Quasi-particle functional Renormalisation Group calculations in the two-dimensional half-filled Hubbard model at finite temperatures

We thank the referee for these valuable and critical comments. Please find our replies to the individual points below.

• The Introduction looks a bit too technical to me. It would be more preferable if the authors overview previous results on 2D Hubbard model, describe open problems, possible applications, etc.

We have extended the introduction in this area. Yet, this work is intentionally on the technical side and a comprehensive overview over open problems and possible applications is beyond of what we can offer here.

• Is there physical justification for the determination of Z-factors from the energy region |w|>2T as shown in Fig. 2? What is accounted and what is neglected by this approximation ? (see also next two points).

We added a further statement on this in the main text. The approach is pragmatic, not rigorous. The physical justification, as stated in the text, consists in the observation that the spectral functions - albeit stemming from a non-Fermi-liquid-like self-energy in the strict sense - can well be approximated by a Fermi-liquid-like parametrisation in a certain range of T and U.

• In Fig. 4 it would be interesting to see the comparison of the obtained critical scale to the results of DDMC approach of Phys. Rev. Lett. 124, 017003 (2020), together with the discussion of the nodal/antinodal dichotomy (see also the next point).

We have included the fit at the anti-node from Phys. Rev. Lett. 124, 017003 (2020) in figure and citation in caption of Uc_T

(Issue nodal/anti-nodal see reply to next point)

• In Fig. 5 the authors show spectral functions, which are split (pseudoagp-like) in the nodal direction, but more quasiparticle like in the antinodal direction. Naively, I would expect the opposite (see DDMC results). The authors assign splitting in the nodal direction to thermal effects, which are similar to SOPT. However, since the considered (U,T) point is close to the crossover line, it is not clear why the nodal spectral function is not affected by correlations. Does that mean that the considered approach underestimates the effect of correlations on the spectral functions in the nodal direction (especially in view of agreement of the obtained crossover line to other approaches, including DDMC, where nodal and antinodal pseudogap appear at close values of T, U)?

We repaired the symbols in Fig. 4, added a Figure for Im(Sigma) and also a discussion of the angular dependence.

• In Fig. 6 the authors plot Z factors as a function of the angle around the Fermi surface. Is the Z factor determined in the way shown in Fig. 2?

Yes. We added this in the caption

• How the true quasiparticle residue (or, better, dReSigma/d\omega if it is positive) determined from small energy range |w|<<T, looks as a function of the angle around the Fermi surface?

There is no "true quasi-particle residue" at omega=0 since there is no quasi-particle when the slope is positive. We could analyse the slope of Re(Sigma) in this region, but we feel it does not add information beyond what is seen in Figure 3 and Figure 5.

• What are the U, T parameters in Fig. 7? Where does the estimate U\approx 1.6 on the top of p. 9 come from?

The x-label ist wrong: The x-axis shows U, not \omega. Temperature is T=0.1 throughout the whole paper, unless indicated otherwise. In turn, the estimate U \approx 1.6 comes from the fact that on the x-axis (which should display U) the flow of the derivative of the self energy along the Fermi surface starts to deviate. That is when numerical criteria for the quality of the approximation begin to worsen, as outlined in the appendix.

We have corrected the x-axis label and added T=0.1 in the caption.

We thank the referee for his comments and suggestions. We agree with most of them, while for some of them we are not sure whether we fully understand the background. Please find our answers below.

Mandatory:

1) The thermal dip in the self-energy was observed already by Katanin and Kampf in their fRG study (Refs. 31 and 32). Their work and a comparison to their results should be discussed more thoroughly.

We believe there is a misunderstanding. What we termed "thermal dip" and what is e.g. shown in our Fig. 1 is not present in our view in [31,32] (we also note that these references do not address the nested half-filled case of the Hubbard model). In SOPT, there is this thermal break-down path, which however fades away at lower temperatures. What may be more relevant is the breakdown of the Fermi liquid due to correlation effects, that is advocated in Refs 31, 32 and also seen near the anti-nodal point in our data. In the revised manuscript we discuss and discriminate these two effects more clearly.

