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arXiv:1909.04784 · Physical review. B./Physical review. B

Deep learning of topological phase transitions from entanglement aspects

Yuan-Hong Tsai, Meng-Zhe Yu, Yu-Hao Hsu, Ming-Chiang Chung

Published Aug 18, 2020• 27 citations
arXivHTMLPDF

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 the 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) originating from Block correlation matrices, 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 information that is too compressed 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.

DOI
10.1103/physrevb.102.054512

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The one-dimensional ppp-wave superconductor proposed by Kitaev has long been a classic example for understanding topological phase transitions through various methods, such as examining the 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) originating from Block correlation matrices, 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 information that is too compressed 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 UUU(1) gauges, which is not reachable by using ES only.
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Machine learning techniques have been shown to be effective to recognize different phases of matter and produce phase diagrams in the parameter space of interest, while they usually require prior labeled data to perform well. Here, we propose a machine learning procedure, mainly in an unsupervised manner, which can first identify topological/nontopological phases and then refine the locations of phase boundaries. By following this proposed procedure, we expand our previous work on the one-dimensional ppp-wave superconductor [Tsai et al., Phys. Rev. B 102, 054512 (2020)] and further on the Su-Schrieffer-Heeger model, with an emphasis on using the quantum-entanglement-based quantities as the input features. We find that our method not only reproduces similar results to the previous work with sharp phase boundaries but importantly it also does not rely on prior knowledge of the phase space, e.g., the number of phases present. We conclude with a few remarks about its potential, limitations, and explanatory power.
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Topological insulators and topological superconductors display various topological phases that are characterized by different Chern numbers or by gapless edge states. In this work we show that various quantum information methods such as the von Neumann entropy, entanglement spectrum, fidelity, and fidelity spectrum may be used to detect and distinguish topological phases and their transitions. As an example we consider a two-dimensional ppp-wave superconductor, with Rashba spin-orbit coupling and a Zeeman term. The nature of the phases and their changes are clarified by the eigenvectors of the kkk-space reduced density matrix. We show that in the topologically nontrivial phases the highest weight eigenvector is fully aligned with the triplet pairing state. A signature of the various phase transitions between two points on the parameter space is encoded in the kkk-space fidelity operator.
Abstract Topologically ordered materials may serve as a platform for new quantum technologies, such as fault-tolerant quantum computers. To fulfil this promise, efficient and general methods are needed to discover and classify new topological phases of matter. We demonstrate that deep neural networks augmented with external memory can use the density profiles formed in quantum walks to efficiently identify properties of a topological phase as well as phase transitions. On a trial topological ordered model, our method’s accuracy of topological phase identification reaches 97.4%, and is shown to be robust to noise on the data. Furthermore, we demonstrate that our trained DNN is able to identify topological phases of a perturbed model, and predict the corresponding shift of topological phase transitions without learning any information about the perturbations in advance. These results demonstrate that our approach is generally applicable and may be used to identify a variety of quantum topological materials.
Entanglement entropy (EE) provides a powerful probe of quantum phases, yet its role in identifying topological phase transitions in disordered systems remains underexplored. We introduce an exact EE-based framework that captures topological phase transitions even in the presence of disorder. Specifically, for a class of Su-Schrieffer-Heeger (SSH) model variants, we show that the difference in EE between half-filled and near-half-filled ground states,ΔSA, vanishes in the topological phase but remains finite in the trivial phase, a direct consequence of edge-state localization. This behavior persists even in the presence of quasiperiodic or binary disorder. By analyzing domain-wall configurations in the SSH chain, we further show how subsystem tuning allows one to distinguish genuine topological zero-energy eigenstates from trivial localized states. Exact phase boundaries, derived from Lyapunov exponents via transfer matrices, agree closely with numerical results fromΔSAand the topological invariant𝓠, with instances whereΔSAoutperforms𝓠. Our results highlight EE as a robust diagnostic tool and a potential bridge between quantum information and condensed matter approaches to topological matter.