Methods similar to the parquet equations were first introduced in the context of many-body theory by de Dominicis and Martin. \cite{DeDominicis_1964a,DeDominicis_1964b}
The parquet formalism consists of a set of coupled equations, i.e. the Bethe-Salpeter, the parquet, and the Schwinger-Dyson equations, which include both single-particle Green's function and two-particle vertex functions.
The parquet equations, in principle, provide a framework for self-consistent determination of the self-energy and the vertex corrections.
The parquet formulation preserves, by construction, crossing symmetry and the Pauli exclusion principle.
Crossing symmetry means that a vertex in one channel can also produce a vertex in all other channels by pulling or crossing the vertex legs and multiplying by appropriate constants.
The parquet equations are essentially a generalization of the Bethe-Salpeter equation.
The Baym-Kadanoff approximation \cite{Baym_1961,Baym_1962} (or $\Phi$-derivable approximations) is similar to the parquet approach and can be used to compute one-body correlation functions but does not include self-consistency at the two-body level.
Baym-Kadanoff is primarily focused on the self-energy and satisfies continuity conditions.
However, it produces two-body correlation functions that violate the Pauli principle.
Parquet does enforce this (crossing) symmetry by construction and attaches primary importance on the two-body scattering vertex $\Gamma$, while $\Sigma$ has a subsidiary role and is obtained via integration of the Dyson-Schwinger equation.
The parquet equations guarantee the self-consistent treatment of both one-body and two-body propagators but break conservation laws.
The infinite-order Baym–Kadanoff approximation is also known as ``fluctuation exchange'' (FLEX) approximation. \cite{Bickers_1989}
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\section{Crossing symmetry}
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Let us introduce the composite variable $1\equiv(\br_1,t_1,\sigma_1)$ so that $\sum_{1}=\sum_{\sigma_1}\sum_{\br_1}\int_0^\beta\frac{dt_1}{\beta}$.
The full vertex function $F =\Gamma+\Gamma_P$ contains the fully irreducible terms produced by the input $\Lambda$ (which is fully irreducible) and the reducible part from the repeated scattering terms (iterations).
At the two-body level, there are three different channels for the reducible vertices: one pp channel $\Gamma^\pp$, and two ph (horizontal and vertical) channels, $\Bar{\Gamma}^\ph$ and $\Gamma^\ph$.
For the pp case, reducible diagrams are defined as the ones that can be separated into two pieces by cutting two fermion lines.
ph irreducibility is tricker as two channels exist.
Diagrams irreducible in the horizontal channel (i.e., those which cannot be split by a vertical cut through two lines) contribute to $\Gamma^\ph$, while diagrams irreducible in the vertical channel contribute to $\Bar{\Gamma}^\ph$.
The quantities $\Gamma^\ph$ and $\Gamma^\pp$ are analogs for ph and pp states of the irreducible self-energy $\Sigma$ for one-body states.
The complete ($\Gamma$ and $\Gamma_P$) and irreducible vertices ($\Gamma^\pp$, $\Gamma^\ph$, and $\Bar{\Gamma}^\ph$) are usually spin adapted (see below).
For example, $\Gamma^\ph$ is written as a sum of its density ($\Gamma^\ph_\text{d}$) and magnetic ($\Gamma^\ph_\text{m}$) components, while $\Gamma^\pp$ is decomposed in its singlet ($\Gamma^\pp_\text{s}$) and triplet ($\Gamma^\pp_\text{t}$) components.
Similar expressions can be found for $\Bar{\Gamma}^\ph$ (crossing relation), and the approximated forms of $\Gamma$ and $\Gamma_P$ (but usually violate crossing conditions).
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\section{Parquet equations}
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Let us first focus on the two-body vertices following the presentation of Bickers. \cite{Bickers_2004}
Thanks to the spin adaptation, we have the following picture: the electrons interacting through the exchange of four varieties of composite ``bosons'' made of electrons and holes which themselves strongly interact through the exchange of other bosons: density, magnetic, singlet-pair, and triplet-pair fluctuations.
Thanks to the crossing symmetry of the complete vertices, the parquet equations automatically build in the nonlinear coupling between dressed electron and boson excitations necessary for full consistency.
Now that we have the parquet equations written in a convenient spin-adapted form, the last step is to write down the self-energy via the Dyson-Schwinger equation.
Doing so displays the four varieties of composite bosons mentioned previously:
Because FLEX is a parquet with non-self-consistent two-body vertices, their expressions are a bit different and two flavours exist.
The first flavour includes the single-fluctuation-exchange diagrams while the second flavor includes also the so-called Aslamazov-Larkin diagrams.
When the vertex functions are spin-diagonalized omitting the Aslamazov-Larkin contributions, we recover the parquet equations where one sets $\Lambda^\irr=\Gamma^\ph=\Gamma^\pp= v$.
It has been shown that the Aslamazov-Larkin diagrams deterioetas the accuray of the two-body vertices obtained via FLEX.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
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%\section*{Data availability statement}
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%The data that supports the findings of this study are available within the article.% and its supplementary material.
