Machine Learning Out-of-Equilibrium Phases of Matter

Neural-network-based machine learning is emerging as a powerful tool for obtaining phase diagrams when traditional regression schemes using local equilibrium order parameters are not available, as in many-body localized (MBL) or topological phases. Nevertheless, instances of machine learning offering new insights have been rare up to now. Here we show that a single feed-forward neural network can decode the defining structures of two distinct MBL phases and a thermalizing phase, using entanglement spectra obtained from individual eigenstates. For this, we introduce a simplicial geometry-based method for extracting multipartite phase boundaries. We find that this method outperforms conventional metrics for identifying MBL phase transitions, revealing a sharper phase boundary and shedding new insight on the topology of the phase diagram. Furthermore, the phase diagram we acquire from a single disorder configuration confirms that the machine-learning-based approach we establish here can enable speedy exploration of large phase spaces that can assist with the discovery of new MBL phases. To our knowledge, this Letter represents the first example of a standard machine learning approach revealing new information on phase transitions.

Figure 1

(a) A depiction of our neural network. (b) The two-simplex codomain of our neural network outputs. Colored circles represent regions with maximal confidence; the white circles represent regions of maximal confusion. The green point represents an example output with its associated $d_{\min }$ marked with a green line.

Figure 2

(a) The phase diagram with average neural network output plotted as an RGB parameter. Cuts marked by white lines in (a) are shown in (b) $\lambda =1.0$, (c) $\overline{\log J}-\overline{\log h}=0.8$, and (d) $\overline{\log J}-\overline{\log h}=-0.8$. The sampling width is 0.1 for each parameter.

Figure 3

(a) Our $\mathcal{C}$ measure for extracting phase boundaries (defined in the main text) and (b) the average standard deviation (Std Dev) of the entanglement entropy. The data in each have been normalized by the largest value in the parameter space for meaningful comparison. (c)–(e) The measures plotted in (a) and (b) along the cuts marked in white lines: (c) $\lambda =1.0$, (d) $\overline{\log J}-\overline{\log h}=+0.8$, and (e) $\overline{\log J}-\overline{\log h}=-0.8$.

Figure 4

The 2D phase diagram where the fully trained network has been tested on a single disorder realization.