This chapter introduces the methods of quantum field theory as it appies to many-particle systems, particularly to superconductors. It focuses the ideas underlying the Green's function method, that is useful in the study of superconductivity. In dealing with an electron-phonon system it is reasonably realistic to approximate the effects of the phonons by an effective interaction between electrons that is of short range in space and retarded in time. It is especially appropriate to study the Green's function method in connection with the theory of superconductivity, because there are two reasons for this. The superconductor is the first system in solid-state physics in which the interaction between electrons qualitatively changes the spectrum of excitations. The other reason is the interaction that causes superconductivity is not describable by an ordinary potential: An electronic charge fluctuation with a frequency just below a characteristic frequency of the ionic lattice excites in the lattice a resonant sympathetic vibration that overcompensates for the electronic charge.
This chapter introduces the methods of quantum field theory as it appies to many-particle systems, particularly to superconductors. It focuses the ideas underlying the Green's function method, that is useful in the study of superconductivity. In dealing with an electron-phonon system it is reasonably realistic to approximate the effects of the phonons by an effective interaction between electrons that is of short range in space and retarded in time. It is especially appropriate to study the Green's function method in connection with the theory of superconductivity, because there are two reasons for this. The superconductor is the first system in solid-state physics in which the interaction between electrons qualitatively changes the spectrum of excitations. The other reason is the interaction that causes superconductivity is not describable by an ordinary potential: An electronic charge fluctuation with a frequency just below a characteristic frequency of the ionic lattice excites in the lattice a resonant sympathetic vibration that overcompensates for the electronic charge.
This chapter presents field-theoretic methods and discusses three well-known "pictures" or representations used in discussing quantum mechanical problems. It explores how to build up the true one-electron and one-phonon Green's functions as a perturbation series involving the corresponding functions for the non-interacting system. In elementary discussions of quantum mechanics one usually works in the "Schrodinger picture" in which the dynamical variables are taken to be time independent so that the wave function contains the time dependence of the problem. The one-electron Green's function gives information about the spin and charge densities and the momentum distribution of the electrons as well as information about the excitation spectrum of the system. When the system consists only of fermions interacting via an instantaneous two-body potential, the one-particle Green's function also suffices to determine the ground-state energy of the system. The electrical conductivity, magnetic susceptibility, and many other non-equilibrium properties can be obtained from the two-particle Green's function.
This chapter reviews the calculation of a number of system properties within the pairing approximation. Since the BCS theory was originally proposed, attempts to justify the pairing correlations basic to the theory have proceeded along two lines. The first approach has been to apply the BCS theory to a wide variety of phenomena in superconductors and check the theoretical predictions of the pairing approximation against experiment. The second approach has been to treat by various approximate methods the residual interactions neglected within the pairing scheme, hoping to show that these residual interactions introduce no major change in the predicted properties of the system. The interaction Hamiltonian, being a one-body operator, can at most change the state of one electron. While the majority of the electron–phonon interaction has been accounted for in forming the wave functions for the normal and superconducting states, there remains the part corresponding to resonant phonon absorption and emission processes.
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
The interaction of electrons with quantized phonons and photons underlies the ultrafast dynamics of systems ranging from molecules to solids, and it gives rise to a plethora of physical phenomena experimentally accessible using time-resolved techniques. Green's function methods offer an invaluable interpretation tool since scattering mechanisms of growing complexity can be selectively incorporated in the theory. Currently, however, real-time Green's function simulations are either prohibitively expensive due to the cubic scaling with the propagation time or do neglect the feedback of electrons on the bosons, thus violating energy conservation. We put forward a computationally efficient Green's function scheme which overcomes both limitations. The numerical effort scales linearly with the propagation time while the simultaneous dressing of electrons and bosons guarantees the fulfillment of all fundamental conservation laws. We present a real-time study of the phonon-driven relaxation dynamics in an optically excited narrow band-gap insulator, highlighting the nonthermal behavior of the phononic degrees of freedom. Our formulation paves the way to first-principles simulations of electron-boson systems with unprecedented long propagation times.
Abstract Chapter 11 employs variational differential techniques and the Schwinger Action Principle to derive coupled-field Green’s function equations for a multi-component system, modeled as an interacting electron-hole-phonon system. The coupled Fermion Green’s function equations involve five interactions (electron-electron, hole-hole, electron-hole, electron-phonon, and hole-phonon). Starting with quantum Hamilton equations of motion for the various electron/hole creation/annihilation operators and their nonequilibrium average/expectation values, variational differentiation with respect to particle sources leads to a chain of coupled Green’s function equations involving differing species of Green’s functions. For example, the 1-electron Green’s function equation is coupled to the 2-electron Green’s function (as earlier), also to the 1-electron/1-hole Green’s function, and to the Green’s function for 1-electron propagation influenced by a nontrivial phonon field. Similar remarks apply to the 1-hole Green’s function equation, and all others. Higher order Green’s function equations are derived by further variational differentiation with respect to sources, yielding additional couplings. Chapter 11 also introduces the 1-phonon Green’s function, emphasizing the role of electron coupling in phonon propagation, leading to dynamic, nonlocal electron screening of the phonon spectrum and hybridization of the ion and electron plasmons, a Bohm-Staver phonon mode, and the Kohn anomaly. Furthermore, the single-electron Green’s function with only phonon coupling can be rewritten, as usual, coupled to the 2-electron Green’s function with an effective time-dependent electron-electron interaction potential mediated by the 1-phonon Green’s function, leading to the polaron as an electron propagating jointly with its induced lattice polarization. An alternative formulation of the coupled Green’s function equations for the electron-hole-phonon model is applied in the development of a generalized shielded potential approximation, analysing its inverse dielectric screening response function and associated hybridized collective modes. A brief discussion of the (theoretical) origin of the exciton-plasmon interaction follows.
A general theory of superconductivity is formulated within the thermodynamic Green function method for vari-ous types of pairing mediated by phonons, spin fluctuations, and strong Coulomb correlations in the Hubbard and t-J models. A rigorous Dyson equation for matrix Green functions is derived in terms of a self-energy as a many-particle Green function. By applying the noncrossing approximation for the self-energy, a closed self-consistent system of equations is obtained, similar to the conventional Eliashberg equations. A brief discus-sion of superconductivity mediated by kinematic interaction with an estimation of a superconducting transition temperature in the Hubbard model is given. Key words: Green functions, theory of superconductivity, strong electron correlations PACS: 74.20.-z, 74.20.Mn, 74.72.-h