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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 342 / 3 / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Two local gapped Hamiltonians are topologically equivalent when a continuous path of local Hamiltonians joins them without closing the bulk gap. The Bogoliubov--de Gennes Hamiltonian has energies

E_{\pm} (k) = \pm (2 t cos k + μ)^{2} + 4 Δ^{2} sin^{2} k .

For

t, Δ > 0

, this gap can close only at

μ = - 2 t (k = 0), μ = 2 t (k = π) .

The whole region

- 2 t < μ < 2 t

is connected and gapped, so its parameters can be continuously deformed to

t = Δ, μ = 0

. Therefore

H ≃ H_{0} for - 2 t < μ < 2 t .

H_{BdG} (k) = d_{y} (k) σ_{y} + d_{z} (k) σ_{z}, d_{y} = - 2Δ sin k, d_{z} = - 2 t cos k - μ,

shows the topological distinction. As

k

crosses the Brillouin zone,

(d_{y}, d_{z})

traces an ellipse. It encloses the origin once when

∣ μ ∣ < 2 t

, giving nonzero winding number of a one-dimensional Bogoliubov--de Gennes Hamiltonian. For

μ < - 2 t

it does not enclose the origin and has winding zero. Changing this integer requires the ellipse to pass through the origin, exactly the bulk gap closing.

Figure 1.
Kitaev-chain ellipse and its winding around the origin
. For chemical potential zero the Bogoliubov--de Gennes vector traces an ellipse enclosing the origin once. For chemical potential minus 3t the translated ellipse misses the origin and has winding number zero. Details.

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