Pseudogap opening in the two-dimensional Hubbard model: A functional renormalization group analysis

We present here results for a prototypical model of correlated fermions, the 2D Hubbard model. In standard second quantization the Hubbard Hamiltonian reads

where $t$ denotes the nearest-neighbor hopping amplitude on a square lattice and $U$ the local Coulomb repulsion. In the following, we define our energies in terms of $t\equiv 1$ and restrict our analysis to half filling $\langle\hat{n}\rangle=1$. This is achieved by an implicit shift of the Hartree part by $U\langle\hat{n}_{\sigma}\rangle$ and setting the chemical potential to $\mu=0$. In this case, the momentum transfer of $(\pi,\pm\pi)$ corresponds to perfect AF nesting on the square-shaped Fermi surface.

The characteristic scale-dependent behavior of numerous strongly correlated electron systems can be treated in a flexible and unbiased way by the fRG, see Refs. Salmhofer (1999); Berges et al. (2002); Kopietz et al. (2010); Metzner et al. (2012) for a review. Its starting point is an exact functional flow equation, which yields the gradual evolution from a microscopic model action to the final effective action as a function of a flow scheme dependent energy scale. By expanding in powers of the fields one obtains an exact hierarchy of flow equations for vertex functions, which in practical implementations is restricted to the one- and two-particle vertex. Neglecting the renormalization of three- and higher order particle vertices yields approximate $1\ell$ flow equations for the self-energy and two-particle vertex. The underlying approximations are devised for the weak to moderate coupling regime, where forefront algorithmic advancements brought the fRG for interacting fermions on 2D lattices to a quantitatively reliable level Tagliavini et al. (2019); Hille et al. (2020).

The substantial improvement with respect to previous fRG-based computation schemes relies on an efficient parametrization of the two-particle vertex which takes into account both the momentum and frequency dependence. Assuming SU(2) spin-rotation symmetry, the one-particle irreducible vertex $V_{\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4}}(k_{1},k_{2},k_{3},k_{4})$ can be expressed by a coupling function depending on three independent generalized momenta $V(k_{1},k_{2},k_{3})$ via Honerkamp et al. (2001)

with $k_{4}=k_{1}+k_{3}-k_{2}$. Then the coupling function can be channel-decomposed by the parquet decomposition Karrasch et al. (2008)

where in the parquet approximation, the fully two-particle irreducible vertex is approximated by $\Lambda_{\mathrm{2PI}}=U$, and the two-particle reducible contributions $\Phi_{pp/ph/\overline{ph}}$ in the particle-particle, particle-hole, and crossed (or transverse) particle-hole channel are parametrized using a single generalized ’bosonic’ transfer momentum/frequency as first argument and two ’fermionic’ ones as second and third argument. In particular, we combine the TU-fRG Husemann and Salmhofer (2009); Wang et al. (2012); Lichtenstein et al. (2017) using a form-factor expansion for the fermionic momentum dependences with the full frequency treatment including the fermionic high-frequency asymptotics Rohringer et al. (2012); Wentzell et al. (2016), see Refs. Tagliavini et al. (2019) and Hille et al. (2020) for the details on the algorithmic implementation.

In addition, we compute the self-energy and its feedback in the fRG flow of the two-particle vertex (see also Refs. Eberlein (2015) and Vilardi et al. (2017)). Instead of the conventional $1\ell$ flow equation for the self-energy, which we recall for completeness

we employ here the scale derivative of the SDE Schwinger (1951); Dyson (1949). This is inspired by its connection to the multiloop extension of the fRG, which allows to sum up all the diagrams of the parquet approximation with their exact weight Kugler and von Delft (2018a, b) and hence yields cutoff-independent results Tagliavini et al. (2019). Specifically, the multiloop equation for the self-energy flow can be derived from the SDE

