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Physical Review A

Planar version of Baym-Kadanoff theory

R. A. Smith

Published Oct 1, 1992• 17 citations
Publisher

Abstract

The Baym-Kadanoff theory of the conserving vertex is discussed within the context of the parquet summation of Feynman diagrams in nonrelativistic many-body theory. Since the parquet language provides a convenient way of characterizing all two-body vertices, the Baym-Kadanoff algorithm for generating a conserving two-body vertex takes a very straightforward form. We present the general theory and then illustrate the results using many of the simple conserving vertices. We then analyze the parquet two-body vertex within the theory, concluding that parquet is not a conserving theory. We show that any conserving theory that generates the direct parquet diagrams must also generate an infinite number of irreducible diagrams and derive integral equations for the minimal set of irreducible diagrams that would be needed in a conserving theory that produced the direct parquet diagrams. We present an algorithm that enables one to determine, for any direct or exchange parquet diagram, the parquet diagram whose contribution to the self-energy, when functionally differentiated, will generate the original parquet diagram. We then show that any set of diagrams for the two-body vertex that includes the bare interaction and satisfies the important properties of being conserving and antisymmetric under exchange of the outgoing particles must necessarily include all diagrams.

DOI
10.1103/physreva.46.4586

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The Baym-Kadanoff theory of the conserving vertex is discussed within the context of the parquet summation of Feynman diagrams in nonrelativistic many-body theory. Since the parquet language provides a convenient way of characterizing all two-body vertices, the Baym-Kadanoff algorithm for generating a conserving two-body vertex takes a very straightforward form. We present the general theory and then illustrate the results using many of the simple conserving vertices. We then analyze the parquet two-body vertex within the theory, concluding that parquet is not a conserving theory. We show that any conserving theory that generates the direct parquet diagrams must also generate an infinite number of irreducible diagrams and derive integral equations for the minimal set of irreducible diagrams that would be needed in a conserving theory that produced the direct parquet diagrams. We present an algorithm that enables one to determine, for any direct or exchange parquet diagram, the parquet diagram whose contribution to the self-energy, when functionally differentiated, will generate the original parquet diagram. We then show that any set of diagrams for the two-body vertex that includes the bare interaction and satisfies the important properties of being conserving and antisymmetric under exchange of the outgoing particles must necessarily include all diagrams.
We exploit the parquet formalism to derive exact flow equations for the two-particle-reducible four-point vertices, the self-energy, and typical response functions, circumventing the reliance on higher-point vertices. This includes a concise, algebraic derivation of the multiloop flow equations, which have previously been obtained by diagrammatic considerations. Integrating the multiloop flow for a given input of the totally irreducible vertex is equivalent to solving the parquet equations with that input. Hence, one can tune systems from solvable limits to complicated situations by variation of one-particle parameters, staying at the fully self-consistent solution of the parquet equations throughout the flow. Furthermore, we use the resulting differential form of the Schwinger–Dyson equation for the self-energy to demonstrate one-particle conservation of the parquet approximation and to construct a conserving two-particle vertex via functional differentiation of the parquet self-energy. Our analysis gives a unified picture of the various many-body relations and exact renormalization group equations.
In many-body theory, the single-particle self-energy can be constructed from the two-body vertex, the single-particle Green's function, and the interaction in a well-known way. Functional differentiation of the self-energy leads back to the two-body vertex. Given a set of two-body vertex diagrams, the process of constructing self-energy diagrams and functionally differentiating them generates a second set of two-body vertex diagrams. If the initial set of diagrams includes the noninteracting two-body Green's function, the two sets will be equal if and only if the initial set includes all diagrams. We discuss this result from the perspective of the parquet theory.
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.
One of the fundamental approaches to microscopic many-body theory is through the use of perturbation theory. This paper presents a clear derivation of the equations that sum the two-body and three-body reducible diagrams that are generated from some input set of irreducible diagrams (typically the bare interaction) in a crossing-symmetric form. The diagrammatic structure is useful for both bosons and fermions at zero and finite temperature. The essential features of the procedure for the two-body case are the following. (i) All constructions involve only direct diagrams. (ii) Diagrams can be joined together by five different ladder, chain, and vertex correction operations. (iii) The sequences in which these operations are applied satisfy certain associativity relations. (iv) These associativity relations are used to insure that each permitted diagram is generated exactly once. (v) The ladder-and-chain operations are used to construct reducible vertex sums ${\mathrm{\ensuremath{\Gamma}}}_{0}$ which have no external vertex corrections. (vi) The full direct vertex \ensuremath{\Gamma} consists of the vertex ${\mathrm{\ensuremath{\Gamma}}}_{0}$ plus ${\mathrm{\ensuremath{\Gamma}}}_{0}$ dressed on either or both sides by the full direct vertex \ensuremath{\Gamma}. (vii) Finally, the full vertex is obtained by adding the exchange of the direct vertex. The extension to the three-body formalism proceeds in much the same way.
We exploit the parquet formalism to derive exact flow equations for the two-particle-reducible four-point vertices, the self-energy, and typical response functions, circumventing the reliance on higher-point vertices. This includes a concise, algebraic derivation of the multiloop flow equations, which have previously been obtained by diagrammatic considerations. Integrating the multiloop flow for a given input of the totally irreducible vertex is equivalent to solving the parquet equations with that input. Hence, one can tune systems from solvable limits to complicated situations by variation of one-particle parameters, staying at the fully self-consistent solution of the parquet equations throughout the flow. Furthermore, we use the resulting differential form of the Schwinger-Dyson equation for the self-energy to demonstrate one-particle conservation of the parquet approximation and to construct a conserving two-particle vertex via functional differentiation of the parquet self-energy. Our analysis gives a unified picture of the various many-body relations and exact renormalization group equations.
We present a general framework for investigating the stability of solutions being a parquet-type extension of the Baym-Kadanoff construction of conserving approximations. To obtain a consistent description of one- and two-particle quantities, needed for the stability analysis, we use explicit equations for one- and two-particle Green functions simultaneously. We formulate a complete set of stability criteria and show that each instability, singularity in a two-particle function, is connected with a symmetry-breaking order parameter, either of density or anomalous type. We explicitly study the Hubbard model at intermediate coupling and demonstrate that approximations with static vertices become unstable with respect to spin flips, before the Mott-Hubbard metal-insulator transition in the paramagnetic phase or the Kondo strong-coupling regime can be reached. Further on, we use the parquet approximation and turn it into a workable scheme with dynamical vertex corrections. We obtain a theory with two-particle self-consistence, the complexity of which is comparable with fluctuation-exchange-type approximations. We show that it is the simplest consistent theory free of spurious phases being able to correctly describe qualitatively, quantum critical points and the transition from weak to strong coupling in correlated electron systems.