带连续对称性的含噪声量子多体动力学的几何：纠缠与关联
Geometry of Noisy Quantum Many-Body Dynamics with Continuous Symmetries: Entanglement and Correlations
浏览论文内容
中文总结 AI 辅助
该研究针对带连续对称性的含噪声量子多体动力学，利用有效复制哈密顿量的量子几何，解释了纠缠增长与关联函数衰减特性，构建了通用几何研究框架。
中文摘要 AI 辅助
我们研究具有全局连续对称性（如$U(1)$和$SU(2)$）的含噪声布朗模型中的幺正量子动力学，重点关注雷尼纠缠熵以及流体动力学和非流体动力学关联函数。通过将平均后的晚期动力学映射到有效复制哈密顿量的低能物理，我们发现该演化由其基态流形的量子几何控制，这与$k$交换子（系统$k$个副本的对称代数）的几何直接相关。在相互作用系统中，这些$k$交换子通常仅由系统的对称性决定，与含噪声演化的微观细节无关。这使我们能够使用含时变分原理（TDVP）为具有连续对称性的相互作用系统中次弹道雷尼纠缠增长和非流体动力学关联函数的反常衰减提供简单的几何解释。我们发现这种行为与$k$交换子流形中的奇点密切相关，这些奇点源于希尔伯特空间中因连续 onsite 对称性而存在的冻结“空态”。这也阐明了空态在这些可观测量动力学中的重要作用，此前该作用已在$U(1)$对称系统中被识别。我们比较了具有阿贝尔和非阿贝尔连续对称性的相互作用系统以及自由费米子系统中的这些行为，它们的差异体现在各自$k$交换子的几何上。最终，这项工作为系统研究具有连续对称性的含噪声系统中的可观测量（包括哈尔随机电路）提供了通用的几何框架。
英文摘要
We study unitary quantum dynamics in noisy Brownian models with global continuous symmetries, such as $U(1)$ and $SU(2)$, focusing on Rényi entanglement entropies and hydrodynamic and non-hydrodynamic correlators. By mapping the averaged late-time dynamics to the low-energy physics of effective replica Hamiltonians, we find that the evolution is controlled by the quantum geometry of their ground-state manifolds, which is directly related to the geometry of $k$-commutants---the symmetry algebra of $k$ replicas of the system. In interacting systems, these $k$-commutants are generically determined solely by the symmetries of the system, independent of microscopic details of the noisy evolution. This allows us to use the time-dependent variational principle (TDVP) to provide simple geometric explanations for the sub-ballistic Rényi entanglement growth and the anomalous decay of non-hydrodynamic correlators in interacting systems with continuous symmetries. We find this behavior to be intimately connected to singularities within the $k$-commutant manifolds, arising from frozen ``void'' states in the Hilbert space that exist due to continuous on-site symmetries. This also demystifies the important role of voids in the dynamics of these observables, previously identified in $U(1)$ symmetric systems. We compare these behaviors in interacting systems with Abelian and non-Abelian continuous symmetries and in free-fermion systems, which differ in the geometry of their $k$-commutants. Ultimately, this work provides a general geometric framework for systematically studying observables in noisy systems with continuous symmetries, including Haar-random circuits.