with $\Sigma(k)=\delta_{\sigma_{1},\sigma_{2}}\Sigma_{\sigma_{1},\sigma_{2}}(k)$, an implicit factor $T$ for each frequency sum, and the normalization of the momentum sum with respect to the first Brillouin zone. Here we do not aim at a quantitative analysis within the multiloop fRG, for which the comparison to determinant Quantum Monte Carlo data showed that the fRG is remarkably accurate up to moderate interaction strengths Hille et al. (2020). We rather address a qualitative description of the spectral properties (with a substantially reduced numerical effort with respect to fully loop converged computations) retaining only the multiloop equation for the self-energy, i.e. the scale derivative of the SDE. Differently to the conventional $1\ell$ flow, the scale derivative of the SDE accounts for the form-factor truncation Hille et al. (2020). Formally, the derivative with respect to the flow parameter $\Lambda$ yields the conventional $1\ell$ contribution as well as the multiloop corrections. The proof thereof reported in Ref. Kugler and von Delft (2018b) assumes the equivalence between the different channel representations of the SDE. However, in the TU-fRG only a finite number of form factors is considered in each channel (typically restricted to just a few). In this approximation, the transfer momentum dependence is not fully reconstructed after a translation from one to another, i.e. the projection between different channel representations is affected by information losses. We cured this by implementing the self-energy flow, considering the most favorable channel representation for each contribution, directly in the derivative of the SDE (see also Ref. Hille et al. (2020))

where the dot represents the $\Lambda$-derivative, and

with

If not otherwise specified, we use the single-scale propagator $\partial_{\Lambda}G^{\Lambda}\approx S^{\Lambda}=\partial_{\Lambda}G^{\Lambda}|_{\Sigma=\textrm{const}}$. Note that in Eqs. (8)-(11), the derivative with respect to $\Lambda$ on the right hand sides yields three contributions each. These depend on the ($1\ell$) flow of the two-particle vertex Tagliavini et al. (2019), that supplements the above Eqs. (7)-(11). In the flow equations for the vertex, the self-energy $\Sigma(k)$ is inserted into the full Green’s functions on the right hand sides without further expansions around the Fermi surface or in small frequencies.

We observe that in principle one could also use directly the SDE (5) replacing its scale derivative in Eqs. (7)-(11). However, we preferred a formulation which remains as close as possible to the original fRG idea involving a differential flow equation for the self-energy. Besides the more straightforward numerical implementation of the derivative in an adaptive differential equation solver, we believe that the close relation to the multiloop fRG flow by which the proposed SDE scheme is inspired makes it more intuitive. In fact, including the Katanin correction or any other higher loop order as discussed in Sec. III.2, requires anyways the computation of the self-energy derivative with Eqs. (7)-(11).

For the two-particle vertex ³³3For the convention on the flow equations, see Ref. Hille, 2020., we consider only a single local $s$-wave form factor and a small number of frequencies, verifying that an additional $d$-wave form factor as well as more frequencies do not qualitatively affect the results in the considered parameter regime. Specifically, we use $8$ fermionic and $17$ bosonic frequencies in the low frequency object depending on all three frequencies. The same numbers are used in the high-frequency asymptotics of a single fermionic frequency for the remaining bosonic and fermionic frequencies. The asymptotic of both fermionic frequencies is described by $513$ bosonic frequencies. Concerning the transfer momentum parametrization, in addition to $16\times 16$ momentum patches distributed on an equally spaced grid in the Brillouin zone, we take into account a finer $5\times 5$ grid around the AF peak at $\bf{q}=(\pi,\pi)$, see also Fig. 7 for a more detailed convergence analysis.

We finally note that we use here the so-called interaction “cutoff” with $G^{0,\Lambda}=\Lambda G_{0}$ which gradually turns on the bare onsite interaction $U$ Honerkamp et al. (2004). This allows one to translate the flowing self-energy and two-particle vertex at scale $\Lambda$ to the solution (at the end of the flow) of a rescaled bare vertex $\Lambda^{2}U$, see Appendix A for the proof. This cutoff choice is motivated by the possibility to trace the onset of the gap opening along the flow and identify the pseudo-critical interaction in correspondence of the vertex divergence. The absolute values depend on the applied flow scheme and for this reason a quantitative comparison to other cutoffs appears difficult. At the same time, the analytical arguments as well as our findings at higher loop order presented below (see also Ref. Tagliavini et al., 2019 for a more detailed discussion on the cutoff independence of the multiloop fRG) support the robustness of the qualitative differences between the conventional and the SDE flow. Therefore, it should be observed independently of the flow scheme.

We present here results for the self-energy together with an analysis of the differences between the fRG calculation using the $1\ell$ flow Eq. (4) with respect to the derivative of the SDE in Eq. (6). In particular, we show how in the latter the long-range AF fluctuations lead to a gap opening, see also Appendix B.

