For the purpose of computer calculation to evaluate time-dependent quantum properties in finite temperature, we propose new numerical method expressed in the forms of simultaneous differential equations. At first we derive the equation of motion in finite temperature, which is found to be same expression as Heisenberg equation of motion except for the c-number. Based on this equation, we construct numerical method to estimate time-dependent physical properties in finite temperature precisely without using analytical procedures such as Keldysh formalism. Since our approach is so simple and is based on the simultaneous differential equations including no terms related to self-energies, computer programming can be easily performed. It is possible to estimate exact time-dependent physical properties, providing that Hamiltonian of the system is taken to be a one-electron picture. Furthermore, we refer to the application to the many body problem and it is numerically possible to calculate physical properties using Hartree Fock approximation. Our numerical method can be applied to the case even when perturbative Hamiltonians are newly introduced or Hamiltonian shows complex time-dependent behavior. In this article, at first, we derive the equation of motion in finite temperature. Secondly, for the purpose of verification and of exhibiting the usefulness, we show the derivation of gap equation of superconductivity and of sum rule of electrical conductivity and the application to the many body problem. Finally we apply this method to these two cases: the first case is most simplified resonance charge transfer neutralization of an ion and the second is the same process but impurity potential is newly introduced as perturbative Hamiltonian. Through both cases, it is found that neutralization process is not so sensitive to temperature, however, impurity potential as small as 10 meV strongly influences the neutralization of ion.
For the purpose of computer calculation to evaluate time-dependent quantum properties in finite temperature, we propose new numerical method expressed in the forms of simultaneous differential equations. At first we derive the equation of motion in finite temperature, which is found to be same expression as Heisenberg equation of motion except for the c-number. Based on this equation, we construct numerical method to estimate time-dependent physical properties in finite temperature precisely without using analytical procedures such as Keldysh formalism. Since our approach is so simple and is based on the simultaneous differential equations including no terms related to self-energies, computer programming can be easily performed. It is possible to estimate exact time-dependent physical properties, providing that Hamiltonian of the system is taken to be a one-electron picture. Furthermore, we refer to the application to the many body problem and it is numerically possible to calculate physical properties using Hartree Fock approximation. Our numerical method can be applied to the case even when perturbative Hamiltonians are newly introduced or Hamiltonian shows complex time-dependent behavior. In this article, at first, we derive the equation of motion in finite temperature. Secondly, for the purpose of verification and of exhibiting the usefulness, we show the derivation of gap equation of superconductivity and of sum rule of electrical conductivity and the application to the many body problem. Finally we apply this method to these two cases: the first case is most simplified resonance charge transfer neutralization of an ion and the second is the same process but impurity potential is newly introduced as perturbative Hamiltonian. Through both cases, it is found that neutralization process is not so sensitive to temperature, however, impurity potential as small as 10 meV strongly influences the neutralization of ion.
To analyze the many-body problem based on differential forms, we apply the Heisenberg equation of motion to the Hamiltonian including the many-body term, focusing on the simplest electron–phonon interaction. Simultaneous differential equations are obtained after applying the Heisenberg equation of motion. To confirm our proposed method, we compare the numerical solutions to the simultaneous differential equations with the analytical results based on the Green's function method. Good agreement is observed between the results, indicating that the analysis based on the Heisenberg equation of motion is suitable for analyzing the steady state. To examine the non-steady state, we investigate weakly pulsed phonon excitation. The numerical calculations suggest that phonons may be conserved if the excitation period is very short. Furthermore, based on the Pauli exclusion principle, the number of simultaneous differential equations that exactly describe electron–electron interactions appears to be finite but becomes extremely large as the number of electrons increases.
In this thesis, an approximate method introduced to solve time-dependent many-body problems known as time-dependent many-body perturbation theory is studied. Many-body perturbation theory for interacting electrons and phonons is reviewed. In particular, the electron propagator G and an unconventional two-component phonon propagator, which satisfy coupled integral Dyson equations, are introduced. In practice, the associated integral kernels known as the electron Σ and phonon self-energies need to be approximated. The conserving approximations known as the Hartree (-Fock) and the first and second Born approximations, which respect the continuity equation between the electron density and current, are considered in this work. Time-dependent systems of interest are studied in this thesis by using the integro-differential forms of the Dyson equations referred to as the Kadanoff-Baym Equations (KBE). The Kadanoff-Baym equations are introduced for the electron and phonon propagators unconventionally as coupled firstorder integro-differential equations. It is reviewed how these equations are solved numerically by describing an integration rule, a class of practical methods and a parallel implementation of the numerical method. In addition, documentation of how the Kadanoff-Baym equations allow to solve the Bethe-Salpeter Equation (BSE) with the kernel δΣ/δG for the density response function, is provided. In two of the enclosed publications, we benchmarked observables obtained by using the Hartree and partially and fully self-consistent Born approximations against numerically exact results for the two- site, two-electron Holstein model. In this model, the two electrons which are constrained to move between two lattice sites interact indirectly via the electron-phonon coupling. It was observed that only the fully self-consistent Born approximation could cope qualitatively correctly with the competition between the delocalizing and localizing effects of the kinetic and interaction energies. For the other two approximations, spurious bifurcative symmetry breaking with an associated unbounded density response was observed. Nevertheless, also the self-consistent Born approximation was concluded to fail in describing the bipolaronic behavior of the true system. In the third publication, we benchmarked the Hartree-Fock and second Born approximations against an exact method for the few-site Hubbard and Pariser-Parr-Pople models in which the underlying lattice is inert and the electrons interact amongst themselves directly. It was found that the second Born approximation is capable of describing the so-called correlation induced doubly-excited states. This is not possible for time-local approximations such as Hartree-Fock. In addition to the qualitative results, which highlight successes of the applied simple self-energy approximations, the approximate and exact results were also compared on a more quantitative level. It is the quantitative and qualitative results combined which are used in this thesis to assess the quality of the many-body approximations used, with the aim to better understand when these methods are predictive.
