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.

from more traditional RG schemes [3,10], which, instead of solely using a scale-dependent propagator, restrict all involved energy-momenta to decreasing energy-momentum shells (often referred to as "mode elimination"). Here, we simply substitute G → G Λ in the well known many-body relations and study the behavior of the solution to these equations upon varying Λ.
As a result, we derive exact flow equations for the two-particle-reducible four-point vertices, the self-energy, and response functions. This provides a concise, algebraic derivation of the multiloop fRG (mfRG) flow equations, which have previously been obtained using diagrammatic arguments [11][12][13]. Our analysis also reveals how one can perform such multiloop flows beyond the parquet approximation (PA), thus including higher-order expressions for the totally irreducible vertex. Moreover, we establish an intimate connection between the functional derivative of the self-energy and the fRG flow equation for the self-energy: the latter constitutes an integration of the former along a specific path in the space of theories.
On a slightly different note, we use our approach to address fundamental questions of (traditional) parquet theory (i.e., without an explicit RG treatment): On the one hand, we demonstrate that the parquet self-energy can be obtained from the Schwinger-Dyson equation (SDE) using either of two possible orderings of the bare and full vertex. According to Baym and Kadanoff [14], it then follows that the PA fulfills one-particle conservation laws. On the other hand, we give an explicit construction to obtain a new, conserving vertex from the parquet self-energy, equivalent to taking the functional derivative. This construction not only allows one to quantify the degree to which the PA violates two-particle conservation laws. It can also be used to modify the PA, which fulfills the SDE but violates two-particle conservation, to obtain a fully conserving solution, albeit violating the SDE. As we show in the appendix, a fulfillment of both the SDE and the functional-derivative relation necessarily amounts to the exact solution of the many-body problem, in agreement with a result by Smith [15].
The paper is structured as follows. In Section II, we first focus (as is typical for RG approaches) on the effective interactions: we derive flow equations for the two-particle-reducible four-point vertices based on the parquet formalism, assuming the one-particle propagator to be given. Then, in Section III, we complement the flow of the four-point vertex by the flow of the self-energy, considering the various relations at hand. In Section IV, we use our approach to discuss conservation properties of the PA. Finally, in Section V, we derive (dependent) flow equations for response functions, i.e., three-point vertices and suceptibilities, used to study collective excitations. In Section VI, we summarize our results.

A. Preliminaries
We consider a general theory of interacting fermions, defined by the action x ,x c x − 1 4 x ,x,y ,y Γ 0;x ,y ;x,ycx c y c y c x , with a bare propagator G 0 and a bare four-point vertex Γ 0 , which is antisymmetric in its first and last two arguments.
The index x denotes all quantum numbers of the Grassmann field c x . Correlation functions of fields, corresponding to time-ordered expectation values of operators, are given by the functional integral where Z ensures normalization, such that 1 = 1. Two-point correlation functions are represented by the full propagator G x,x = − c xcx ; four-point correlation functions can be expressed via the full (one-particle-irreducible) four-point vertex Γ: The notation given so far is identical to the one in Ref. 12; all formulae further needed in this paper are defined in Appendix A. In the following derivation of flow equations, we use a compact notation of contractions and need not write quantum numbers (such as x, x , etc.) explicitly.

