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 tonian (H) over capS = (H) over capB circle times I thorn (x) over cap1 circle times sigma 1 + circle times (x) over cap2 circle times sigma 3 of the bosonic Hamil-(H) over capB = (p) over cap 21 thorn (p) over cap 22 + (x) over cap2 1 (x) over cap2 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 Banks-Fischler-Shenker-Susskind (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, a property that is also expected for the BFSS model from holographic duality. We find that this growth is saturated by low-energy eigenstates with effectively one-dimensional wave functions and a completely nonchaotic energy level distribution. We observe a sharp boundary separating these low-energy states from the bulk of chaotic high-energy states. Our data suggest, although with 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.

Automated summary

This study used exact diagonalization to investigate energy level statistics and out-of-time-order correlators (OTOCs) for the simplest supersymmetric extension of the Hamiltonian. The continuous energy spectrum of the supersymmetric model leads to monotonous growth of OTOCs down to the lowest temperatures, with a sharp boundary separating low-energy states from chaotic high-energy states.