Supervised and unsupervised learning of the many-body critical phase, phase transitions, and critical exponents in disordered quantum systems

Abstract: In this work, we begin by questioning the existence of a new kind of nonergodic extended phase, namely, the many-body critical (MBC) phase in finite systems of an interacting quasiperiodic system. We find that this phase can be separately detected from the other phases such as the many-body ergodic (ME) and many-body localized (MBL) phases in the model through supervised neural networks made for both binary and multi-class classification tasks, utilizing, rather un-preprocessed, eigenvalue spacings and eigenvector probability densities as input features. Moreover, the output of our trained neural networks can also indicate the critical points separating ME, MBC and MBL phases, which are consistent with the same obtained from other conventional methods. We also employ unsupervised learning techniques, particularly principal component analysis (PCA) of eigenvector probability densities to investigate how this framework, without any training, captures the, rather unknown, many-body phases (ME, MBL and MBC) and single particle phases (delocalized, localized and critical) of the interacting and non-interacting systems, respectively. Our findings reveal that PCA entropy serves as an effective indicator (order parameter) for detecting phase transitions in the single-particle systems. Moreover, this method proves applicable to many-body systems when the data undergoes a suitable pre-processing. Interestingly, when it comes to extraction of critical (correlation length) exponents through a finite size-scaling, we find that for single-particle systems, scaling collapse of neural network outputs is obtained using components of inverse participation ratio (IPR) as input data. Remarkably, we observe identical critical exponents as obtained from scaling collapse of the IPR directly for different single-particle phase transitions.