This work reviews recent advances in the analytical treatment of the continuum spectrum of correlated few-body non-relativistic Coulomb systems. The exactly solvable two-body problem serves as an introduction to the non-separable three-particle system. For the latter case we discuss the existence of an approximate separability of the long and the short-range dynamics which is exposed in an appropriately chosen curvilinear coordinates. The three-body wave functions of the long-ranged part of the Hamiltonian are derived and methods are presented to account approximately for the short-ranged dynamics. Furthermore, we present a generalization of the methods employed for the derivation of the three-body wave functions to the scattering states of $N$ charged particles. To deal with thermodynamic properties of finite systems we develop and discuss a recent Green function methodology designed for the non-perturbative regime. In addition, we give a brief account on how thermodynamic properties and critical phenomena can be exposed in finite interacting systems.
This work reviews recent advances in the analytical treatment of the continuum spectrum of correlated few-body non-relativistic Coulomb systems. The exactly solvable two-body problem serves as an introduction to the non-separable three-particle system. For the latter case we discuss the existence of an approximate separability of the long and the short-range dynamics which is exposed in an appropriately chosen curvilinear coordinates. The three-body wave functions of the long-ranged part of the Hamiltonian are derived and methods are presented to account approximately for the short-ranged dynamics. Furthermore, we present a generalization of the methods employed for the derivation of the three-body wave functions to the scattering states of N charged particles. To deal with thermodynamic properties of finite systems we develop and discuss a recent Green function methodology designed for the non-perturbative regime. In addition, we give a brief account on how thermodynamic properties and critical phenomena can be exposed in finite interacting systems.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
The quantum theory of many-particle systems is deeply connected to the mathematical formalism and techniques of quantum field theory, which are crucial in order describe the emergent phenomena that arise in condensed matter physics. The main goal of this thesis is to provide a unified description of quantum field theory methods in the context of many-particle physics in order to extend and develop these techniques to better understand the different quantum properties of matter. This both furthers the conceptual understanding of quantum many-body correlation and also allows for the development of novel theoretical techniques that combine the different formalisms employed in condensed matter theory, quantum chemistry and nuclear physics. To achieve this aim, the thesis is composed of two parts. Part I deals with many-body fermion-fermion interactions and is physically motivated by the non-relativistic electronic structure problem. Here, we introduce the Keldysh contour formulation of quantum field theory to provide a unified presentation of quantum systems both in and out-ofequilibrium. Then, we extend the conventional quantum theory of many-particle systems, based on hermitian quantum mechanics, to describe systems governed by non-hermitan Hamiltonians. This allows us to present a natural unification of two major pillars in the ab initio quantum theory of many-particle systems: coupled-cluster theory and the Green’s function formalism. This unification is further demonstrated through the emergence of the coupled-cluster amplitude equations from the Green’s function formalism. In particular, this analysis has direct bearing on the development of novel theoretical approaches to the electronic structure and nuclear many-body problems. Our findings are also relevant to the study of general non-hermitian as well as open quantum systems. Part II focuses on the quantum field theory of electron-nuclear interactions in infinite solids. Here, we use the Keldysh contour representation presented in Part I to understand excited state dynamics that include electron-electron and electron-lattice interactions in materials. We also derive the analytic continuation of the Green’s function for systems that have continuous eigenspectra. This analysis demonstrates the relationship between the Green’s function formalism and scattering theory, making the role of the continuum and causality conditions explicitly clear in the context of the pole condition of the Dyson equation. Using the analytic continuation of the propagator, we provide an in-depth analysis of electron-phonon and exciton-phonon scattering channels, with a particular focus on phonon-driven exciton dissociation scattering. Our theoretical analysis then firmly demonstrates the connection between the extension of the Bethe-Salpeter equation approach to include electron-phonon interactions and effective field theories based on composite exciton-phonon coupling. We subsequently combine the techniques presented in Part I, based on the Dyson supermatrix formulation, with the analysis presented in Part II to calculate, from first principles, the effects of including electron-phonon interactions on the excited states of heterogeneous semiconductors. Finally, we provide a summary of the fundamental contributions presented in this thesis and the potential further research directions to be pursued.
Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.
