(9710) Two particles in two sites with interaction

Consider two identical but distinguished particles in an \( N \) site system. There is a contact interaction \( u \) if the two particles occupy the same site. The Hamiltonian commutes with the transposition operator, and therefore the eigenstates have definite symmetry. Explain why an anti-symmetric state is not affected by the interaction. In the following assume a two site system \( N=2 \). This is formally the same problem as two interacting qubits.

(1) Write the anti-symmetric state \( |A\rangle \) in the standard representation and specify its energy \( E_A \). Write the remaining \( 3 \times 3 \) Hamiltonian \( H^{[S]} \) in the \( \{11, 22, 12_S \} \) basis. Assume that the on-site energies are \(\varepsilon_1=0 \) and \(\varepsilon_2=\varepsilon \), and that the hopping frequency between the sites is \( c \).

(2) Regard \( c \) as a perturbation. Plot level diagram for \( u \ll \varepsilon \) and for \( u \gg \varepsilon \). Explain why the \( u \sim \varepsilon \) regime is problematic for treatment by perturbation theory.

(3) Use second-order perturbation theory in \( c \) in order to find the perturbed energies \( E_{11}, E_{22}, E_S \). In particular conclude what is the splitting \( \Delta = E_S-E_A \).

(4) The two-site system has two orbitals \( |a\rangle \) and \( |b\rangle \). Write the representation of the the two orbitals in the standard basis. Use the notation \( \tan(\theta) = 2c/\varepsilon \), and write the explicit expressions for \( \varepsilon_a \) and \( \varepsilon_b \).

(5) Write the \( 4 \times 4 \) Hamiltonian in the orbital-basis \( \{ aa, bb, ab, ba \} \). Find the exact eigenstates and spectrum for \( \varepsilon = 0 \).

(6) For \( \varepsilon \neq 0 \) use perturbation theory in \( u \) in order to calculate the splitting \( \Delta = E_S-E_A \). For this purpose define the direct and exchange matrix elements \( (K, J) \), and write \( 2 \times 2 \) matrix for the \( \{ ab, ba \} \) subspace.

(7) Verify consistency of the results of items 3 and 6 for the splitting \( \Delta \).