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Physics Letters B

Crossing-symmetric rings, ladders, and exchanges

Alexander Lande, R. A. Smith

Published Nov 1, 1983• 22 citations
Publisher

Abstract

A new approach is used in the description of fermion and boson systems at zero or finite temperature. We generalize the familiar ladder and chaining operations to construct a crossing-symmetric approximation to the two-particle vertex from the bare interaction. The explicit rules for this construction are given in terms of Feynman propagators. The high level of symmetry of the vertex simplifies the theory. In one form, the structure appears as a generalization of parquet diagrams. The energy, self-energy, and vertex satisfy a number of consistency relations.

DOI
10.1016/0370-2693(83)90492-6

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A new approach is used in the description of fermion and boson systems at zero or finite temperature. We generalize the familiar ladder and chaining operations to construct a crossing-symmetric approximation to the two-particle vertex from the bare interaction. The explicit rules for this construction are given in terms of Feynman propagators. The high level of symmetry of the vertex simplifies the theory. In one form, the structure appears as a generalization of parquet diagrams. The energy, self-energy, and vertex satisfy a number of consistency relations.
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Parquet diagrams sum self-consistently Feynman graphs for the vertex function with all two-particle multiple scatterings. We show how the complete parquet equations for the Hubbard-like models can be integrated to a generating functional from which all thermodynamic quantities are derived via (functional) derivatives. An explicit Luttinger-Ward functional Φ[G;Λ,K;U]Φ[G;Λ,{\cal K};U]Φ[G;Λ,K;U] is constructed containing only the renormalized one-particle, GGG, irreducible, ΛΛΛ, and reducible, K{\cal K}K, two-particle propagators as independent variational functions. The parquet approximation is proven to be a thermodynamically consistent, ΦΦΦ-derivable theory obeying the necessary conservation laws.
We address the problem of finding self-consistent parquet equations for two-body correlated Greens functions that arise out of a cumulant expansion. The general theory as developed for non-relativistic electron systems reproduces the conventional two-body parquet diagrammatic technique for the vertex functions. As an application, the crossing-invariant linearized basic parquet equations are treated for the correlated two-particle Greens function. It is pointed out that besides the Cooperon there are two additional prominent crossing-symmetric two-particle channels in many-particle systems with pair interaction.
We address the problem of finding self-consistent parquet equations for two-body correlated Greens functions that arise out of a cumulant expansion. The general theory as developed for non-relativistic electron systems reproduces the conventional two-body parquet diagrammatic technique for the vertex functions. As an application, the crossing-invariant linearized basic parquet equations are treated for the correlated two-particle Greens function. It is pointed out that besides the Cooperon there are two additional prominent crossing-symmetric two-particle channels in many-particle systems with pair interaction.
Abstract The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here, we present first results of the numerical implementation of this method by applying it to the half-filled Hubbard model on the square lattice at weak coupling. The improved algorithmic performance allows us to evaluate the parquet approximation for a 16×1616\times 1616×16 16 × 16 lattice, retaining the full momentum and frequency structure of the various vertex functions. We discuss their symmetries and consider parametrizations of their momentum dependence using the truncated-unity approximation. Graphical abstract
A quantum-field approach for describing many-particle Fermi systems at finite temperatures and with spontaneously broken symmetry has been proposed. A generalized model of self-consistent field (SCF), which allows one to describe the states eligible for this system with various symmetries, is used as the initial approximation. A perturbation theory has been developed, and a diagram technique for temperature Green's functions (GFs) has been constructed. The Dyson's equation for the self-energy and vertex parts has been deduced.