The ways of introducing and handling renormalizations in the many-body perturbation theory are reviewed. We stress the indispensable role the technique of Green functions plays in extrapolating the weak-coupling perturbative approaches to intermediate and strong couplings. We separately discuss mass and charge renor- malizations. The former is incorporated in a self-consistent equation for the self-energy derived explicitly from generating Feynman diagrams within the Baym and Kadanoff approach. The latter amounts to self-consistent equations for two-particle irreducible vertices. We analyze the charge renormalization initiated by De Domi- nicis and Martin and demonstrate that its realization via the parquet approach may become a powerful and viable way of using the many-body diagrammatic approach reliably in non-perturbative regimes with coopera- tive phenomena induced by either strong interaction or strong static randomness. Electric and magnetic properties of solids are determined by the behavior of valence electrons weakly bound to the atoms forming the crystalline lattice. The behavior of the gas of valence electrons in solids is most profound in metals with an open Fermi surface. Conduction electrons in metals can be described in a number of situations and for various purposes quite accurately by Bloch waves, eigenstates of the Fermi-gas Hamiltonian. We know, however, that electrons even in metals are not a noninteracting gas. They are generally exposed to forces driving them out of the equilibrium state of the Fermi gas. Firstly, no metal is a pure crystal and hence the electrons do not move freely in space. They are scattered on randomly distributed impurities and lattice defects. Secondly, the electrons are neither free and their mutual repulsion cannot be neglected particularly for transition and heavy metals. It is the most important and long standing task of condensed-matter theorists to comprehend and qualitatively capture the deviations in the behavior of electrons in real metals from their idealized representation via Bloch waves. The problem with quantification of the behavior of electrons beyond the almost free-electron picture is in paucity of available tools for accomplishing this task. Even if we can reduce the description of the electron gas in metals to rather simple and generic models, we are unable to solve them exactly, except for a few limiting cases that mostly do not correspond to situations of physical interest. With the increasing power of modern computers the role of numerical solutions and numerical brute force approaches has increased. Numerical solutions are usually unbiased and can reach a rather high level of quantitative precision. In some aspects they substitute the missing exact analytic solutions. Numerical solutions are suitable and even indispensable in showing and displaying the trends and some global features of the models studied. They, however, fail in critical regions of phase transitions in the vicinity of singularities in correlation functions. In these
We especially need imagination in science. aria Mitchell Summary The many-body problem consists of two parts: the first is the non-interacting system in a materials-specific external potential; the second is the Coulomb interaction that makes the problem so hard to solve. The most straightforward idea is to use perturbation theory, with the Coulomb interaction as perturbation. This is conceptually simple, but it turns out to be difficult in practice, since the Coulomb interaction is often not small compared with typical energy differences, it is long-ranged and in the thermodynamic limit there is an infinite number of particles, contributing with an infinite number of mutual interaction processes. The present chapter outlines how one can deal with this problem. It contains an overview of facts that one can also find in many standard textbooks on the many-body problem, but that are useful to keep in mind in order to look at later chapters from a sound and well-established perspective. The many-body problem is a tough one, and it has many facets. Sorting it out is like putting together a huge puzzle. The eight introductory chapters of this book provide pieces of the puzzle, and ideas on what one might do about it. In the present chapter we choose to go in one of the possible directions, in order to arrive at something tangible. The chapter gives the general framework and the main ideas; specific approximations are the topic of Chs. 10–15. The idea is to start from an independent-particle problem and add the Coulomb interaction as a perturbation. This is not easy: first, the interaction is responsible for a rich variety of phenomena that are completely absent otherwise, such as the finite lifetime of quasiparticles, or additional structures in spectra due to the fact that a quasi-particle excitation may transfer its energy to other elementary excitations, for example plasmons. Second, because of the two-body Coulomb interaction, the problem scales badly with the number of electrons, and straightforward perturbation theory for the many-body hamiltonian with the Coulomb interaction as perturbation rapidly becomes intractable or even useless, especially in large systems. To get started, Sec. 9.1 recalls why things are not so easy. The following sections try to solve one problem after the other, starting from Sec. 9.2 where the Green's function is reformulated in a way that is appropriate for a perturbation expansion.
Introduction to integrable many-body systems I This is the first volume of a three-volume introductory course about integrable (exactly solvable) systems of interacting bodies. The aim of the course is to derive and analyze, on an elementary mathematical and physical level, the Bethe ansatz solutions, ground-state properties and the thermodynamics of integrable many-body systems in many domains of physics: Nonrelativistic one-dimensional continuum Fermi and Bose gases; One-dimensional quantum models of condensed matter physics like the Heisenberg, Hubbard and Kondo models; Relativistic models of the (1+1)-dimensional Quantum Field Theory like the Luttinger model, the sine-Gordon model and its fermionic analog the Thirring model; Two-dimensional classical models, especially the symmetric Coulomb gas. In the first part of this volume, we deal with nonrelativistic one-dimensional continuum Fermi and Bose quantum gases of spinless (identical) particles with specific types of pairwise interactions like the short-range δ-function and hard-core interactions, and the long-range 1/