2) In 1.2 it is claimed that Ref. 16 is the only fRG implementation where the self-energy is computed directly on the real axis. This may be true for studies of the 2D Hubbard model, but certainly not in general. Hence, the authors should restrict the statement to the two-dimensional Hubbard model.

We agree. We have softened our statement in the manuscript. Further, we added two works by Förchinger on general fRG formalism and by Enss with a real-frequency self-energy analysis in other systems. Beyond this, we would also be grateful to get hints on concrete references where this has been done since we are not aware of these works and have not been able to find them.

3) In the introduction the authors write that ''qualitatively there are hardly any issues'' (for the 2D Hubbard model). This is not true and should therefore be removed. In particular, none of the existing fRG versions is able to capture the pseudogap, and no method at all is able to deal with strange metal transport behavior.

We were imprecise and will improve on this section. We now write 'Qualitatively, the method has been able to provide relevant insights, but after more than 20 years the goal now is to also reach a better definition of its quantitative character.' What we meant to say was that the methods qualitatively agree amongst each other in most respects. Of course there are effects that are captured by none of them. Yet, some fRG-works do capture pseudogap-like features or the onset thereof, e.g. references [31,32] and [16].

4) The authors should not try to sell a one-loop truncation with a static vertex as a step toward ''quantitative maturity'' (in introduction), as it is clear that higher loops and frequency dependencies of the vertex change results a lot already at moderate interaction strength. The partial agreement with results from more accurate methods might be just accidental.

Again, we were imprecise and changed the wording, no longer using 'maturity', see answer to 3) above. The line of thinking in our motivation was to include aspects which have shown to improve fRG results quantitatively, in particular the feedback of the self-energy in the Katanin-corrected version (reference [14]). Of course there are effects that remain out of scope such as higher-loop corrections, and also the fact that we use a static vertex together with a frequency-dependent self-energy violates the general line of though in reference [14]. That is the reason why we did calculations for both cases, with and without Katanin-correction, to compare them to results from other methods. Whether agreement is accidental or not is a matter of discussion, of course. Here, we present data, no more no less.

Optional:

1) I strongly encourage the authors to include results away from half-filling in the present paper, instead of postponing this to a future publication. This would allow one to assess to what extent the unusual self-energy dip is associated with nesting and the van Hove singularity. It shouldn't be hard to produce the data and add a few figures. The method remains the same, so that the paper would not become much longer.

The computation does require substantial compute time. We can include a few results for these cases, but a comprehensive study is out of scope for this work. We would prefer to disentangle the peculiar case of the half-filled perfectly nested case from the general, non-half filled non-nested cases to avoid a loss in oversight.

Minor points and typos:

1) There are words missing in the last sentence of the first paragraph on page 3.

Confirmed. We repaired this in the revised manuscript.

2) Why do the authors refer to their results in Fig. 1 as raw data''? What does it mean?

We were incomplete here: "Raw" means these are the frequencies at which the self-energy is actually computed. To calculate the real part, the Z-factor and the spectral function we need to compute a Kramers-Kronig-transform. This requires a much finer resolution of the data, which is at the numerical level achieved by an Akima-spline interpolation. The result of the latter is the "post-processed" data as contrasted to the "raw" data. We make this clearer now in the revised version.

3) There is something missing in the condition ''omega = ...'' in line 11 on page 6. As written, the equation doesn't make sense.

Confirmed. '\Sigma' is missing. Has been fixed.

4) ''FEA'' is an unusual acronym for the fluctuation-exchange approximation. Why not ''FLEX'', which is the standard one?

The authors of that very reference used this acronym. We are happy to use FLEX.

5) I don't see the QMC data points from Rost 2012 in Fig. 4. Were they forgotten?

It is present but lies on top of the point for Vekic 1993. That makes it hard to tell them apart. We will try if using different symbols helps.