B.5 Imaginary Time Operators
At finite temperature under thermodynamic equilibrium the state of a system is
described by the equilibrium density operator
. For a given
the ensemble average of any operator
can be calculated as
(see (3.11))
![\begin{displaymath}\begin{array}{ll}\displaystyle \langle \hat{O}\rangle \ &\dis...
...{K}} \hat{O}]} {\mathrm{Tr}[e^{-\beta\hat{K}}]} \ , \end{array}\end{displaymath}](img1118.png) |
(B.25) |
where
may be interpreted as a grand
canonical HAMILTONian. For any SCHRÖDINGER operator
, the
so called modified HEISENBERG and interaction pictures can be introduced as
 |
(B.26) |
where
includes only the non-interacting part of
. It
should be noticed that
is not
the adjoint of
as long as
is real. If
is interpreted as a complex variable, however, it may be analytically continued
to a pure imaginary value
. The resulting expression
then becomes the true adjoint of
and is formally identical with the original
HEISENBERG picture defined in (B.12), apart from
the substitution of
for
. For this reason
(B.26) are sometimes called imaginary-time operators.
The modified HEISENBERG and interaction pictures are related by (compare
(B.13) and
(B.14))
 |
(B.27) |
where the operator
is defined by (compare (B.17))
 |
(B.28) |
Note that
is not unitary, but it still
satisfies the group property
 |
(B.29) |
and the boundary condition
 |
(B.30) |
In addition, the equation of motion of
is calculated as
 |
(B.31) |
where
 |
(B.32) |
It follows that the operator
obeys essentially the same
differential equation as the unitary operator introduced in (B.15), and one
may immediately write down the solution (compare (B.24))
 |
(B.33) |
If
is set equal to
, (B.28) may be rewritten as
 |
(B.34) |
which relates the many particle density operator to the single-particle
density operator by means of an imaginary time-evolution operator.
M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors