(9755) Bogolyubov transformation for fermions - Kitaev chain

Consider the the Hamiltonian
\( H_G = H_{\text{hopping}} + H_{\text{pairing}} -\mu N \)
where \( \mathcal{H}_{\text{hopping}} = -K ( a_{j{+}1}^{\dagger} a_j + a_j^{\dagger} a_{j{+}1} ) \)
\( \mathcal{H}_{\text{pairing}} = \Delta a_j a_{j{+}1} + \Delta^* a_{j{+}1}^{\dagger} a_j^{\dagger} ) \)
(1) Write the Hamiltonian using Majorana operators.
(2) TBD.
paper