Many-body mobility edges in one and two dimensions revealed by convolutional neural networks

We adapt a machine-learning approach to study the many-body localization transition in interacting fermionic systems on disordered one-dimensional (1D) and two-dimensional (2D) lattices. We perform supervised training of convolutional neural networks (CNNs) using labeled many-body wave functions at weak and strong disorder. In these limits, the average validation accuracy of the trained CNNs exceeds 99.95%. We use the disorder-averaged predictions of the CNNs to generate energy-resolved phase diagrams, which exhibit many-body mobility edges. We provide finite-size estimates of the critical disorder strengths at ${W}_{c}\ensuremath{\sim}2.8$ and 9.8 for 1D and 2D systems of 16 sites, respectively. Our results agree with the analysis of energy-level statistics and inverse participation ratio. By examining the convolutional layer, we unveil its feature extraction mechanism which highlights the pronounced peaks in localized many-body wave functions while rendering delocalized wave functions nearly featureless.