Automated discovery of characteristic features of phase transitions in many-body localization

We identify a new “order parameter” for the disorder-driven many-body localization transition by leveraging machine learning. Contrary to previous studies, our method is almost entirely unsupervised. A game theoretic process between neural networks defines an adversarial setup with conflicting objectives to identify what characteristic features to base efficient predictions on. This reduces the numerical effort for mapping out the phase diagram by a factor of $100\times$ and allows us to pin down the transition, as the point at which the physics changes qualitatively, in an objective and cleaner way than is possible with the existing diverse array of quantities. Our approach of automated discovery is applicable specifically to poorly understood phase transitions and is a starting point for a research program leveraging the potential of machine learning–assisted research in physics.

Figure 1

By using a contemporary neural network architecture, we automate feature extraction and drastically reduce computational cost at the same time. To achieve this higher level of automation, a pair of neural networks shares a pipeline for feature extraction. They compete in a game theoretic framework to achieve conflicting goals: one network has to classify states according to their phase, and the other is supposed to tell from how deep in the phase they are. The equilibrium of the game tells what features to base predictions on and allows determination of the phase boundary in a largely unsupervised way.

Figure 2

The output of the neural network directly provides a meaningful estimate of the phase diagram for a finite system $N=12$ [background in (a)] from just 50 disorder realizations, while traditional quantities, like the gap statistics [background in (b)] are still far too noisy. The dots in (a) are the extrapolated phase boundary in the thermodynamic limit obtained from 100 disorder realizations via the data collapse for systems up to $N=18$, shown exemplary for $\epsilon =0.5$ in Fig. 3. The symbols in (b) are the phase boundaries found in [13] based on the average adjacent gap ratio $r$ (triangles) and the dynamical spin fraction $f$ (triangles pointing downwards) for systems of size up to $N=22$ and vastly more disorder realizations. (The data collapse plots for all values of $\epsilon$ are shown in the Appendix in Fig. 4.)

Figure 3

Exemplary output (a) of the neural network at normalized energy $\epsilon =0.5$ averaged over 50 disorder realizations and the data collapse (inset) to determine the position of the phase transition $h_c$ in the thermodynamic limit. The average adjacent gap ratio $r$ is still far too noisy (b) to obtain a good collapse (inset) for the same amount of averaging. The error bands show the ensemble standard deviation $s={[{\sum }_i^N{(x_i-\widehat{x})}^2/(N-1)]}^{1/2}$ of the disorder average.

Figure 4

Output of the neural network at energies $\epsilon =0.2\text{–}0.8$. $N=12$ and 14 are averaged over 500 disorder realizations and $N=16$ and 18 over 100 realizations. The data collapse (inset) determines the position of the phase transition $h_c$ in the thermodynamic limit. The error bands show the ensemble standard deviation $s={[{\sum }_i^N{(x_i-\widehat{x})}^2/(N-1)]}^{1/2}$ of the disorder average.