Homework 2.

Consider the atomic limit of the Hubbard Model

H = −μ ∑_{σ} f_{σ}^{†}f_{σ} + Uf_{↑}^{†}f_{↑}f_{↓}^{†}f_{↓}

(1) Compute the retarded Green’s function and the spectral function.

(2) Consider the Hubbard operators

X_{↑0} = f_{↑}^{†}(1 − n_{↓})
X_{d↓} = f_{↑}^{†}n_{↓}

referring to atomic states |0⟩ , |↑⟩ , |↓⟩ and |d⟩ = |↑↓⟩ ≡ f_{↑}^{†}f_{↓}^{†} |0⟩. Express the operator f_{↑}^{†} in terms of Hubbard operators.

(3) Repeat (2) for f_{↓}^{†}.

(4) Evaluate ⟨{X_{↑0}X_{0↑}}⟩ and ⟨{X_{d↓}X_{↓d}}⟩. What is the physical meaning of these quantities ?

(5) Evaluate ⟨{X_{↑0} [HX_{0↑}]}⟩ and ⟨{X_{d↓} [HX_{↓d}]}⟩. What is the physical meaning of these averages ?