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

Using the recently introduced multiloop extension of the functional renormalization group, we compute the frequency- and momentum-dependent self-energy of the two-dimensional Hubbard model at half filling and weak coupling. We show that, in the truncated-unity approach for the vertex, it is essential to adopt the Schwinger-Dyson form of the self-energy flow equation in order to capture the pseudogap opening. We provide an analytic understanding of the key role played by the flow scheme in correctly accounting for the impact of the antiferromagnetic fluctuations. For the resulting pseudogap, we present a detailed numerical analysis of its evolution with temperature, interaction strength, and loop order.

Figure 1

Self-energy as a function of the flowing interaction $U$ for $1/T=10$. A comparison of the nodal (circles, solid lines) and antinodal points (diamonds, dashed lines) for the first and second Matsubara frequencies (closed and open symbols, respectively) are shown (a) in the conventional $1\ell$ scheme (b) with the derivative of the SDE for the self-energy. The crossings in the latter (inset) indicate the gap opening, occurring first at the antinodal point and then at the nodal point.

Figure 2

Plot of ${\partial }_{i\nu }Im\mathrm{\Sigma }(\mathbf{k},i\nu )$ evaluated at $i\nu =i\pi T$ as a function of the flowing interaction $U$ for $1/T=10$. The conventional $1\ell$ self-energy flow (green) is compared to the one of the derivative of the SDE (blue). The latter crosses zero for both momentum points indicating the opening of a gap, while the conventional flow exhibits a monotonic behavior.

Figure 3

Self-energy as a function of the Matsubara frequency $i{\nu }_n$ for $1/T=10$ and (a) $U=1.55$, (b) $U=1.6$, (c) $U=1.65$, and (d) $U=1.675$ just below and above the gap opening at the nodal (circles) and antinodal (diamonds) points. The results obtained in the conventional $1\ell$ flow are shown in green and the ones from the derivative of the SDE in blue. Note that for $U=1.65$ the $1\ell$ flow has already diverged. For comparison, also the second-order perturbation theory for $U=1.55$ is shown in gray.

Figure 4

Plot of ${\partial }_{i\nu }Im\mathrm{\Sigma }(\mathbf{k},i\nu )$ evaluated at $i\nu =i\pi T$ as a function of the momentum on the Fermi surface.

Figure 5

Plot of ${\partial }_{i\nu }Im\mathrm{\Sigma }(\mathbf{k},i\nu )$ evaluated at $i\nu =i\pi T$ as a function of the flowing interaction $U$ for different temperatures. Using (a) the conventional $1\ell$ self-energy flow, no gap opening occurs for any temperature, while (b) the derivative of the SDE yields a gap opening which for decreasing $T$ sets in at lower values of the effective $U$.

Figure 6

Flowing interaction $U$ at which the gap opens as a function of the temperature. The gap opening is shown for the antinodal point (blue diamonds) and the nodal one (blue circles), which occurs in the proximity of the AF divergence (gray squares).

Figure 7

Flowing interaction $U$ at which the gap opens as a function of the number of momenta $k_x$ (with $k_y=k_x$ and the subscript $f$ for an enlarged fine patching region) accounted for in the two-particle vertex, for $1/T=10$. The gap at the antinodal point (blue diamonds) always opens before the AF divergence (gray squares) sets in, while the gap at the nodal point (blue circles) vanishes with increasing resolution of the Brillouin zone (see the text for details). The light blue line corresponds to the parametrization of all other computations.

Figure 8

Plot of ${\partial }_{i\nu }Im\mathrm{\Sigma }(\mathbf{k},i\nu )$ evaluated at $i\nu =i\pi T$ as a function of the flowing interaction $U$, for $1/T=10$ and different loop orders $\ell$, using both (a) the conventional fRG and (b) the SDE approach for the self-energy flow, which takes into account the form-factor projections in the different channels. In the conventional fRG a tendency towards gap opening is observed only very close to the pseudocritical temperature without actually leading to a crossing of the self-energy in the first Matsubara frequencies. In the SDE approach, the gap opening at the antinodal point is observed also at higher loop order, while the gap at the nodal point vanishes with increasing $\ell$. For a direct comparison of the gap opening scales and the onset of the AF divergence we refer to Fig. 9, which includes higher loop orders.