B. Parquet equations for the four-point vertex
The fRG flow equation for the four-point vertex, Γ (4) ≡ Γ, contains the six-point vertex, Γ (6) , which poses great difficulty for a numerical treatment. Similarly, the SDE (equation of motion) for Γ contains Γ (6) and therefore is 1. The Bethe-Salpeter equations for the channels r = a, p, t are solved in an RG approach by introducing a scale (Λ) dependence to the propagators connecting the vertices. Consequently, γr, Γ, and Ir inherit a scale dependence while the totally irreducible vertex R remains as given input. (See Appendix A for details on the diagrammatic notation.) As prime example for the scale dependence, one can multiply the frequency-dependent propagator by a step function, G Λ (ω) = Θ(|ω| − Λ)G(ω), such that the many-body relations are trivially solved at Λi = ∞ and reproduce the desired solution at Λ f = 0.
likewise impractical. To circumvent the calculation of Γ (6) , we revert to the parquet formalism [2,9], which provides self-consistent equations for the two-particle-reducible contributions to the four-point vertex Γ but assumes as input a given, totally irreducible four-point vertex R. In a diagrammatic expansion, R is given by the bare vertex, Γ 0 , with corrections starting at fourth order. The famous parquet approximation [16][17][18] (see Section IV) consists of using R = Γ 0 and allows one to sum up all leading logarithmic contributions in logarithmically divergent perturbation theories [9,19]. Importantly, however, the parquet equations can be used more generally as an exact classification of all diagrams of the four-point vertex.
In the parquet formalism, one decomposes the full four-point vertex, Γ, into the totally irreducible vertex, R, and the three two-particle-reducible vertices γ r , r ∈ {a, p, t} [20]. Diagrams belonging to γ r are reducible in channel r, i.e., they can be separated into two parts by cutting two antiparallel, parallel, or transverse antiparallel lines, respectively. Diagrams that cannot be separated in this way belong to R. (For exemplary diagrams, see Fig. 9 in Appendix A.) While the γ r are subject to further equations, this set of coupled equations closes only for a fixed choice of R.
Let us assume a given expression for the totally irreducible vertex, R. Furthermore, we will for now assume the one-particle propagator, G, to be given; computation of G via the self-energy will be discussed later. The parquet equations, involving the two-particle-reducible vertices, γ r , and two-particle-irreducible vertices, I r , read For given R, these equations must be solved self-consistently to obtain the appropriate reducible vertices, γ r , that complement the full vertex, Γ. In Eq. (4a), we use the notationr for the complementary channel of a given channel r, such that γr = r =r γ r . The Bethe-Salpeter equation (BSE) (4b) describes two vertices, I r and Γ, connected by a bubble, Π r , of two dressed propagators in channel r (see also Fig. 1). This bubble of vertices can be expressed as a matrix multiplication (given a suitable parametrization depending on the channel r, cf. Appendix A), as indicated by the symbol • attached to Π r . Note that Π p and Π t implicitly contain a factor of 1/2 and (−1), respectively. In the following, we list relations that can be easily deduced from the parquet equations (4) and will be used repeatedly in the derivations of flow equations. The combination of Eqs. (4a) and (4b) directly yields Γ = I r + I r • Π r • Γ (for all channels r). Exploiting the multiplicative structure, we can isolate Γ on the l.h.s. to obtain the inverted BSE, Here, we assume that Γ can be determined from I r , i.e., (1 − I r • Π r ) is indeed invertible. A further straightforward manipulation yields an extended BSE, Using the inverted BSE (5), one directly sees (by isolating γ r ) that the order of the vertices in the BSE (4b) is irrelevant: C. Flow of the four-point vertex The central aspect of our RG treatment is incorporated by attaching a scale (Λ) dependence to the propagator, G → G Λ , appearing in the self-consistent many-body relations. The physical picture is that Λ separates high-and 2. (a) Exact mfRG flow equation for the reducible vertex γa, involving the differentiated propagator,Ġ, (line with two vertical dashes) and the differentiated irreducible vertex given byİr = r =rγ r ≡γr (asṘ = 0 in our construction). (b) Exact fRG flow equation for γa involving the single-scale propagator, S = ∂ΛG | Σ=const , (line with one vertical dash) and the six-point vertex, whose contribution is (for conceptual purposes) reduced to the part reducible in the a channel via the projector Pa.
low-energy degrees of freedom, and by using G Λ we allow for successive renormalization of the low-energy (< Λ) theory by high-energy (> Λ) degrees of freedom as Λ is decreased. However, one can also simply consider Λ as some additional dependence in the propagators connecting the vertices in the BSEs: G → G Λ , Π r → Π Λ r (cf. Fig. 1). Hence, the reducible vertices γ Λ r -and consequently Γ Λ and I Λ r -will inherit a scale (Λ) dependence, obtained from the requirement that the parquet relations be fulfilled for each value of Λ, while R remains as given input.
The scale dependence is auxiliary in the sense that we are ultimately interested in the fully renormalized theory: we are interested in γ Λ f r = γ r where (at the final scale) G Λ f = G. Suppose we know the vertices at the initial scale, i.e., we can solve the BSEs using G Λi . Then, we can obtain γ Λ f r by solving a differential equation specified by the initial condition together with the flow ∂ Λ γ Λ r ≡γ Λ r , which is induced by the scale dependence of G Λ in the BSEs. We remark that it is natural to exclude the totally irreducible vertex R from the renormalization flow, as it constitutes precisely the part of the vertex that cannot be constructed iteratively and therefore does not have a flow equation that allows for an efficient (i.e., iterative one-loop) calculation.

Flow equation
To find the scale dependence of the two-particle-reducible vertices, γ r , we start by differentiating the BSEs w.r.t. Λ (suppressing the Λ dependence to lighten the notation) according to the product rule and decomposing the full vertex via the parquet equation (4a):γ Similar to the manipulations in Eq. (7), we bringγ r to the l.h.s. and subsequently multiply by (1 − I r • Π r ) −1 from the left. According to the inverted BSE (5), we geṫ and, resolving the remaining inverse by the extended BSE (6), we finḋ The algebraic derivation of this exact flow equation, as the differential form of the BSE (4b), is our first main result. It is depicted diagrammatically in Fig. 2(a) (exemplified by the a channel) and contrasted with the corresponding standard fRG flow equation [ Fig. 2(b)]. It describes the flow of the reducible vertices, γ r ; the totally irreducible vertex, R, does not have an efficient flow equation and remains as input. SinceṘ = 0, we haveİ r = r =rγ r ≡γr, and Eq. (10) constitutes a closed, coupled set of differential equations for all reducible vertices γ r . The natural way to solve these equations is to start by computing the independent, one-loop part,γ From the given derivation, it is clear that, if the scale dependence of G is chosen such that we are initially able to solve the BSEs (using G Λi ) and finally revert to the original theory (G Λ f = G), then solving the mfRG vertex flow (10) is equivalent to solving the BSEs (4b). In the same way that any solution of the BSEs depends on a certain choice of R, so do results of mfRG. Note, however, that the multiloop flow equation only requires the initial condition of the full vertex Γ Λi , containing R, and not of the individual reducible vertices; the decomposition intoγ r is only performed on the differential level. The degree of approximation in this approach is encoded in the underlying expression for R, which can range from the simplest approximation, R = Γ 0 , to the exact object, R ex .

