The inverse definitions are
$$
a_j=\frac12(c_{2j-1}+ic_{2j}),
\qquad
a_j^\dagger=\frac12(c_{2j-1}-ic_{2j}).
$$
Substitution 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, while
$$
c_jc_k=-c_kc_j\qquad(j\ne k)
$$
removes the symmetric part; diagonal terms are constants because $c_j^2=1$. Therefore, up to that additive constant,
$$
\boxed{
H=\frac i4\sum_{j,k=1}^{2M}A_{jk}c_jc_k},
$$
where
$$
\boxed{A_{jk}\in\mathbb R,\qquad A^T=-A}.
$$
Conversely every real antisymmetric $A$ makes this expression Hermitian, so this is the general Majorana form of a <quadratic fermion Hamiltonian>.

Solution

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

 Solution

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