We explore the phase transitions at the onset of time-crystalline order in $\mathrm{O}(N)$ models driven out of equilibrium. The spontaneous breaking of time translation symmetry and its Goldstone mode are captured by an effective description with $\mathrm{O}(N)\times \mathrm{SO}(2)$ symmetry, where the emergent external SO(2) results from a transmutation of the internal symmetry of time translations. Using the renormalization group and the $\epsilon =4-d$ expansion in a leading two-loop analysis, we identify a new nonequilibrium universality class. Strikingly, it controls the long-distance physics no matter how small the microscopic breaking of equilibrium conditions is. The $\mathrm{O}(N=2)\times \mathrm{SO}(2)$ symmetry group is realized for magnon condensation in pumped yttrium iron garnet films and in exciton-polariton systems with a polarization degree of freedom.

中文翻译：

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时间晶体有序开始时的非平衡临界

我们探索了时间结晶秩序开始时的相变$\mathrm{氧}（氮）$模型失去平衡。时间平移对称性的自发破缺及其戈德斯通模式被有效的描述所捕获$\mathrm{氧}（氮）\times \mathrm{所以}（2）$对称性，其中出现的外部 SO(2) 来自时间平移的内部对称性的嬗变。使用重整化群和$\epsilon =4-d$在领先的双环分析中进行扩展，我们确定了一个新的非平衡普遍性类别。引人注目的是，无论平衡条件的微观破坏有多小，它都控制着长距离物理。这$\mathrm{氧}（氮=2）\times \mathrm{所以}（2）$在泵浦钇铁石榴石薄膜和具有极化自由度的激子极化系统中实现了磁振子凝聚的对称群。