Examples
Let us give some examples for possible flows which are specified by the input R and the choice of G Λi initializing the progression towards G Λ f = G.
(i) The BSEs at the initial scale can be trivially solved if G Λi = 0: Due to Π Λi r = 0, the corresponding initial condition for the reducible vertices is γ Λi r = 0. As we introduce the scale dependence only for the propagators connecting the vertices in the BSEs but leave the totally irreducible vertex R-the input to the parquet equationsunchanged, the initial condition for the full vertex is given by Γ Λi = R [25]. Hence, the mfRG flow generates all two-particle-reducible diagrams given the irreducible building block R; the special case of R = Γ 0 yields all diagrams of the parquet approximation (PA) [11,12].
(ii) The mfRG flow (10) is an exact flow equation for the two-particle-reducible vertices and thus gives us full control over the vertices corresponding to given propagators G Λ . Immediate consequences are that (a) for given boundary conditions G Λi , G Λ f , we are completely free to choose any specific Λ dependence in G Λ -the results of the flow do not depend on this choice; and (b) that we can perform loops in theory space, going from G Λi to G Λ f = G Λi without any loss of information. Conceptually, this underlines the power of the mfRG flow; practically, it can also be used as a consistency check for a numerical implementation (which might employ approximate parametrizations of the vertex functions, etc.). We emphasize that, while both properties directly follow from the given derivation based on the BSEs, they are violated in the widely used one-loop form (γ r ≈γ If we assume (from a conceptual point of view) we were to possess the exact solution of the many-body problem in the form of G Λi = G ex , Γ Λi = Γ ex , then such a vertex flow would return to the exact result, too. This corresponds to the fact that solving the parquet equations with G ex and R ex yields the exact full vertex, Γ ex : At each value of Λ, the reducible vertices γ Λ r solve the BSEs with propagators G Λ and fixed totally irreducible vertex R = R Λ . At Λ f , the BSEs with G Λ f = G ex reproduce γ ex r and thus Γ ex = R ex + r γ ex r .
(iii) As a highly correlated and, yet, numerically tractable initial condition [26], one can choose the solution of dynamical mean-field theory (DMFT) [27] and use the mfRG flow to generate nonlocal correlations [21,28], thus extending the DMF 2 RG idea [28] to multiloop DMF 2 RG [11,12]. A related approach that gives diagrammatic, nonlocal corrections to DMFT is given by the dynamical vertex approximation (DΓA) [29][30][31]. This approach directly employs the parquet equations, using as input R DMFT , the totally irreducible vertex from the local DMFT solution [22]. If we used the same initial propagator G Λi = 0 as in example (i) above, we would start the vertex flow from Γ Λi = R DMFT in perfect analogy to the DΓA algorithm. However, at this point we can leverage the flexibility of the RG framework: the parquet equations are solved by the mfRG flow (10) independent of the specific Λ dependence in G Λ and independent of the initial form of G Λi . If we use G Λi = G DMFT (as opposed to G Λi = 0), we start the vertex flow not from R DMFT but from the full vertex Γ DMFT [28]. [Recall that the decomposition into two-particle channels in Eq. (10) occurs only for differentiated verticesγ r , which are ultimately combined to giveΓ = rγ r .] Consequently, the multiloop flow is not affected by the (likely) unphysical divergences of the totally irreducible vertex, R, that have been observed in strongly correlated systems [32][33][34][35][36]; it can thus be used to analyze such systems in wider regimes of the phase diagram.
So far, we have assumed the dressed propagator, G, to be known. However, as this is in general not the case, we now combine Eq. (10) with a self-energy flow,Σ Λ , to generate G Λ during the flow. Via the Dyson equation, we then

FIG. 3. (a) Schwinger-Dyson equation (SDE)
for the self-energy, where the second term contains two equivalent lines connected to antisymmetric vertices and hence requires a factor of 1/2. One notes that the three propagators in the second summand can be both viewed as contracting a parallel and antiparallel bubble of the vertices Γ0 and Γ. (b) Multiloop fRG self-energy flow [12], derived from the SDE in the parquet approximation. The first term,Σ std , constitutes the standard fRG self-energy flow.

III. DERIVATION OF THE SELF-ENERGY FLOW
First, let us mention that the straightforward derivation of the vertex flow was based on the parquet equations (for given input R). These merely represent a classification of diagrams, reducing the need for an explicit input expression to the most fundamental building block. We did not use equations which provide a construction of the four-point vertex from higher-point vertices, such as the SDE involving Γ (6) , or a functional derivative connecting four-and six-point vertices.
By contrast, we next want to construct the self-energy, Σ, from the four-point vertex, Γ. For this purpose, three equations are available: (i) the SDE relating Σ to Γ, typically used in the parquet formalism [2], (ii) a functional derivative between self-energy and two-particle-irreducible vertex, known from Hedin's equations [1] and Φ-derivable approaches [37,38], and (iii) the fRG flow equation for Σ [5]. In Section III B, we show that the fRG flow for Σ can be easily derived from the functional derivative (as a necessary condition). In Appendix B, we show that the SDE and the functional derivative are complementary in the sense that any solution that fulfills both equations must be the exact solution. It is therefore not surprising that it is complicated to relate a self-energy flow to the SDE for Σ. Nevertheless, we will use the SDE to derive a self-energy flow (different from the standard fRG flow), which is well-suited for the parquet approximation (PA) and allows us to gain insight into its conservation properties (see Section IV). While this multiloop flow deduced from the SDE indeed proves beneficial in the PA [12], the general advantages and disadvantages of the different starting points (i) and (ii) are not entirely clear (see also Section III A 2).

A. Self-energy flow from the Schwinger-Dyson equation
Deriving a flow equation from the SDE of the self-energy is a difficult task since (as already mentioned) SDEs and differential equations are of fundamentally different nature-for instance, SDEs always contain the bare interaction whereas differential equations are typically phrased with renormalized objects only. In Ref 8, the SDE was used to derive the fRG self-energy flow up to terms O (Γ) 3 ; here, we demonstrate agreement up to O (Γ) 4 . In fact, we derive the mfRG self-energy flow from Ref. 12, which includes important terms that would be neglected if one simply inserts the approximate parquet vertex into the standard fRG self-energy flow equation [12]. The calculation with the main results given in Eqs. (26) and (30) (see also Fig. 3) is presented in detail in the following Section III A 1 and interpreted in Section III A 2.

