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Numerical study of higher order topological insulators by machine learning and Berry phases

荒木, 広夢 筑波大学 DOI:10.15068/00160436

2020.07.21

概要

1.1 Background
Since the discovery of the quantum Hall effect by K. von Klitzing et al. in 1980 [1], the topological materials have attracted great deals of attentions in condensed matter physics. One of the remarkable features of the topological materials is that the topological property is determined by the bulk wavefunctions [2]. Another remarkable feature is the bulk-edge correspondence [3–6], which claims that gapless edge states appear when the insulator has non-trivial bulk topological property.

In 2005, the quantum spin Hall effect for the insulator under the time-reversal symmetry was proposed by C. Kane and E. Mele [7,8]. It is a topological material that has a finite spin current on edges protected by non-trivial topological invariant. The concept of the quantum spin Hall effect was brought in three dimensions, which is called the topological insulator [9]. After the epoch-making study, the topological insulators had been extensively studied. One of the prominent discovery might be the periodic table of the topological insulators and su- perconductors [10–12], which is also called the Altland-Zirnbauer classification [13]. It gives the comprehensive classification of non-interacting topological insulators and superconduc- tors under time-reversal symmetry, chiral symmetry and particle-hole symmetry in arbitrary spacial dimensions. The topological classifications are extended to include the crystalline symmetries, such as inversion, mirror and glide symmetries [14, 15, 15–19].

While the topological insulator phases appear in non-interacting systems, the electron- electron interactions enrich the physics of topological insulators. One of the effect of in- teractions is known to be the reduction of the classification [20–23]. For example, the Z classification of one-dimensional BDI symmetry class is broken to Z8 with interactions. Fur- ther, the interactions induce the new topological physics – such as the topological Mott insulator [24, 25] and the fractional topological insulator [26–29]. Also, it is revealed that the short-range entangled states of many-body systems protected by symmetries can host the nontrivial topological state, which are now understood as the symmetry-protected topological (SPT) phases [30–34].

The Berry phase is known to be an useful tool to characterize the SPT phases [35–44]. The Berry phase get a quantized value because of the symmetry of the system, such as the chiral symmetry, the time-reversal symmetry, the inversion symmetry and the SU(N) symmetry of the spins.

In the couple of years, the higher-order topological insulators (HOTI) are intensely stud- ied [45–56]. The HOTIs in d-dimensions has topologically protected boundary states in d − n dimensional boundaries (n ≥ 2), e.g., hinge states in three dimensions and corner states in two and three dimensions. Since the proposal of the HOTIs, the higher-order topo- logical materials are experimentally realized one after another, such as bismuth crystals [57], mechanical systems [58], electrical circuits [59, 60], photonic crystals [61, 62] and acoustic systems [63].

On the other hand, the machine learning, especially the artificial neural network, is widely used in the various fields in physics, including astrophysics [64], high-energy physics [65, 66], Monte Carlo simulations [67] and quantum many-body systems [68–72]. As for the research of topological states, the first-order topological insulators and the topological superconductors have been successfully classified in the presence of disorders, and the resulting phase diagrams reproduce those obtained by the other methods [73–75].

1.2 Purpose
The primary purpose of this thesis is the theoretical investigation of the properties of the higher-order topological insulator phases by numerical analysis of the boundary states and the topological invariants. We investigate the higher-order topological insulator phases by both the Berry phases and the machine learning. By the Berry phase, we investigate the higher-order topological insulator phases for the system with and without interactions and the spin models. By the machine learning, we investigate the phase transition of the higher- order topological insulator models with disorders.

1.3 Outline
In Chapter 2, we briefly introduce the topological phases and their topological invariants. Firstly, we introduce the quantum Hall effect and its topological invariants, the Chern num- ber. Secondly, we introduce the quantum spin Hall effect and the Z2 index. Thirdly, we introduce the topological insulator phase and the Z2 indices. Lastly, we review the related studies about the entanglement Chern numbers, which characterizes these topological phases.

Chapter 3 reviews the higher-order topological insulators. Firstly, we show the theory of the higher-order topological insulators in both hyper-cubic lattices and hyper-tetrahedral lattices. Then, we briefly review the experimental realizations of the higher-order topological materials.

Chapter 4 describes the Berry phase and its quantization. Firstly we show the definition of the Berry phase by introducing a local bond twists. Then we show the symmetries and the quantization of the Berry phase.

Chapter 5 proposes the quantized Berry phase with the bond-twists as a topological in- variant, which characterizes the higher-order topological insulator phases. The quantized Berry phase is topologically stable even with the electron-electron interactions unless the en- ergy gap is closed. To demonstrate it for a concrete model, we have shown the quantization of the Berry phase in Z4 and the characterization of the higher-order topological insulator phases in several extended Benalcazar-Bernevig-Hughes (BBH) models that contain the in- tersite Coulomb interactions and the next-nearest neighbor hopping. In addition, we show the quantized Berry phase for the quantum spin analog of the BBH model. Further, we show that the BBH model in three-dimensions also has the Berry phase quantized in Z4. We also confirm the bulk-corner correspondence between the quantized Berry phase and the corner states in the higher-order topological insulator phases.

In Chapter 6, we investigate the higher-order topological insulators by using machine learning technique. We applied the image recognition method of machine learning, which detects the boundary states of topological materials. Focusing on the higher-order topological insulator model on a breathing kagome lattice, we studied the robustness of the higher- order topological insulator phases against disorders. We have successfully generated a phase diagram by machine learning which is consistent with the other analytical method. We have also numerically found that the higher-order topological insulator phases are robust against disorder as far as the disorder strength does not exceed the energy gap.

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