Quantum chaos in supersymmetric quantum mechanics: An exact diagonalization study

We use exact diagonalization to study energy level statistics and out-of-time-order correlators (OTOCs) for the simplest supersymmetric extension $\hat{H}_S = \hat{H}_B \otimes I + \hat{x}_1 \sigma_1 \hat{x}_2 \sigma_3$ of 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 was considered one systems which exhibit dynamical chaos both classically quantum-mechanically. Its structure closely resembles that spatially compactified pure Yang-Mills theory. Correspondingly, our is similar theory, also known as BFSS model. present numerical evidence continuous spectrum model leads monotonous growth OTOCs down lowest temperatures, expected from holographic duality. This saturated by low-energy eigenstates with effectively one-dimensional wave functions completely non-chaotic distribution. observe sharp boundary separating these states bulk chaotic high-energy states. Our data suggests, although limited confidence, at low temperatures OTOC might be exponential over finite range corresponding Lyapunov exponent scaling linearly temperature. In contrast, gapped oscillating without any signatures growth. find are never sufficiently close classical distance. On other hand, system agree limit reasonably well evolution times.