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/nonequilibr ium.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 Heisenberg picture Interaction picture S-matrix

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 −∞ +∞ τ(t,C) Contour variables 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

Keldysh formulation Dyson equation 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.

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