Many-body theory is largely based on self-consistent equations that are constructed in terms of the physical quantity of interest itself, for example the density. Therefore, the calculation of important properties such as total energies or photoemission spectra requires the solution of nonlinear equations that have unphysical and physical solutions. In this work we show in which circumstances one runs into an unphysical solution, and we indicate how one can overcome this problem. Moreover, we solve the puzzle of when and why the interacting Green's function does not unambiguously determine the underlying system, given in terms of its potential, or non-interacting Green's function. Our results are general since they originate from the fundamental structure of the equations. The absorption spectrum of lithium fluoride is shown as one illustration, and observations in the literature for some widely used models are explained by our approach. Our findings apply to both the weak and strong-correlation regimes. For the strong-correlation regime we show that one cannot use the expressions that are obtained from standard perturbation theory, and we suggest a different approach that is exact in the limit of strong interaction.
Many-body theory is largely based on self-consistent equations that are constructed in terms of the physical quantity of interest itself, for example the density. Therefore, the calculation of important properties such as total energies or photoemission spectra requires the solution of nonlinear equations that have unphysical and physical solutions. In this work we show in which circumstances one runs into an unphysical solution, and we indicate how one can overcome this problem. Moreover, we solve the puzzle of when and why the interacting Green's function does not unambiguously determine the underlying system, given in terms of its potential, or non-interacting Green's function. Our results are general since they originate from the fundamental structure of the equations. The absorption spectrum of lithium fluoride is shown as one illustration, and observations in the literature for some widely used models are explained by our approach. Our findings apply to both the weak and strong-correlation regimes. For the strong-correlation regime we show that one cannot use the expressions that are obtained from standard perturbation theory, and we suggest a different approach that is exact in the limit of strong interaction.
Many-body theory is largely based on self-consistent equations that are constructed in terms of the physical quantity of interest itself, for example the density. Therefore, the calculation of important properties such as total energies or photoemission spectra requires the solution of non-linear equations that have unphysical and physical solutions. In this work we show in which circumstances one runs into an unphysical solution, and we indicate how one can overcome this problem. Moreover, we solve the puzzle of when and why the interacting Green's function does not unambiguously determine the underlying system, given in terms of its potential, or non-interacting Green's function. Our results are general since they originate from the fundamental structure of the equations. The absorption spectrum of lithium fluoride is shown as one illustration, and observations in the literature for some widely used models are explained by our approach. Our findings apply to both the weak and strong-correlation regimes. For the strong-correlation regime we show that one cannot use the expressions that are obtained from standard perturbation theory, and we suggest a different approach that is exact in the limit of strong interaction.
A review of electronic dynamics of single-impurity and many-impurity Anderson models is contained in this report. Those models are used widely for many of the applications in diverse fields of interest, such as surface physics, theory of chemisorption and adsorbate reactions on metal surfaces, physics of intermediate valence systems, theory of heavy fermions, physics of quantum dots and other nanostructures. While standard treatments are generally based on perturbation methods, our approach is based on the non-perturbative technique for the thermodynamic Green functions. The method of the irreducible Green functions is used as the basic tool. This irreducible Green functions method allows one to describe the quasiparticle spectra with damping of the strongly correlated electron systems in a very general and natural way and to construct the relevant dynamical solution in a self-consistent way on the level of Dyson equation without decoupling the chain of the equations of motion for the Green functions. The subject matter includes the improved interpolating solution of the Anderson model. It was shown that an interpolating approximation, which simultaneously reproduces the weak-coupling limit up to second order in the interaction strength U and the strong-coupling limit up to second order in the hybridization V (and thus also fulfils the atomic limit) can be formulated self-consistently. This approach offers a new way for the systematic construction of approximate interpolation dynamical solutions of strongly correlated electron systems.
