I am working on the first problem on self-consistent T-matrix approximation in Chapter 5 of Condensed Matter Field Theory by Altland and Simons. This is on page 234 of the textbook. I have some questions regarding part (a).

The Hamiltonian of the problem is $\hat H = \hat H_0 + \hat H_{imp}$, where

$\hat H_0 = \sum_{\mathbf{k}} \epsilon_{\mathbf{k}} c_{\mathbf{k}}^\dagger c_{\mathbf{k}}$,

$\hat H_{imp} = v_0 a^d \displaystyle\sum_{i=1}^{N_{imp}} c^{\dagger}(\mathbf{R_i})c(\mathbf{R_i})$, $N_{imp}$ is the number of impurities.

The goal is to compute the single-particle Green function $G_n = \langle \langle c^{\dagger}_n (\mathbf{r}) c_n(\mathbf{r'}) \rangle \rangle_{imp}$, where $n$ is the Matsubara frequency index and $\langle \cdots \rangle_{imp} \equiv \frac{1}{L^d} \int \prod_i d^d\mathbf{R_i}$ is the configurational average over all impurity coordinates.

Here is the problem statement of part (a)

Consider the scattering from a single impurity. By developing a perturbative expansion in the impurity potential, show that the Green function can be written as $\hat G_n=\hat G_{0,n} + \hat G_{0,n} \hat T_n \hat G_{0,n}$, where

$\hat T_n = \langle \hat H_{imp}+ \hat H_{imp} \hat G_{0,n} \hat H_{imp} + \hat H_{imp} \hat G_{0,n}\hat H_{imp} \hat G_{0,n} \hat H_{imp}+\cdots \rangle_{imp}$

denotes the T-matrix. Show that the T-matrix equation is solved by $T_n(\mathbf{r},\mathbf{r'})=\delta(\mathbf{r}-\mathbf{r'})L^{-d}((v_0 a^d)^{-1}-G_{0,n}(0))^{-1} $.

Here are my questions:

(1) The answer gives a hint about approaching the problem by representing the Green function $G_n = \langle \langle c^{\dagger}_n (\mathbf{r}) c_n(\mathbf{r'}) \rangle \rangle_{imp}$ in coherent state path integral, where $n$ is the Matsubara frequency index. The textbook states that the formal result after integrating over Grassmann field is $\hat G_{n} = (i \omega_n-\hat H_0-\hat H_{imp})^{-1}$. In my understanding, the coherent state path integral always yields a number. Is there a formal procedure for reverting the scalar back to operator? Or after obtaining the actual number, how should I proceed with the perturbation expansion?

(2) When evaluating the path integral, do I treat $c^{\dagger}_n (\mathbf{r}) c_n(\mathbf{r'})$ as Grassmann variable $ \bar\psi_n(\mathbf{r}) \psi_n(\mathbf{r'}) $? Should I fourier transform $\hat H_{imp}$ first?

(3) I don't see why the diagram in Fig. 5.11 is relevant for a single impurity.It looks more like the diagram for scattering off multiple impurities. Is this an error?

The diagram is shown in the image.

This post imported from StackExchange Physics at 2014-12-10 17:33 (UTC), posted by SE-user chicane