Entanglement Chern Number of the Kane–Mele Model with Ferromagnetism

Abstract

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...

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