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

Supervised and unsupervised machine-learning-based frameworks have been implemented in the past to classify various dynamical phases of disordered quantum systems using rather sophisticated (highly preprocessed or difficult to generate) input data. In this paper, we begin by questioning the existence of a 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 multiclass classification tasks, utilizing, rather unpreprocessed, 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 noninteracting 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 preprocessing. Interestingly, when it comes to the 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 the scaling collapse of the IPR directly for different single-particle phase transitions. However, the current finite-size scaling approach to the network output cannot reliably extract critical exponents for the many-body system that needs further investigation.