一般无序离散时间晶体的鲁棒性条件，以及由不同 n 元 DTC 之间相变产生的子空间热 DTC
The robustness condition for general disordered discrete time crystals, and subspace-thermal DTCs from phase transitions between different n-tuple DTCs
AI总结：
本文提出可任意倍周期响应的 n 元无序 Floquet DTC，揭示不同周期 DTC 间相变形成的子空间热 DTC，并基于 2π/n 准能隙与混合长度阈值建立严格鲁棒性框架。
摘要：
我们提出了一种新的 Floquet 时间晶体模型，它能够以驱动周期的任意倍数作出响应。这种 n 元离散时间晶体（n-tuple discrete time crystal）在理论上通过置换无序链中的自旋而构建，并且非常适合实验实现。这些具有不同周期的时间晶体之间的跃迁产生了一种新的物相，我们称之为子空间热离散时间晶体（subspace-thermal discrete time crystals）；在该相中，确定电荷的子空间内的状态会在早期完全热化。然而，整个系统仍然对周期性驱动表现出鲁棒的次谐波响应，其周期为原来两个周期的最大公约数。来自多体局域化的现有理论分析不能用于理解这种子空间热时间晶体相的刚性。为解决这一问题，我们从鲁棒的 2π/n 准能隙视角出发，发展了一个关于 DTC 鲁棒性的新理论框架。如果系统满足某个条件，即由对称电荷的 Hamming 距离定义的混合长度不超过全局阈值，则其鲁棒性可得到严格证明。该证明不仅适用于这里考虑的模型，也适用于通过踢动无序系统实现的其他现有 DTC；在这些情形中，传统 MBL-DTC 可被视为子空间维度为一的子空间热 DTC 的特例，从而为构建新的离散时间晶体模型提供了一种系统方法。我们还引入了 DTC-charge 的概念，使我们能够在常规离散时间晶体和子空间热离散时间晶体中探测自发破缺时间平移对称性的可观测量。此外，我们的离散时间晶体模型可以推广到具有更高自旋大小的系统或 qudit，以及更高空间维度的系统。
英文摘要：
We propose a new Floquet time crystal model that responds in arbitrary multiples of the driving period. Such an $n$-tuple discrete time crystal is theoretically constructed by permuting spins in a disordered chain and is well suited for experimental implementations. Transitions between these time crystals with different periods give rise to a novel phase of matter that we call subspace-thermal discrete time crystals, where states within subspaces of definite charges are fully thermalized at an early time. However, the whole system still robustly responds to the periodic driving subharmonically, with a period being the greatest common divisor of the original two periods. Existing theoretical analysis from many-body localization cannot be used to understand the rigidity of such subspace-thermal time crystal phases. To resolve this, we develop a new theoretical framework for the robustness of DTCs from the perspective of the robust $2π/n$ quasi-energy gap. Its robustness is rigorously proved if the system satisfies a certain condition where the mixing length, defined by the Hamming distance of the symmetry charges, does not exceed a global threshold. The proof applies beyond the models considered here to other existing DTCs realized by kicking disordered systems, where conventional MBL-DTCs can be regarded as a special case of the subspace-thermal DTC with the subspace dimension being one, thus offering a systematic way to construct new discrete time crystal models. We also introduce the notion of DTC-charges that allow us to probe the observables that spontaneously break the time-translation symmetry in both the regular discrete time crystals and subspace-thermal discrete time crystals. Moreover, our discrete time crystal models can be generalized to systems with higher spin magnitudes or qudits, as well as to higher spatial dimensions.
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