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Published byRobyn Pearson Modified over 8 years ago
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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," Joseph Maciejko , “An Introduction to Nonequilibrium Many-Body Theory,” ium.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 Heisenberg picture Interaction picture S-matrix
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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 −∞ +∞ 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.
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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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Keldysh formulation Dyson equation 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! Comments & Questions?
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