To avoid confusion, it is necessary to be quite explicit about what is assumed and what is to be proved. A. Klein, Phys. Rev. 121, 950 (1961) Summary The topic of this chapter is functionals that provide a concise formulation for thermodynamic quantities and Green's function methods in interacting many-body systems. Building upon the instructive examples of density functional theory and the Hartree– Fock approximation, functionals of the Green's function are developed to provide a framework for practical methods used extensively in many-body perturbation theory (Chs. 9–15), dynamical mean-field theory (Chs. 16–20), and their combination in Ch. 21. Functional concepts give a firm foundation for those methods, for example, they give the conditions for approximations to obey conservation laws. Additional background and specific aspects are in App. H. Many physical quantities can be expressed as functionals , i.e., they depend on an entire function.1 For example, in the quantum variational method to determine a wavefunction, the total energy of a system of many electrons is a functional of a many-body trial wavefunction ( r 1 , r 2 , …): it is the expectation value of the hamiltonian. At the variational minimum, assuming the trial wavefunction is completely flexible, one finds the many-body Schrödinger equation for the ground state. Many of the properties we are interested in are, formally, simple functionals of the ground- or excited-state wavefunctions. In the search for feasible ways to calculate properties such as excitation spectra, we are led to different types of functional. The idea is to deal only with a physical, measurable quantity Q of interest and a variational functional E [ Q ] of this quantity. The variational character of E allows one to derive equations that determine Q 0 , the value of Q for the system in a desired state, typically the ground state or equilibrium state at T ≠ 0. Other properties should be different functionals F m [ Q ] that can then be evaluated at Q = Q 0 . This strategy is outlined in Fig. 8.1. To appreciate the power of such approaches, one can consider the impact of density functional theory, which is formulated as a functional of the density n ( r ) of electrons; it is defined for a range of densities and the variational minimum for the energy functional E [ n ] leads to equations to determine the actual ground-state density n 0 ( r ) and total energy E 0 .
Several widely used methods for the calculation of band structures and photo emission spectra, such as the GW approximation, rely on many-body perturbation theory. They can be obtained by iterating a set of functional differential equations (DEs) relating the one-particle Green's function (GF) to its functional derivative with respect to an external perturbing potential. In this work, we apply a linear response expansion in order to obtain insights into various approximations for GF calculations. The expansion leads to an effective screening while keeping the effects of the interaction to all orders. In order to study various aspects of the resulting equations, we discretize them and retain only one point in space, spin and time for all variables. Within this one-point model we obtain an explicit solution for the GF, which allows us to explore the structure of the general family of solutions and to determine the specific solution that corresponds to the physical one. Moreover, we analyze the performances of established approaches like GW over the whole range of interaction strength, and we explore alternative approximations. Finally, we link certain approximations for the exact solution to the corresponding manipulations of the DE which produces them. This link is crucial in view of a generalization of our findings to the real (multidimensional functional) case where only the DE is known.
In modern quantum many-body physics, the Kadanoff-Baym equations have become a crucial component in the treatment of strongly and weakly correlated systems far from equilibrium. From the one-particle Green function G(t,${t}^{\ensuremath{'}}$), they allow for the calculation of time-dependent expectation values of all one-particle observables and the total energy. In this work, for isolated Coulomb systems, the numerical behavior of the Kadanoff-Baym equations is investigated. The electron density dynamics are damped to an unphysical homogeneous density distribution, across both the linear and nonlinear response regimes. Unphysical features are shown to exist for $\mathrm{\ensuremath{\Phi}}$-derivable self-energy approximations, such as Hartree-Fock, second-Born, or $G\phantom{\rule{0}{0ex}}W$, in Hubbard and Coulomb systems, irrespective of interaction strength. With this degree of universality, these findings are pertinent to all two-time formalisms, and suggest the need for a different approach to the dynamics of quantum systems.
We report an implementation of self-consistent Green's function many-body theory within a second-order approximation (GF2) for application with molecular systems. This is done by iterative solution of the Dyson equation expressed in matrix form in an atomic orbital basis, where the Green's function and self-energy are built on the imaginary frequency and imaginary time domain, respectively, and fast Fourier transform is used to efficiently transform these quantities as needed. We apply this method to several archetypical examples of strong correlation, such as a H32 finite lattice that displays a highly multireference electronic ground state even at equilibrium lattice spacing. In all cases, GF2 gives a physically meaningful description of the metal to insulator transition in these systems, without resorting to spin-symmetry breaking. Our results show that self-consistent Green's function many-body theory offers a viable route to describing strong correlations while remaining within a computationally tractable single-particle formalism.