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

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)

Interaction representation change to interaction representation

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

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:

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

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

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 (-)

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

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

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

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

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

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

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

Wick’s theorem: order 1 Instantaneous potential Keldish space

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

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

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

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

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

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

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

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):

This makes possible to eliminate and work only with

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

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

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:

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

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:

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

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

In summary Using the relations between all these functions:

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

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

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:

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

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

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

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

The current through a metallic atomic contact d Tunnel regime Contact regime STM MCBJ

Simple model t Model Hamiltonian Left lead Right lead average current

Simplest model: wide-band approximation gR Assuming left-right symmetry: Keldish space Then, from Dyson equation

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

With the abbreviations

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

Problem: asymmetric contact perfect transmission for