Introduction to the Keldysh non-equilibrium

Green function technique

Reporter: Chen Jianxiong

2015/3/30

Outline

• Background

• Review of equilibrium theory

• Introduction to non-equilibrium theory

• Discussions

References

• A. P. Jauho , "Introduction to the Keldysh Nonequilibrium Green Function Technique," https://nanohub.org/resources/1877.

• Joseph Maciejko , “An Introduction to Nonequilibrium Many-Body Theory,” http://www.physics.arizona.edu/~stafford/Courses/560A/nonequilibrium.pdf

• G. D. Mahan , “Many-Particle Physics”, second edition.

Background

Non-equilibrium Transport phenomena

Mesoscopic systems Quantum mechanics

Important quantities Green functions

Review of equilibrium theory

Hamiltonian

Green function

S-matrix

Heisenberg picture

Interaction picture

After some algebraic manipulations

Using a trick

Standard result

Equilibrium & Non-equilibrium

Non-equilibrium theory

Rewind back to avoid any reference to future state

Substituting it into

Then

Keldysh contour

+∞−∞

Contour variables τ(t,C)

Contour-ordering operator

Any time residing on the first part is early in the contour sense to any time residing on the latter part.

Contour S-matrix

Contour-ordered Green’s function

Satisfying Dyson equation

Contour representation: Impractical in calculations !!!

Six Green’s Functions

+∞−∞

Time-ordered Green function

Antitime-ordered Green function

The “greater” function

The “lesser” function

Relation

Advanced and retarded functions

Advanced function

Retarded function

Relation

Langreth Theorem

where

Matrix form

Dyson equation

Keldysh formulation

Langreth Theorem

Infinite order iteration

Discussion

• Non-equilibrium formulism can be applied to handle equilibrium problem;

• Generalization to finite temperature case

h is the time-independent part of the total Hamiltonian.

Thanks for your time!

Comments & Questions?