Probing many-body localization with neural networks

Abstract

We show that a simple artificial neural network trained on entanglement spectra of individual states of a many-body quantum system can be used to determine the transition between a many-body localized and a thermalizing regime. Specifically, we study the Heisenberg spin-1/2 chain in a random external field. We employ a multilayer perceptron with a single hidden layer, which is trained on labeled entanglement spectra pertaining to the fully localized and fully thermal regimes. We then apply this network to classify spectra belonging to states in the transition region. For training, we use a cost function that contains, in addition to the usual error and regularization parts, a term that favors a confident classification of the transition region states. The resulting phase diagram is in good agreement with the one obtained by more conventional methods and can be computed for small systems. In particular, the neural network outperforms conventional methods in classifying individual eigenstates pertaining to a single disorder realization. It allows us to map out the structure of these eigenstates across the transition with spatial resolution. Furthermore, we analyze the network operation using the dreaming technique to show that the neural network correctly learns by itself the power-law structure of the entanglement spectra in the many-body localized regime.

Learning phase transitions from dynamics

“…2, and shows good agreement with the phase diagram obtained from static entanglement spectra in Ref. [11].…”

“…So far, these methods have relied only on static properties of the underlying physical systems, such as raw state configurations sampled from Monte Carlo simulations [1,15] or entanglement spectra obtained using exact diagonalization [3,11,17]. However, dynamics of physical observables are often more accessible experimentally.…”

“…Machine learning is emerging as a novel tool for identifying phases of matter [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. At its core, this problem can be cast as a classification problem in which data obtained from physical systems are assigned a class (i.e., a phase) using machine learning methods.…”

“…This training was done on the ε = 0.5 data, and uses the confidence enhancement introduced in Ref. [11]. In order to gain a better understanding of what the LSTM neurons are doing, we analyze the case of a single LSTM neuron trained on a single-spin subsystem in the Supplemental Material [57].…”

“…Machine learning methods applied on entanglement spectra of eigenstates were used to obtain a phase diagram of the same model [11], as well as on a slightly different model featuring two distinct MBL phases [17]. Here, we insist on using only experimentally relevant (i.e., measurable) quantities such as the magnetization of individual spins.…”