物质的类似激发的行为：具有自发时间平移对称性破坏的离散时间晶体相。结果表明，缓慢的倾斜可能会引起两个之间的动态过渡。${ž}_2$对称性打破了时域中的时间晶体相，可以认为是聚乙炔中空间上被两个简并电荷密度波态夹在中间的孤子激发的时间模拟。通过调整上升速率，我们观察到一个临界值，在该点过渡持续时间有所不同，类似于非平衡统计物理学中的临界减速现象。我们还讨论了这种相变过程的随机序列的影响及其对离散时间晶体相位对噪声扰动的稳定性的影响。