Machine Learning Many-Body Localization: Search for the Elusive Nonergodic Metal

The breaking of ergodicity in isolated quantum systems with a single-particle mobility edge is an intriguing subject that has not yet been fully understood. In particular, whether a nonergodic but metallic phase exists or not in the presence of a one-dimensional quasiperiodic potential is currently under active debate. In this Letter, we develop a neural-network-based approach to investigate the existence of this nonergodic metallic phase in a prototype model using many-body entanglement spectra as the sole diagnostic. We find that such a method identifies with high confidence the existence of a nonergodic metallic phase in the midspectrum at an intermediate quasiperiodic potential strength. Our neural-network-based approach shows how supervised machine learning can be applied not only in locating phase boundaries but also in providing a way to definitively examine the existence or not of a novel phase.

Figure 1

Schematic results from applying our NN approach to toy examples identifying (a) a falsely assumed phase $C$ in a system with only two phases $A$ and $B$, and (b) an unnoticed hidden phase $C$ in a system with three phases $A$, $B$, and $C$. (a) shows schematics illustrating situations described in the text, where the upper panel shows the falsely positive result produced by a three-phase classifier for $A$, $B$, and $C$, and the lower panel shows the correct result produced by a two-phase classifier for $A$ and $B$. (b) shows results from calculation using MNIST data as input, where we associate $A$, $B$, and $C$ with digits 1, 6, and 3, respectively. The upper panel shows the falsely negative result produced by a two-phase classifier for only $A$ and $B$. The lower panel shows the correct result produced by a three-phase classifier for $A$, $B$, and $C$. Each tick on the horizontal axis corresponds to a group of 160 MNIST images of the associated digit. In both plots, $p_i$ is the confidence for identifying certain input data as phase $i=A$, $B$, $C$, and the horizontal axis with background color blue, yellow, and red correspond to phase $A$, $B$, and $C$, respectively.

Figure 2

(a) The schematics of the building block in our NN approach, a general $M$-phase classifier for phase $i=1,\dots ,M$. The phase diagrams of the GAA model with $L=30$ sites, potential strength $\lambda =0.3$, and interaction strength $V=1$ produced by (b) a two-phase classifier, (c) a three-phase classifier, and (d) a four-phase classifier. Here, $p_i(E)$ is the energy-dependent confidence at which the corresponding classifier identifies the eigenstates to be in each of the studied phases. The training data for the classifiers are collected from energy bins $E_1,\dots ,E_4$ and ${\tilde{E}}_4$ labeled in (c) for corresponding phases as discussed in the text.

Figure 3

The comparison between the $L=30$ (upper) and $L=24$ (lower) phase diagrams of the GAA model focused on the NEM regimes. Here the parameter choices are the same as those in Fig. 2.