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Quasi-particle functional Renormalisation Group calculations in the two-dimensional half-filled Hubbard model at finite temperatures
by D. Rohe, C. Honerkamp
This is not the current version.
|As Contributors:||Daniel Rohe|
|Arxiv Link:||https://arxiv.org/abs/2005.00827v1 (pdf)|
|Date submitted:||2020-05-05 02:00|
|Submitted by:||Rohe, Daniel|
|Submitted to:||SciPost Physics|
We present a highly parallelisable scheme for treating functional Renormalisation Group equations which incorporates a quasi-particle-based feedback on the flow and provides direct access to real-frequency self-energy data. This allows to map out the boundaries of Fermi-liquid regimes and to study the effect of quasi-particle degradation near Fermi liquid instabilities. As a first application, selected results for the two-dimensional half-filled perfectly nested Hubbard model are shown.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 3 on 2020-7-31 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2005.00827v1, delivered 2020-07-31, doi: 10.21468/SciPost.Report.1874
This article presents a method that extends the standard one-loop functional renormalization group approach by including quasiparticle-based feedback on the flow and allows the direct determination of the real frequency dependence of the single-particle self-energy.
Results are presented for the frequency dependence of the single-particle spectral function compared with second order perturbation theory, as well as the crossover scale between weak and strong correlations, compared with various other implementations of the fRG.
This work is one of the few examples which presents real frequency behavior obtained with fRG and thus will be of interest to the community. I believe that this work potentially opens a new pathway for future work in the study of correlated electron systems and therefore recommend publication, but ask the authors to address the following points:
(1) The determination of the quasiparticle weight Z factors by neglecting the change in the low frequency slope of the real part of the self-energy is concerning. Can the authors come up with a better physical justification for this approximation?
(2) The spectral function in Fig. 5 seems to indicate the opening of a pseudogap at the node, and a quasiparticle peak at the anti-node. This is contrary to the results from other numerical studies of the half-filled Hubbard model. For recent work co-authored by one of the present authors, see e.g. arXiv:2006.10769. There it is shown, within numerically exact techniques, that a pseudogap first opens at the anti-node, and then at the node.
(3) Related to (2), arXiv:2006.10769 also presents fRG results for the self-energy on the imaginary frequency axis. Down to the temperatures accessible by fRG, these results are shown to compare well with the numerically exact QMC results. To better assess the quality of the present fRG scheme, I think it would be useful to add a figure showing results for the imaginary frequency behavior of the self-energy. This should be relatively easy to obtain from the current real frequency results and could potentially resolve the issue in (2).
Anonymous Report 2 on 2020-7-14 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2005.00827v1, delivered 2020-07-14, doi: 10.21468/SciPost.Report.1826
The manuscript addresses properties of the 2D Hubbard model, e.g. phase diagram, self-energy and spectral functions, which are still actively discussed. The authors propose new fRG approach, which yield reasonable results despite considering static vertices and one-loop approximation.
Some parts of the text (Introduction, discussion of the results) need to be extended (see Report).
The paper analyses phase diagram of 2D Hubbard model. The method and the results are certainly interesting, but some points require clarification.
- 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.
- 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).
- 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).
- 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)?
- 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? 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?
- 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?
Report 1 by Walter Metzner on 2020-5-27 (Invited Report)
- Cite as: Walter Metzner, Report on arXiv:2005.00827v1, delivered 2020-05-27, doi: 10.21468/SciPost.Report.1716
The authors present a one-loop functional renormalization group (fRG) study of the two-dimensional half-filled Hubbard model at finite temperature, with a focus on the frequency-dependence of the self-energy. Using a static approximation for the two-particle vertex, the flow of the self-energy can be computed directly on the real frequency axis. While numerous fRG studies of the Hubbard model have already been published, there are only few results for the real-frequency self-energy.
The authors observe an unconventional dip of the imaginary part of the self-energy around zero frequency, which is present only at finite temperature.This feature was discovered previously within plain second order perturbation theory (as the authors correctly point out), but it didn't receive the attention it deserves. They thoroughly discuss the consequences of the dip for the spectral function and Fermi liquid behavior. Signatures of pseudogap behavior are also discussed.
I recommend publication after the points under ''Requested changes'' have been taken into account.
I see several points that should or could be improved, which I divide in ''mandatory'' and ''optional''.
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.
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.
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.
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.
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.
Minor points and typos:
1) There are words missing in the last sentence of the first paragraph on page 3.
2) Why do the authors refer to their results in Fig. 1 as ``raw data''? What does it mean?
3) There is something missing in the condition ''omega = ...'' in line 11 on page 6. As written, the equation doesn't make sense.
4) ''FEA'' is an unusual acronym for the fluctuation-exchange approximation. Why not ''FLEX'', which is the standard one?
5) I don't see the QMC data points from Rost 2012 in Fig. 4. Were they forgotten?