Let us first look at the flow of the self-energy displayed in Fig. 1, for an inverse temperature of $1/T=10$. The panel (a) corresponds to the fRG calculation with the $1\ell$ flow of the self-energy and (b) to the scheme with the derivative of the SDE. We follow the flow at the nodal ${\bf k}_{n}=(\pi/2,\pi/2)$ (circles) and antinodal ${\bf k}_{an}=(\pi,0)$ (diamonds, solid lines) momentum point, both for the first (closed symbols) and second (open symbols, dashed lines) Matsubara frequencies. The imaginary part of the self-energy is shown as a function of the flowing interaction $U$. The end point on the right defines the so-called pseudo-critical interaction at which the maximal component in one of the channels in the vertex exceeds $10^{3}$. Due to the self-energy feedback into the flow of the vertex, and vice versa, it depends on the flow scheme applied to the self-energy. In the SDE scheme this divergence sets in at $U=1.69$, which is larger than in $1\ell$ where the flow diverges at $U=1.64$.

This is consistent as the self-energy is larger in the SDE scheme and therefore its screening of the dominating $\overline{ph}$-excitations in the 2D Hubbard model at half filling is stronger (see also Ref. Tagliavini et al., 2019). In the inset of (b) a zoom on the crossings of the imaginary parts of the self-energy at the first and second Matsubara frequency is shown, which at sufficiently low temperatures is associated with a smooth, non-critical transition between a Fermi-liquid and an insulating behavior Simkovic et al. (2020). In particular, this transition occurs first at the antinodal momentum point and only for large values of the bare interaction at the nodal point. The region in between identifies the pseudogap regime which we will discuss in the following.

The presence of quasiparticles in a Fermi-liquid is equivalent to a nonzero quasiparticle weight Mahan (2000)

where $\nu$ is a real frequency. In a Fermi liquid, we have a nonzero $Z({\bf k})<1$. Non-Fermi-liquid behavior can be signaled by deviations from this, e.g. $Z({\bf k})\to 0$ or $Z({\bf k})>1$. $Z({\bf k})\to 0$ amounts to an infinitely steep slope of $\operatorname{Re}\Sigma(\nu,{\bf k})$ at $\nu=0$ and thus a vanishing of the quasi-particle weight without any dip or pseudogap feature in the spectral function, while $Z({\bf k})>1$ corresponds to a positive slope of $\operatorname{Re}\Sigma(\nu,{\bf k})$ at $\nu=0$ and implies the emergence of a double-peak structure of the spectral function when this slope is large enough. In the low temperature limit, $\partial_{\nu}\operatorname{Re}\Sigma(\nu,{\bf k})|_{\nu\rightarrow 0}$ can be translated to Matsubara frequencies. Then, the gap opening can be observed directly in the imaginary part of the self-energy. For a Fermi liquid, the imaginary part goes linearly through zero imaginary frequency. In the gapped regime, $\operatorname{Im}\Sigma$ bends towards positive large values approaching the zero (Matsubara) frequency limit from below and towards negative large values from above while in the Fermi liquid regime the bending is always towards small values. The onset of a pseudo-gap can hence be detected by the crossing of the imaginary parts of the self-energy at the first and second Matsubara frequency (see also Refs. Schäfer et al. (2015, 2016); Simkovic et al. (2020) and Schäfer et al. (2020)). Therefore, we study the difference between the latter via

with $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)|_{i\nu=i\pi T}\leq 0$ corresponding to the Fermi-liquid-like regime and $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)|_{i\nu=i\pi T}>0$ to a pseudogap at momentum $\bf{k}$.

We report $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)|_{i\nu=i\pi T}$ in Fig. 2, again for both flow schemes. The zeros corresponding to $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)|_{i\nu=i\pi T}=0$ in the SDE scheme (corresponding to the crossings observed in Fig. 1) can be directly read off. The gap opening at the antinodal momentum point sets in first, followed by the one at the nodal point close to the vertex divergence. Note that the $1\ell$ flow does not exhibit any zeros. Using the derivative of the SDE replacing the conventional $1\ell$ flow of the self-energy is therefore crucial for the description of spectral properties, i.e. for seeing the gap open up.

A more conventional representation of the self-energy as a function of the Matsubara frequency is provided in Fig. 3, for bare interactions close to the gap opening. Below the gap opening (for $U=1.55$), we observe a Fermi-liquid behavior with a pronounced upturn towards zero at low frequencies in both the SDE and the $1\ell$ flow. At the antinodal momentum point, the self-energy resulting from the SDE flow already exhibits a tendency to a non Fermi-liquid behavior. Once the gap is opened, this behavior turns first into a slight (for $U=1.6$) and then to a more pronounced (for $U=1.65$) downturn at the first Matsubara frequency, while the $1\ell$ results remain Fermi-liquid like for all values of $U$ (for comparison also the second-order perturbation theory is shown). At the nodal point, we still find a Fermi-liquid behavior in agreement with a picture in which the pseudogap opens at the anti-nodal points first. For the observation of the gap opening at the nodal point, we would have to tune the bare interaction very close to the pseudo-critical interaction.

