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

The inverse definitions areSubstitution into the hopping and pairing terms expresses the Hamiltonian as a bilinear in the Majorana fermion operators. Hermiticity makes its off-diagonal coefficients purely imaginary in the Majorana bilinear, whileremoves the symmetric part; diagonal terms are constants because $c_j^2=1$. Therefore, up to that additive constant,whereConversely every real antisymmetric $A$ makes this expression Hermitian, so this is the general Majorana form of a quadratic fermion Hamiltonian.

DefineBecause $O$ is real orthogonal, the ${\chi }_j$ are again self-adjoint and satisfy the Majorana anticommutation relations. Using $A=O^T\epsilon O$,Each two-dimensional block contributes twice the same ordered bilinear:Henceup to the original additive constant.

At $t=\Delta$ and $\mu =0$, direct substitution ofinto the bond Hamiltonian makes the hopping and pairing terms cancel except forThusThe terms pair disjoint Majoranas, so this is already Majorana diagonal, with nonzero single-particle values ${ϵ}_j=2t$. For periodic boundaries the final pair closes around the chain; for open boundaries $c_1$ and $c_{2M}$ remain unpaired.

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.

A Majorana zero mode is a normalized self-adjoint fermion operator localized near a boundary and commuting with the Hamiltonian. With open boundaries,Neither $c_1$ nor $c_{2M}$ appears. Each anticommutes with both factors in every displayed bilinear and consequently commutes with their product. Thusand $c_1,c_{2M}$ are Majorana zero modes localized exactly at the two end sites.

Let $U$ be the quasi-local unitary carrying the ground space of $H_0$ to that of a topologically equivalent $H$. DefineUnitary conjugation preserves self-adjointness and the Majorana algebra. Quasi-locality spreads each endpoint operator only into an exponentially decaying tail, and the two operators preserve the ground space because $c_1,c_{2M}$ preserve the ground space of $H_0$.

Diagonalize the quadratic Hamiltonian asThe hint gives a real expansion $\gamma ={\sum }_j{g}_j{\chi }_j$. Any coefficient along a pair with ${ϵ}_j>0$ would create or remove a positive-energy quasiparticle and send some ground state outside the two-lowest-state subspace. Ground-space preservation therefore forces ${\gamma }_1$ and ${\gamma }_{2M}$ to have support only in the zero-energy Majorana subspace. Hencein the ideal infinite-chain limit, with only exponentially small finite-size corrections. They are the exponentially localized endpoint Majorana zero modes throughout the topological phase.