Correction to: Single-boson exchange functional renormalization group application to the two-dimensional Hubbard model at weak coupling

Correction to: Eur. Phys. J. B (2022) 95 :202  https://doi.org/10.1140/epjb/s10051-022-00438-2

We discovered an error in the code used to generate the data of our study that slightly affects the flow of the filling. Only our data at finite doping is altered by this issue. However, this problem does not affect the analysis of the paper: besides the plots, mainly a few numbers need to be updated in captions and in the main text. The corrected versions of the figures to be modified, together with their updated captions, are given in the following pages and have been used to replace the corresponding figures in the original article. In the main text, the following corrections should be made:

• In page 12: “below $2.2 \%$” and “does not exceed $6.2 \cdot 10^{-5}$” should be respectively replaced by “below $2.3 \%$” and “does not exceed $5 \cdot 10^{-5}$”.

• In page 13: “a relative difference of $18\%$ ($16\%$)” should be replaced by “a relative difference of $15\%$ ($13\%$)”.

These last two corrections have also been implemented in the original article.

Fig. 8

Momentum dependence of the static Yukawa couplings for the magnetic $\lambda _{\nu =\pi T}^\textrm{M}(\textbf{Q},0)$, density $\lambda _{\nu =\pi T}^\textrm{D}(\textbf{Q},0)$, and both $\lambda _{\nu =\pi T}^{\textrm{SC},s}(\textbf{Q},0)$ and $\lambda _{\nu =\pi T}^{\textrm{SC},d}(\textbf{Q},0)$ superconducting channels as obtained from the SBE formulation of the fRG with (blue symbols) and without rest function (red symbols), for $U=2$, $t^\prime =-0.2$, $\mu =4t'$, and different values of the temperature. At the end of the flow, the filling equals 0.44 for all temperatures (with 0.5 corresponding to half filling), with or without rest function

Fig. 9

Frequency dependence of the Yukawa couplings for the magnetic $\lambda _{\nu =\pi T}^\textrm{M}((\pi ,\pi ),\varOmega )$, density $\lambda _{\nu =\pi T}^\textrm{D}((0,0),\varOmega )$, and s-wave superconducting $\lambda _{\nu =\pi T}^{\textrm{SC},s}((0,0),\varOmega )$ channels for the same parameters as in Fig. 8. Note that $\lambda ^{\textrm{SC},d}(\textbf{Q}=\textbf{0},\varOmega )=0$

Fig. 10

Magnetic $\chi ^\textrm{M}(\textbf{Q},0)$, density $\chi ^\textrm{D}(\textbf{Q},0)$, and s-wave superconducting $\chi ^{\textrm{SC},s}(\textbf{Q},0)$ static susceptibilities as obtained from the SBE formulation of the fRG with and without rest function, for the same parameters as in Fig. 8. The relative difference between the results with and without rest function is below $1\%$ for all temperatures in the magnetic, density and s-wave superconducting channels

Fig. 11

Momentum dependence of the d-wave superconducting susceptibility $\chi ^{\textrm{SC},d}$, determined from post-processing with and without rest function, for the same parameters as in Fig. 8. The relative difference between the results with and without rest function is below $1 \%$ for all temperatures. Additionally, results for the bare bubble contribution $\chi _{0}$ are provided for $T=0.2$. There, the relative difference between the full d-wave superconducting susceptibility and its bare bubble contribution reaches $5\%$ at the $\mathrm {\Gamma }$-point (with or without rest function), which decreases for larger temperatures

For more details on this study, we refer to Sarah Heinzelmann’s PhD thesis [1].

Fig. 12

Magnetic $\chi ^\textrm{M}(\textbf{Q},0)$ and d-wave superconducting $\chi ^{\textrm{SC},d}(\textbf{Q},0)$ static susceptibilities including both the results obtained from the SBE and the conventional fermionic formulations of the fRG with and without rest function, for $U=3$ and $t'=-0.2$ (left panels) and $t'=-0.25$ (right panels), with $\mu =4t'$, and $T=0.2$. At the end of the flow, the filling equals 0.45 and 0.44 at $t'=-0.2$ and $t'=-0.25$ respectively, for both the SBE and conventional fermionic decompositions, with or without rest function. We note that $\chi ^\textrm{M}(\textbf{Q},0)$ is determined from the screened interaction, whereas $\chi ^{\textrm{SC},d}(\textbf{Q},0)$ from post-processing. The relative difference between the data with and without rest function is always below $4\%$ for the SBE decomposition. For the conventional fermionic fRG, however, this relative difference reaches $20\%$ ($10\%$) in the magnetic and $6\%$ ($6\%$) in the superconducting channel for $t'=-0.2$ ($t'=-0.25$). Results for the bare bubble contribution $\chi _{0}$ to the susceptibilities are shown as well

Fig. 18

Bosonic frequency dependence of the magnetic $\chi ^\textrm{M}$ and s-wave superconducting $\chi ^{\textrm{SC},s}$ susceptibilities for the same parameters as in Fig. 8. The relative difference between the results with and without rest function is always below $4\%$ in both the magnetic and s-wave superconducting channels

Fig. 22

Rest function $M^{\textrm{SC},d}_{\nu \nu '}((0,0),\varOmega )=\phi ^{\textrm{SC},d}_{\nu \nu '}((0,0),\varOmega )$ in the d-wave pairing channel, with n and $n^{\prime }$ labeling the fermionic Matsubara frequencies according to $\nu ^{(\prime )}=(2n^{(\prime )}+1)\pi T$, for $U=2$, $t^\prime =-0.2$, $\mu =4t'$, and different choices of the bosonic frequency $\varOmega$

Fig. 23

Same as Fig. 22 with $U=3$ instead. Note the increase of one order of magnitude in the absolute values as compared to Fig. 22