Flow equation
The starting point of our calculation is the Schwinger-Dyson equation for the self-energy [cf. Fig. 3(a)]: Here, we have used bubbles in either the a or the p channel, as well as the contraction of two vertex legs with a propagator [denoted by Γ · G, cf. Appendix A, Eq. (A5)]. As we can freely choose the specific propagator for the final contraction, we can write the SDE with a bubble in either the p or the a channel-the factor of 1/2 is implicitly contained in Π p and must be explicitly written when using Π a . The presence of two equivalent lines [i.e., parallel lines connected to (anti)symmetric vertices] in the second summand of the SDE opens the possibility for further manipulations. For this, let us explicitly denote the propagators contained in a bubble by Π r;G1,G2 ; the standard bubble is then simply given by Π r ≡ Π r;G,G . In the SDE, we can not only freely choose the propagator used in the final contraction [Eq. (12a)], we can also switch the equivalent lines by crossing two We will use the contracted crossing relations extensively on the relevant vertices, which obey the crossing symmetrieŝ Note that the vertices in the particle-hole channels a, t are mapped onto each other upon crossing two external legs. For this reason, we will often combine contributions from the a and t channel in the following calculations. The SDE yields a scale dependent self-energy if we attach a Λ dependence to every propagator connecting the vertices in Eq. (11) and account for the Λ dependence of the four-point vertex, Γ, as discussed in Section II. In light of the functional derivative δΣ/δG = −I t (see Section III B below), we aim at generating the irreducible vertex I t , for which we need the totally irreducible vertex, R, instead of the bare vertex, Γ 0 . Hence, we define R = R − Γ 0 , and, since Eq. (11) is linear in Γ 0 , we obtain We now consider the flow of Σ SD (R, Γ, G) and organize our computation according to [cf. Figs. 4(b) and 4(c)] Here, we have subtracted and added a term such that the first bracket,Σ 1 , contains only those terms of the differentiated SDE in which the derivative is explicitly applied to propagators. The second part,Σ 2 , accounts for the differentiated vertex for which we will insert the vertex flow (10). Finally,Σ 3 contains all remaining contributions proportional to R . In the PA, one has R = Γ 0 ⇔ R = 0; thus,Σ 3 will only be relevant in calculations that go beyond the PA. In fact, from Eq. (14), we see that the role ofΣ 3 is to cancel the extra terms that have been added toΣ 1 +Σ 2 by using Σ SD (R, Γ, G) instead of Σ SD (Γ 0 , Γ, G). We begin our calculations withΣ 1 . Generate I t ·Ġ-As already mentioned, we want to single out the two-particle-irreducible vertex I t (since it constitutes the functional derivative of the self-energy). The first summand in Eq. (11) (using R instead of Γ 0 withṘ = 0) is easily differentiated as −R ·Ġ. In the remaining part ofΣ 1 , we have three propagators to differentiate. Two of the resulting terms can be combined to factor outĠ if we use the contracted crossing symmetry (12) on R and Γ: Next, we collect the terms for Use differentiated bubbles-The extra terms accompanying I t ·Ġ in Eq. (17) will later be combined with contributions fromΣ 2 . SinceΣ 2 contains the differentiated vertex, which itself is built from differentiated bubblesΠ r , we rewrite these contributions in terms ofΠ r . Using the contracted crossing symmetry (12), we find This leads to the final expression forΣ 1 [illustrated in Fig. 4(b)]: Organize vertex derivative-The second contribution to Eq. (15),Σ 2 , contains the differentiated vertex. Inserting the decompositionΓ = rγ r , we can combine the contributions from both particle-hole channels, a and t, by applying the contracted crossing symmetry (12) on R andγ t : Once we insert the flow equation (10) forγ a andγ p in Eq. (20), R will be connected to further bubbles of vertices. These connections can be simplified if we have I r instead of R. Hence, we rewrite Eq. (20), using I r = R + γr, as The next step consists of repeated use of the contracted crossing symmetry (12): After usingİ r =γr, we then obtainΣ Insert vertex flow -Whereas the previous manipulations were possible due to the contracted crossing symmetry, the following insertion of the vertex flow forγ r , given by Eq. (10), can be simplified already on the vertex level. In fact, using the parquet equations (4) with γ r = I r • Π r • Γ and Γ = I r + γ r , we get The first term also occurs (with opposite sign) in Eq. (19), the second term reproducesγ (C) r , and the third term gets canceled in Eq. (23). Hence,Σ 2 can be simplified [as summarized in Fig. 4(c)] tȯ With the definitionγ p , the full derivative of the self-energy is given bẏ This result forΣ ≡ ∂ Λ Σ SD in skeleton form (i.e., phrased with dressed propagators G,Ġ only) will be considered more closely in Section IV. Here, we move on by noting that Eq. (26) still containsΣ on both the l.h.s. and the r.h.s. (viaĠ).
IsolateΣ-At this point in our derivation, we specify how the Λ dependence is supposed to enter G: it shall be incorporated in the bare propagator G 0 such that the Dyson equation, Once we insert this expression forĠ into Eq. (26), we will face the contraction of a vertex with a composite line G ·Σ · G. In such a case, one can equivalently attribute the two propagators to either the self-energy or the vertex, such that we have the following equality for a composite contraction [recall the minus sign in Π t ; see Eq. (A6) for details]: We insert Eq. (27) into Eq. (26) to isolateΣ: Next, we use the inverted BSE (5) as well as the extended BSE (6) to express this through Γ and 1 + Γ • Π t , respectively: For convenience, we finally write the contraction of (Γ • Π t ) with both summands as composite contractions [using Eq. (27) for a general vertex and self-energy] and obtaiṅ This is our final result for the mfRG self-energy flow deduced from the SDE. It constitutes the bare ("nonskeleton") form of Eq. (26) as it involves G and S instead of G andĠ. The first term in Eq. (30),Σ std , is the standard fRG self-energy flow. The next two terms,Σt andΣ t , constitute the multiloop corrections to the self-energy flow [cf. Fig. 3(b)], which have been derived diagrammatically in Ref. 12. These contributions are needed to ensure that the self-energy flow generates all contributions to the self-energy arising within the PA. Finally, the two terms involvingΣ 3 remain in our final result and-in calculations beyond the PA-are required to cancel doubly counted terms coming from the replacement Σ SD (Γ 0 , Γ, G) → Σ SD (R, Γ, G) in Eq. (14). We remark thatΣ 3 constitutes precisely the part that cannot be simplified further with our parquet tools, as it originates from the appearance of a bare instead of renormalized vertex in the SDE.

