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The pseudospin α, which labels the eigenstates, is different from the spin in the presence of the spin-orbit coupling.
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By considering all possible 2D magnetic point groups, we found some exceptional models with degeneracy at k = G /2, even without I symmetry. These cases result from a rather high symmetry of the system. It is expected that by including some perturbation to lower the symmetry the Hamiltonian will reduce to the case (a) in Fig. 1.
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By considering all possible 2D magnetic point groups, we found some exceptional models with degeneracy at k = G /2, even without I symmetry. These cases result from a rather high symmetry of the system. It is expected that by including some perturbation to lower the symmetry the Hamiltonian will reduce to the case (a) in Fig. 1.
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In some exceptional cases with high point-group symmetry, the gap between two doubly degenerate bands can close even when k ≠ G /2. In 2D this occurs for the magnetic point-group symmetries 3̄ ′, 3̄ ′ m, 3̄ ′ m′, 6′ /m, 6′/m m′ m, which are in trigonal system or hexagonal systems. It is expected that the closing of the gap in these cases does not correspond to phase transition between the QSH and the SHI, as some perturbation can circumvent this degeneracy.
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In some exceptional cases with high point-group symmetry, the gap between two doubly degenerate bands can close even when k ≠ G /2. In 2D this occurs for the magnetic point-group symmetries 3̄ ′, 3̄ ′ m, 3̄ ′ m′, 6′ /m, 6′ /m m′ m, which are in trigonal system or hexagonal systems. It is expected that the closing of the gap in these cases does not correspond to phase transition between the QSH and the SHI, as some perturbation can circumvent this degeneracy.
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A similar problem has been discussed in the context of fractional fermion number. See PRPLCM 0370-1573 10.1016/0370-1573(86)90167-5
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