Consider a particle in a 2D periodic square lattice. The lattice constant is \( a \) and consists of two sub-lattices (A,B). Accordingly the unit-cell is composed of two sites whose binding energy difference is \( \epsilon \) . All the hoping amplitudes are \( c \) . Find the energy dispersion \( E_{\pm}(\varphi_1,\varphi_2) \) , where the \( \varphi \) s are the component of the Bloch quasi-momentum.

(1) Define translation operators \( (D_1, D_2) \) that generate the symmetry group of the lattice. This requires to identify principal axes.

(2) Write the wavefunction in terms of two amplitudes \( (\psi_A,\psi_B) \) given that it belongs to the \( (\varphi_1, \varphi_2) \) subspace of the symmetry group.

(3) Write the reduced 2x2 Hamiltonian within the \( (\varphi_1, \varphi_2) \) symmetry subspace.

(4) Find the eigen-energies \( E_{\pm}(\varphi_1,\varphi_2) \)

(5) Explain how your answer reduces to the usual expression for square lattice if \( \epsilon=0 \) .