Figure 9

Flowing interaction $U$ at which the gap opens as a function of the loop order $\ell$, for $1/T=10$. The gap opening at the antinodal point (blue diamonds) always occurs before the AF divergence (gray squares), while the gap opening at the nodal point (blue circles) disappears into the pseudo-ordered phase at higher loop order. The oscillatory behavior is characteristic of the loop convergence [29].

Figure 10

Lowest-order diagrammatic contributions to the SDE self-energy flow, where solid (dashed) lines carry spin up (down) and the orange bubbles are projected to the blue ones. The colored boxes facilitate the comparison to the conventional flow shown in Fig. 11 (note that due to the restriction to bare Green's functions not all corresponding gray boxed diagrams are included).

Figure 11

Lowest-order diagrams for the conventional self-energy flow, with the (a) $F_{\uparrow \downarrow }$ and (b) $F_{\uparrow \uparrow }$ contributions. All propagators are bare ones (neglecting any self-energy correction). In order to facilitate the comparison with the SDE flow shown in Fig. 10, we group them in tadpole diagrams (gray boxes), equivalent second-order diagrams (red boxes), and third-order diagrams which are formally equivalent to the third-order diagrams from ${\mathrm{\Phi }}_{pp}$ (${\mathrm{\Phi }}_{\overline{ph}}$) in the SDE flow [yellow (green) boxes] but partly affected by approximations due to the form-factor expansion and their projections in the different channels.

Figure 12

Comparison of the lowest-order diagrams describing the contribution of the crossed particle-hole channel to the self-energy, in (a) the SDE and (b) the conventional $1\ell$ flow. The results are shown in Figs. 13 and 14.

Figure 13

Plot of ${\partial }_{i\nu }Im\mathrm{\Sigma }(\mathbf{k},i\nu )$ evaluated at $i\nu =i\pi T$ resulting from the contributions of the crossed particle-hole channel up to the (a) fourth and (b) seventh order in $U$, as illustrated in Fig. 12. While at the second order no gap opens, starting from the third both approaches lead to a gap opening at the nodal and antinodal points at some large value of $U$ (not shown). At the fourth order, the SDE flow opens a gap at the antinodal point first. At the seventh order one can already see that both flow schemes open a gap first at the antinodal point and then at the nodal one. Note that in the SDE flow both gaps open before the first gap opening in the conventional $1\ell$ flow sets in. For a study of the gap opening as a function of order in $U$ see Fig. 14.

Figure 14

Gap opening as a function of order in $U$ at $1/T=10$ as extracted from ${\partial }_{i\nu }Im\mathrm{\Sigma }(\mathbf{k},i\nu )$ evaluated at $i\nu =i\pi T$, resulting from the contributions of the crossed particle-hole channel only. With increasing order, the gap opening occurs at lower values of $U$. The gap at the antinodal point opens first and then the gap at the nodal point, consistently with the full calculation in Fig. 2. While the qualitative behavior of the SDE and $1\ell$ flow is the same, the gaps in the former open first, followed by the ones in the latter. Note that the pseudocritical interaction is not captured in this approach as a divergence of the vertex occurs only at infinite order.

Figure 15

Plot of ${\partial }_{i\nu }Im\mathrm{\Sigma }(\mathbf{k},i\nu )$ evaluated at $i\nu =i\pi T$ as a function of the flowing interaction $U$ for $1/T=10$ and different channel approximations with both (a) the conventional fRG and (b) the SDE approach for the self-energy flow. Neglecting the $pp$ and $ph$ channels does not qualitatively affect the appearance of the gap opening, except for the reduction of the critical scale, which eventually prevents its observation at $k_n$.