We introduce a family of local models of dynamics based on “word problems” from computer science and group theory, for which we can place rigorous lower bounds on relaxation timescales. These models can be regarded either as random circuit or local Hamiltonian dynamics and include many familiar examples of constrained dynamics as special cases. The configuration space of these models splits into dynamically disconnected sectors, and for initial states to relax, they must “work out” the other states in the sector to which they belong. When this problem has a high time complexity, relaxation is slow. In some of the cases we study, this problem also has high space complexity. When the space complexity is larger than the system size, an unconventional type of jamming transition can occur, whereby a system of a fixed size is not ergodic but can be made ergodic by appending a large reservoir of sites in a trivial product state. This finding manifests itself in a new type of Hilbert space fragmentation that we call fragile fragmentation. We present explicit examples where slow relaxation and jamming strongly modify the hydrodynamics of conserved densities. In one example, density modulations of wave vector $q$ exhibit almost no relaxation until times $O\mathbf{(}\exp (1/q)\mathbf{)}$, at which point they abruptly collapse. We also comment on extensions of our results to higher dimensions.

中文翻译：

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玻璃字问题：超慢松弛、希尔伯特空间干扰和计算复杂性


我们引入了一系列基于计算机科学和群论中的“文字问题”的局部动力学模型，我们可以在松弛时间尺度上设置严格的下限。这些模型可以被视为随机电路或局部哈密顿动力学，并且包括许多熟悉的约束动力学示例作为特殊情况。这些模型的配置空间分为动态断开的扇区，为了放松初始状态，它们必须“计算出”它们所属扇区中的其他状态。当这个问题的时间复杂度很高时，松弛速度很慢。在我们研究的一些案例中，这个问题也具有很高的空间复杂度。当空间复杂度大于系统尺寸时，可能会发生非常规类型的干扰转换，其中固定尺寸的系统不是遍历的，但可以通过在平凡的产品状态中附加大量站点来使其遍历。这一发现体现在一种新型的希尔伯特空间碎片中，我们称之为脆弱碎片。我们提出了明确的例子，其中缓慢松弛和干扰强烈改变了守恒密度的流体动力学。在一个示例中，波矢量 $q$ 的密度调制在时间 $O\mathbf{(}\exp (1/q)\mathbf{)}$ 之前几乎没有松弛，此时它们突然崩溃。我们还评论了我们的结果扩展到更高维度的情况。