Interpretation
Let us interpret the flow equation (30) step by step: and R [and henceΣ 3 = ∂ Λ Σ SD (R , Γ, G)] are of order O (Γ) 4 , we have explicitly shown how to derive the standard fRG self-energy flow,Σ std , from the SDE up to and including terms of fourth order in the (effective) interaction. If we were in the standard fRG setting where every line is Λ-dependent, further terms coming fromṘ = 0 would arise in our derivation. However, as these terms are similarly of order O (Γ) 4 , the result ∂ Λ Σ SD =Σ std + O (Γ) 4 would remain unchanged.
(ii) The above derivation is applicable with any expression for the totally irreducible vertex, R. Unfortunately, as discussed above, we cannot efficiently keep track of its Λ evolution, and thus use R = R Λ ⇔Ṙ = 0. Hence, even if we were to possess the exact object R ex , we would have R = R Λ during the flow, and the (in principle exact) standard self-energy flowΣ std [see also Eq. (31) below] would not be exact for points of the flow where G Λ 0 = G 0 . (iii) In the PA, the totally irreducible vertex is reduced to its simplest approximation, such that R = Γ 0 ⇔ R = 0 and thusΣ 3 = 0. In this case, Eq. (30) reproduces the mfRG self-energy flow from Ref. 12 including the correctionṡ Σt andΣ t [cf. Fig. 3(b)], necessary to provide a total derivative of the SDE using the approximate parquet vertex.
(iv) Let us come back to the idea of a loop in theory space, which-including the self-energy flow-is now driven by the bare propagator G Λ 0 . A possible realization is given by If we start the flow from the solution in the PA (R = Γ 0 ) with Σ Λi = Σ PA and Γ Λi = Γ PA , the combination of the mfRG vertex flow (10) and self-energy flow (30) (usingΣ 3 = 0) gives the corresponding result in the PA for all Λ (as R = Γ 0 throughout) and returns to the original solution at Λ f . However, if we started the flow from the exact solution (R = R ex ) initialized by Σ Λi = Σ ex and Γ Λi = Γ ex , we would have to includeΣ 3 in the self-energy flow (30) in order to return to the exact solution for both the self-energy, Σ ex , and the vertex, Γ ex (dressed by Σ ex ) at Λ f ; settingΣ 3 would introduce an approximation in the full derivative of the SDE. Conversely, we can apply this idea to simplified models, where the exact solution is indeed available, and compare the final result of the flow to the initial value in order to gauge the importance of the individual terms in Eq. (30).
(v) To better understand the effect ofΣ 3 , we recall thatΣt andΣ t were originally derived diagrammatically to compensate for missing diagrams ofΣ PA when using the parquet vertex inΣ std [12]. With this perspective oṅ Σt +Σ t in mind, it is intuitive that higher-order corrections to R (i.e., R = 0) generate doubly counted terms betweenΣ std andΣt +Σ t . Yet, as Eq. (30) is exact, these overcounted terms are precisely canceled by the parts involvingΣ 3 .
For illustration, consider the (parquet) self-energy at fourth order in the interaction, which contains no approximation and whose flow is fully described byΣ std +Σt +Σ t using vertices in the PA. Now, fourth-order diagrams of R = 0 generate fourth-order terms inΣ std but not inΣt andΣ t (due to their structure involving further vertices that raise the interaction order). The additional fourth-order contributions ofΣ std are precisely canceled by R ·Ġ (containing only one Λ-dependent line) as part ofΣ 3 . Generally, we believe that, for situations where R = 0, the overcounting of differentiated diagrams inΣ std +Σt +Σ t has rather small weight and that, even if usingΣ 3 ≈ 0, the multiloop additionsΣt +Σ t provide an improvement of the standard self-energy flow,Σ std .
(vi) A very interesting application with R = 0 is the previously mentioned multiloop DMF 2 RG approach starting from the local solution of DMFT. In its full form, combining the flow equations of the vertex (10) and self-energy (30), the mfRG flow is controlled by the bare propagator G Λ 0 , which interpolates between the local theory of DMFT and the actual lattice problem. The simplest realization [28] in a flow from Λ i = 1 to Λ f = 0, formulated in terms of Matsubara frequencies iω and momentum k, is given by Here, ∆(iω) is the self-consistently determined hybridization function of the auxiliary Anderson impurity model [27] and k the lattice dispersion. We have already explained the advantages of the mfRG vertex flow initialized by Γ Λi = Γ DMFT [see Section II C 2, example (iii)]. For the self-energy flow, starting from Σ Λi = Σ DMFT , it remains to be seen whether the standard fRG flow,Σ std , with or without the multiloop correctionsΣt +Σ t , or other realizations corresponding to certain expressions forΣ 3 in Eq. (30) lead to optimal results.
(vii) Generally, a fRG flow can be used to tune or turn on one-particle parameters via G Λ 0 if the physics at G Λi 0 is known [26]; the mfRG approach guarantees that one stays at the fully self-consistent solution of the parquet equations for all values of Λ. It is then possible to deform solutions of the parquet equations from known, limiting cases to more complicated situations. As two applications, where the most relevant properties should already be captured within the PA (R = 0), let us mention Fermi polarons [39,40], where one can tune the chemical potential of the majority species, and nonequilibrium transport (requiring a Keldysh fRG description [24,41]), where one can gradually increase the bias voltage.

