A generating functional approach to the Hubbard model

We combine the recently developed many-body Green's function theory for electrons and nuclei with the exact factorization of the wave function. The existing Born-Oppenheimer Green's functions are shown to be special cases of our exact approach. We consider the limitations of the laboratory frame formulation of the Green's function theory and discuss why the body-fixed frame formulation is needed in order to go beyond the Born-Oppenheimer theory. We give exact forms of the electronic and nuclear Green's functions written in terms of the exact factorized states, providing a systematic approach beyond the Born-Oppenheimer approximation. The lowest order approximation to the exact electronic Green's function is found to be an expected value of the Born-Oppenheimer electronic Green's function with respect to the nuclear density.

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.

We develop nonequilibrium Green's function-based transport theory, which includes effects of nonadiabatic nuclear motion in the calculation of the electric current in molecular junctions. Our approach is based on the separation of slow and fast time scales in the equations of motion for Green's functions by means of the Wigner representation. Time derivatives with respect to central time serve as a small parameter in the perturbative expansion enabling the computation of nonadiabatic corrections to molecular Green's functions. Consequently, we produce a series of analytic expressions for non-adiabatic electronic Green's functions (up to the second order in the central time derivatives), which depend not solely on the instantaneous molecular geometry but likewise on nuclear velocities and accelerations. An extended formula for electric current is derived which accounts for the non-adiabatic corrections. This theory is concisely illustrated by the calculations on a model molecular junction.

A numerical method for calculation electronic structure of a nanosystem\ncomposed of a pseudoisocyanine (PIC) molecule assembled on a silver\nnanoparticle is developed. The electronic structure of the silver nanoparticle\ncontaining 125 atoms is calculated within the local density version of the\ndensity functional method. A model of an Ag atom embedded in the center of a\nspherical jellium cluster is used. The host electron Green function is\ncalculated by means of the spherically symmetric expansion. The principal\ntheoretical tool is the scattering theory using the Green function method. The\nmolecule -- silver nanoparticle interaction is studied using the approach\nsimilar to that of the Anderson model for transition metal impurities in\nsolids. Localized levels are shown to split off from the top of the band of the\nnanosystem. The electronic structure calculations yield information on the\ncharacter of chemical bonding in the PIC molecule --- silver particle\nnanosystem.\n

Here, we present numerical solutions of the spectral functions of large dimension t -J models with random nearest neighbor exchange and global SU(M) spin rotation symmetry. The solutions are obtained from the saddle-point equations of the large dimension limit, followed by the large M limit. The same saddle point equations also apply to the model with all-to-all and random hopping and exchange. Such a t -J model realizes a deconfined critical state proposed as a model for the optimally doped cuprates. The large M theory involves Green’s functions for fractionalized spinons and holons carrying emergent U(1) gauge charges, obeying relations similar to those of the Sachdev-Ye-Kitaev (SYK) models. The low frequency spectral functions are compared with an analytic analysis of the operator scaling dimensions with good agreement. We also compute the low frequency and temperature behavior of experimentally observable gauge-invariant observables: the electron Green’s function, the local spin susceptibility and the optical conductivity, along with the temperature dependence of the d.c. resistivity. The time reparameterization soft mode (equivalent to the boundary graviton in holographically dual models of two-dimensional quantum gravity) makes important contributions to all observables and provides a linear-in-temperature contribution to the d.c. resistivity.

In this chapter, we present two versions of the multiple scattering (MS) theory in the real-space electronic Green’s function (GF) formalism: single-channel (MS-GF) and multichannel (MCMS-GF). While the first one based on the single-particle picture provides a tool for a precise description of MS processes, the second one allows us to take into account not only MS effects but also electron correlations and spin-orbit coupling on the same footing. Multichannel generalization of the MS-GF method relies on the Dyson integral equation relating the GF of a perturbed system with the GF of the corresponding unperturbed system. The second basic feature of the MCMS-GF approach is the use of the close-coupling method, which via Kohn variational principle for the reactance K-matrix gives rise to a set of the coupled integro-differential equations with the matrix of a potential. An iterative algorithm for solving this system has been developed to evaluate single-site multichannel scattering t-matrices through which the GF of the total many-atom system is expressed. In addition, some numerical aspects concerning the application of both versions are discussed with a focus on x-ray absorption spectroscopy.

