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. arXiv:1809.09865v2 [cond-mat.dis-nn]

We propose the ZQ Berry phase as a topological invariant for higher-order symmetry-protected topological (HOSPT) phases. It is topologically stable for electron-electron interactions assuming the gap remains open. As a concrete example, we show that the Berry phase is quantized in Z4 and characterizes the HOSPT phase of the extended Benalcazar-Bernevig-Hughes (BBH) model, which contains the next-nearest neighbor hopping and the intersite Coulomb interactions. Furthermore, we introduce the Z4 Berry phase for the spin-model-analog of the BBH model, whose topological invariant has not been found so far. We also confirm the bulk-corner correspondence between the Z4 Berry phase and the corner states in the HOSPT phases.

We propose a realization of higher-order topological phases in a spring-mass model with a breathing kagome structure. To demonstrate the existence of the higher-order topological phases, we characterize the topological properties with the Z3 Berry phase, and show that the corner states appear under the fixed boundary condition. Moreover, we suggest that the corner states can be detected experimentally through a forced vibration.

Higher-order topological insulators have attracted considerable interests as a novel topological phase of matter, where topologically non-trivial nature of bulk protects boundary states whose co-dimension is larger than one. It has been revealed that the alternating pattern of hopping amplitudes in two-dimensional lattices provides a promising route to realization of the higher-order topological insulators. In this paper, we propose that a honeycomb-lattice model with anti-Kekulé distortion hosts a higher-order topological phase. Here, the term anti-Kekulé distortion means that the pattern of strong and weak hoppings is opposite to that for the conventional Kekulé distortion. We demonstrate the existence of the higher-order topological phase by calculating the Z 6 Berry phase that serves as a bulk topological invariant of the higher-order topological phase, and by showing the existence of corner states.

The entanglement Chern number, the Chern number for the entanglement Hamiltonian, is used to characterize the Kane-Mele model, which is a typical model of the quantum spin Hall phase with time-reversal symmetry. We first obtain the global phase diagram of the Kane-Mele model in terms of the entanglement spin Chern number, which is defined by using a spin subspace as a subspace to be traced out in preparing the entanglement Hamiltonian. We further demonstrate the effectiveness of the entanglement Chern number without time-reversal symmetry by extending the Kane-Mele model to include the Zeeman term. The numerical results confirm that the sum of the entanglement spin Chern number is equal to the Chern number.Symmetry enriches topological classification of material phases.1 For free fermion systems, the fundamental symmetries, i.e., time-reversal symmetry, charge conjugation symmetry, and chiral symmetry, are essential to obtain the so-called periodic table of topological insulators and superconductors.2-5 The classification has been farther refined to include some crystalline point group symmetries.6-8 In some cases, the physical and intuitive construction of topologically nontrivial phases with higher symmetry is possible by assembling two or multiple copies of topologically nontrivial phases with lower symmetry so that the symmetry of the assembled system is restored. A typical example is a quantum spin Hall (QSH) insulator with time-reversal symmetry. 9, 10Physically, it is constructed by making each spin subsystem (up and down) a quantum Hall state.11, 12 The point is that, even when the whole system has time-reversal symmetry, the symmetry is effectively broken and the Chern number is finite for each spin subspace.The "entanglement" Chern number has recently been introduced to characterize various topological ground states.13 The entanglement Chern number is the Chern number 14-16 for the entanglement Hamiltonian, and the entanglement Hamiltonian is constructed by tracing out certain subspaces of a given system. [17][18][19] This means that the entanglement Chern number is suitable for analyzing the topological properties of a high-symmetry system composed of multiple copies of lower-symmetry systems. That is, we can focus on a specific subsystem by tracing out the others. For instance, if the up-or down-spin sector is chosen as a subspace to be traced out, the obtained entanglement Chern number, which we name as the entanglement spin Chern number, should be useful for characterizing the QSH state. It is worth noting that the choice of the subspace is not limited to spin sectors and that entanglement Chern number potentially has wide applications. Also, the entanglement Chern number can be defined regardless of the symmetry of the system or the details of the Hamiltonian provided we can choose a subsystem to be traced out.In this paper, we first briefly explain the idea behind the entanglement (spin) Chern number. Then, we extend the arguments in Ref. 13 to cover the global phase diagram of the Kane-Mele mode...