Interpretable Machine Learning Study of Many-Body Localization Transition in Disordered Quantum Ising Spin Chains

We apply support vector machine (SVM) to study the phase transition between many-body localized and thermal phases in a disordered quantum Ising chain in a transverse external field. The many-body eigenstate energy $E$ is bounded by a bandwidth $W=E_{max}-E_{min}$. The transition takes place on a phase diagram spanned by the energy density $\epsilon=2(E-E_{min})/W$ and the disorder strength $\delta J$ of the spin interaction uniformly distributed within $[-\delta J, \delta J]$, formally parallel to the mobility edge in Anderson localization. In our study we use the labeled probability density of eigenstate wavefunctions belonging to the deeply localized and thermal regimes at two different energy densities ($\epsilon$'s) as the training set, i.e., providing labeled data at four corners of the phase diagram. Then we employ the trained SVM to predict the whole phase diagram. The obtained phase boundary qualitatively agrees with previous work using entanglement entropy to characterize these two phases. We further analyze the decision function of the SVM to interpret its physical meaning and find that it is analogous to the inverse participation ratio in configuration space. Our findings demonstrate the ability of the SVM to capture potential quantities that may characterize the many-body localization phase transition.