Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9806118

Generating functional for the full parquet approximation

V. Janiš

Published Jun 9, 1998• 1 citation
arXivHTMLPDF

Abstract

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 $Φ[G;Λ,{\cal K};U]$ is constructed containing only the renormalized one-particle, $G$, irreducible, $Λ$, and reducible, ${\cal K}$, two-particle propagators as independent variational functions. The parquet approximation is proven to be a thermodynamically consistent, $Φ$-derivable theory obeying the necessary conservation laws.

DOI
10.48550/arxiv.cond-mat/9806118

Related papers

Derivation of exact flow equations from the self-consistent parquet relations

Kugler, F., von Delft, J.

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.

•Dec 1, 2018•21 citations

Derivation of exact flow equations from the self-consistent parquet relations

Fabian B. Kugler, Jan von Delft

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.

arXiv:1807.02898•New Journal of Physics•Dec 6, 2018•54 citations

Planar version of Baym-Kadanoff theory

R. A. Smith

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.

Physical Review A•Oct 1, 1992•17 citations

A functional-analysis derivation of the parquet equation

Christian Eckhardt, Patrick Kappl, Anna Kauch +1 more

The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green's function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely employing techniques of functional analysis namely functional Legendre transformations and functional derivatives. The advantage of a derivation in terms of a straightforward calculation is twofold: (i) the quantities appearing in the calculation have a clear mathematical definition and interpretation as derivatives of the Luttinger--Ward functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higher-order Green's functions potentially leading to a classification of these in terms of their (ir)reducible components.

arXiv:2305.16050•May 25, 2023•0 citations

A functional-analysis derivation of the parquet equation

Christian Eckhardt, Patrick Kappl, Anna Kauch +1 more

The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green’s function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely employing techniques of functional analysis namely functional Legendre transformations and functional derivatives. The advantage of a derivation in terms of a straightforward calculation is twofold: (i) the quantities appearing in the calculation have a clear mathematical definition and interpretation as derivatives of the Luttinger-Ward functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higher-order Green’s functions potentially leading to a classification of these in terms of their (ir)reducible components.

SciPost Physics•Nov 24, 2023•1 citation

Strongly correlated electrons: Dynamical vertex renormalization

V. Janiš

Single-band Hubbard model at criticality of the metal-insulator transition is studied using approximations derived from parquet theory. It is argued that only the electron-hole and interaction two-particle channels in the parquet algebra are relevant. A scheme is proposed how to reduce the parquet equations to a manageable form the complexity of which is comparable with single-channel approximations such as the renormalized RPA. The newly derived approximation, however, contains dynamical vertex corrections, remains self-consistent at the two-particle level and allows only for integrable singularities. A qualitatively new approach for studying two-particle singularities at zero temperature is obtained.

arXiv:cond-mat/9810175•Oct 15, 1998•0 citations
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 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 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.
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.
The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green's function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely employing techniques of functional analysis namely functional Legendre transformations and functional derivatives. The advantage of a derivation in terms of a straightforward calculation is twofold: (i) the quantities appearing in the calculation have a clear mathematical definition and interpretation as derivatives of the Luttinger--Ward functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higher-order Green's functions potentially leading to a classification of these in terms of their (ir)reducible components.
The parquet equation is an exact field-theoretic equation known since the 60s that underlies numerous approximations to solve strongly correlated Fermion systems. Its derivation previously relied on combinatorial arguments classifying all diagrams of the two-particle Green’s function in terms of their (ir)reducibility properties. In this work we provide a derivation of the parquet equation solely employing techniques of functional analysis namely functional Legendre transformations and functional derivatives. The advantage of a derivation in terms of a straightforward calculation is twofold: (i) the quantities appearing in the calculation have a clear mathematical definition and interpretation as derivatives of the Luttinger-Ward functional; (ii) analogous calculations to the ones that lead to the parquet equation may be performed for higher-order Green’s functions potentially leading to a classification of these in terms of their (ir)reducible components.
Single-band Hubbard model at criticality of the metal-insulator transition is studied using approximations derived from parquet theory. It is argued that only the electron-hole and interaction two-particle channels in the parquet algebra are relevant. A scheme is proposed how to reduce the parquet equations to a manageable form the complexity of which is comparable with single-channel approximations such as the renormalized RPA. The newly derived approximation, however, contains dynamical vertex corrections, remains self-consistent at the two-particle level and allows only for integrable singularities. A qualitatively new approach for studying two-particle singularities at zero temperature is obtained.