The momentum anisotropy can also be studied in the evolution of $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)|_{i\nu=i\pi T}$ along the Fermi surface, shown in Fig. 4 for the same parameters. We note that the spread with momentum is significantly larger in the SDE flow where the monotonic decrease from the antinodal to the nodal momentum point crosses zero and opens the gap with increasing bare interaction first for $U=1.6$ in proximity of the antinodal point. The $1\ell$ fRG flow instead leads to Fermi-liquid behavior on the whole Fermi surface.

We now investigate the temperature dependence of the gap opening, starting from the considered $1/T=10$ in Figs. 1-4 down to $1/T=18$, see Fig. 5 for the respective results for $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)|_{i\nu=i\pi T}$. The detected behavior is qualitatively the same: no gap opening occurs for any temperature in the $1\ell$ flow, whereas the derivative of the SDE yields a gap which sets in first at the antinodal and subsequently at the nodal point. For decreasing $T$, the pseudo-critical interaction and with it the gap opening is shifted to lower values, reducing at the same time the distance between the zeros of $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)|_{i\nu=i\pi T}$ at the antinodal and nodal point and the pseudo-critical scale. This can be seen in Fig. 6, where we show the temperature dependence of the interaction at which the gaps open and of the pseudo-critical interaction (gray squares) at which the $\overline{ph}$-channel exceeds the critical value and the flow is stopped. We find that the antinodal gap opening is clearly distinct from the pseudo-critical interaction even for the lowest temperature considered here. In contrast, the pseudo-critical interaction almost coincides with the gap opening at the nodal point that sets in only a little below.

At the same time, the presented fRG data is sensitive to the employed parametrization of the two-particle vertex. In particular, a convergence in frequencies (with minor impact, see discussion in Sec. II.2) and momenta becomes numerically challenging for lower values of $T$ and is not fully reached yet. The latter is illustrated in Fig. 7. Upon increasing the number of patches, the gap opening at the nodal point is shifted closer towards the pseudo-critical interaction until it eventually merges with the latter around 20 $k_{x}$ momentum points. With a further refinement around $\bf{q}=(\pi,\pi)$ including $15\times 15$ instead of the former $5\times 5$ patches and covering a larger momentum area (indicated by the subscript $f$ on the right side of Fig. 7), the gap opening can be resolved again as a precursor of the pseudo-criticality. In contrast, the gap opening at the antinodal point does not approach the pseudo-critical interaction. The disappearance of the nodal gap as a precursor is due to the decreasing region covered by the fine patching and the consequently reduced resolution of the AF peak for larger values of $k_{x}$, i.e. the region becomes smaller than the peak width. When turning to a larger fine-patching region (indicated by the subscript $f$), the peak width is again covered, explaining the nonmonotonic behavior.

Hence, within the present analysis, the question whether in the low-temperature regime the gap opening at the nodal point is stable as a distinct feature from the occurrence of long-range correlations cannot be answered conclusively. Nevertheless, the data shown here are consistent with the physical picture emerging from various other state-of-the-art many-body techniques for the weak- or moderate-coupling regime of the 2D Hubbard model at half filling, as, e.g., in Huscroft et al. (2001); Sénéchal and Tremblay (2004); Gull et al. (2010), or currently collected in Schäfer et al. (2020). The AF correlations (associated to an exponentially increasing correlation length) increase at lower temperatures and eventually diverge at $T=0$ when the ground-state characterized by an AF long-range order is reached. In this low-temperature regime, long-wavelength AF fluctuations lead to an enhanced quasiparticle scattering rate and to the formation of a pseudogap in the single-particle spectrum (which evolves into a sharp gap in the Slater-like insulator at $T=0$). These fluctuations gradually destroy the coherent quasiparticles of the Fermi liquid. The crossover temperature corresponding to the pseudogap opening is not uniform along the Fermi surface: it is higher at the antinodal and lower at the nodal points. Eventually, all states of the Fermi surface are suppressed by AF fluctuations, resulting in a full gap.

