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Chinese Physics B

Disorder induced phase transition in magnetic higher-order topological insulator: A machine learning study*

Zixian Su, Yan-Zhuo Kang, Bofeng Zhang, Zhiqiang Zhang, Hua Wei Jiang

Published Sep 18, 2019• 31 citations
Publisher

Abstract

Previous studies presented the phase diagram induced by the disorder existing separately either in the higher-order topological states or in the topological trivial states, respectively. However, the influence of disorder on the system with the coexistence of the higher-order topological states and other traditional topological states has not been investigated. In this paper, we investigate the disorder induced phase transition in the magnetic higher-order topological insulator. By using the convolutional neural network and non-commutative geometry methods, two independent phase diagrams are calculated. With the comparison between these two diagrams, a topological transition from the normal insulator to the Chern insulator is confirmed. Furthermore, the network based on eigenstate wavefunction studies also presents a transition between the higher-order topological insulator and the Chern insulator.

DOI
10.1088/1674-1056/ab4582

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Previous studies presented the phase diagram induced by the disorder existing separately either in the higher-order topological states or in the topological trivial states, respectively. However, the influence of disorder on the system with the coexistence of the higher-order topological states and other traditional topological states has not been investigated. In this paper, we investigate the disorder induced phase transition in the magnetic higher-order topological insulator. By using the convolutional neural network and non-commutative geometry methods, two independent phase diagrams are calculated. With the comparison between these two diagrams, a topological transition from the normal insulator to the Chern insulator is confirmed. Furthermore, the network based on eigenstate wavefunction studies also presents a transition between the higher-order topological insulator and the Chern insulator.
A higher-order topological insulator is a new concept of topological states of matter, which is characterized by the emergent boundary states whose dimensionality is lower by more than two compared with that of the bulk, and draws a considerable interest. Yet, its robustness against disorders is still unclear. In this work, we investigate a phase diagram of higher-order topological insulator phases in a breathing kagome model in the presence of disorders by using a state-of-the-art machine learning technique. We find that the corner states survive against the finite strength of disorder potential as long as the energy gap is not closed, indicating the stability of the higher-order topological phases against the disorders.
Quantum material phases such as the Anderson insulator, diffusive metal, and Weyl/Dirac semimetal as well as topological insulators show specific wave functions both in real and Fourier spaces. These features are well captured by convolutional neural networks, and the phase diagrams have been obtained, where standard methods are not applicable. One of these examples is the cases of random lattices such as quantum percolation. Here, we study the topological insulators with random vacancies, namely, the quantum percolation in topological insulators, by analyzing the wave functions via a convolutional neural network. The vacancies in topological insulators are especially interesting since peculiar bound states are formed around the vacancies. We show that only a few percent of vacancies are required for a topological phase transition. The results are confirmed by independent calculations of localization length, density of states, and wave packet dynamics.
The topological Anderson and Mott insulators are two phases that have so far been separately and widely explored beyond topological band insulators. Here we combine the two seemingly different topological phases into a system of spin-1/2 interacting fermionic atoms in a disordered optical lattice. We find that the topological Anderson and Mott insulators in the noninteracting and clean limits can be adiabatically connected without gap closing in the phase diagram of our model. Lying between the two phases, we uncover a disordered correlated topological insulator, which is induced from a trivial band insulator by the combination of disorder and interaction, as the generalization of topological Anderson insulators to the many-body interacting regime. The phase diagram is determined by computing various topological properties and confirmed by unsupervised and automated machine learning. We develop an approach to provide a unified and clear description of topological phase transitions driven by interaction and disorder. The topological phases can be detected from disorder-/interaction-induced edge excitations and charge pumping in optical lattices.
In this paper we study the phase diagram of a disordered, spin-orbit coupled superconductor with sss-wave or d+idd+idd+id-wave pairing symmetry in symmetry class DDD. We analyze the topological phase transitions by applying three different methods which include a disorder averaged entanglement entropy, a disorder averaged real-space Chern number, and an evaluation of the momentum space Chern number in a disorder averaged effective model. We find evidence for a disorder-induced topological state. While in the clean limit there is a single phase transition from a trivial phase with a Chern number C=4C=4C=4 to a topological phase with C=1C=1C=1, in the disordered system there is an intermediate phase with C=3C=3C=3. The phase transition from the trivial C=4C=4C=4 phase into the intermediate phase with C=3C=3C=3 is seen in the real-space calculation of the Chern number. In spite of this, this phase transition is not detectable in the entanglement entropy. A second phase transition from the disorder induced C=3C=3C=3 into the C=1C=1C=1 phase is seen in all three quantities.
We study topological insulators under dephasing noise. With examples of both a 2d2d2d Chern insulator and a 3d3d3d topological insulator protected by time-reversal symmetry, we demonstrate that there is a phase transition at finite dephasing strength between phases with nontrivial and trivial topological indices. Here, the topological index is defined through the correlation matrix. The transition can be diagnosed through the spectrum of the whole correlation matrix or of a local subsystem. Interestingly, even if the topological insulator is very close to the topological-trivial critical point in its Hamiltonian, it still takes finite strength of dephasing to change the topological index, suggesting the robustness of topological insulators under dephasing. We further consider Chern insulators in the presence of real-space disorder, which exhibit a ground-state transition between topological and Anderson insulating phases. We find that even strongly disordered Chern insulators, close to the critical disorder strength, exhibit robustness with respect to dephasing.
We investigate the behavior of a topological Chern insulator in the presence of disorder, with a focus on its entanglement spectrum (EtS) constructed from the ground state. For systems with symmetries, the EtS was shown to contain explicit information about the topological universality class revealed by sorting the EtS against the conserved quantum numbers. In the absence of any symmetry, we demonstrate that statistical methods such as the level statistics of the EtS can be equally insightful, allowing us to distinguish when an insulator is in a topological or trivial phase and to map the boundary between the two phases. The phase diagram of a Chern insulator is explicitly computed as function of Fermi level (EF) and disorder strength using the level statistics of the EtS and energy spectrum, together with a computation of the Chern number (C) via a new, efficient real-space formula.