多体局域系统的动力学：对数光锥和α-Rényi熵的log t律
Dynamics of many-body localized systems: logarithmic lightcones and $\log \, t$-law of $α$-Rényi entropies
AI总结：
研究多体局域系统中纠缠的缓慢增长，通过假设对数光锥的Lieb-Robinson界，推导出α-Rényi熵的log t律，并证明存在局域积分运动时冯·诺依曼熵也呈log t增长。
摘要：
在多体局域化现象学背景下，我们考虑任意大的一维局域自旋系统，XXZ模型加随机磁场是一个典型例子。不假设存在指数局域的积分运动（LIOM），而是假设系统动力学产生具有对数光锥的Lieb-Robinson界（L-R），我们严格评估从一般乘积态出发的α-Rényi熵（α接近1）的动态生成，得到log t律，表示纠缠的缓慢传播。这与安德森局域相形成鲜明对比，后者没有动态生成的纠缠。为了证明这一结果，我们应用了我们最近在arXiv:2408.00743中发展的一般理论，该理论定量地将局域哈密顿量的L-R界与纠缠的动态生成联系起来。假设存在LIOM，我们提供了新的独立证明，证明系统动力学的L-R界具有对数光锥，并表明冯·诺依曼熵的动态生成在大时间下呈log t形状。L-R界量化了局域算符的动态传播，与纠缠等全局量相比，可能在实验中更容易测量。
英文摘要：
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 $ α$-Rényi entropies, with $ α$ 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 lightcone 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.
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