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.
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.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
Ab initio electron propagator methods are developed to study electronic properties of molecular systems with strong electron-electron and electron-phonon interactions. For the calculation of electron Green's functions we apply a canonical small polaron transformation that intrinsically contains strong electron-phonon effects. In the transformed Hamiltonian, the energy levels for the noninteracting particles are shifted down by the relaxation (solvation) energies. The Coulomb integrals are also renormalized by the electron-phonon interaction. For certain values of the electron-phonon coupling constants, the renormalized Coulomb integrals can be negative which implies the attraction between two electrons. Within the small polaron transformation we develop a diagrammatic technique for the calculation of electron Green's function in which the electron-phonon interaction is already included into the multiple phonon correlation functions. Since the decoupling of the phonon correlation functions is impossible, and therefore, a Wick's theorem for such correlation functions is invalid, there is no Dyson equation for the electron Green's function. To find the electron Green's function, we use different approximations. One of them is a link-cluster approximation that includes diagonal transitions for the renormalized zeroth Green's function. In the linked-cluster approach the Dyson equation is derived in the most general case, where the self-energy operator is an arbitrary functional (not only in the Hartree-Fock approximation). It is shown that even a Hartree-Fock electron (hole) is not a particle any longer. It is a quasiparticle with a finite lifetime that depends on energy of particle and hole states in different ways. As a consequence of this, a standard description of a Hartree-Fock approximation in terms of wave functions becomes inappropriate in this problem. To challenge the linked-cluster approximation we develop a different approach: a sequential propagation approximation where scattering events occur only for sequential transitions. A self-consistent Hartree-Fock equation for a four-index Green's function matrix is derived. In conclusion, the proposed schemes can be considered for future method developments for quantum chemical calculations for large molecules with strong nonadiabatic effects, e-e correlated electron transfer reactions, and electron transport in molecular transport junctions.
Abstract The methods of coupled quantum field theory, which had great initial success in relativistic elementary particle physics and have subsequently played a major role in the extensive development of non-relativistic quantum many-particle theory and condensed matter physics, are at the core of this book. As an introduction to the subject, this presentation is intended to facilitate delivery of the material in an easily digestible form to students at a relatively early stage of their scientific development, specifically advanced undergraduates (rather than second or third year graduate students), who are mathematically strong physics majors. The mechanism to accomplish this is the early introduction of variational calculus with particle sources and the Schwinger Action Principle, accompanied by Green’s functions, and, in addition, a brief derivation of quantum mechanical ensemble theory introducing statistical thermodynamics. Important achievements of the theory in condensed matter and quantum statistical physics are reviewed in detail to help develop research capability. These include the derivation of coupled field Green’s function equations of motion for a model electron-hole-phonon system, extensive discussions of retarded, thermodynamic and non-equilibrium Green’s functions, and their associated spectral representations and approximation procedures. Phenomenology emerging in these discussions includes quantum plasma dynamic, nonlocal screening, plasmons, polaritons, linear electromagnetic response, excitons, polarons, phonons, magnetic Landau quantization, van der Waals interactions, chemisorption, etc. Considerable attention is also given to low-dimensional and nanostructured systems, including quantum wells, wires, dots and superlattices, as well as materials having exceptional conduction properties such as superconductors, superfluids and graphene.
A new technique is presented for evaluating the electron Green function Gk(t) in an insulating electron–phonon system. The technique is particularly suitable for those electron–phonon interactions in which the long wavelength phonons are of prime importance such as the polaron and piezoelectric interactions.An exact formal expression is obtained for the time and temperature dependent Green function in the form[Formula: see text]where S is a functional of the differential operator d/dk. The Green function is then expressed as [Formula: see text], and from the exact result a series expansion is derived for the action function Ak(t).This series expansion has a very special property: to whatever order Ak(t) is evaluated the resulting expression for Gk(t) includes, in some way, every perturbation theory diagram. By this we mean that as in conventional techniques (the Hartree–Fock approximation, for example) an infinite subset of diagrams are summed exactly and in contrast to these techniques the remaining infinity of diagrams are not discarded but are evaluated approximately.The technique is applied to the polaron interaction.
Abstract Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
Abstract Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.