Machine learning wave functions to identify fractal phases

Image recommendation can be used to identify the localization-delocalization transition (Anderson transition) (T. Ohtsuki et al., J. Phys. Soc. Jpn. 85, 123706 (2016)). 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 that 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. We also apply the trained neural network to the quantum sum model.