3.3.2 Contour-Ordered GREEN's Function

To express the field operators in the interaction representation an operator $ \hat{S}$ is defined (see Appendix B.4) and applied for calculating the GREEN's functions as in Section 3.1.1. The time in (3.8) is taken over the interval $ (-\infty,\infty)$. The state at $ t \rightarrow -\infty$ is well defined as the ground-state of the non-interacting system $ \vert\phi_{0}\rangle$. The interactions are turned on slowly. At $ t=0$ the fully interacting ground state is $ \vert\Psi(0)\rangle =
\hat{S}(0,-\infty)\vert\phi_{0}\rangle$. The state at $ t\rightarrow\infty$ must be defined carefully. If the interactions remain on, then this state is not well described by the non-interacting ground state. Alternatively, one could require that the interactions are turned off at large times, which returns the system to the ground-state $ \vert\phi_{0}\rangle$.

SCHWINGER [92] suggested another method of handling the asymptotic limit $ t \rightarrow -\infty$. He proposed that the time integral in the $ \hat{S}$ operator has two parts; one goes from $ (-\infty,t)$ while the second goes from $ (t,-\infty)$. The integration path is a contour, which starts and ends at $ -\infty$. The advantage of this method is that one starts and ends the $ S$ operator expansion with a known state $ \vert\Psi(-\infty)\rangle=\vert\phi_{0}\rangle$. Instead of the time-ordering operator (B.21), a contour-ordering operator can be employed. The contour-ordering operator $ T_\mathrm{C}$ orders the time labels according to their order on the contour $ {C}$. Under equilibrium condition the contour-ordered method gives results that are identical to the time-ordered method described in Section 3.1.1. The main advantage of the contour-ordered method is in describing non-equilibrium phenomena using GREEN's functions. Non-equilibrium theory is entirely based upon this formalism, or equivalent methods.

Any operator $ \hat{O}_\mathcal{H}$ in the HEISENBERG picture can be transformed into the interaction picture (see (B.13))

 \hat{O}_\mathcal{H} \ = \ \hat{S}(t_0,t)\ \hat{O}_\mathrm{I}\
 \hat{S}(t,t_0)\ .
 \end{array}\end{displaymath} (3.19)

Analogous to the derivation of (B.24), it can be shown that the $ \hat{S}$ operator is given by

\begin{displaymath}\begin{array}{l} \displaystyle \hat{S}(t,t_{0}) \ = \ T_\math...
 \} \ ,
 \end{array}\end{displaymath} (3.20)

where the operators are in the interaction representation. The ordinary time-ordering can also be written as ordering along contour branches $ {C}_1$ and $ {C}_2$ as depicted in Fig. 3.1

 \hat{S}(t,t_0) \ \displaystyle...
 \hat{H}^\mathrm{int}_\mathrm{I}(t)}\right)\} \ .
 \end{array}\end{displaymath} (3.21)

By combining two contour branches, $ {C}={C}_1\cup {C}_2$, (3.19) can be rewritten as

 \hat{O}_\mathcal{H}(t) \ &\di...
...{C}\{\hat{S}_\mathrm{C}\ \hat{O}_\mathrm{I}\} \
 \end{array}\end{displaymath} (3.22)


 \hat{S}_\mathrm{C}\ &\displaystyle = \ \ex...
...xt}_\mathrm{C}\ \hat{S}^\mathrm{int}_\mathrm{C}\ .
 \end{array}\end{displaymath} (3.23)

Figure 3.1: The contour $ {C}={C_1}\cup {C_2}$ runs on the real axis, but for clarity its two branches $ C_1$ and $ C_2$ are shown slightly away from the real axis. The contour $ {C_i}$ runs from $ t_0$ to $ t_0-i\beta $.

In equation (3.17) $ \hat{\rho}$ describes the equilibrium state of the system before the external perturbation $ \hat{H}^\mathrm{ext}$ is turned on. Interactions $ \hat{H}^\mathrm{int}$, which are switched on adiabatically at $ -\infty$, are present in $ \hat{\rho}$. However, to apply WICK's theorem (Section 3.4.1), one has to work with non-interacting operators. A methodology similar to the MATSUBARA theory can be applied to express the many-particle density operator $ \hat{\rho}$ in terms of the single-particle density operator $ \hat{\rho}_0$, see Appendix B.5. If the contour $ {C_i}=[t_0,t_0-i\beta]$ is chosen (Fig. 3.1), then (B.34) takes the form

 e^{-\beta \hat{K}} \displaystyle = \ e^{-\beta \hat{K}_0} \ {\hat{S}}_{C_i}
 \ .
 \end{array}\end{displaymath} (3.24)

Therefore, (3.17) can be rewritten as

$\displaystyle \langle \hat{O}_\mathscr{H}(t) \rangle \ = \frac{\mathrm{Tr}[e^{-...
...}_\mathscr{H}(t) ]}
 {\mathrm{Tr}[e^{-\beta \hat{K}_0}T_{C_i}\hat{S}_{C_i}]}\ ,$ (3.25)

Using the relations (3.22) and (3.25), the GREEN's function in (3.18) becomes [196]

 G({\bf {r}},t,{\bf {r'}},t') \...
...}_0}T_{C_i} \hat{S}_{C_i}\ T_{C} \hat{S}_{C}]} \ .
 \end{array}\end{displaymath} (3.26)

The twofold expansion of the density operator and the field operators may conveniently be combined to a single expansion. The two contours $ {C_i}$ and $ {C}$ can be combined together, $ {C^*}={C}\cup {C_i}$ (Fig. 3.2), and a contour-ordering operator $ T_{C^*}=T_{C_i}T_{C}$, which orders along $ {C^*}$, can be introduced. Hence, a point on $ {C}$ is always earlier than a point on $ {C_i}$. Furthermore, we define an interaction representation with respect to $ \hat{H}_0$ on $ {C}$ and with respect to $ \hat{K}_0$ on $ {C_i}$. Therefore, the GREEN's function in (3.18) is given by

 G({\bf {r}},t,{\bf {r'}},t') ...
...dagger_{\mathrm{I}}({\bf {r}'},t')\} \rangle_0 \ ,
 \end{array}\end{displaymath} (3.27)

where $ \langle\ldots\rangle_0$ represents the statistical average with respect to $ \hat{\rho}_0$. From here we assume that all statistical averages are with respect to $ \hat{\rho}_0$ and drop the 0 from the brackets $ \langle\ldots\rangle_0$.

Figure 3.2: The contour $ {C^*}={C_i}\cup {C}$, runs from $ t_0$ to $ t_0$ and from $ t_0$ to $ t_0-i\beta $.
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors