3.1.1 Definition of the GREEN's Function

The time-ordered single-particle GREEN's function at zero temperature is defined as [189]

$$G({\bf {r}},t;{\bf {r'}},t')\ =\ -\frac{i}{\hbar} \frac{\langle\... ...\bf {r'}},t')\}\vert\Psi_{0}\rangle} {\langle\Psi_{0}\vert\Psi_{0}\rangle} \ ,$$ (3.2)

where $\vert\Psi_{0}\rangle$ is the ground-state of the interacting system in the HEISENBERG picture (Appendix B) and $T_\mathrm{t}$ is the time-ordering operator defined in (B.21). The field operator $\hat{\psi}_\mathrm{H}({\bf {r}},t)$ in the HEISENBERG picture is given by

$$\hat{\psi}_\mathrm{H}({\bf {r}},t)\ = \ e^{i\hat{H}t/\hbar} \hat{\psi}({\bf {r}}) e^{-i\hat{H}t/\hbar} \ .$$ (3.3)

Inserting (3.3) into (3.2), the physical interpretation of the GREEN's function becomes obvious

$$G({\bf {r}},t;{\bf {r'}},t')\ =\ -\frac{i}{\hbar} \frac{\langle\... ...t{H}t'/\hbar}\}\vert\Psi_{0}\rangle} {\langle\Psi_{0}\vert\Psi_{0}\rangle} \ ,$$ (3.4)

If $t>t'$, the GREEN's function $G({\bf {r}},t;{\bf {r'}},t')$ is the probability amplitude that a particle created at time $t'$ at place ${\bf {r'}}$ moves to time $t$ and place ${\bf {r}}$. This follows from the definition of $G({\bf {r}},t;{\bf {r'}},t')$. At zero time the system is at the ground-state $\Psi_{0}$. The system then evolves to time $t'$ with the operator $e^{-i\hat{H}t'/\hbar}$. At this time $\hat{\psi}^{\dagger}({\bf {r'}},t')$ creates a particle at place ${\bf {r'}}$. Then, the system continues its evolution from $t'$ to $t$ with the operator $e^{-i\hat{H}(t-t')/\hbar}$, after which $\hat{\psi}({\bf {r}},t)$ annihilates the particle at place ${\bf {r}}$. The system returns to the initial ground-state with the operator $e^{i\hat{H}t/\hbar}$. In a similar way, if $t'>t$, the field operator creates a hole at time $t$, and the system then propagates according to the HAMILTONian $\hat{H}$. These holes can be interpreted as particles traveling backward in time [191]. The probability amplitude that a hole created at time $t$ at place ${\bf {r}}$ moves to time $t'$ and place ${\bf {r'}}$ is again just the GREEN's function for $t<t'$.

To calculate $G({\bf {r}},t;{\bf {r'}},t')$, a perturbation expansion is very useful. However, the definition of the GREEN's function in (3.2) does not allow a direct solution, since it involves the exact ground-states of the interacting HAMILTONian $\hat{H}$, which is one of the things to be calculated. In the interaction representation the HAMILTON ian is expressed in terms of the non-interacting and interacting parts, see the equation (3.1). The ground state of the non-interacting part, $\hat{H}_{0}$, can be calculated easily. Therefore, one tries to express the ground state of the interacting system $\vert\Psi_{0}\rangle$ in terms of the ground state of the non-interacting one $\vert\phi_{0}\rangle$. For that purpose, in equation (B.18) one adds to the operator $\hat{H}^\mathrm{int}_\mathrm{I}(t)$ a factor $e^{-\vert\epsilon\vert t}$, which switches the interaction off at $t\rightarrow\pm\infty$ [189]. The non-interacting ground state $\vert\phi_{0}\rangle$ is assigned to the system at $t \rightarrow -\infty$ and the connection to $\vert\Psi_{0}\rangle$ is formed by the GELL-MANN and Low theorem [192]

$$\begin{array}{ll} \displaystyle \vert\Psi(0)\rangle\ &\displ... ...le = \ \hat{S}(0,-\infty)\vert\phi_{0}\rangle \ , \end{array}$$ (3.5)

where the $\hat{S}$ operator in defined in Appendix B.4. The traditional argument is that one starts from $t \rightarrow -\infty$ with a wave function $\phi_{0}$ which does not contain the effects of the interaction $\hat{H}^\mathrm{int}$. The operator $\hat{S}(0,-\infty)$ brings this wave function up to the present, $t=0$ [189]. Thus one has the wave function which contains the effects of the interaction $\hat{H}^\mathrm{int}$, so that it is an eigenstate of $\hat{H}$. As $t\rightarrow\infty$, one gets

$$\displaystyle \vert\Psi(\infty)\rangle\ = \ \hat{S}(\infty,-\infty)\vert\phi_{0}\rangle \ .$$ (3.6)

One possible assumption is that $\vert\Psi(\infty)\rangle$ must be related to $\phi_{0}$. The system returns to its ground state for $t\rightarrow\infty$ except for a phase factor [190] $\vert\Psi(\infty)\rangle= e^{iL}\vert\phi_{0}\rangle$ which implies that,

$$\displaystyle \langle \phi_{0}\vert\hat{S}(\infty,-\infty)\vert\phi_{0}\rangle \ = \ e^{iL}.$$ (3.7)

An alternative to this assumption is discussed in Section 3.3.2.

Using the relation (3.5) for the ground-state, equation (3.2) becomes

$$\begin{array}{ll} \displaystyle G({\bf {r}},t;{\bf {r'}},t')... ... {\langle \hat{S}(\infty,-\infty)\rangle_{0}} \ , \end{array}$$ (3.8)

where the short-hand notation $\langle\ldots\rangle_{0}=\langle\phi_{0}\vert\ldots\vert\phi_{0}\rangle$ is introduced to represent the expectation value over the ground-state of the non-interacting system at zero temperature. The transition from the first to the second line is achieved by using (B.13) for converting the HEISENBERG representation of operators into the interaction representation. The second step is obtained by taking account of the properties of the $\hat{S}$ operators described in Appendix B.4 and the return of the system to its ground-state as $t\rightarrow\infty$. In the forth line the operator $\hat{S}(\infty,-\infty)$ contains several time intervals $(\infty,t)$, $(t,t')$, and $(t',-\infty)$. The $T_\mathrm{t}$ operator automatically sorts these intervals so that they act in their proper sequences. Replacing operator $\hat{S}$ with its formal definition (see (B.24)) one gets

$$\begin{array}{l} \displaystyle G({\bf {r}},t;{\bf {r'}},t') ... ...m{int}_\mathrm{I}(t_1)\right) } \} \rangle_0} \ . \end{array}$$ (3.9)

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors