introduction to the keldysh non-equilibrium green function technique reporter: chen jianxiong...
TRANSCRIPT
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Introduction to the Keldysh non-equilibrium
Green function technique
Reporter: Chen Jianxiong
2015/3/30
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Outline
• Background
• Review of equilibrium theory
• Introduction to non-equilibrium theory
• Discussions
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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.
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Background
Non-equilibrium Transport phenomena
Mesoscopic systems Quantum mechanics
Important quantities Green functions
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Review of equilibrium theory
Hamiltonian
Green function
S-matrix
Heisenberg picture
Interaction picture
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After some algebraic manipulations
Using a trick
Standard result
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Equilibrium & Non-equilibrium
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Non-equilibrium theory
Rewind back to avoid any reference to future state
Substituting it into
Then
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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.
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Contour S-matrix
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Contour-ordered Green’s function
Satisfying Dyson equation
Contour representation: Impractical in calculations !!!
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Six Green’s Functions
+∞−∞
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Time-ordered Green function
Antitime-ordered Green function
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The “greater” function
The “lesser” function
Relation
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Advanced and retarded functions
Advanced function
Retarded function
Relation
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Langreth Theorem
where
Matrix form
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Dyson equation
Keldysh formulation
Langreth Theorem
Infinite order iteration
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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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Thanks for your time!
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