Green function for single impurity

Question

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

Comments

Since not everyone has that book, it might be more useful to include more information (e.g., copy question verbatim, include Fig. 5.11) than just limiting the potential answerers to those who have a copy of the book.

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

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I can recommend looking at the book "Green's functions in quantum physics" by Economou, I think he discusses this problem and the underlying theory in terms of resolvent operators in quite some detail. What you have written above is called the resolvent operator. The Green's functions are simply the expectation values of this in a suitable single-particle (or two-particle etc.) basis.

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

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@KyleKanos Thanks for the advice. I have included the problem statement and the diagram.

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

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@dimension10, the picture is not properly sized.

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@JiaYiyang Thanks, fixed.

Answer 1

You may notice that they are re-deriving the Dyson equation (or whatever it is called):

$$\begin{equation}G(b,a) = G_0(b,a) + \int dc G(b,c)V(c)G_0(c,a),\end{equation}$$ where $a,b,c$ denote complete sets of indices, say in this case $(\mathbf{r}, \tau)$, $G(b,a) =- \langle T \psi(b)\psi^{\dagger}(a)\rangle$, and $V$ is an external potential. This equation is one that can be used to derive Feynman's rules and it can be easily derived just by equation of motion of the Hamiltonian operator. (If you are interested, check Chapter 11, Many-body quantum theory in condensed matter physics. ) Anyway, let me try to answer your questions:

(1) Yes, $G_n(\mathbf{r},\mathbf{r'}) = \langle \langle c_n^{\dagger}(\mathbf{r})c_n(\mathbf{r'})\rangle \rangle_{imp}$ is a scalar for fixed $\mathbf{r}$ and $\mathbf{r'}$, but if we treat $\mathbf{r}$ and $\mathbf{r'}$ as indices, $G_n(\mathbf{r},\mathbf{r'})$ can also be regarded an infinite-dimensional matrix (an operator if you prefer, but in a different sense, depending on which space it acts.) For clarification, let me show all indices explicitly in the relation:

$$\begin{align}\hat{G}(\mathbf{r},\mathbf{r'} )&= \frac{1}{i\omega_n \delta(\mathbf{r}-\mathbf{r'})-\hat{H}_0(\mathbf{r},\mathbf{r'}) - \hat{H}_{imp}( \mathbf{r},\mathbf{r'})}\\ &=\frac{1}{\hat{G}^{-1}_0(\mathbf{r},\mathbf{r''} )[1- \hat{G}_0(\mathbf{r},\mathbf{r''} )\hat{H}_{imp}( \mathbf{r''},\mathbf{r'})]}. \end{align}$$ Summation over repeated indices is assumed. The expansion should also carry these indices with caution. After that, you can get the expression for $\hat{G}(\mathbf{r},\mathbf{r'} )$.

(2) Notice that there are always two common ways to evaluate Green's functions (correlation functions, propagators...), namely, using operators and using functional integration (path integral). If you do not know the difference, compare Eq(4.31) with Eq(9.18) in Peskin and Schroeder. In path integral, these fields are scalars.

(3) Yes. You are right. It seems to me that the ''scattering lines'' should connect to a single point. Hope this can help :~)

Comments

Minorly edited:

1.Enclosed punctuations in begin{equation} end{equation}

2.Embedded links into texts.(To do this you can first highlight the targeted texts and then click on the "link" icon on top of the text editor box, the icon looks a bit like $\infty$.)

3. Splitted points (1)(2)(3) to different paragraphs.

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Thank you. Technique learned.