A <Majorana zero mode> is a normalized self-adjoint fermion operator localized near a boundary and commuting with the Hamiltonian. With open boundaries,
$$
H_0=it\sum_{j=1}^{M-1}c_{2j}c_{2j+1}.
$$
Neither $c_1$ nor $c_{2M}$ appears. Each anticommutes with both factors in every displayed bilinear and consequently commutes with their product. Thus
$$
\boxed{[H_0,c_1]=[H_0,c_{2M}]=0},
$$
and $c_1,c_{2M}$ are Majorana zero modes localized exactly at the two end sites.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-342/3/e/solution