We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.
We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numerically exact diagonalization method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations.
The problem of quantum dynamics in open systems has gained attention in recent decades and not the least due to the advances made in quantum transport in molecular systems. The main motivation behind quantum transport and molecular electronics is the futuristic goal to be able at some point to replace, or to complement, the silicon-based technology and to make the electronic devices faster. On a fundamental level, one has to deal with time-dependent processes where electron-electron or electron- phonon interactions are of great importance, and they can cause profound quantitative and qualitative changes on the physical and dynamical properties of electronic systems compared to the non-interacting case. Most of the studies of quantum transport have been focused on the steady-state description while neglecting the short-time dynamics. However, the dynamical effects are of great importance since fast- switching processes play a pivotal role in the operation of future devices. We studied the problem of time-dependent electron transport through the Anderson impurity model by using many-body perturbation theory (MBPT) together with Keldysh Green’s functions as well as with time-dependent density functional theory (TDDFT). These methods were compared with numerically exact time-dependent density-matrix renormalization group (tDMRG) method. We found that the many-body perturbation theory results beyond Hartree-Fock approximation were in close agreement with tDMRG results. In addition we studied the possibility of multistablity in the density and current of an interacting nanoscale junction as well as how to reversibly switch between the multiple solutions in time domain. An accurate theoretical treatment of electron correlation even in as simple model as an interacting electron gas at metallic densities still continues to be a challenge; especially description of features in the photoemission spectra due to electron correlations provides a theoretical challenge. The many-body perturbation theory yields a systematic way to study electron-electron (electron-phonon) correlations in various systems. One of the widely used approximations in MBPT is the GW approximation in which the bare interaction line is replaced with screened interaction line in the first order exchange diagram. The GW approximation gives good estimates for the band gap values close to experimental ones but especially the self-consistent GW approximation has a number of deficiencies like washing out of plasmon features and overestimation of bandwidths compared to experiment. One way to improve GW calculations is to include vertex corrections. Unfortunately, the straightforward inclusion of vertex corrections yields negative spectra in some frequency regions. We developed a diagrammatic approach to construct self-energy approximations with positive spectral properties. Our approach consists of expressing a self-energy of response diagram as a product of half-diagrams after which a minimal set of additional diagrams is identified to construct a perfect square. We applied this method to study vertex corrections in a homogeneous electron gas. In addition we analyzed the diagrammatic content of photocurrent with density functional theory. The expression for the photocurrent was obtained as an integral over the Kohn-Sham spectral function renormalized by effective potentials that depend on the exchange correlation kernel of current density functional theory. The expression for the photocurrent gives us the angular dependence of the photocurrent but it does not provide a direct access to the kinetic energy distribution of the photoelectrons.
The Green function formalism introduced by Keldysh for irreversible processes is applied to the calculation of the time-dependent charge exchange probabilities in scattering problems. Considering the initial state of two non-interacting subsystems described by an independent particle model, their interaction is taken in a first stage, as evolving in time according to a time-dependent Hartree - Fock (TDHF) scheme. The equations of motion for the two-times Green functions are subsequently obtained in Dyson-like form by including the residual electronic correlations expanded perturbatively up to second order. It is shown that, by solving these equations of motion, this in turn allows one to calculate the two-particle correlation functions at equal times, as required to describe the fractions of charge in each subsystem. This procedure contrasts with that of considering expectation values of the correlation functions taken between the corrected TDHF state up to an equivalent order in the perturbation. In order to test the advantages of this method we have applied this scheme to the scattering of an atom from a three-substrate atom chain described by an Anderson Hamiltonian, where comparisons with exact solutions can be easily established. We found that, in this case, the fractions of charges carried by the scattered particle obtained with our proposal compare fairly well with exact results, within an ample range of parameter selection.