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

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.

arXiv:1807.02898New Journal of PhysicsDec 6, 201854 citations

Multiloop functional renormalization group for general models

Fabian B. Kugler, Jan von Delft

We present multiloop flow equations in the functional renormalization group (fRG) framework for the four-point vertex and self-energy, formulated for a general fermionic many-body problem. This generalizes the previously introduced vertex flow [F. B. Kugler and J. von Delft, Phys. Rev. Lett. 120, 057403 (2018)] and provides the necessary corrections to the self-energy flow in order to complete the derivative of all diagrams involved in the truncated fRG flow. Due to its iterative one-loop structure, the multiloop flow is well suited for numerical algorithms, enabling improvement of many fRG computations. We demonstrate its equivalence to a solution of the (first-order) parquet equations in conjunction with the Schwinger-Dyson equation for the self-energy.We present multiloop flow equations in the functional renormalization group (fRG) framework for the four-point vertex and self-energy, formulated for a general fermionic many-body problem. This generalizes the previously introduced vertex flow [F. B. Kugler and J. von Delft, Phys. Rev. Lett. 120, 057403 (2018)] and provides the necessary corrections to the self-energy flow in order to complete the derivative of all diagrams involved in the truncated fRG flow. Due to its iterative one-loop structure, the multiloop flow is well suited for numerical algorithms, enabling improvement of many fRG computations. We demonstrate its equivalence to a solution of the (first-order) parquet equations in conjunction with the Schwinger-Dyson equation for the self-energy.

arXiv:1707.04536Physical review. B./Physical review. BJan 31, 201888 citations

Multiloop flow equations for single-boson exchange fRG

Marcel Gievers, Elias Walter, Anxiang Ge +2 more

Abstract The recently introduced single-boson exchange (SBE) decomposition of the four-point vertex of interacting fermionic many-body systems is a conceptually and computationally appealing parametrization of the vertex. It relies on the notion of reducibility of vertex diagrams with respect to the bare interaction U, instead of a classification based on two-particle reducibility within the widely used parquet decomposition. Here, we re-derive the SBE decomposition in a generalized framework (suitable for extensions to, e.g., inhomogeneous systems or real-frequency treatments) following from the parquet equations. We then derive multiloop functional renormalization group (mfRG) flow equations for the ingredients of this SBE decomposition, both in the parquet approximation, where the fully two-particle irreducible vertex is treated as an input, and in the more restrictive SBE approximation, where this role is taken by the fully U-irreducible vertex. Moreover, we give mfRG flow equations for the popular parametrization of the vertex in terms of asymptotic classes of the two-particle reducible vertices. Since the parquet and SBE decompositions are closely related, their mfRG flow equations are very similar in structure. Graphic abstractAbstract The recently introduced single-boson exchange (SBE) decomposition of the four-point vertex of interacting fermionic many-body systems is a conceptually and computationally appealing parametrization of the vertex. It relies on the notion of reducibility of vertex diagrams with respect to the bare interaction U, instead of a classification based on two-particle reducibility within the widely used parquet decomposition. Here, we re-derive the SBE decomposition in a generalized framework (suitable for extensions to, e.g., inhomogeneous systems or real-frequency treatments) following from the parquet equations. We then derive multiloop functional renormalization group (mfRG) flow equations for the ingredients of this SBE decomposition, both in the parquet approximation, where the fully two-particle irreducible vertex is treated as an input, and in the more restrictive SBE approximation, where this role is taken by the fully U-irreducible vertex. Moreover, we give mfRG flow equations for the popular parametrization of the vertex in terms of asymptotic classes of the two-particle reducible vertices. Since the parquet and SBE decompositions are closely related, their mfRG flow equations are very similar in structure. Graphic abstract

arXiv:2201.04878The European Physical Journal BJul 1, 202223 citations

Multiloop Functional Renormalization Group That Sums Up All Parquet Diagrams

Fabian B. Kugler, Jan von Delft

We present a multiloop flow equation for the four-point vertex in the functional renormalization group (FRG) framework. The multiloop flow consists of successive one-loop calculations and sums up all parquet diagrams to arbitrary order. This provides substantial improvement of FRG computations for the four-point vertex and, consequently, the self-energy. Using the x-ray-edge singularity as an example, we show that solving the multiloop FRG flow is equivalent to solving the (first-order) parquet equations and illustrate this with numerical results.We present a multiloop flow equation for the four-point vertex in the functional renormalization group (FRG) framework. The multiloop flow consists of successive one-loop calculations and sums up all parquet diagrams to arbitrary order. This provides substantial improvement of FRG computations for the four-point vertex and, consequently, the self-energy. Using the x-ray-edge singularity as an example, we show that solving the multiloop FRG flow is equivalent to solving the (first-order) parquet equations and illustrate this with numerical results.

arXiv:1703.06505Physical Review LettersJan 31, 201887 citations

Generating functional for the full parquet approximation

V. Janiš

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.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.

arXiv:cond-mat/9806118Jun 9, 19981 citation

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.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 AOct 1, 199217 citations

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.