论文标题
超对称量子力学中的量子混乱：一项精确的对角研究
Quantum chaos in supersymmetric quantum mechanics: an exact diagonalization study
论文作者
论文摘要
我们使用精确的对角度化来研究最简单的超对称扩展$ \ hat {h} _s = \ hat { hamiltonian $ \ hat {h} _b = \ hat {p} _1^2 + \ hat {p} _2^2 + \ hat {x} _1^2 \，\ hat {x} _2 _2 _2^2 $。很长一段时间以来，这种骨髓的哈密顿量被认为是最简单的系统之一，它们在经典和机械上都表现出动态混乱。它的结构与空间压缩的纯阳米尔斯理论相似。相应地，我们的超对称性哈密顿量的结构与空间压缩的超对称阳米尔斯理论（也称为BFSS模型）相似。我们提供了数值证据，表明超对称模型的连续能谱会导致OTOC的单调生长至最低的温度，这对于全息二元性的BFSS模型也可以预期。这种生长被低能特征状态饱和，具有有效的一维波函数和完全非骨质的能级分布。我们观察到一个尖锐的边界，将这些低能状态与大部分混沌高能状态分开。我们的数据表明，尽管置信度有限，但在低温下，OTOC的生长可能在有限的时间范围内指数呈指数，而相应的Lyapunov指数随温度线性缩放。相比之下，玻色剂哈密顿量的低能频谱会导致在低温下振荡的OTOC，而没有任何指数增长的特征。我们还发现，骨髓哈密顿量的OTOC永远不会足够接近经典的Lyapunov距离。另一方面，超对称系统的OTOC在有限的温度和进化时间范围内相当符合经典极限。
We use exact diagonalization to study energy level statistics and out-of-time-order correlators (OTOCs) for the simplest supersymmetric extension $\hat{H}_S = \hat{H}_B \otimes I + \hat{x}_1 \otimes σ_1 + \hat{x}_2 \otimes σ_3$ of the bosonic Hamiltonian $\hat{H}_B = \hat{p}_1^2 + \hat{p}_2^2 + \hat{x}_1^2 \, \hat{x}_2^2$. For a long time, this bosonic Hamiltonian was considered one of the simplest systems which exhibit dynamical chaos both classically and quantum-mechanically. Its structure closely resembles that of spatially compactified pure Yang-Mills theory. Correspondingly, the structure of our supersymmetric Hamiltonian is similar to that of spatially compactified supersymmetric Yang-Mills theory, also known as the BFSS model. We present numerical evidence that a continuous energy spectrum of the supersymmetric model leads to monotonous growth of OTOCs down to the lowest temperatures, which is also expected for the BFSS model from holographic duality. This growth is saturated by low-energy eigenstates with effectively one-dimensional wave functions and a completely non-chaotic energy level distribution. We observe a sharp boundary separating these low-energy states from the bulk of chaotic high-energy states. Our data suggests, although with a limited confidence, that at low temperatures the OTOC growth might be exponential over a finite range of time, with the corresponding Lyapunov exponent scaling linearly with temperature. In contrast, the gapped low-energy spectrum of the bosonic Hamiltonian leads to oscillating OTOCs at low temperatures without any signatures of exponential growth. We also find that the OTOCs for the bosonic Hamiltonian are never sufficiently close to the classical Lyapunov distance. On the other hand, the OTOCs for the supersymmetric system agree with the classical limit reasonably well over a finite range of temperatures and evolution times.
