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

TL;DR

This paper derives exact flow equations from the parquet formalism, enabling self-consistent solutions of many-body problems without reliance on higher-point vertices, and demonstrates conservation laws within this framework.

Contribution

It provides a novel algebraic derivation of multiloop flow equations from the parquet relations, unifying many-body relations and renormalization group equations.

Findings

Derived exact flow equations for two-particle vertices and self-energy.

Established one-particle conservation within the parquet approximation.

Unified various many-body relations and renormalization group equations.

Abstract

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.