Title: Dynamics of many-body localized systems: logarithmic lightcones and $\log \, t$-law of $α$-Rényi entropies Show Chinese title

Title: 多体局域化系统的动力学：对数光锥和$\log \, t$-定律的$α$-Rényi 熵

Abstract: In the context of the Many-Body-Localization phenomenology we consider arbitrarily large one-dimensional local spin systems, the XXZ model with random magnetic field is a prototypical example. Without assuming the existence of exponentially localized integrals of motion (LIOM), but assuming instead that the system's dynamics gives rise to a Lieb-Robinson bound (L-R) with a logarithmic lightcone, we rigorously evaluate the dynamical generation, starting from a generic product state, of $ \alpha$-R\'enyi entropies, with $ \alpha $ close to one, obtaining a $\log \, t$-law, that denotes a slow spread of entanglement. This is in sharp contrast with Anderson localized phases that show no dynamically generated entanglement. To prove this result we apply a general theory recently developed by us in arXiv:2408.00743 that quantitatively relates the L-R bounds of a local Hamiltonian with the dynamical generation of entanglement. Assuming instead the existence of LIOM we provide new independent proofs of the known facts that the L-R bound of the system's dynamics has a logarithmic light cone and show that the dynamical generation of the von Neumann entropy has for large times a $ \log \, t$-shape. L-R bounds, that quantify the dynamical spreading of local operators, may be easier to measure in experiments in comparison to global quantities such as entanglement. Show Chinese abstract

Abstract: 在多体局域化现象学的背景下，我们考虑任意大的一维局部自旋系统，具有随机磁场的XXZ模型是一个典型的例子。 不假设存在指数局域化的守恒量（LIOM），而是假设系统的动力学导致了具有对数光锥的Lieb-Robinson界限（L-R），我们严格评估了从一般乘积态开始的动力学生成的$ \alpha$-黎曼熵，其中$ \alpha $接近于一，得到一个$\log \, t$定律，表示纠缠的缓慢传播。 这与展示没有动态生成纠缠的安德森局域相位形成鲜明对比。 为了证明这个结果，我们应用了我们最近在arXiv:2408.00743中开发的一般理论，该理论定量地将局部哈密顿量的L-R界限与纠缠的动力学生成联系起来。 假设LIOM的存在，我们提供了新的独立证明，已知事实是系统的L-R界限具有对数光锥，并表明冯·诺依曼熵的动力学生成在长时间下具有$ \log \, t$形状。 量化局部算符动力学扩展的L-R界限，在实验中可能比纠缠等全局量更容易测量。