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.
Highlights
Going beyond Ginzburg-Landau theory of phase transitions [1], a topological phase transition (TPT) can occur when no symmetry is broken in a physical system
We take aforementioned deep learning (DL) approach to study topological phase transitions occurring in 1D p-SC and examine various quantum information-inspired input features in order to provide a better compressed representation of the naive ground state wave function
Given the correlation function matrices, the entanglement spectra or entanglement eigenstates as a possible form of inputs, the transition point λ = 1 is still stood out via the proposed deep learning approach to distinguish between the ferromagnetic phase (λ > 1) and the paramagnetic one (λ < 1)
Summary
Going beyond Ginzburg-Landau theory of phase transitions [1], a topological phase transition (TPT) can occur when no symmetry is broken in a physical system. The one-dimensional (1D) topological p-wave superconductor (pSC) proposed by Kitaev [5] has become one of the most interesting proposals due to the fact that the edge modes in these superconductors can be viewed as “Majorana fermions”, whose antiparticle is the particle itself They are essential components in forming practical fault-tolerant quantum computers [6]. Since quantum information is known to be useful when there is no local order parameter available in a system, one pioneering work takes the entanglement spectrum (ES), used to compress the ground state information, as the input data and trains a neural network to distinguish the topological phase from the trivial one [21]. The eigenvalues and their corresponding eigenvectors of BCM could be considered as potential representations of quantum information in the system for our deep learning purpose
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