B. Self-energy flow from the functional derivative
We now show how the standard fRG self-energy flow,Σ std , can be directly derived from the equality between the functional derivative of the self-energy and the (particle-hole) two-particle-irreducible vertex. To be in perfect accordance with the standard fRG setup, we have to require that every G line be Λ-dependent-even those in the totally irreducible vertex, R = R Λ . Incorporating the Λ dependence in the bare propagator G 0 , we again relate the differentiated propagator,Ġ, to the single-scale propagator, S viaĠ = S + G ·Σ · G.
The functional derivative between self-energy and vertex, δΣ/δG = −I t , holds for any variation of G. If this variation is realized by having a scale-dependent propagator G Λ and varying the scale parameter Λ, this equation impliesΣ = −I t ·Ġ. Starting from this, we can perform the same steps as above: To obtain the standard fRG flow equation for the self-energy, it remains to insertĠ = S + G ·Σ · G, express the composite contraction I t · (G ·Σ · G) as −I t • Π t ·Σ [cf. Eq. (27)], and use the inverted BSE (5): · G, we get a contribution to the new, two-particle-irreducible vertex I t . The lowest-order realization of this, obtained by inserting the bare vertex for Γ, constitutes an envelope vertex, which is not contained in the initial It in the PA.
Solving for Σ in a specific fRG flow via Eq. (31) amounts to integrating δΣ = −I t · δG along a specific path in the space of theories defined by the bare propagator G 0 = G Λ 0 [and the bare interaction Γ 0 , cf. Eq. (1)]. Only if this integration is independent of the path, i.e., ifΣ contains a total derivative of diagrams, the flowΣ = −Γ · S yields results consistent with the functional derivative. This is neither the case in the truncated fRG flow (without Γ (6) ) nor in the mffRG flow of Fig. 3 with R = Γ 0 , which forms a total derivative of diagrams but deviates from Eq. (31) by the additionsΣt andΣ t . (In fact, the latter reproduces precisely the self-energy diagrams generated by the SDE using the vertex in the PA. However, as shown in Appendix B, the requirement of fulfilling both the functional derivative and the SDE necessitates the exact solution.) As a direct application of the above calculations, we derive an fRG flow which is equivalent to self-consistent Hartree-Fock (HF), in agreement with a result by Katanin [42]. This conserving fRG flow provides a simple example for which the integration of δΣ = −I t · δG is indeed independent of the path. In HF theory, the functional derivative of the self-energy is given by the bare vertex, δΣ HF /δG = −Γ 0 . By replacing I t → Γ 0 in Eq. (31), we immediately finḋ Equation (32c) describes the vertex flow in the truncated Katanin form [43], restricted to the t channel. If the same vertex is used for the standard self-energy flow [Eq. (32a)], the fRG flow yields the Hartree-Fock self-energy together with a particle-hole ladder vertex (noteΓ lad t = −Γ lad a ). As this vertex consists of ladder diagrams in only one channel, it clearly violates crossing symmetry.

IV. CONSERVATION LAWS IN THE PARQUET APPROXIMATION
In this section, we take a slightly different perspective and are not concerned with RG flows. Instead, we use our insight into the structure of the many-body relations gained from the above derivations to address conceptual questions of many-body (parquet) theory. First, we derive two technical results: (i) We show how one can construct a two-particle-irreducible vertex which equals the functional derivative of the parquet self-energy. Evidently, the operation δΣ/δG can be performed in an analytical study of Feynman diagrams [44]. However, in a numerical treatment, one never has access to the self-energy as a functional of the full propagator. Instead, one only has its value for the specific, given propagator, and the general construction for such a vertex remains unknown [15]. Here, we provide its construction for the case of the parquet self-energy. (ii) We demonstrate that the parquet self-energy can be obtained from the SDE using either of two possible orderings of the bare and full vertex. While it is believed that most approximations for Σ obtained from the SDE obey this property [14], it has (to our knowledge) not been shown for the PA. These results can then be interpreted in the context of conservation laws in the PA using arguments from Baym and Kadanoff [14]. 6. Illustration for the relation between (skeleton) diagrams of the vertex and the self-energy at fourth order in the interaction: Inserting the first (parquet) vertex diagram into the Schwinger-Dyson equation, we generate the second self-energy diagram of Σ PA . Upon taking the functional derivative w.r.t. to the full propagator, this self-energy diagram relates to multiple diagrams of the two-particle-irreducible vertex It. Among those, the third diagram, obtained by cutting the (light) red line, is an envelope diagram and not part of It in the parquet approximation. However, the fourth diagram, obtained by cutting the blue line, belongs to it. Note that we ignore signs and prefactors in these diagrams.
The final manipulations can be made in complete analogy to obtaiṅ i.e., the identical differential equation (34). Since, for the specific propagator G = 0, one has Σ SD (Γ 0 , Γ, 0) = 0 = Σ SD (Γ, Γ 0 , 0), it follows that the self-energy in the PA can indeed be obtained from any of the two versions of the SDE. The strategy of generating, first, a self-energy via the SDE and, then, obtaining a vertex by functional differentiation has been famously put forward by Baym and Kadanoff [14]. They showed that, if the self-energy can equivalently be constructed via the SDE with either order of the vertices, then, the one-particle propagator is conserving. Thus, using this argument together with Eq. (35), one finds that the PA fulfills one-particle conservation laws. Baym and Kadanoff further showed that, if the vertices are subsequently constructed from I t = −δΣ/δG and Γ = I t + I t • Π t • Γ , two-particle conservation laws are fulfilled as well. As is well known, the PA does not fulfill two-particle conservation laws. In fact, Eq. (34) shows how the parquet vertex I t needs to be modified to be conserving; in other words, the correction termγ (C) t · G allows one to quantify to what degree the vertex I t in the PA violates conservation laws. Furthermore, Eq. (34) provides a construction how to generate a fully conserving solution originating from the parquet self-energy. After both the vertex I t and the self-energy Σ PA in the PA have been obtained, one computes γ (C) t · G and adds this to I t to get a conserving vertex I t . Note that the original parquet self-energy need not be modified. Similarly as one computes Γ = I t + I t • Π t • Γ with the original Π t (containing Σ PA ), physical quantities (such as susceptibilities, conductivities, etc.) are computed using I t (or Γ ) together with Σ PA . The resulting solution fulfills one-and two-particle conservation laws, but, clearly, it does not fulfill the SDE anymore. This is not surprising since, as shown in Appendix B, a solution that fulfills both the SDE and the functional derivative must be the exact solution. The preferential choice between Γ and Γ will surely depend on the physical application.
We remark that there have also been suggestions of how to keep the vertex I t in the PA but modify the self-energy, Σ PA , to obtain a thermodynamically consistent description [45]. While these ideas might be useful in practical situations, it is, however, not possible to construct a combination of the skeleton two-particle-irreducible vertex I t [G] in the PA together with any skeleton self-energyΣ[G], such that the functional derivative I t = −δΣ/δG is fulfilled. The reason is that the functional derivative generates from any diagram ofΣ a multitude of diagrams for I t -the same self-energy diagram related to missing diagrams of I t in the PA also relates to diagrams that are contained in I t (cf. Fig. 6). Therefore, the functional derivative cannot be fulfilled by starting from the PA and simply removing diagrams from the self-energy.

