Integral equations of the Watson-Faddeev type, developed in the context of few-body nonrelativistic scattering theory, are shown to play an important role in the structure analysis of many-body Green's functions and in variational techniques for the approximate determination of these Green's functions. Such equations sum up an infinite chain of few-body correlations. Those correlations which are not of this form, i.e., the intrinsic many-body correlations, are accounted for by the appearance of an effective (or "renormalized") interaction, as can be seen by direct transformation of the exact Martin-Schwinger equations for the Green's functions. For a given choice of effective interaction, the few-body correlation functions must be determined self-consistently. The Schwinger variational principle, originally developed for linear integral equations, is nevertheless the key element in a variational formulation of the nonlinear equations encountered here. For simplicity we confine our attention to equations which determine the two-body Green's function, although the entire procedure can be generalized to higher orders.
Integral equations of the Watson-Faddeev type, developed in the context of few-body nonrelativistic scattering theory, are shown to play an important role in the structure analysis of many-body Green's functions and in variational techniques for the approximate determination of these Green's functions. Such equations sum up an infinite chain of few-body correlations. Those correlations which are not of this form, i.e., the intrinsic many-body correlations, are accounted for by the appearance of an effective (or "renormalized") interaction, as can be seen by direct transformation of the exact Martin-Schwinger equations for the Green's functions. For a given choice of effective interaction, the few-body correlation functions must be determined self-consistently. The Schwinger variational principle, originally developed for linear integral equations, is nevertheless the key element in a variational formulation of the nonlinear equations encountered here. For simplicity we confine our attention to equations which determine the two-body Green's function, although the entire procedure can be generalized to higher orders.
A variational approach to the approximate solution of a wide variety of self-consistency problems is described. Attention is focused on the variational formulation of the nonlinear integral equation, which determines the single-particle Green's function of a many-particle system; such an equation arises from a particular truncation of the hierarchy of equations which dynamically couple the complete set of Green's functions. To illustrate the method, we have applied it, with encouraging results, to the numerical determination of the self-energy function for the helium atom in the Hartree-Fock approximation. The applicability of this approach to the general problem of electron-atom scattering is pointed out.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
We are concerned with few-particle correlations in a fermionic system at finite temperature and density. Within the many-body Green functions formalism the description of correlations is provided by the Dyson equation approach that leads to effective few-body equations. They contain the dominant medium effects, which are self energy corrections and the Pauli blocking. Hence the effective two-body interactions between quasiparticles are momentum/energy-dependent and therefore they can be usesed in the medium modified, momentum space, integral AGS equations for three- and four-body systems. To investigate correlations and clusters beyond four-body, we employ, instead, the configuration space two-variable integro-differential equations (IDEA) for A-body bound systems which are based on Hyperspherical Harmonics and the Faddeev decomposition of the wave function in two-body amplitudes. This requires the transformation of the energy dependent two-body interactions to equivalent local, energy independent, ones. To achieve this we use inverse scattering techniques the resulting interactions being, on-- and (to all practical purposes) off--shell equivalent to the energy dependent potentials. In this way we obtain binding energy results for the 2--, 3--, 4--, and 16--particle in a medium at a finite temperature and various densities. Several aspects of the problem are discussed and the behavior of the potential surfaces obtained in the extreme adiabatic approximation, below and above the Mott transition, is investigated.
The generalization of the pair-labeled Rosenberg equations for many-particle scattering are found in the case where there are arbitrary multiparticle interactions. These are called interaction-set equations because they involve auxiliary transition operators which are labeled by the same set of partitions which characterizes the various connectivities of the interparticle interactions. The technique which we employ also provides the analogous extension of a recently proposed set of connected-kernel multiple scattering equations for the Watson-type transition operators. Further, the structure of the interaction-set equations leads to the identification of an entire class of interaction-set connected-kernel scattering integral equations, each of which is based upon a distinct choice of unperturbed Green’s function and its associated connectivity structure. The generalized Rosenberg equations and the connected-kernel Watson-type multiple scattering equations, which are limiting members of this class, correspond to the choice of the N-free-particle and two-cluster-channel unperturbed Green’s functions, respectively.
