We pinpoint the conditions necessary for discrete time crystal (DTC) formation in fully connected spin-cavity systems from the perspective of parametric resonance by mapping these systems onto oscillatorlike models. We elucidate the role of nonlinearity and dissipation by mapping the periodically driven open Dicke model onto effective linear and nonlinear oscillator models, while we analyze the effect of global symmetry breaking using the Lipkin-Meshkov-Glick model with tunable anisotropy. We show that the system's nonlinearity restrains the dynamics from becoming unbounded when driven resonantly. On the other hand, dissipation keeps the oscillation amplitude of the period-doubling instability fixed, which is a key feature of DTCs. The presence of global symmetry breaking in the absence of driving is found to be crucial in the parametric resonant activation of period-doubling response. We provide analytic predictions for the resonant frequencies and amplitudes leading to DTC formation for both systems using their respective oscillator models.

中文翻译：

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全连接自旋腔系统中离散时间晶体的参数共振理论

我们通过将这些系统映射到类振荡器模型上，从参数共振的角度确定了在完全连接的自旋腔系统中形成离散时间晶体（DTC）所需的条件。我们通过将周期性驱动的开放 Dicke 模型映射到有效的线性和非线性振荡器模型来阐明非线性和耗散的作用，同时我们使用具有可调各向异性的 Lipkin-Meshkov-Glick 模型分析全局对称性破缺的影响。我们表明，系统的非线性会限制谐振驱动时动力学变得无界。另一方面，耗散使倍周期不稳定性的振荡幅度保持固定，这是 DTC 的一个关键特征。发现在没有驱动的情况下全局对称性破缺的存在对于倍周期响应的参数共振激活至关重要。我们使用各自的振荡器模型为两个系统提供导致 DTC 形成的谐振频率和振幅的分析预测。