1. L2:Non-equilibrium theory: Keldish formalism
Non equilibrium perturbation theory
Keldish space & Keldish propagators
Application to transport phenomena:
Simple model of a metallic atomic contact

2. Non equilibrium perturbation theory
Keldish formalism
Assume an electronic system described by:
explicit time dependence
non-interacting electrons in equilibrium
Keldish formalism: non-equilibrium theory with a structure formally
identical to the equilibrium case
The diagrammatic expansion is formally identical (with a twist)

3. Interaction representation
change to interaction representation

4. Adiabatic hypothesis (theorem)
We could still switch on an off the perturbation at
this is no longer true !
The temporal symmetry is broken

5. Therefore, the following will still be true (up to a phase)
But, we have however
The following expression then still holds true
But, due to the breaking of time symmetry:

6. Time symmetry (equilibrium)
And the rest of the theory breaks down
Thus, step 3 of the equilibrium theory is not valid
Time symmetry (equilibrium)
Keldish suggestion: although the times are no longer ordered in
along the time axis

7. they can be thought as being ordered along the time contour:
upper branch (+)
lower branch (-)
Keldish temporal contour
Defining the operator that order times along this contour
This implies that time gets a label

8. An expression similar to the equilibrium case is obtained:
Using the operator
An expression similar to the equilibrium case is obtained:
Where is the time evolution operator along the whole
Keldish contour:
upper branch (+)
lower branch (-)

9. All the remaining steps of the equilibrium theory are still valid:
Expansion of Sc
Substituting in <A> we have its perturbation expansion

10. Wick’s theorem Reminder
The averages appearing in the expansion are calculated in
ground state of H0 (non-interacting electrons in equilibrium)
Reminder
The Keldish formalism is a perturbation theory over a system of
non-interacting electrons in equilibrium

11. The unit in the Wick’s theorem factorization will be
Difference
The unit in the Wick’s theorem factorization will be
Instead of
Equilibrium case
This difference makes it necessary to modify the definition of propagator
Keldish

12. Consequence Difference
Expansion of the Green function: the Dyson equation
Consequence
The diagrammatic structure of the expansion of the Green function
Is analogous to the equilibrium case
Difference
Necessity of distinguishing between times in both branches
of the Keldish contour

13. Consequence in practice
additional index in G
Keldish 2x2 space
equilibrium
Keldish
There is an effective doubling of the Hilbert space size
All the diagrammatic expansion remains invariant
The diagrams are the same with the extra Keldish index

14. Therefore the propagators still verify the Dyson equation
in Keldish space
Real difficulty: explicit time dependence:
However in a stationary situation (stationary currents)

15. Example: Time-dependent one-electron potential as a perturbation
The diagrams are the same (with an extra index):
Self-energy?

16. Wick’s theorem: order 1
Instantaneous potential
Keldish space

17. Keldish Green functions (propagators)
The four Keldish Green functions:
upper branch (+)
lower branch (-)
it is the usual
causal function

18. upper branch (+)
lower branch (-)
anti-causal function
related to G++

19. Keldish Green function
upper branch (+)
lower branch (-)

20. It is related to the electron distribution function
The diagonal function at equal times:
For instance at equilibrium
Fermi-Dirac distribution

21. t e0 It is directly related to the electron current Example: 1D chain
The current between sites I and i+1 is related to

22. It is related to the distribution of empty states
upper branch (+)
lower branch (-)
It is related to the distribution of empty states
It is related to

23. Properties of the Keldish Green functions
Just by inspection on their definitions:
The 4 Keldish Green functions are not independent
and

24. Relation to the advanced and retarded functions
only 3 are necessary
Relation to the advanced and retarded functions
The functions G+,+ and G-,- are cumbersome objects in time space
The Keldish Green functions are closely related to the retarded (advanced):

25. This makes possible to eliminate
and work only with

26. Triangular representation
The most compact and elegant way of eliminating one of the 4 functions
Transformation (rotation) in Keldish space
Dyson equation in standard Keldish space
Dyson equation in triangular representation

27. For analyzing transport problems we will use
from
triangular representation
satisfy Dyson equations on their own
Equation for calculating
an useful result
Similar expression for

28. Applying this rule to the Dyson equation:
For a system of non-interacting electrons
This equation could be written also, using
This two equations can be combined:

29. Summary of important equations
Dyson equation in standard Keldish space
Dyson equation in triangular representation

30. Keldish Green functions in equilibrium
Keldish formalism is a perturbation theory over an equilibrium
situation
A previous knowledge of the equilibrium functions is therefore
needed
In equilibrium it is possible to work in frequency space:
Let us analyze the expression of the Keldish Green functions:

31. The Keldish formalism is in principle a T=0 theory
It is possible to relate these functions to the Fermi distribution
function by the following ansatz
Fermi function

32. From their definitions:
From these expressions :
spectral distribution of occupied states
spectral distribution of empty states

33. In summary
Using the relations between all these functions:

34. Example: Non-interacting electrons in equilibrium
site (tight-binding) basis
local density of states

35. Simple case: local Green functions in the “wide-band approx.”
Wide-band approximation:
Keldish space

36. Application to transport phenomena
The current operator
Let us analyze first a simple 1D problem: t e0
it is plausible that the current between sites I and i+1 have the
functional form:

37. t e0 A B from the continuity equation we can calculate the pre-factor:
where
and its derivative can be calculated from

38. A B The generalization to a general (tight-binding) system is
immediate: A B

39. The current in terms of Keldish Green functions
The average current will be then
which is simply given by

40. For a stationary current
Which can be calculated by purely algebraic means (Dyson Eq.)

41. The current through a metallic atomic contactd
Tunnel regime
Contact regime
STM
MCBJ

42. Simple model t Model Hamiltonian Left lead Right lead average current

43. Simplest model: wide-band approximationgR
Assuming left-right symmetry:
Keldish space
Then, from Dyson equation

44. Then, by means of simple matrix algebra
Keldish space
perturbation
one electron perturbation
Then, by means of simple matrix algebra

45. With the abbreviations

46. Identifying the transmission of the contact:
Landauer formula
Identifying the transmission of the contact:
0.0
0.5
1.0
1.5
2.0
0.2
0.4
0.6
0.8
Transmission ( a )
t / W
perfect transmission condition for
conductance

47. Problem: asymmetric contact
perfect transmission for