Deep learning of topological phase transitions from entanglement aspects

TL;DR

This paper explores the use of deep learning to identify topological phase transitions in a 1D p-wave superconductor model by analyzing entanglement-related data, revealing that Majorana correlation matrices and entanglement eigenvectors are most informative.

Contribution

It demonstrates that deep neural networks can effectively distinguish topological phases using entanglement data, highlighting the superior information content of MCM and EE over ES.

Findings

MCM and EE provide richer information than ES for phase recognition.

Deep learning can identify phases with different U(1) gauges.

ES alone is too compressed to capture all topological features.

Abstract

The one-dimensional $p$-wave superconductor proposed by Kitaev has long been a classic example for understanding topological phase transitions through various methods, such as examining Berry phase, edge states of open chains and, in particular, aspects from quantum entanglement of ground states. In order to understand the amount of information carried in the entanglement-related quantities, here we study topological phase transitions of the model with emphasis of using the deep learning approach. We feed different quantities, including Majorana correlation matrices (MCMs), entanglement spectra (ES) or entanglement eigenvectors (EE) originated from Block correlation matrices (BCMs), into the deep neural networks for training, and investigate which one could be the most useful input format in this approach. We find that ES is indeed too compressed information compared to MCM or EE. MCM and EE can provide us abundant information to recognize not only the topological phase transitions in the model but also phases of matter with different $U$(1) gauges, which is not reachable by using ES only.