How to apply Wick's theorem in Anderson model

Question

I'm trying to solve the non-interacting single impurity Anderson model where we consider free electrons in a conduction band: $$H_{cond} =\sum_k \varepsilon_k c_k^\dagger c_k$$ and an impurity with Hamiltonian $$H_{trap} =\varepsilon_c b^\dagger b. $$ Also, there is some interaction between the impurity and the conduction band described by the Hamiltonian $$H_{int} =\sum_k A_k \left( c_k^\dagger b +b^\dagger c_k \right),$$ where the second term describes an electron being removed from the conduction band and getting trapped by the impurity, while the first term is when the electron gets off the impurity and returns to the conduction band.

Now I'm interested in calculated the propagator for the trapped state. $$G_{trap} (t) = -\langle \Psi |T[b(t)b^\dagger(0)]|\Psi\rangle$$ However to calculate the second correction $$G^{(2)}_{trap} = -i\frac{1}{2!i^2}\int dt_1 \int dt_2 \langle \Psi_0 | T[b(t)H_{int}(t_1)H_{int}(t_2)b^\dagger (0) | \Psi_0 \rangle$$ I need to evaluate this time-ordered product using Wick's theorem. However, I'm struggling with applying Wick's theorem, e.g. can I separate the operators of the impurity and the electron or not, because they are both fermionic? If someone could give me some hints on how to apply Wick's theorem here. Thank you.

Answer 1

The electrons on the impurity and the conduction electrons are totally different, so they do not suffer from Pauli's principle and anti-commute. Then, you can write $H_{\text{int}} (t_n)$ in function of these operators and use Wick's theorem to split your expectation value into the usual some of products of expectation values of pairs of operators.

I guess your conduction electrons need to paired together, otherwise the expectation value vanishes, so that restricts a lot. It will give you a product of a Green function for the the conduction electrons, with two Green functions for the impurity.