Quantum Spin Hall Effect — Kane-Mele Model
Z₂ Topological Invariant · Edge States · Topological Phase Transition · SOC
Bulk bands (↑)
Bulk bands (↓)
Edge states
Kane-Mele Parameters
Hopping t
1.0
SOC (spin-orbit) λ_SO
0.3
Rashba coupling λ_R
0.1
Staggered potential M
0.2
Z₂ = 1 (Topological)
Kane-Mele Hamiltonian (honeycomb):
H = H_t + H_SO + H_R + H_M
SOC term (next-nearest-neighbor):
H_SO = iλ_SO Σ ν_ij c†_i σ_z c_j
ν_ij = ±1 (right/left turn)
Topological condition:
|M| < 3√3 λ_SO
→ Z₂=1 (QSH)
|M| > 3√3 λ_SO
→ Z₂=0 (trivial)
Edge states:
Helical — spin-up moves
right on top, spin-down on bottom
Protected by time-reversal symmetry
Z₂ invariant:
parity of Wannier
centers; ν=0 trivial, ν=1 topological