V. RESPONSE FUNCTIONS
Finally, we use our results from Section II to derive dependent, mfRG flow equations for response functions. In fact, the (fermionic) four-point vertex, Γ, and the self-energy, Σ, give us full control over correlation functions up to the fourpoint level, and thus they suffice to compute response functions such as three-point vertices, Γ (3) , and susceptibilities, χ. If Γ and Σ are obtained by an RG flow, the response functions can be deduced from the scale-dependent Γ Λ , Σ Λ at any stage during the flow. Alternatively, the response functions Γ (3),Λ and χ Λ are often deduced from their own RG flows [5]. In this case, the flow equations provided by the standard fRG hierarchy again require knowledge about unknown, higher-point vertices (namely a five-point vertex for the flow of Γ (3) and a boson-fermion four-point vertex for χ) [6]. In particular, the inevitable truncation in the fRG hierarchy leads to ambiguities in the computation of the response function [13,46]. These ambiguities have been recently resolved by a diagrammatic derivation of the mfRG flow equations for the response functions [13]. Here, we provide algebraic derivations of these flow equations. We find that one can circumvent the influence of unknown, higher-point vertices by using exact flow equations for the response functions, which follow from the standard relations between the response functions and the (known) fermionic four-point vertex and self-energy.

A. Three-point vertex
The Schwinger-Dyson equation relating the (full) three-point vertex to the bare three-point vertex (often taken to be unity) and the four-point vertex [6] is given by (cf. Fig. 7) Employing the scale dependence described in the previous sections, we can differentiate Eq. (40) to geṫ We insert the mfRG vertex flow (10), combine several terms according to Eq. (40), and obtaiṅ The first term occurs similarly in the fRG flow equation (with the typical replacementĠ ↔ S). However, the remaining part of our flow equation successfully replaces the contributions from the unknown five-point vertex in the fRG flow.

B. Susceptibility
The susceptibility is fully determined by the three-point vertex or [via Eq. (40)] the four-point vertex [6], according to (cf. Fig. 8) We can differentiate either relation; choosing the first one, we insert the mfRG flow (42) of Γ (3) to find the mfRG flow of the susceptibility: Again, the first term occurs similarly in the fRG flow equation (withĠ ↔ S), and the remaining terms in our flow equation replace the contributions from the unknown boson-fermion four-point vertex in the fRG flow. Let us briefly summarize: The response functions Γ (3) , χ can be deduced from the four-point vertex, Γ, and the self-energy, Σ, at any point of the RG fow. As Γ and Σ evolve with Λ, so do Γ (3) and χ. With the above derivation, we have cast this evolution into exact, mfRG flow equations for the response function, each containing the vertex flow from the complementary channel (İ r =γr). The two-particle-reducible vertices still obey the mfRG flow (10); approximations come from the chosen expression for the totally irreducible vertex, R, which affects the initial conditions but is itself not part of the flow.

