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CDMFT+HFD : an extension of dynamical mean field theory for nonlocal interactions applied to the single band extended Hubbard model
by Sarbajaya Kundu, David Senechal
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Authors (as registered SciPost users):  Sarbajaya Kundu 
Submission information  

Preprint Link:  scipost_202312_00023v1 (pdf) 
Date submitted:  20231214 13:10 
Submitted by:  Kundu, Sarbajaya 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We examine the phase diagram of the extended Hubbard model on a square lattice, for both attractive and repulsive nearestneighbor interactions, using CDMFT+HFD, a combination of Cluster Dynamical Mean Field theory (CDMFT) and a HartreeFock meanfield decoupling of the intercluster extended interaction. For attractive nonlocal interactions, this model exhibits a region of phase separation near halffilling, in the vicinity of which we find pockets of dwave superconductivity, decaying rapidly as a function of doping, with disconnected patches of extended swave order at smaller (higher) electron densities. On the other hand, when the extended interaction is repulsive, a Mott insulating state at halffilling is destabilized by hole doping, in the strongcoupling limit, in favor of dwave superconductivity. At the particlehole invariant chemical potential, we find a firstorder phase transition from antiferromagnetism (AF) to dwave superconductivity as a function of the attractive nearestneighbor interaction, along with a deviation of the density from the halffilled limit. A repulsive extended interaction instead favors chargedensity wave (CDW) order at halffilling.
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Reports on this Submission
Report #2 by Anonymous (Referee 2) on 202425 (Invited Report)
 Cite as: Anonymous, Report on arXiv:scipost_202312_00023v1, delivered 20240205, doi: 10.21468/SciPost.Report.8492
Report
I have read with interest the manuscript by Kundu and Senechal, where the Authors present a systematic investigation of the effects of nonlocal attractive as well as repulsive interaction terms to the groundstate properties of the (extended) Hubbard model of an (unfrustrated) square lattice.
Their investigation has been carried out by means of a reliable manybody approach, the Cluster (or Cellular)dynamical meanfield theory (CDMFT), which captures nonperturbatively the electronic correlations within the clusters considered and which has been further supplemented by a conventional meanfield treatment of the intercluster effects.
The results presented in the manuscript, while essentially confirming some of the qualitative trends emerged in previous, more sporadic investigations of the literature, allow the Authors to outline a quite general picture of the physics driven by nonlocal electronic interactions in twodimensional lattice systems.
In particular, I have appreciated the well conceived organization of the presentation of the data and of the associated physical discussions, and the interrelation the Authors make between the different regimes considered. This renders the reading of the whole manuscript quite pleasant and instructive.
On the basis of these considerations, I recommend the publication of the manuscript by Kundu and Senechal on SciPost PhysicsCore, after the Authors have (optionally) considered the specific observations I am including below, aimed at further improving the clarity and the impact of their paper.
Requested changes
Optional observations/suggestions for changes:
1) In the presentation of the method, I find a bit confusing the description of the number of (bath) parameters. In particular, while the fact that they should be 6 for the simple impurity case, illustrated in Fig. 1 is quite evident, that their number rises for the general impurity model up to 18 is not immediately clear understandable by looking at Fig. 2, as well as from the related discussion at p. 6, lines 165171. Maybe the Authors could extend/make the connections between the Figure and manuscript more transparent on that specific point.
2) As the model considered has no frustration, I would have expected a completely symmetric shape of the different phases appearing in Fig. 7, as a function of filling (with the possible exception of first order coexistence features). This expectation seems indeed to be met except for the small swave region present at the extreme band edge (n=0) but not on the opposite side (n=2) in the first panel of the Figure (U=0, V= 0.4) or displaying an apparently different shape in the fourth panel (U=2, V= 0.7). Is this just a numerical effect, or is such generic expectation not fully correct?
