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Introduction to the Keldysh non-equilibrium Green function technique

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Presentation on theme: "Introduction to the Keldysh non-equilibrium Green function technique"— Presentation transcript:

1 Introduction to the Keldysh non-equilibrium Green function technique
Reporter: Chen Jianxiong 2015/3/30

2 Outline Background Review of equilibrium theory
Introduction to non-equilibrium theory Discussions

3 References A. P. Jauho ,  "Introduction to the Keldysh Nonequilibrium Green Function Technique," Joseph Maciejko , “An Introduction to Nonequilibrium Many-Body Theory,” ium.pdf G. D. Mahan , “Many-Particle Physics”, second edition.

4 Background Non-equilibrium Transport phenomena Mesoscopic systems
Quantum mechanics Important quantities Green functions

5 Review of equilibrium theory
Hamiltonian Green function Heisenberg picture Interaction picture S-matrix

6

7 After some algebraic manipulations
Using a trick Standard result

8 Equilibrium & Non-equilibrium

9 Non-equilibrium theory
Rewind back to avoid any reference to future state Substituting it into Then

10 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.

11 Contour S-matrix

12 Contour-ordered Green’s function
Satisfying Dyson equation Contour representation: Impractical in calculations !!!

13 Six Green’s Functions +∞ −∞

14 Time-ordered Green function
Antitime-ordered Green function

15 The “greater” function
The “lesser” function Relation

16 Advanced and retarded functions
Advanced function Retarded function Relation

17 Langreth Theorem where Matrix form

18 Keldysh formulation Dyson equation Langreth Theorem
Infinite order iteration

19 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.

20 Thanks for your time! Comments & Questions?


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