# Strongly correlated electrons: Analytic mean-field theories with two-particle self-consistency

## TL;DR

This paper develops an analytic mean-field framework incorporating two-particle self-consistency via parquet equations, enabling accurate modeling of strongly correlated electrons and avoiding unphysical artifacts.

## Contribution

It introduces a general scheme for self-consistent approximations using parquet equations, ensuring quantum criticality is preserved in strong coupling regimes.

## Key findings

- Applied to the single-impurity Anderson model demonstrating Kondo behavior.
- Ensured compliance with Ward identity and Schwinger-Dyson equation.
- Provided a static approximation capturing essential physics of strong correlations.

## Abstract

A two-particle self-consistency is rarely part of mean-field theories. It is, however, essential for avoiding spurious critical transitions and unphysical behavior. We present a general scheme for constructing analytically controllable approximations with self-consistent equations for the two-particle vertices based on the parquet equations. We explain in detail how to reduce the full set of parquet equations not to miss quantum criticality in strong coupling. We further introduce a decoupling of convolutions of the dynamical variables in the Bethe-Salpeter equations to make them analytically solvable. We connect the self-energy with the two-particle vertices to satisfy the Ward identity and the Schwinger-Dyson equation. We discuss the role of the one-particle self-consistency in making the approximations reliable in the whole spectrum of the input parameters. We demonstrate the general construction on the simplest static approximation that we apply to the Kondo behavior of the single-impurity Anderson model.