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.
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.
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 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 A general method for studying the properties of free and bound polarons at arbitrary strength of electron‐phonon coupling is given from a unified point of view by first performing a suitable canonical transformation of the electron and phonon field operators and subsequently applying the double‐time Green's function formalism to the transformed polaron‐phonon Hamiltonian. Calculating the one‐polaron Green's function in the lowest approximation and employing the variational principle for the free energy, a pair of coupled differential equations is derived for the effective wave function and a trial function as introduced by the canonical transformation. From these equations, the weakly excited polaron energy levels at low temperatures can be found in the entire range of coupling constants when external fields are present. In this paper, detailed calculations are confined to the optical‐polaron problem in the zero‐field case. Intermediate‐ and strong‐coupling formulas for temperature‐dependent polaron self‐energy and effective mass are given. The strong‐coupling regime appears to increase with temperature.
The method of moments is employed to determine the ground state of the phonon-polaron system. The trial function employed in this calculation is a bare-electron wave function. It appears that the method is convergent for this problem. Although the energy is in general a complicated function of the parameter and must be determined numerically, it is possible to formally construct a power series expansion for the energy about the unperturbed energy in powers of the coupling constant. The first term in this series has been determined and agrees with the result of conventional perturbation theory. The second term has not yet been evaluated to all orders but it appears that it will have the opposite sign from the corresponding term in perturbation theory. The second and higher terms in some orders of approximation are singular for small values of the cutoff.
This book will introduce advanced concepts and topics of solid-state theory. To this end we need a tool box that enables us to treat electron–electron interactions, and possibly also electron–phonon or phonon–phonon interactions in some well-defined, approximate way. The most advanced, yet still feasible approach, is the Green's function technique and the perturbation expansion using Feynman diagrams. This technique allows the interacting system to be treated as 'solid' within well-characterizable approximations, still widely used today, but also provides the language used in modern computer-based simulations as well as for the interpretation of experiments.
I compute the spectral functions of a 1D Holstein polaron moving in a modulated potential, to examine the effects of electron-phonon interactions on band gaps. The imaginary time Green function is computed using diagrammatic quantum Monte Carlo, which exactly sums the diagrammatic series. From the imaginary time Green function, spectral functions are computed. The electron-phonon interaction flattens the electronic dispersion and leads to an increase in the gap at momentum k = π/2. At strong coupling, polaron sidebands form in the gap. These results demonstrate the strong effect that electron-phonon interactions can have on band gaps.