We derive an exact renormalization group recursion relation for the Loschmidt amplitude of the quantum $Q$-state clock model and the quantum $Q$-state Potts model in one dimension. The renormalization group flow is discussed in detail. The fixed points of the renormalization group flow are found to be complex in general. These fixed points control the dynamical phases of the two models, giving rise to nonanalyticities in its Loschmidt rate function, for both the pure and the disordered system. For the quench protocols studied, dynamical quantum phase transitions are found to occur in the clock model for all $Q\mathrm{s}$ considered, while in the Potts model, they only occur when $Q<4$.

中文翻译：

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量子时钟链和量子Potts链的非平衡重归一化组不动点

我们为量子的Loschmidt幅度推导了一个精确的重归一化组递归关系 $问$状态时钟模型和量子 $问$状态一维的Potts模型。重新规范化组流程将详细讨论。通常，重归一化组流的不动点很复杂。这些不动点控制着两个模型的动力学阶段，从而导致纯系统和无序系统的Loschmidt速率函数都无法进行分析。对于研究的猝灭协议，发现动态量子相变发生在所有模型的时钟模型中$问\mathrm{s}$ 考虑到，在Potts模型中，它们仅在 $问<4$。