We consider quantum systems with causal dynamics in discrete spacetimes, also known as quantum cellular automata (QCA). Due to time-discreteness this type of dynamics is not characterized by a Hamiltonian but by a one-time-step unitary. This can be written as the exponential of a Hamiltonian but in a highly non-unique way. We ask if any of the Hamiltonians generating a QCA unitary is local in some sense, and we obtain two very different answers. On one hand, we present an example of QCA for which all generating Hamiltonians are fully non-local, in the sense that interactions do not decay with the distance. We expect this result to have relevant consequences for the classification of topological phases in Floquet systems, given that this relies on the effective Hamiltonian.

On the other hand, we show that all one-dimensional quasi-free fermionic QCAs have quasi-local generating Hamiltonians, with interactions decaying exponentially in the massive case and algebraically in the critical case. We also prove that some integrable systems do not have local, quasi-local nor low-weight constants of motion; a result that challenges the standard definition of integrability.

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因果动力学是否意味着局部相互作用？

我们考虑在离散时空中具有因果动力学的量子系统，也称为量子元胞自动机 (QCA)。由于时间离散性，这种类型的动力学不是以哈密顿量为特征，而是以单时间步酉为特征。这可以写成哈密顿量的指数，但以一种高度非唯一的方式。我们询问生成 QCA 酉的任何哈密顿量是否在某种意义上是局部的，我们得到了两个截然不同的答案。一方面，我们提出了一个 QCA 的例子，其中所有生成的哈密顿量都是完全非局部的，因为相互作用不会随着距离而衰减。鉴于这依赖于有效的哈密顿量，我们预计该结果将对 Floquet 系统中的拓扑相分类产生相关影响。

另一方面，我们表明所有一维准自由费米子 QCA 都具有准局部生成哈密顿量，在大规模情况下相互作用呈指数衰减，而在临界情况下相互作用呈代数衰减。我们还证明了一些可积系统没有局部、准局部或低权重的运动常数；这一结果对可积性的标准定义提出了挑战。