3) While the overall trend of suppressing/enhancing the longrange ordered phases by excluding/including the (corresponding) selfconsistent meanfield parameters (E_s, E_d, etc ...) is clear, the data shown in Fig. 13 seem to display also some exception, such as for the swave order without E_s, which appears between 0.6 <n < 0.65. Further, it looks to me, that although the strongest mismatch between the calculations w/wo E_s is found for the case of V=0.7 in the density range of 0 <n < 0.3, this is not explicitly remarked in the Caption as well as in the manuscript text.
4) Somewhat related to this: The Authors correctly mention in their discussions, that the inclusion of intracluster correlation reduces the ordering tendency of the system w.r.t. the pure meanfield, while they find (which is quite reasonable) that the intercluster meanfield selfconsistency partially "reenhances back" the ordering tendency. On this basis, would it be reasonable, then, to expect that the (unknown) exact ground state properties lie somewhat in between those two description (CDMFT) and (CDMFT + mean field) ? Or the limitation to small cluster size would prevent to draw such conclusion?
5) In the calculations based on diagrammatic extensions of DMFT (such as the Dynamical Vertex Approximations, the Dual Fermion/Dual Boson approach, etc.), it was shown that for the unfrustrated 2D Hubbard model for V=0 the metalinsulator transitions described by the DMFT MIT was shifted down to U > 0 for T> 0, with some hints that its 1st order nature might be dissolved in a sharp crossover at finite T. Would the Authors expect a similar trend might affect their 1st order transitions, if considering correlation on larger cluster size?
6) At the end of p. 12 the Authors state that the region of phaseseparation shrinks with increasing onsite repulsion U. However, while this is intuitively understandable in the weakcoupling limit, there might be exceptions at stronger coupling. I refer here to the finite temperature analysis of , e.g., PRB 99, 035161 (2019) and PRL 125, 196403 (2020), where it was shown how nonperturbative scattering processes arising from the strong onsite repulsion might appear, to a certain extent, as an effective attraction.
An extension of the discussion on this point might be then considered (see also below).
7) In the interest of the manuscript impact, one could consider to extend the final part of the conclusion, by presenting more concrete outlooks for this work. For instance, would it be technically possible to perform such systematic investigation for single orbital Hubbard model on triangular lattices? If yes, this might worth to be mentioned, as the importance of nonlocal interactions (even longrange ones!) have been pointed out in the literature (see e.g., PRL 110, 166401 (2013).
Further, in the light of 6), it could be interesting to extend, in the future, the Authors' calculations to the case with V<0 and larger U values to estimate their intertwined effect in the nonperturbative regime in driving/suppressing the tendency to phaseseparation.
Finally, one could mention the insertion of a geometrical frustration (eg, via a next to nearest hopping term), which would be also important for make a more precise connection to the physics of cuprates, in the case this might be incorporated within the ED implementation (for the foursite cluster) adopted by the Authors.
8) Eventually, I would replace the word "pockets" in the abstract with some equivalent one, to avoid confusion with the usual meaning (referred to the Fermi Surface) with which this term is used to describe the physics of cuprates.
Report #1 by Anonymous (Referee 1) on 202414 (Contributed Report)
 Cite as: Anonymous, Report on arXiv:scipost_202312_00023v1, delivered 20240104, doi: 10.21468/SciPost.Report.8364
Strengths
1 The authors provide a detailed and wellcited presentation of the different studies previously performed on the Extended Hubbard Model (EHM).
2 The manuscript is written in a clear way and it leaves no ambiguities.
3 This work illustrates the significance of the nonlocal (intercluster) interactions by comparing the results of CDMFT+HFD to those of simple CDMFT. It is shown that certain regions of superconducting order appear only upon inclusion of the intercluster interactions.
4 The method appears to be agile with no significant restriction in its applicability and can be used in the future for studies of more complicated systems (e.g. the authors suggest the study of systems with spinorbit interactions).
Weaknesses
1 The results are in good qualitative agreement with previous works, however there is not significant originality in the findings.
