(9752) Bogolyubov transformation for fermions - two sites

Consider the two site Hamiltonian with pairing interaction:

\( \mathcal{H} = (\varepsilon_1 a_1^{\dagger} a_1 + \varepsilon_2 a_2^{\dagger} a_2) + \Delta (a_2 a_1 + a_1^{\dagger} a_2^{\dagger}) \)

(1) Write the antisymmetric matrix \( H \) of the quadratic form.

(2) Write the matrix \( JH \) - does it come symmetric or antisymmetric? Is it expected?

(3) Perform symplectic diagonalization. Find expressions for the Bogolyubov frequencies.

(4) Write expressions for \( |u_j|^2 \) and \( |v_j|^2 \) of the transformation.

Guidance:
See Lectures notes [51.6] and [37.2].
Use the convention \( c_{j}^{\dagger} = u_j a_{1}^{\dagger} + v_j a_{2} \).
Express the results of the diagonalization in terms of \( \theta \) defined via \( \tan(\theta)=\Delta/\varepsilon \).
Using trigonometric identities derive explicit expression for \(|u_j|^2\) and \(|v_j|^2\).