Z2 Topological Order and the Quantum Spin Hall Effect

Z2 Topological Order and the Quantum Spin Hall Effect

27 Jun 2005 | C.L. Kane and E.J. Mele
The paper by C.L. Kane and E.J. Mele introduces a novel topological classification, the Z2 invariant, for the quantum spin Hall (QSH) phase, which is a time-reversal invariant electronic state with a bulk band gap and gapless edge states. This classification distinguishes the QSH phase from ordinary insulators and is analogous to the Chern number classification in the quantum Hall effect. The authors demonstrate that the QSH phase in graphene, characterized by a conserved perpendicular spin component, is robust even when this conservation is broken due to perturbations. They derive the Z2 index from the Bloch wavefunctions and show that it can be used to distinguish the QSH phase from the insulator. The Z2 index is determined by counting the number of pairs of complex zeros of the Pfaffian of the overlap matrix, which is a measure of the twist of the Bloch wavefunction bundle under time-reversal symmetry. This classification is also formulated in terms of the sensitivity of the ground state to phase-twisted periodic boundary conditions, providing a generalization applicable to multi-band and interacting systems. The Z2 classification highlights the topological stability of the QSH phase against weak disorder and interactions.The paper by C.L. Kane and E.J. Mele introduces a novel topological classification, the Z2 invariant, for the quantum spin Hall (QSH) phase, which is a time-reversal invariant electronic state with a bulk band gap and gapless edge states. This classification distinguishes the QSH phase from ordinary insulators and is analogous to the Chern number classification in the quantum Hall effect. The authors demonstrate that the QSH phase in graphene, characterized by a conserved perpendicular spin component, is robust even when this conservation is broken due to perturbations. They derive the Z2 index from the Bloch wavefunctions and show that it can be used to distinguish the QSH phase from the insulator. The Z2 index is determined by counting the number of pairs of complex zeros of the Pfaffian of the overlap matrix, which is a measure of the twist of the Bloch wavefunction bundle under time-reversal symmetry. This classification is also formulated in terms of the sensitivity of the ground state to phase-twisted periodic boundary conditions, providing a generalization applicable to multi-band and interacting systems. The Z2 classification highlights the topological stability of the QSH phase against weak disorder and interactions.
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Understanding Z2 topological order and the quantum spin Hall effect.