Quantum chaos in supersymmetric quantum mechanics: an exact diagonalization study

TL;DR

This study uses exact diagonalization to analyze energy level statistics and out-of-time-order correlators in a supersymmetric quantum system, revealing insights into quantum chaos, spectral properties, and potential holographic duality implications.

Contribution

It provides the first detailed numerical analysis of supersymmetric quantum chaos using exact diagonalization, connecting spectral features with chaotic behavior and holographic models.

Findings

Continuous spectrum leads to monotonous OTOC growth at low temperatures.

Low-energy states are non-chaotic with effectively one-dimensional wave functions.

A sharp boundary separates low-energy non-chaotic states from high-energy chaotic states.

Abstract

We use exact diagonalization to study energy level statistics and out-of-time-order correlators (OTOCs) for the simplest supersymmetric extension $\hat{H}_S = \hat{H}_B \otimes I + \hat{x}_1 \otimes \sigma_1 + \hat{x}_2 \otimes \sigma_3$ 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.