Assuming that, in the narrow-band region, the exact Green's function of the Hubbard model essentially retains the two-peak structure proposed by Hubbard, we derive its consequences as rigorously as possible. When a quasielectron is added to the ground state of $N$ electrons in the narrow-band region of the Hubbard model, the whole system is shown to relax due to the strong interaction $I$, thus yielding a relaxation energy $\ensuremath{\Delta}E$ between the quasielectron and the original $N$ electrons. This explains the conclusion in the preceding paper that a gap $\ensuremath{\Delta}E$ analogous to the energy gap in superconductivity appears between the quasielectron spectrum and the chemical potential for any occupation $n$ regardless of whether the lower band is exactly filled or not and that the system cannot be a normal metal. We further found that the ground state is, in fact, a bound state, and electrons involved do not obey the Landau theory of quasiparticles. However, the electron Green's function describes the motion of an electron added to this $N$-electron ground state without the relaxation of the original $N$ electrons, and hence this added electron behaves exactly like a Landau quasiparticle and can be represented by the quasiparticle operator ${\mathrm{A}}_{k\ensuremath{\sigma}}^{\ifmmode\dagger\else\textdagger\fi{}}$ introduced by Luttinger and Nozi\`eres. Only if the original $N$ electrons are allowed to relax is the energy of the system reduced by $\ensuremath{\Delta}E$, yielding the energy gap $\ensuremath{\Delta}E$. When a quasielectron is removed from the system of $N+1$ electrons, the same relaxation energy $\ensuremath{\Delta}E$ is removed, and again the gap $\ensuremath{\Delta}E$ is introduced between the chemical potential $\ensuremath{\mu}$ and the quasihole spectrum, thus creating a gap $2\ensuremath{\Delta}E$ between the quasielectron and quasihole spectra for any occupation $n$. The Fermi level, lying at the center of the gap, never intersects the quasiparticle spectra and therefore, Luttinger's theorem that the volume within the Fermi surface is independent of interactions and equal to the value in the noninteracting limit does not apply to the Hubbard model in the narrow-band region. However, this conclusion does not immediately imply that the system is an insulator. Instead, it is shown that a current-carrying state analogous to the condensate state found by Fr\ohlich appears under certain conditions. When the number of electrons $N$ is nearly or exactly equal to the number of atoms ${N}_{a}$, quasiparticlelike behavior disappears completely, making the system insulating. The only exception is the pathological limit $I=\ensuremath{\infty}$, where a metallic state with a Fermi volume twice that in the noninteracting limit is found.

Free carriers in thin films was analysed using Green's function method including quantum size effect and effect of boundaries to Hamiltonian parameters. We calculate diagonal components of electron Green's functions, local densities of states, Fermi energy and spatial distribution of carriers concentration in thin film. The numerical calculations performed for GaAs films show that spatial distribution of free carriers may be manipulated by varying the surface parameters which is significant for operation of devices based on thin films.

