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Multiloop functional renormalization group for the twodimensional Hubbard model: Loop convergence of the response functions
by Agnese Tagliavini, Cornelia Hille, Fabian B. Kugler, Sabine Andergassen, Alessandro Toschi, Carsten Honerkamp
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Submission summary
Authors (as registered SciPost users):  Sabine Andergassen · Cornelia Hille 
Submission information  

Preprint Link:  https://arxiv.org/abs/1807.02697v2 (pdf) 
Date submitted:  20180810 02:00 
Submitted by:  Hille, Cornelia 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We present a functional renormalization group (fRG) study of the two dimensional Hubbard model, performed with an algorithmic implementation which lifts some of the common approximations made in fRG calculations. In particular, in our fRG flow; (i) we take explicitly into account the momentum and the frequency dependence of the vertex functions; (ii) we include the feedback effect of the selfenergy; (iii) we implement the recently introduced multiloop extension which allows us to sum up {\sl all} the diagrams of the parquet approximation with their exact weight. Due to its iterative structure based on successive oneloop computations, the loop convergence of the fRG results can be obtained with an affordable numerical effort. In particular, focusing on the analysis of the physical response functions, we show that the results become {\sl independent} from the chosen cutoff scheme and from the way the fRG susceptibilities are computed, i.e., either through flowing couplings to external fields, or through a "postprocessing" contraction of the interaction vertex at the end of the flow. The presented substantial refinement of fRGbased computation schemes paves a promising route towards future quantitative fRG analyses of more challenging systems and/or parameter regimes.
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Reports on this Submission
Anonymous Report 1 on 2018910 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1807.02697v2, delivered 20180910, doi: 10.21468/SciPost.Report.575
Strengths
(1) Manuscript develops and discusses a very advanced manybody method
(2) This method is applied to a complicated model (the 2D Hubbard model)
(3) Many convergence and benchmark checks are performed
(4) The developed method has much potential for future applications
Weaknesses
(1) Some technical aspects could be better illustrated (see below)
(2) The violation of the MerminWagner theorem could be better discussed
(3) Possible qualitatively new results for the 2D Hubbard model could be better highlighted (see below)
Report
This manuscript is about the development and application of a fRG algorithm for twodimensional Hubbard models, improving previous implementations in various different aspects. Firstly, the full frequency and momentum dependence of the vertex functions are explicitly accounted for and, secondly, the feedback of the self energy into the twoparticle vertex flow is included. Furthermore, the multiloop fRG scheme is implemented which successively adds higher loop orders to the diagrammatic approximation and allows to systematically improve the scheme. The implementation of the multiloop fRG also represents the main advancement compared to previous fRG studies of twodimensional Hubbard models. Another focus lies on response functions which are computed in two different ways, via a postprocessing of the vertex functions and via explicitly solving the flow equations for the susceptibilities. In the latter case, the authors show how these flow equations can be generalized to a multiloop structure. As a test system, the authors apply their technique to the two dimensional Hubbard model on the square lattice at half filling. Extensive convergence and benchmark checks concerning the number of fermionic frequencies and momentum patching points, the impact of the selfenergy insertion, and the number of loops are performed. As a main result, the authors find a rapid convergence of the susceptibility when increasing the number of loops. This convergence also applies to the aforementioned two methods of computing the susceptibility.
Overall, I find it quite impressive how this work improves the fRG on so many different fronts. Even though the paper has a clear methodological focus I am also impressed that the authors manage to apply their technique to a quite nontrivial system such as the two dimensional Hubbard model. Indeed, what they call their "test model" is already a very complex system. Finally, the comprehensive and careful convergence checks add a significant amount of trust in the validity of their analysis. I see significant potential of this method for future applications to other (possibly more complicated) Hubbard models. For these reasons I recommend the publication of this work after the authors have considered the points/questions listed below.
Requested changes
(1) The Section 2.2 about the derivation of the multiloop fRG equations for the susceptibility is rather technical and I had problems understanding how this multiloop extension works. The section would profit much from a diagrammatic illustration of the scheme. In Fig. 1 an illustration is already given, but I didn't find it very enlightening.
(2) As shown in Fig. 6 the real part of the self energy vanishes at two points at the Fermi surface; hence at these points the Fermi surface remains unchanged by the self energy feedback. Does this apply to the whole Fermi surface and for higher loop orders as well? In other words, does the perfect nesting effect always remain intact?
(3) The authors argue that in the limit of large loop numbers, the MerminWagner theorem should be fulfilled. On the other hand, their numerical results in Fig. 10 still show a significant violation of MerminWagner even at l=8. The authors provide arguments why it is very challenging to suppress the pseudocritical temperature in the absence of extremely longrange fluctuations. However, given the fact that the fRG is often criticized because of the violation of MerminWagner, it would be desirable to see a more detailed discussion here. What I mean is, if a method that is known to fulfill MerminWagner still finds a finite ordering temperature, then this temperature should have a physical meaning. For example, it should be related to the extent of correlations taken into account and it should maybe even be possible to calculate/estimate it without solving the fRG equations. Can the authors comment on this?
(4) Besides quantitative changes in the results when applying their extended fRG scheme to the twodimensional Hubbard model, do the authors also find qualitatively new properties of this system? If yes, I think it should be better highlighted.
(5) It would be very interesting to know about the numerical efforts of these calculations. How long did they take and on how many CPUs did they run? Do these calculations require massive parallelization and supercomputing facilities?
(6) The numbering of the subsections in Sec. 4 is very strange as it starts with a zeroth subsection.