Machine learning wave functions to identify fractal phases

We demonstrate that an image recognition algorithm based on a convolutional neural network provides a powerful procedure to differentiate between ergodic, nonergodic extended (fractal), and localized phases in various systems: Single-particle models, including random-matrix and random-graph models, and many-body quantum systems. We propose an efficient procedure in which the network is successfully trained on a small data set of only 500 wave functions (images) per class for a single model which exhibits these phases. The trained network is then used to classify phases in the other models. We discuss the strengths and limitations of the approach.