This chapter proves the Jonson-Mahan theorem, which is a generalization of the result for noninteracting systems that all transport coefficients can be derived from integrals over frequency ω of a single transport function multiplied by different powers of ω. The Jonson-Mahan theorem extends this to many-body systems with a local (on-site) interaction, as contained, for example, in the Hubbard, Falicov-Kimball, and periodic Anderson models. This theorem is of great importance, because correlation functions involving the energy current density operator are extremely difficult to obtain by other means for interacting electrons. The proof starts from the observation that they are given by appropriate limits and derivatives of a time-ordered correlation function involving four variables, which are then calculated explicitly.
This chapter proves the Jonson-Mahan theorem, which is a generalization of the result for noninteracting systems that all transport coefficients can be derived from integrals over frequency ω of a single transport function multiplied by different powers of ω. The Jonson-Mahan theorem extends this to many-body systems with a local (on-site) interaction, as contained, for example, in the Hubbard, Falicov-Kimball, and periodic Anderson models. This theorem is of great importance, because correlation functions involving the energy current density operator are extremely difficult to obtain by other means for interacting electrons. The proof starts from the observation that they are given by appropriate limits and derivatives of a time-ordered correlation function involving four variables, which are then calculated explicitly.
We prove the Jonson-Mahan theorem for the thermopower of the Falicov-Kimball model by solving explicitly for correlation functions in the large dimensional limit. We prove a similar result for the thermal conductivity. We separate the results for thermal transport into the pieces of the heat current that arise from the kinetic energy and those that arise from the potential energy. Our method of proof is specific to the Falicov-Kimball model, but illustrates the near cancellations between the kinetic- and potential-energy pieces of the heat current implied by the Jonson-Mahan theorem.
Preparation, I have often said, is rightly two-thirds of any venture. Amelia Earhart Summary Correlation functions provide a direct way to characterize and analyze many-body systems, both theoretically and experimentally. In this chapter we review the properties of one- and two-body correlation functions in quantum systems, with emphasis on several key quantities: static density correlations that determine the energy and thermodynamic potentials, dynamic correlation functions such as response functions that describe excitations of the system, and Green's functions that are basic tools in the theory of interacting many-body systems. Correlation functions are central quantities in the description of interacting many-body systems, both in the theoretical formulation and in the analysis of experiments. In contrast to single numbers like the total energy, correlation functions reveal far more information about the electrons, how they arrange themselves, and the spectra of their excitations. In contrast to the many-body wavefunctions that contain all the information on the system, correlation functions extract the information most directly relevant to experimentally measurable properties. Dynamic current–current correlation functions are sufficient to determine the electrical and optical properties: one-body Green's functions describe the spectra of excitations when one electron is added to or removed from the system, static and dynamic correlations are measured using scattering techniques, and so forth. In this chapter we present the basic definitions and properties of correlation functions and Green's functions that are the basis for much of the developments in the following chapters. In general, a correlation function quantifies the correlation between two or more quantities at different points in space r , time t , or spin σ . Very often the correlation function can be specified as a function of the Fourier-transformed variables, momentum (wavevector) k , and frequency ω . It is useful to distinguish between a dynamic correlation function which describes the correlation between events at different times and a static or equal-time correlation function, by which we mean that of a property measured or computed with “snapshots” of the system. Also, the different correlation functions can be classified by the number of particles and/or fields involved.
This chapter shows how, for noninteracting particles, the charge and energy current density operators are easily obtained from the continuity equation, with the time derivative of the corresponding density operators given by their commutator with the Hamiltonian in the presence of the external fields. This can be generalized to include spin-orbit scattering from heavy impurities, which gives rise to a spin current. For interacting particles, the commutator is very hard to calculate, and the uniform current operators are derived from the rate of change of the charge and energy polarization operators, which are defined as the first moments of the charge and energy densities, respectively. This is done for continuous models, as well as for the Hubbard, periodic Anderson, and Falicov-Kimball lattice models.
By studying the Falicov--Kimball lattice model, we demonstrate that the correlation functions of local electrons can be written directly in the form of the partition function of conduction electrons with a time dependent external potential. These general expressions of the correlation functions are equivalent to the exact ones obtained by the dynamical mean field theory in infinite dimension limit. Moreover, we also prove that the asymptotic power-law behavior of the local electron correlation functions is universal in any dimensions, as long as the conduction electron excitation spectrum is gapless, regardless whether they are Landau Fermi liquid or not. This conclusion is also valid for other quantum many-particle systems, if the coupling between the conduction and local electrons is on-site or short-range Coulomb interaction.
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.
In this Thesis, we develop and apply an accurate method to determine the ground-state properties of strongly-correlated electrons in the framework of lattice model Hamiltonians. In lattice density-functional theory (LDFT) the basic variable is the one-particle density matrix γ. From a generalized HK theorem, the ground state energy Egs[γgs] = minγ E[γ] is obtained by minimizing the energy-functional E[γ] over all the physical or representable γ. The energy functional can be divided into two contributions: the kinetic-energy functional T [γ], which linear dependence on γ is exactly known, and the correlation-energy functional W [γ], which dependence on γ is not explicitly known. Finding accurate approximations for W [γ] constitutes the actual challenge of this thesis. Part of this work is based on previous studies which derive an approximation of W [γ] for the Hubbard Hamiltonian based on scaling hypothesis and an exact analytical results for the Hubbard dimer. However, this approach is limited to spin independent and homogeneous systems. In order to extend the scope of LDFT, we have developed three different approaches to derive W [γ] in order to study broken symmetry problems. First we extend the previous scaling functional to system having charge transfer. By a systematic study of the dependence of W [γ] on the charge distribution we find similar scaling properties as for the homogeneous case. An extension to the Hubbard model on bipartite lattices is then derived and applied to finite and infinite systems with repulsive and attractive interactions. The high accuracy of this functional is demonstrated. Nevertheless, this approach is difficult to transfer to more complex systems, since the calculation of W [γ] applies to the system as a whole. To overcome this problem, we derive another approximation based on local scaling properties. This functional is formulated on-site so that it can be applied to any ordered or disordered Hamiltonian with on-site interaction. As applications we study the metal-insulator transition in the ionic Hubbard model in oneand two-dimensions, as well as the one-dimensional Hubbard chain with first and second nearest neighbor hoppings. Finally, we develop a numerical approach of W [γ], on the basis on exact diagonalizations of an effective many-body Hamiltonian corresponding to a cluster surrounded by an effective medium. This effective Hamiltonian depends on the density matrix γ and allows to derive approximation to W [γ] which improve systematically with increasing cluster size. The formulation is spin-dependent and allows a straightforward generalization to multi-orbital correlated systems such spd-Hamiltonians. In addition it take into account the effects of short-range charge and spin fluctuations in the functional. The accuracy of the method is demonstrated for the Hubbard model by comparison with the Bethe-Ansatz solution (1D) and quantum Monte Carlo simulations (2D). Finally, an outlook or relevant future developments of the theory is provided.