Two mass-imbalanced atoms in a hard-wall trap: Deep learning integrability of many-body systems

The study of integrable systems has led to significant advancements in our understanding of many-body physics. We design a series of numerical experiments to analyze the integrability of a mass-imbalanced two-body system through energy-level statistics and deep learning of wave functions. The level spacing distributions are fitted by a Brody distribution and the fitting parameter $\omega$ is found to separate the integrable and nonintegrable mass ratios by a critical line $\omega =0$. The convolutional neural network built from the probability density images could identify the transition points between integrable and nonintegrable systems with high accuracy, yet in a much shorter computation time. A brilliant example of the network's ability is to identify a new integrable mass ratio $1/3$ by learning from the known integrable case of equal mass, with a remarkable network confidence of $98.22\%$. The robustness of our neural networks is further enhanced by adversarial learning, where samples are generated by standard and quantum perturbations mixed in the probability density images and the wave functions, respectively.