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

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- Solution e

Solution

A Majorana zero mode is a normalized self-adjoint fermion operator localized near a boundary and commuting with the Hamiltonian. With open boundaries,

H_{0} = i t \sum_{j = 1}^{M - 1} c_{2 j} c_{2 j + 1} .

Neither

c_{1}

nor

c_{2 M}

appears. Each anticommutes with both factors in every displayed bilinear and consequently commutes with their product. Thus

[H_{0}, c_{1}] = [H_{0}, c_{2 M}] = 0,

and

c_{1}, c_{2 M}

are Majorana zero modes localized exactly at the two end sites.

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