Disordered interacting spin chains that undergo a many-body localization transition are characterized by two limiting behaviors where the dynamics are chaotic and integrable. However, the transition region between them is not fully understood yet. We propose here a possible finite-size precursor of a critical point that shows a typical finite-size scaling and distinguishes between two different dynamical phases. The kurtosis excess of the diagonal fluctuations of the full one-dimensional momentum distribution from its microcanonical average is maximum at this singular point in the paradigmatic disordered $J_1$-$J_2$ model. For system sizes accessible to exact diagonalization, both the position and the size of this maximum scale linearly with the system size. Furthermore, we show that this singular point is found at the same disorder strength at which the Thouless and the Heisenberg energies coincide. Below this point, the spectral statistics follow the universal random matrix behavior up to the Thouless energy. Above it, no traces of chaotic behavior remain, and the spectral statistics are well described by a generalized semi-Poissonian model, eventually leading to the integrable Poissonian behavior. We provide, thus, an integrated scenario for the many-body localization transition, conjecturing that the critical point in the thermodynamic limit, if it exists, should be given by this value of disorder strength.

中文翻译：

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多体本地化过渡中关键点的签名

经历多体定位转变的无序相互作用自旋链的特征是动力学具有混沌性和可积分性的两个极限行为。但是，它们之间的过渡区域尚未完全理解。我们在这里提出临界点的可能的有限大小的前体，该临界点显示典型的有限大小的缩放比例，并区分两个不同的动力学阶段。在范式无序的$ J_1 $-$ J_2 $模型中，此奇异点上，一维动量分布的全部对角波动从其微典范平均的峰度过量最大。对于可精确对角线化的系统尺寸，此最大刻度的位置和尺寸均与系统尺寸成线性关系。此外，我们表明，这个奇异点是在相同的无序强度下找到的，在无序强度下，Thouless和Heisenberg能量重合。在此点以下，频谱统计将遵循通用随机矩阵的行为直至Thouless能量。在其上方，没有任何混沌行为的痕迹，并且频谱统计被广义的半泊松模型很好地描述，最终导致了可积的泊松行为。因此，我们提供了多体定位过渡的综合方案，并推测热力学极限中的临界点（如果存在）应由无序强度的该值给出。不会留下任何混沌行为的痕迹，并且频谱统计可以通过广义的半泊松模型很好地描述，最终导致可积的泊松行为。因此，我们提供了一个多体定位过渡的综合方案，并推测热力学极限中的临界点（如果存在）应该由无序强度的这个值给出。不会留下任何混沌行为的痕迹，并且频谱统计可以通过广义的半泊松模型很好地描述，最终导致可积的泊松行为。因此，我们提供了一个多体定位过渡的综合方案，并推测热力学极限中的临界点（如果存在）应该由无序强度的这个值给出。