The $1\ell$ description truncates the infinite hierarchy of flow equations Metzner et al. (2012) after the two-particle vertex. In the multiloop extension Kugler and von Delft (2018a, b), a part of the higher order vertex contributions is taken into account effectively by including higher loop contributions to the one- and two-particle vertex flows. The inclusion of all loop orders yields the parquet approximation enhancing the pseudo-critical interactions and lowering the pseudo-critical temperatures Tagliavini et al. (2019) with respect to the $1\ell$ truncation. According to the Mermin-Wagner theorem Mermin and Wagner (1966), which is fulfilled by the parquet approximation, the instability eventually disappears (this regime is however extremely difficult to reach numerically). We note that at infinite loop order, the results no longer depend on the employed flow scheme. We hence expect the trend observed for the interaction flow to be qualitatively reproduced also by other flow schemes at finite loop order.

For the precise form of the multiloop equations, we refer to Refs. Kugler and von Delft (2018b) and Tagliavini et al. (2019). In the SDE-scheme for the self-energy, we replaced both the $1\ell$ equation and multiloop corrections of the self-energy by Eq. (5). Note that for the Katanin replacement and in any higher loop order the differentiated propagator becomes $\partial_{\Lambda}G^{\Lambda}=S^{\Lambda}+G^{\Lambda}\dot{\Sigma}^{\Lambda}G^{\Lambda}$. For the second part of $\partial_{\Lambda}G^{\Lambda}$, the self-energy change has to be known before the calculation of the vertex flow and, in the SDE-scheme, inside the self-energy flow itself. Here, we replace it with the $1\ell$ flow in Eq. (4) independently of the self-energy scheme used. For a full feedback of the self-energy change according to Eq. (6) to the Katanin replacement, further iterations within the same $\Lambda$-step should be performed. As the correction is only of quantitative nature Hille et al. (2020), we neglect here these iterations.

In Fig. 8 we present the results for $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)$ evaluated at $i\nu=i\pi T$ as a function of the flowing interaction $U$, for $1/T=10$ and different loop orders $\ell$. While in the conventional fRG no gap opens at any loop order, in the SDE-approach, the gap opening at the antinodal point is observed also at higher loop order, before the flow has to be stopped and pseudo-criticality is reached. In Appendix A, we show that also for the multiloop flow equations, the scale at which the interaction flow diverges can be translated into a pseudo-critical interaction.

A direct comparison of the gap opening interaction and the onset of the AF divergence is shown in Fig. 9, as a function of the loop order (for $1/T=10$). A trend towards higher pseudo-critical interactions can be observed already at the first loop orders, while the oscillatory behavior is characteristic for the loop convergence Tagliavini et al. (2019). The larger pseudo-critical interactions could in principle leave more space for the pseudogap to develop. However, the gap at the nodal point vanishes at higher loop order, while at the antinodal point sets in at a rather small but constant distance from the pseudo-critical line. Therefore, the gap opening tendency is not loop-order dependent and the only remaining difference between the $1\ell$ and SDE scheme is the form-factor truncation. Further, this indicates indirectly that the gap opening is driven by strong AF fluctuations, see also Appendix B.

Our results do not contradict the findings of the parquet approximation Eckhardt et al. (2020). There, for $U=2$ a pseudogap occurs at $1/T=26$ and a full gap at $1/T=30$. These temperatures are presently difficult to access within a multiloop fRG calculation, due to the required refinement of the momentum and frequency dependence of the two-particle vertex and the high number of loop orders needed for convergence.

In this section we will discuss the origin of the different gap opening tendencies in the $1\ell$ and SDE flow and in particular address the question why the long-range AF correlations lead to a gap opening in the flow of the SDE, while its effect is weakened in the conventional $1\ell$ flow of the self-energy.

In the multiloop expansion of the fRG, the two schemes are formally equivalent. Differences are introduced only by truncating the form-factor expansion, which prevents the full reconstruction of the SDE at loop convergence. The results and discussion in Section III.2 rule out the finite loop order as being responsible for the qualitatively different physical behavior between the conventional and the SDE flow.

