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

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 energiesFor $t,\Delta >0$, this gap can close only atThe whole region $-2t<\mu <2t$ is connected and gapped, so its parameters can be continuously deformed to $t=\Delta ,\mu =0$. Therefore

Writingshows the topological distinction. As $k$ crosses the Brillouin zone, $(d_y,d_z)$ traces an ellipse. It encloses the origin once when $|\mu |<2t$, giving nonzero winding number of a one-dimensional Bogoliubov--de Gennes Hamiltonian. For $\mu <-2t$ 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.