Topology
Chern Topology
Haldane-model topological invariants, Berry curvature, and bulk-boundary correspondence.
Research views described in the portal
- Berry curvature integration and the first Chern number.
- Band structure along Γ → K → M → K′ → Γ.
- Phase diagrams, edge states, and convergence.
- Quantum anomalous Hall effect and bulk-boundary correspondence.
Model parameters
The source interface exposes nearest-neighbor hopping t₁, next-nearest-neighbor hopping t₂, the Haldane flux phase φ, staggered potential M, a Berry-curvature integration grid, and band-path sampling.
Physics reference from the portal
| Quantity | Formula / Expression | Physical meaning |
|---|---|---|
| Bloch Hamiltonian | H(k) = d₀I + dₓσₓ + dᵧσᵧ + dᵩσᵩ | 2×2 spinor per k-point; d-vector on Bloch sphere |
| NNN complex hopping | t₂ e±iφ | Breaks TRS without net B-field (Haldane 1988) |
| Berry curvature | Ω⁻(k) = ½ 𝒏 ⋅ (∂𝒏/∂kₓ × ∂𝒏/∂kᵧ) | Gauge-invariant geometric curvature of lower band |
| First Chern number | C₁ = (1/2π) ∫BZ Ω⁻ d²k ∈ ℤ | Topological invariant; integer by Gauss-Bonnet theorem |
| Phase boundary | |M| = M_c = √3 ⋅ t₂ ⋅ |sinφ| | Gap closes at K or K′; C₁ changes by ±1 |
| Hall conductance | σxy = C₁ ⋅ e²/h | Quantised anomalous Hall conductance (QAHE) |
| Bulk-boundary | Nedge = |C₁| | Chiral edge modes = Chern number (robust to disorder) |
| Dirac gap at K | Δ_K = 2|d_z(K)| = 2|M − √3 t₂ sinφ| | Opens at Dirac point; closes at phase transition |
| Kubo formula | Ω⁻ = Im ∑m≠n ⟨n|∂H/∂kₓ|m⟩⟨m|∂H/∂kᵧ|n⟩ / (E_m−E_n)² | Equivalent gauge-invariant expression (used in code) |
| BZ integration | C₁ = (1/2π) ∫ d²k Ω⁻(k) | Gauss-Bonnet: integral over closed 2-torus BZ = 2πC₁ |