Evaluation of Green function for two site system?

Question

Let's consider I have two site system whose hamiltonian has $2\times2$ matrix form. In general we can write the Green function for above Hamiltonian as $G^{-1}=i \omega-H $ or $G=[i\omega-H]^{-1}$ and Green function will have form

$$G=\begin{pmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{pmatrix} \, .$$

Does $g_{11}$ correspond to the Fourier transform in time domain of

$$g_{11}=G^{r}=-i\theta(t)\langle\{c_{1},c^{\dagger}_{1}\}\rangle$$

and similarly

$$g_{12}=G^{r}=-i\theta(t)\langle\{c_{1},c^{\dagger}_{2}\}\rangle$$

where $G^{r} $ corresponds to retarded green function?

@L.Su let consider simple hamiltonian $H=\epsilon_0a_1 ^{\dagger}a_1+\epsilon_0a_2 ^{\dagger}a_2+ta_1 ^{\dagger}a_2+\epsilon_0a_2 ^{\dagger}a_1$

so,$H=\begin{pmatrix}\epsilon_0&t\\t&\epsilon_0\end{pmatrix}$ than using relation $G=[iw-H]^{-1}$

Greens function will have form $G=\begin{pmatrix}\frac{1/2}{(iw-\epsilon_0)-t}+\frac{1/2}{(iw-\epsilon_0)+t}&\frac{1/2}{(iw-\epsilon_0)-t}-\frac{1/2}{(iw-\epsilon_0)+t}\\\frac{1/2}{(iw-\epsilon_0)-t}-\frac{1/2}{(iw-\epsilon_0)+t}&\frac{1/2}{(iw-\epsilon_0)-t}+\frac{1/2}{(iw-\epsilon_0)+t}\end{pmatrix}$

so $G_{11}=\frac{1/2}{(iw-\epsilon_0)-t}+\frac{1/2}{(iw-\epsilon_0)+t}$

My question is can I calculate $G_{11} ,G_{12}$ components of above matrix using relation $G_{11}=-i\theta(t)\langle\{c_{1},c^{\dagger}_{1}\}\rangle$ and $G_{12}=-i\theta(t)\langle\{c_{1},c^{\dagger}_{2}\}\rangle$ respectively?

Answer 1

The answer is yes. Your Hamiltonian is $$H = t_{11}a^{\dagger}_1a_1 + t_{12}a^{\dagger}_1a_2 + t_{21}a^{\dagger}_2a_1 + t_{22}a^{\dagger}_2a_2,$$ and the retarded Green's function is defined as $$G^{R}(\nu t,\nu't')= -i \theta(t-t')\langle {a_{\nu}(t),a^{\dagger}_{\nu'}(t')}\rangle ,$$ for each $\nu,\nu' =1,2$. Notice that, for emphasis, I will use $G^{R}$ to denote a matrix indexed by $\nu$ and $\nu'$. Explicitly, $$G^{R} = \begin{pmatrix} G^{R}(1 t,1 t') & G^{R}(1 t,2 t') \\ G^{R}(2 t,1 t') & G^{R}(2 t,1 t') \\ \end{pmatrix}.$$ Using equation of motion, you can obtain $$\sum_{\nu''}[\delta_{\nu\nu''}(\omega+i\eta) -t_{\nu\nu''}]G^{R}(\nu''\nu,\omega)= \delta_{\nu\nu''}.$$ $G^{R}(\nu''\nu,\omega)$ is an element of the matrix $G^{R}$.

Written in terms of matrices, we have $$[(\omega+i\eta)I - \tilde{H} ]G^{R} = I,$$ and $$G^{R} = [(\omega+i\eta)I - \tilde{H} ]^{-1}$$ Here $$\tilde{H} = \begin{pmatrix} t_{11}& t_{12} \\ t_{21} &t_{22} \\ \end{pmatrix}.$$