2 This approach is not suitable for studying incommensurate magnetic orders, which are found in the literature to be realized at certain regimes of the phase diagram.
Report
The manuscript investigates the phase diagram of the Extended Hubbard Model (EHM) using Cluster Dynamical MeanField Theory (CDMFT) + HartreeFock meanfield Decoupling (HFD). This method is advantageous because it treats the interactions within a cluster nonperturbatively and it further goes beyond that by accounting for intercluster interactions in a meanfield manner.
The authors study the competition between different phases for a repulsive local Coulomb interaction ($U>0$) and for both a repulsive and an attractive nearestneighbor interaction ($V>0$ and $V<0$ respectively). They work on two different impurity models, a more general one and a simplified one, and they compare the relevant results. Different regimes of the phase diagram are investigated (with respect to interaction strength and doping) and both spin/charge ordered phases as well as superconducting phases are investigated.
The authors find that for repulsive nearestneighbor interactions and strong local repulsion a firstorder transition from an antiferromagnetic to a charge density wave state is observed. Upon holedoping a $d$wave superconducting dome is observed. For attractive nearestneighbor interactions, at the particlehole symmetric chemical potential, a firstorder transition takes place from antiferromagnetism to $d$wave superconductivity, accompanied by phase separation in the vicinity of halffilling. Away from halffilling, regions of $d_{x^2y^2}$wave superconducting order are found as well as disconnected pockets of extended $s$wave order.
The results are in good qualitative agreement with previous works using other approaches. The CDMFT+HFD method is novel and it has the advantage of combining a nonperturbative treatment of local and shortrange correlations with the inclusion of spatial fluctuations, in a meanfield manner. As the authors state in the Introduction "This work constitutes a test of the CDMFT+HFD method". To this end, even though there is no significant originality in the obtained results, since there is originality in the method, I believe the goal of the paper has been reached. In my view, the manuscript meets the criteria for publication in SciPost Physics Core and its publication would be beneficial to the community, as it validates the use of this new technique (I would be glad, however, to see the points raised in the "Requested changes" section addressed).
Requested changes
1 I believe it would be beneficial if the authors made a comment on the effect of the cluster size on the validity of the results. They only discuss a 4site cluster, but they do not state if there is a significant technical limitation that prevents the increase of the cluster size. Given for example the observed jumps in the number density $n$ close to quarterfilling, at the attractive $V$ case, that are not well understood, I believe such a discussion might be enlightening.
2 It appears that the comparison between the behavior of the simple and general model as a function of density for $V<0$ (Fig. 7 and Fig. 8) suggests that the simple model can, in certain cases, give misleading results (see $s$wave patches at $U>0$). Maybe the authors could make a comment on this conclusion, since it appears to be an important remark to keep in mind for future applications of the method.
Author: Sarbajaya Kundu on 20240417 [id 4427]
(in reply to Report 1 on 20240104)

Regarding the issue of cluster and bath size: The cluster and bath size are limited by the exact diagonalization solver: the upper limit for the total number of cluster and bath orbitals that one can realistically consider is 4+8=12, given that the ground state and Green function must be computed repeatedly in this approach. Of course, the same remark applies to singlesite DMFT, with a cluster of size of 1! It is therefore impossible to perform a meaningful finitesize scaling analysis in this approach. The assumption is that any changes observed for larger system sizes would be quantitative in nature and that the basic dependence on model parameters or filling would be the same. We comment on this issue in the revised version. An additional remark is that we do not believe that the jumps observed in the number density close to quarterfilling are a result of the cluster size alone, as no such jumps are observed for the simple bath model even with the same cluster size.

Regarding the comparison between the simple and general bath models: The simple model provides a submanifold of the parameter space of the general model. Indeed, we find that this simple model can sometimes give rise to additional phases and the only way to verify the correctness of these results is to confirm them by using the general model. Unfortunately, we do not have a systematic understanding of why such differences appear. However, the results are for the most part qualitatively similar for both models, and we believe that the differences observed in the extended swave order for U>0 serve to provide an illustration of the advantages of considering a larger number of bath parameters in the CDMFT procedure. Indeed, this is a reason for testing two different bath models: to check the robustness of the results, and by and large the results are robust, with the exceptions pointed out by the referee.