VI. CONCLUSION
We have used the well-known self-consistent relations of the parquet formalism to derive exact flow equations for various vertex and correlation functions. Compared to the standard fRG framework, these multiloop fRG (mfRG) flow equations can be advantageous as they circumvent the reliance on higher-point vertices. In fact, our calculations include concise, algebraic derivations of the mfRG flow equations that have previously been derived diagrammatically [11][12][13] and have already been used [11,13] to improve the approximations of the truncated fRG flow (see Refs. 23,47,and 48 for results of two-loop fRG).
The analysis presented in this paper puts the mfRG approach on a general basis. The algebraic derivations open the route to RG flows beyond the diagrams of the parquet approximation (PA). Since the totally irreducible vertex, R, is precisely the part of the vertex that cannot be efficiently included in the flow, the focus can now shift to systematic ways of computing R. If one chooses a scale dependence in the propagators that starts from G Λi 0 = 0, all reducible contributions built on R will be fully included by the mfRG flow. Other starting points for the flow are a possible as well. In particular, if one uses as initial propagator the solution from dynamical mean-field theory, G Λi 0 = G DMFT 0 , the nonlocal correlations not contained in DMFT will be added by a flow that starts from the self-energy Σ DMFT and the full vertex Γ DMFT [28]. Similarly, if the systems in question is related to another, solvable reference system [23] by variation of one-particle parameters, mfRG can be used to tune between these systems via G Λ 0 , with the guarantee that the parquet equations are fulfilled throughout the flow. Our computations also provide a basis for setting up mfRG flows for more complicated theories, including, for instance, further bosonic degrees of freedom. Generally, we believe that the insights presented in this paper will be useful for further development of quantum-field-theoretical RG techniques.
Additionally, we have demonstrated an intimate relation between the functional derivative of the self-energy (inducing a conserving solution) and the (standard) fRG self-energy flow: The flow equation directly follows from the functional derivative for the case that the propagator is varied through a scale parameter. However, a solution of the fRG flow is consistent with the functional derivative only if the flow is independent of the specific scale dependence, i.e., only if Γ · S constitutes a total derivative of diagrams. A simple example for which this is indeed the case is given by a truncated fRG flow with a (particle-hole) ladder vertex that reproduces self-consistent Hartree-Fock. Building on this, it would be worthwhile to devise other approximate flows that comply with the functional derivative but go beyond Hartree-Fock, thereby including an interplay between different two-particle channels.
Lastly, we have used our approach to address important general questions of (traditional) parquet theory. Using an argument of Baym and Kadanoff [14], we have demonstrated that the PA fulfills one-particle conservation laws. Furthermore, we have shown how to construct a two-particle-irreducible vertex equivalent to taking the functional derivative of the parquet self-energy. With this, one can quantify to what extent the PA violates two-particle conservation laws, and one can modify the PA to obtain a fully conserving approximation. It would be interesting to apply this modified parquet approach in situations where conservation properties are crucial, such as studies of transport phenomena.
The generality of our formalism opens up a vast field of applications. Multiloop fRG flows have already yielded impressive results for the prototypical 2D Hubbard model [13] (see Ref. 47 for results using two-loop fRG) and promise a better understanding of strongly correlated electron systems [5,12,21]. In the study of quantum magnetism, the pseudo-fermion fRG approach [49] has become a competing method, and first calculations with two-loop corrections [48] suggest that a full multiloop treatment would yield further improvements. Moreover, mfRG could be applied to the various forms of mobile impurity problems [40,50], or to one-dimensional fermion systems [51] beyond the Luttinger liquid paradigm [52]; for both classes of problems, the PA is believed to capture the leading power laws, hence they should also emerge within a mfRG treatment. In the field of transport phenomena in disordered systems, our mfRG approach could provide unprecedented insight into many-body localization in large systems [53,54] or interaction effects on the Anderson localization transition [55]. Finally, we remark that mfRG flows can also be naturally set up within the Keldysh formalism [24,41] to provide real-frequency information, both in and out of equilibrium.
If we combine two fermionic indices into one bosonic index, the above equations directly translate to three-point vertices. For instance, one could combine the two external legs of the first vertex in the a bubble according to some Furthermore, one can contract a four-point vertex with a one-particle propagator to obtain another one-particle object. We define the symbol · between vertex and propagator to be such a contraction applied to the "upper" external legs of the vertex [i.e., legs 2 and 2 in Fig. 9(c)]. In Ref. 12, this has been dubbed a "self-energy loop", L, defined as −L(Γ, G) x ,x = y ,y Γ x ,y ;x,y G y,y = (y ,y)Γ t;(x ,x),(y ,y)G(y ,y) ≡ (Γ · G) x ,x . (A5) If the contracting line is a composite object of the type G · Σ · G, we can view the G lines as a t bubble attached to the vertex, according to Γ · (G · Σ · G) x ,x = y ,y,z ,z Γ x ,y ;x,y G y,z Σ z ,z G z,y = − (y ,y),(z ,z)Γ t;(x ,x),(y ,y)Πt;(y ,y),(z ,z)Σ(z ,z) ≡ −(Γ • Π t · Σ) x ,x .
The Schwinger-Dyson equation for the self-energy contains a contraction of three propagators. Using the bubble functions defined above, this can equivalently be written with Π p and Π a : −Σ x ,x = y ,y Γ 0;x ,y ;x,y G y,y + 1 2 y ,y,z ,z,w ,w Γ 0 x ,z ;y,w G y,y G z,z G w,w Γ y ,w ;x,z The functional derivative between self-energy and two-particle-irreducible vertex (in the t or a channel) is given by δΣ x ,x δG y,y = −I t;x ,y ;x,y = I a;x ,y ;y,x . (A8)

Appendix B: Schwinger-Dyson equation and functional derivative
We consider the Schwinger-Dyson equation (SDE) for the self-energy as well as the functional derivative between self-energy and vertex, and show that a solution for Σ and Γ that fulfills both Eqs. (B1a) and (B1b) must necessarily be the exact solution. In essence, this proof has already been given by Smith [15]. However, we find it useful to present it here in our notation, which exclusively consists of properly symmetrized objects. In fact, this proof puts on solid ground what has long been known to the community [2]: In any approximate solution to the many-body problem, one has to decide whether to comply with either conservation laws or crossing symmetry; achieving both amounts to finding the exact solution.
To be able to apply the functional derivative, we consider the self-energy as a functional of the full propagator, Σ[G]. This is perfectly compatible with the SDE (B1a), which is formulated using full propagators only. Furthermore, all vertex functions depend on the given theory's bare vertex Γ 0 (which we here label Γ 0 = U for ease of notation); in particular, this holds for Σ[G, U ] and Γ[G, U ]. Since U is the bare vertex, we have Γ[G, U ] = U + O(G 2 , U 2 ); by use of either the SDE (B1a) or the functional derivative (B1b), it is clear that Σ[G, U ] = U · G + O(G 3 , U 2 ).
Assume that we know the exact vertex up to terms of order n ≥ 2 in both G and U , i.e., Γ = Γ ex + O(G n , U n ). If we apply the SDE (B1a), we obtain (inserting into the second term) Σ = Σ ex + O(G n+3 , U n+1 ). Now, we apply the functional derivative (B1b) and get I t = I ex t + O(G n+2 , U n+1 ). Finally, using the BSE (B1b) yields Γ = Γ ex + O(G n+2 , U n+1 ), i.e., the exact vertex one order higher in G 2 and U than we started with. Since we do know the exact vertex up to terms of second order, Γ[G, U ] = U + O(G 2 , U 2 ), it follows by induction that a solution which fulfills both Eq. (B1a) and (B1b) consists of the exact functionals Σ ex [G, U ], Γ ex [G, U ].
We remark that this proof applies equivalently to finite-order approximations of Σ and Γ as well as to approximations of infinite order in U . As soon as an expression for Γ contains the bare vertex U [15], the combination of Eq. (B1a) and (B1b) requires all expansion coefficients of Σ and Γ to be the ones of the exact solution.