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The non-ergodic extended (NEE) regime in physical and random matrix (RM) models has attracted a lot of attention in recent years. Formally, NEE regime is characterized by its fractal wavefunctions and long-range spectral correlations such as number variance or spectral form factor. More recently, it's proposed that this regime can be conveniently revealed through the eigenvalue spectra by means of singular-value-decomposition (SVD), whose results display a super-Poissonian behavior that reflects the minibands structure of NEE regime. In this work, we employ SVD to a number of RM models, and show it not only qualitatively reveals the NEE regime, but also quantitatively locates the ergodic-NEE transition point. With SVD, we further suggest the NEE regime in a new RM model-the sparse RM model., (© 2023. The Author(s).)

We employ a convolutional neural network to explore the distinct phases in random spin systems with the aim to understand the specific features that the neural network chooses to identify the phases. With the energy spectrum normalized to the bandwidth as the input data, we demonstrate that a network of the smallest nontrivial kernel width selects level spacing as the signature to distinguish the many-body localized phase from the thermal phase. We also study the performance of the neural network with an increased kernel width, based on which we find an alternative diagnostic to detect phases from the raw energy spectrum of such a disordered interacting system., (© 2020 IOP Publishing Ltd.)

The transition between thermal and many-body localized phases in isolated random spin systems are typically identified by the distribution of nearest level spacings. In this work, by employing machine learning methodology, we show this transition can be learnt through raw energy spectrum without any pre-processing. After achieving so in conventional random spin chain with differentiable level spacing distributions, we further construct novel models with misleading signatures of level spacing, and show machine can defeat the latter when training data is raw energy spectrum. Our work shows the low-level energy spectrum contains more information than level spacings, and can be captured by machine in a direct while efficient way, which makes it a promising new tool for studying a variety of isolated quantum systems.