Author: Sarbajaya Kundu on 20240417 [id 4428]
(in reply to Report 2 on 20240205)We thank the referee for this extensive and stimulating report.
Concerning the number of independent bath parameters. We agree that the statement about the number of bath parameters in the general model can be confusing for the reader without a more explicit explanation. We have added a sentence in the caption of Fig. 2, as well as in lines 177178 of the main text, explaining that for the general model, we have three bath parameters per bath orbital, which leads to 24 parameters in total, and then we subtract 6 of those due to the presence of a C4v rotational symmetry, giving us 18 bath independent parameters in this case.
Concerning the symmetry of the phase diagram with respect to halffilling. Indeed, the phase diagram must be symmetric about halffilling. Based on our simulations, we find that the observed asymmetry in the swave region present near the extreme band edges in the figure for U=0,V=0.4 is a consequence of insufficient accuracy in the CDMFT computations, as opposed to any new physics. We now comment on this observation in the caption of Fig.7. We have also now replaced Fig.7 of the main text with results obtained with a higher accuracy for the CDMFT procedure, wherever possible. However, we now find some asymmetry between the dwave orders on both sides of halffilling for U=0,V=0.4, which we clearly identify to be the result of lack of convergence and consequently, a lower accuracy being used, in that particular region. This being said, this symmetry may be broken depending on the sweep direction of the chemical potential, but is restored when we sweep in the opposite direction, with possible hysteresis, hence the necessity of covering the full range from n=0 to n=2. This also serves as a sanity check on the accuracy of the method.
Regarding the presence or not of the pairing meanfield E_s. While it is true that there are some additional points obtained in the range 0.6<n<0.65 that are found only in the absence of the extended swave anomalous meanfield parameters E_s, the values of the swave order parameter at these points are found to be below the accuracy of the CDMFT procedure for these computations; hence we do not comment on these differences in the paper. We agree with the referee that the strongest mismatch between the calculations with and without E_s is found for V=0.7 in the density range 0<n<0.3; we have now added a remark about this in the caption of Fig.13.
On the issue of whether one can say that the exact result would lie somewhere between the CDMFT and CDMFT + meanfield results. This is an interesting point. We believe it is too simplistic to say that the exact ground state properties lie in between the two descriptions, since increasing the cluster size might further suppress the ordering tendency due to longerrange fluctuations within the cluster. Hence, we are unable to take a firm stand on this point.
Regarding the eventual disappearing of firstorder transitions upon increasing the cluster size. In general, we do expect that increasing the cluster size will lead to smaller hysteresis regions because of enhanced fluctuations. However, it is impossible to say a priori whether these effects will turn first order transitions into continuous transitions. After all, firstorder transitions between states that share the same symmetry do exist in nature. Dimensionality is probably an important factor. Our results still have a meanfield character; the effect of increasing the cluster size within a purely twodimensional model (more fluctuations) might be compensated by adding a weak hopping term in the third dimension (less fluctuations), when considering a threedimensional model.
Regarding an effective attraction coming from nonperturbative scattering processes at strong coupling. We agree that it is important to discuss the change in the behavior of the phase separation when we consider a stronger onsite repulsion, and it is possible that an analogous situation does arise at strong coupling, even though there is no evidence for continuity between the weak and strongcoupling limits. Nevertheless, it would be interesting to explore the physics in the strongcoupling limit using our method and we have commented on this in the revised version.
About extending the conclusion. We thank the referee for these valuable suggestions, which not only provide us with new perspectives for possible applications of our method, but also contribute to the usefulness of the paper for readers. We have added the suggested remarks to the final part of the conclusion.
We have now replaced the word “pockets” in the abstract and in some other parts of the paper with ”regions/islands”.