Previous studies revealed a crucial effect of symmetries on the properties of a single particle moving in a disorder potential. More recently, a phenomenon of many-body localization (MBL) has been attracting much theoretical and experimental interest. MBL systems are characterized by the emergence of quasi-local integrals of motion, and by the area-law entanglement entropy scaling of its eigenstates. In this paper, we investigate the effect of a non-Abelian $SU(2)$ symmetry on the dynamical properties of a disordered Heisenberg chain. While $SU(2)$ symmetry is inconsistent with conventional MBL, a new non-ergodic regime is possible. In this regime, the eigenstates exhibit faster than area-law, but still strongly sub-thermal scaling of the entanglement entropy. Using extensive exact diagonalization simulations, we establish that this non-ergodic regime is indeed realized in the strongly disordered Heisenberg chains. We use real-space renormalization group (RSRG) to construct approximate excited eigenstates by tree tensor networks, and demonstrate the accuracy of this procedure for systems of size up to $L=26$. As the effective disorder strength is decreased, a crossover to the thermalizing phase occurs. To establish the ultimate fate of the non-ergodic regime in the thermodynamic limit, we develop a novel approach for describing many-body processes that are usually neglected by RSRG. This approach is capable of describing systems of size $L\approx 2000$. We characterize the resonances that arise due to such processes, finding that they involve an ever growing number of spins as the system size is increased. Crucially, the probability of finding resonances grows with the system’s size. Even at strong disorder, we can identify a large length scale beyond which resonances proliferate. Presumably, this would eventually drive the system to a thermalizing phase. However, the extremely long thermalization time scales indicate that a broad non-ergodic regime will be observable experimentally. Our study demonstrates that, similar to the case of single-particle localization, symmetries control dynamical properties of disordered, many-body systems. The approach introduced here provides a versatile tool for describing a broad range of disordered many-body systems, well beyond sizes accessible in previous studies.

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非阿贝尔对称性和无序性：广泛的非遍历模式和异常热化

先前的研究揭示了对称性对以无序电位运动的单个粒子的性质的关键影响。最近，多体定位（MBL）现象吸引了许多理论和实验兴趣。MBL系统的特征在于运动的准局部积分的出现，以及其本征态的面积定律纠缠熵定标。在本文中，我们研究了非阿贝尔人的影响$\mathrm{小号}ü（\mathrm{2个}）$Heisenberg链的动力学性质的对称性。尽管$\mathrm{小号}ü（\mathrm{2个}）$对称性与常规MBL不一致，因此有可能采用新的非遍历模式。在这种情况下，本征态显示出比面积定律快的速度，但是仍然强烈地表现出纠缠熵的亚热定标。使用广泛的精确对角线化模拟，我们确定在强无序的海森堡链中确实实现了这种非遍历机制。我们使用实空间重归一化组（RSRG）通过树张量网络构造近似的激发本征态，并证明该程序对系统规模最大的精度$\mathrm{大号}=26$。随着有效无序强度的降低，发生了进入热化阶段的过渡。为了在热力学极限中确定非遍历系统的最终命运，我们开发了一种新颖的方法来描述通常被RSRG忽略的多体过程。这种方法能够描述规模系统$\mathrm{大号}\approx 2000$。我们对由于这种过程而引起的共振进行了表征，发现随着系统规模的增加，它们涉及的旋转次数越来越多。至关重要的是，发现共振的可能性随系统的大小而增加。即使在严重的疾病中，我们也可以识别出较大的音阶，超过该音阶，共振会扩散。据推测，这最终将使系统进入热化阶段。但是，极长的热化时间尺度表明，可以通过实验观察到广泛的非遍历模式。我们的研究表明，类似于单粒子定位的情况，对称性控制着无序的多体系统的动力学特性。本文介绍的方法提供了一种用于描述各种无序多体系统的通用工具，