The renormalized phonon spectrum beyond the random phase approximation in ferroelectric thin films is obtained using a Green's function technique and the transverse Ising model including anharmonic spin-phonon and phonon-phonon interactions. The temperature and film thickness dependence of the soft phonon mode energy $\ensuremath{\omega}$ is calculated for the case of positive and negative anharmonic spin-phonon interaction constants $R$ and for different surface constants ${R}_{s}$. For $Rg0$ the phonon mode energy increases with increasing temperature, $T\ensuremath{\rightarrow}{T}_{C}$, whereas for $Rl0$ the phonon mode decreases with increasing temperature. For the case of smaller surface constant compared to the bulk, ${R}_{s}l{R}_{b}$, we obtain that $\ensuremath{\omega}$ of the thin film is larger in comparison to that of the bulk, i.e., we obtain hardening of the soft mode. In the other case, ${R}_{s}g{R}_{b}$, the soft phonon mode energy of the thin films is smaller compared to the bulk, i.e., we obtain softening of the phonon mode. The first case could explain the behavior of the phonon mode, soft mode hardening in ${\mathrm{SrTiO}}_{3}$ and $(\mathrm{Ba},\mathrm{Sr}){\mathrm{TiO}}_{3}$ thin films obtained experimentally, whereas the second case is responsible for ${\mathrm{PbTiO}}_{3}$ thin films. The dependence of the phase transition temperature on film thickness and different surface spin-spin ${J}_{s}$ and spin-phonon interaction constants ${R}_{s}$ is discussed, too. The spin-phonon interaction enhances the phonon damping of the thin film. The damping increases near ${T}_{c}$ for the two cases $Rg0$ and $Rl0$. It is larger for thin films in comparison to the bulk. Taking into account the influence of an external electric field we obtain hardening of the soft phonon modes for $Rg0$ and $Rl0$.

We study electron transmission through two impurities in a narrow quantum wire by solving Dyson’s equations for single electron Green functions. We have verified that, for the delta-function potential of two impurities, the Green function can be factorized into a product of the ‘free’ Green function and current transmission amplitude. Meanwhile Green function and current transmission amplitude obey Fisher-Lee’s relation. An analytical expression of the electron transmission amplitude for intrasubband and intersubband is obtained as a function of Fermi energy and the distance between two impurities. The resonant behavior of the current transmission amplitude are detail discussed.

A coordinate-independent formulation of the chiral-symmetric pion theory is developed using a dreibein field which introduces a redundant gauge field into the field equations. There is a particular gauge, where the field variables transform linearly under chiral transformations and the field equations reduce to those of the Sugawara-model. Quantization of Sugawara's field equations is obtained in terms of Green's functions. The field equations for the coordinate-independent Green's functions are translated into equations for coordinate Green's functions by introduction of the coordinates as auxiliary variables. The latter field equations incorporate the additional Feynman graphs discovered before.

The lowest-order vacuum polarization in homogeneous magnetic fields is calculated exactly, using a momentum representation of the electron Green's function obtained by Schwinger in 1951.

The electron-phonon vertex part Λ is investigated as a function of real frequencies in a narrow-band electron system coupled with local phonons. Even though the bare electron-phonon coupling is not very strong, Λ is found to diverge at some characteristic frequencies ώl. The divergence induces the vanishment of the electron Green's function at ώl. This phenomenon can be interpreted as the locking of an electron by such local phonons. By including this effect in the theory of superconductivity, we suggest an anomalous behavior in the tunneling spectroscopy.

The interaction of an intense coherent photon beam with free electrons is discussed. The photon beam is treated as a classical external electromagnetic field. The discussion is exact within the approximation of neglecting radiative corrections and the restriction to the case of a plane-wave field or arbitrary spectral composition and polarization properties. The scattering of a single photon out of a monochromatic beam by an isolated free electron is considered in detail. The cross sections corresponding to the scattering of the various harmonics of the incident beam are evaluated. These cross sections display a complicated dependence upon the intensity of the incident beam, at least for very intense beams. It is found that a mass change induced in the electron by the external field shifts the wavelength of the scattered photons by an amount depending on the intensity of the incident beam. Other processes involving free electrons in the final state are also considered briefly, and a discussion of the magnitude of the effects depending upon the intensity is given. Two Appendices are concerned with the electron Green's function and the vacuum-vacuum transformation function in the presence of a plane-wave field. In the course of the discussion of the latter, the problem of the correct definition of the vacuum current is encountered, and it is shown that a very careful procedure is necessary to obtain a covariant result.

We evaluate the Delbr\"uck scattering amplitude to all orders of the interaction with the external field of a nucleus employing nonperturbative electron Green's functions. The results are given analytically in form of a multipole expansion.