The general form of the energy of the ground state of a many-fermion system is shown to be exactly of the form proposed by Brueckner and Bethe, without approximation. In a variational treatment, if the trial wave function is picked containing only pair correlations, together with all possible unlinked pairs, it is described by a two-body excitation matrix 〈m1m2∣A∣p1p2〉. Variation of this matrix in the Ritz-Rayleigh principle yields a set of integral equations of the scattering type for the matrix A. Hole-state energies are given self-consistently in terms of the matrix A, but particle-state energies are Hartree-Fock energies. This may be corrected for by widely enlarging the class of terms admitted into the wave function. If the approximation is then made of omitting a class of terms, defined as cross-linked clusters in $〈\ensuremath{\psi}|H|\ensuremath{\psi}〉$, the particle-state energies are easily renormalized. Variation then leads to an infinite hierarchy of integral equations.
This dissertation concerns the quantum many-body problem, which is the problem of predicting the properties of systems of several quantum particles from the first principles of quantum mechanics. Included under this umbrella are various problems of fundamental importance in quantum chemistry, condensed matter physics, and materials science. Of particular interest is the electronic structure problem, the problem of determining the state of the electrons in a system with fixed atomic nuclei. Since direct numerical solution of the many-body Schrödinger equation is intractable even for systems of moderate size, a diverse array of approximate methods has been developed. The broad goals of this dissertation are to improve the mathematical understanding of certain widely-used approximations, as well as to propose new methods. Roughly speaking, we consider three (overlapping) categories of methods: Green's function methods, embedding methods, and variational methods.One can understand Green's function methods in terms of many-body perturbation theory, which computes series expansions of physical quantities about a non-interacting reference system. These expansions can be expressed graphically in terms of Feynman diagrams, which can in turn be reorganized, in some cases, into an expansion in terms of so-called bold diagrams. Green's function methods can be specified by choosing a subset of bold diagrams to approximate the sum. At the same time, such methods can be understood in terms of an object known as the Luttinger-Ward (LW) functional, which admits a representation in terms of the bold diagrams. Many aspects of these constructions are purely formal, and indeed the existence of the fermionic LW functional as a single-valued functional has recently been called into question. To contribute to the understanding of these issues, we provide rigorous proofs of the combinatorial construction and analytic interpretation of the bold diagrams in the simplified setting of a classical field theory. In this setting we also provide a rigorous non-perturbative construction of the LW functional via convex duality and prove several key properties, including continuity up to the boundary of its domain and asymptotics in the limit of large interaction.Quantum embedding methods, meanwhile, view a large system as being composed of smaller fragments that are treated with high accuracy and embedded in the larger system in a mutually consistent way. Inspired by a connection between the boundary analysis of the LW functional and embedding, we perform similar analysis for the 1-RDM theory for fermionic systems, which is also developed via convex duality, illustrating a relation to fermionic embedding methods such as the density matrix embedding theory (DMET).Another embedding method of note is the dynamical mean-field theory (DMFT), which is at the same time a Green's function method that can be understood in terms of the LW functional. DMFT relies on the solution of impurity problems, which specify the embedding of an interacting system into a non-interacting bath. Underlying DMFT is a result about the sparsity pattern of the self-energy matrix for impurity problems, which to our knowledge has not been proved in the literature. We provide a rigorous proof of this result in various classical and quantum settings. We go on to investigate the fermionic DMFT in depth, identifying the key mathematical structures that appear in the algorithmic loop for solving it and using these to prove the well-posedness of this loop, in a certain sense.Finally, we introduce a suite of new approaches to the quantum many-body problem that provide variational lower bounds to the ground-state energy. These methods, which combine the themes of convexity and embedding, are based on novel convex relaxations of the variational principles for the ground-state energies of many-body systems. To begin, we recover a second-quantized version of the formalism of strictly correlated electrons (SCE), which yields an exact expression for the exchange-correlation functional in Kohn-Sham density functional theory in the limit of infinite Coulomb repulsion in terms of the solution of a multi-marginal optimal transport problem. We introduce a semidefinite relaxation method for approximately solving this problem and obtaining a lower bound for the ground-state energy. The ideas underlying this relaxation are generalized considerably, outside the context of SCE, to yield much tighter lower bounds, which we validate numerically for both quantum spin systems and fermionic systems. We also describe how these relaxation methods can be interpreted as embedding methods via convex duality.