At $t=\Delta$ and $\mu=0$, direct substitution of
$$
a_j=\frac12(c_{2j-1}+ic_{2j})
$$
into the bond Hamiltonian makes the hopping and pairing terms cancel except for
$$
\boxed{h_j=it\,c_{2j}c_{2j+1}}.
$$
Thus
$$
\boxed{H_0=it\sum_jc_{2j}c_{2j+1}}.
$$
The terms pair disjoint Majoranas, so this is already Majorana diagonal, with nonzero single-particle values $\epsilon_j=2t$. For periodic boundaries the final pair closes around the chain; for open boundaries $c_1$ and $c_{2M}$ remain unpaired.


Solution

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

 Solution

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