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WORLD SCIENTIFIC eBooks

Lectures on the Non-Equilibrium Theory of Condensed Matter

L. Bányai

Published Jun 8, 2020• 8 citations
Publisher

Abstract

Time-Reversal and Evolution Toward Equilibrium Schrodinger Equation Versus Master Equation Naive Quantum Kinetics The Boltzmann Equation The Nyquist Theorem Excitonic Absorption in a Semiconductor Charged Particles Interacting with the Quantized Electromagnetic Field Two-Level Atom Interacting with Light Real-Time Experiments: Ultra-Short-Time Spectroscopy Keldysh-Green Functions and Time-Reversal and other papers.

DOI
10.1142/11923

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Within the Landauer approach to conduction discussed in the previous chapter, electron interactions have been included only at the mean-field level. This is quite a strong approximation, especially in nanojunctions, where large current densities are common. I therefore want to go beyond this level of description. I can follow two different routes. (1) I can develop a functional theory of quantum correlations via time-dependent effective single-particle equations, as I will do in Chapter 7 where I will use dynamical density-functional theories within the micro-canonical approach to conduction. (2) I can try to solve the time-dependent Schrödinger equation 1.16 – or its mixed-state version, the Liouville-von Neumann equation 1.60 – directly. Written this way, this last proposition seems hopeless. In reality, we can employ a many-body technique, known as the non-equilibrium Green's function formalism (NEGF), also referred to as the Keldysh formalism (Keldysh, 1964; Kadanoff and Baym, 1962), which allows us, at least in principle, to do just that: solve the time-dependent Schrödinger equation for an interacting many-body system exactly, from which one can, in principle, calculate the time-dependent current. This is done by solving equations of motion for specific time-dependent single-particle Green's functions, from which the physical properties of interest, such as the charge and current densities, can be obtained. I have stressed the term “time-dependent” several times, because, as I will show in a moment, the NEGF is “exact” only when one solves the time-dependent Schrödinger equation for a closed quantum system, subject to deterministic perturbations: the system is closed but not necessarily isolated. These perturbations may drive the system far away from its initial state of thermodynamic equilibrium (Keldysh, 1964).

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Time-Reversal and Evolution Toward Equilibrium Schrodinger Equation Versus Master Equation Naive Quantum Kinetics The Boltzmann Equation The Nyquist Theorem Excitonic Absorption in a Semiconductor Charged Particles Interacting with the Quantized Electromagnetic Field Two-Level Atom Interacting with Light Real-Time Experiments: Ultra-Short-Time Spectroscopy Keldysh-Green Functions and Time-Reversal and other papers.
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Preface 1. Quantum fields 2. Operators on the multi-particle state space 3. Quantum dynamics and Green's functions 4. Non-equilibrium theory 5. Real-time formalism 6. Linear response theory 7. Quantum kinetic equations 8. Non-equilibrium superconductivity 9. Diagrammatics and generating functionals 10. Effective action 11. Disordered conductors 12. Classical statistical dynamics Appendices: A. Path integrals B. Retarded and advanced propagators C. Analytic properties of Green's functions Bibliography Index.
Nonequilibrium Green's Functions: History, General Problems, Plasmas: Real-Time Nonequilibrium Green's Functions (L V Keldysh) Theory of High-Tc Superconductivity in Layered Cuprates (A A Abrikosov) Correlated Quantum Plasmas in Strong Laser Fields (M Schlanges et al.) Quantum Transport in Semiconductors: Non-Equilibrium Green's Functions in Semiconductor Device Modeling (R Lake et al.) Nonequilibrium Green Function Modelling of Transport in Mesoscopic Systems (A-P Jauho) Bohm Trajectories in Quantum Transport (J R Barker) Optical and Electronic Properties of Semiconductors: Coulomb Correlations in the Quantum Kinetics of Electron Phonon Systems (W Schafer et al.) Excitonic Quantum Coherences and Correlations in Semiconductor Quantum Wells (R Binder et al.) Nuclear Matter and High Energy Physics: Renormalization of Self-Consistent F-Derivable Approximations (H van Hees & J Knoll) Crossover from Neutron-Proton Superfluidity to Bose-Einstein Condensation of Deuterons in Nuclear Matter (A A Isayev) Kinetics of Vacuum Pair Creation in Strong Electromagnetic Fields (A V Tarakanov et al.) Alternative Computational Methods: Time-Dependent Density Functional Theory from a Practitioners Perspective (K Andrae et al.) Combination of Quantum Kinetic Theory and First-Principle Simulations for Strongly Correlated Quantum Plasmas (M Bonitz et al.) Tunneling of Interacting Identical Particles (Yu E Lozovik et al.) and other papers.
1. First and second quantization 2. The electron gas 3. Phonons: coupling to electrons 4. Mean field theory 5. Time evolution pictures 6. Linear response theory 7. Transport in mesoscopic systems 8. Green's functions 9. Equation of motion theory 10. Transport in interacting mesoscopic systems 11. Imaginary time Green's functions 12. Feynman diagrams and external potentials 13. Feynman diagrams and pair interactions 14. The interacting electron gas 15. Fermi liquid theory 16. Impurity scattering and conductivity 17. Green's functions and phonons 18. Superconductivity 19. 1D electron gases and Luttinger liquids A. Fourier transformations B. Exercises C. Index
Within the Landauer approach to conduction discussed in the previous chapter, electron interactions have been included only at the mean-field level. This is quite a strong approximation, especially in nanojunctions, where large current densities are common. I therefore want to go beyond this level of description. I can follow two different routes. (1) I can develop a functional theory of quantum correlations via time-dependent effective single-particle equations, as I will do in Chapter 7 where I will use dynamical density-functional theories within the micro-canonical approach to conduction. (2) I can try to solve the time-dependent Schrödinger equation 1.16 – or its mixed-state version, the Liouville-von Neumann equation 1.60 – directly. Written this way, this last proposition seems hopeless. In reality, we can employ a many-body technique, known as the non-equilibrium Green's function formalism (NEGF), also referred to as the Keldysh formalism (Keldysh, 1964; Kadanoff and Baym, 1962), which allows us, at least in principle, to do just that: solve the time-dependent Schrödinger equation for an interacting many-body system exactly, from which one can, in principle, calculate the time-dependent current. This is done by solving equations of motion for specific time-dependent single-particle Green's functions, from which the physical properties of interest, such as the charge and current densities, can be obtained. I have stressed the term “time-dependent” several times, because, as I will show in a moment, the NEGF is “exact” only when one solves the time-dependent Schrödinger equation for a closed quantum system, subject to deterministic perturbations: the system is closed but not necessarily isolated. These perturbations may drive the system far away from its initial state of thermodynamic equilibrium (Keldysh, 1964).