The difference between the self-energy results as obtained from the two flow schemes is hence to be attributed to the truncation of the form-factor expansion. As discussed in Section III.2, the approximations introduced by the projections between the different channel representations imply that the $1\ell$ and SDE flow are not equivalent any more. We first focus on the SDE (5) together with the corresponding flow Eq. (6) and consider the lowest orders in the bare interaction $U$, see Fig. 10. We decompose the vertex into the different channel contributions and for simplicity neglect the self-energy corrections in the propagators. The second order diagram is not associated to any specific channel and can be computed by using Fast-Fourier-transforms, see also Eq. (8). For a better comparison to the conventional $1\ell$ flow, the different colors of the boxes indicate the topology while the ones of the Green’s functions refers to the projections, i.e. the order in which the (truncated to $s$-wave) bubbles are inserted. In the present convention, the orange bubble is inserted into the blue one which is then closed by a single Green’s function (black). We note that for the diagrams shown in Fig. 10, the restriction to $s$-wave does not introduce any approximation in the projection of the different channel representations.

Now we compare these diagrams with those of the standard $1\ell$ flow shown in Fig. 11. We insert the parquet decomposition of the two-particle vertex into the flow equation for the self-energy, being aware that in the $1\ell$ approximation this holds only up to second order, and group the different contributions according to their topology (indicated by the colored boxes), in analogy to Fig. 10. The gray box at first order (a) is the tadpole diagram, while the gray box at second order (b) is a self-energy correction thereof, which is included in the SDE-flow once the self-energy corrections in the Green’s functions are accounted for (despite not being displayed in Fig. 10). The red ones at the second order include three contributions. These amount to exactly the same diagram in the SDE (see Fig. 10), as the blue bubble has no other contributions than $s$-wave. At third order, the red boxes with internal gray boxes are associated to self-energy corrections of the second order diagram, while the gray ones with and without internal red boxes correspond to self-energy corrections of the first order diagram. Most importantly, there are five different yellow and green boxes at third order which represent the contributions to the particle-particle and crossed particle-hole diagram in the SDE (see Fig. 11). Here only a single one of each is correctly accounted for in $s$-wave. That the others are only correct in the infinite form factor limit, can be seen from the example of the first green diagram in the second line for $F_{\uparrow\downarrow}$: here the blue bubble gets non-$s$-wave contributions due to the insertion of the orange bubble. In the $s$-wave (or any few form factor) truncation, these are neglected. This explains why we obtain different results for the two self-energy flow schemes.

In order to underline that the different form factor approximation of the crossed particle-hole channel is responsible for the flow-scheme-dependent gap opening tendencies, we analyze the lowest order contributions of the latter (e.g. the green boxes in Fig. 10 represent the second order). The corresponding self-energy diagrams up to fourth order are shown in Fig. 12, the extension to higher orders is straightforward. The resulting $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)$ is reported in Fig. 13 and 14 for the 4th and 7th order in $U$. The downturn of $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)$ at small $U$ is due to the second order diagram in Fig. 12. This is compensated by higher order contributions leading to a zero in $\partial_{i\nu}\operatorname{Im}\Sigma({\bf k},i\nu)$. The effect of the crossed particle-hole channel contributions is more pronounced at the antinodal point, see Fig. 13. Concerning the two flow schemes, in the SDE flow the zero crossing is observed first at 4th order at roughly $U=4.8$, while all other crossings occur only at much larger values of $U$. For higher orders, the zero crossings shift to smaller interactions. At the 7th order both flows open a gap first at the antinodal point and then at the nodal one, consistently with the physical picture obtained by the fRG results of Fig.  2. Note that in the SDE flow both gaps open before the first gap opening in the conventional $1\ell$ flow sets in. This can seen also in the gap opening as a function of order in $U$, see Fig. 14. With increasing order in $U$, the gap opening occurs at lower values of $U$. While the qualitative behavior of the SDE and $1\ell$ flow is the same, the gaps at the antinodal and then at the nodal point open first in the former, followed by the ones in the latter.

We note that in the fRG calculation the gap opening sets in at lower interactions due to the resummation of all orders. In fact, the perturbation theory does not capture the AF divergence. Despite these limitations, the present analysis illustrates the order in which the gap opening occurs: first in SDE-flow at the antinodal and then nodal point; secondly, if we ignore the vertex divergency, in the $1\ell$ flow in the antinodal and last at the nodal point.

Finally, we emphasize that the two schemes yield different results only in a truncated form-factor expansion. A straightforward patching of all three momentum dependencies on the same grid involves no information loss in the projection from one channel to another. At the same time, the fine grid required to resolve the AF divergence in the magnetic channel, implies a huge numerical cost. Similarly, also in the TU-fRG scheme both schemes should converge to the same result for an increasing number of form factors. However, this number may be as large as the bosonic patching points.