Abstract
We show that quantum spin Hall (QSH) effect does not occur in a square lattice model due to cancellation of the intrinsic spin-orbit coupling coming from different hopping paths. However, we show that QSH effect can be induced by the presence of staggered magnetic fluxes alternating directions square by square. When the resulting Peierls phase takes a special value , the system has a composite symmetry ΘΡ− with Θ the time-reversal operator and Ρ− transforming the Peierls phase from γ to γ − , which protects the gapless edge states. Once the phase deviates from , the edge states open a gap, as the composite symmetry is broken. We further investigate the effect of a Zeeman field on the QSH state, and find that the edge states remain gapless for . This indicates that the QSH effect is immune to the magnetic perturbation.
Highlights
We show that quantum spin Hall (QSH) effect does not occur in a square lattice model due to cancellation of the intrinsic spin-orbit coupling coming from different hopping paths
A QSH system can no longer be divided into two quantum Hall (QH) subsystems, and the existence of the gapless edge states has been attributed to the nontrivial topological properties of bulk energy bands, which can be characterized by a nonzero Z2 index[16] or a nonzero spin Chern number[17,18]
We further investigate the effect of a Zeeman field on the Remarkably, QSH state, which it is found that the veidogleatsetsattehseinΘtPh−eπ2QsSyHmmpheatsrey no matter what value remain to be gapless, if the the Peierls Peierls phase γtakes. phase takes the special value γ = π
Summary
We show that quantum spin Hall (QSH) effect does not occur in a square lattice model due to cancellation of the intrinsic spin-orbit coupling coming from different hopping paths. OernacteorthaendphΡa−sπ2e transforming the Peierls phase deviates from γ = π , the edge states open a gap, as the composite symmetry is broken. We further investigate the effect of a Zeeman field on the QSH state, and find that the edge states remain gapless for γ
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