Floquet dynamical quantum phase transitions (FDQPTs) are signified by recurrent nonanalytic behaviors of observables in time. In this work, we introduce a quench-free and generic approach to engineer and control FDQPTs for both pure and mixed Floquet states. By applying time-periodic modulations with two driving frequencies to a general class of spin chain model, we find multiple FDQPTs within each driving period. The model is investigated with equal, commensurate and incommensurate driving frequencies. The nonanalytic cusps of return probability form sublattice structures in time domain. Notably, the number and time locations of these cusps can be flexibly controlled by tuning the Hamiltonian parameter and the frequencies of the drive. We further employ the dynamical topological order parameter, which shows a quantized jump whenever a DQPT happens, to identify the topological feature of FDQPTs. Our findings reveal the advantage of engineering nonequilibrium phase transitions with multifrequency driving fields.

中文翻译：

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工程 Floquet 动态量子相变

Floquet 动态量子相变 (FDQPT) 由可观测物在时间上反复出现的非解析行为来表示。在这项工作中，我们引入了一种无淬火和通用的方法来设计和控制纯和混合 Floquet 状态的 FDQPT。通过将具有两个驱动频率的时间周期调制应用于一般类的自旋链模型，我们在每个驱动周期内发现了多个 FDQPT。该模型以相等、相称和不相称的驱动频率进行研究。返回概率的非解析尖点在时域中形成亚晶格结构。值得注意的是，这些尖点的数量和时间位置可以通过调整哈密顿参数和驱动频率来灵活控制。我们进一步采用动态拓扑序参数，它显示了每当 DQPT 发生时的量化跳跃，以识别 FDQPT 的拓扑特征。我们的研究结果揭示了利用多频驱动场设计非平衡相变的优势。