CDMFT+HFD : an extension of dynamical mean field theory for nonlocal interactions applied to the single band extended Hubbard model

I have read with interest the manuscript by Kundu and Senechal, where the Authors present a systematic investigation of the effects of non-local attractive as well as repulsive interaction terms to the ground-state properties of the (extended) Hubbard model of an (unfrustrated) square lattice.
Their investigation has been carried out by means of a reliable many-body approach, the Cluster (or Cellular)-dynamical mean-field theory (C-DMFT), which captures non-perturbatively the electronic correlations within the clusters considered and which has been further supplemented by a conventional mean-field treatment of the inter-cluster 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 non-local electronic interactions in two-dimensional 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 Physics-Core, 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.

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 165-171. 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 s-wave 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 long-range ordered phases by excluding/including the (corresponding) self-consistent mean-field parameters (E_s, E_d, etc ...) is clear, the data shown in Fig. 13 seem to display also some exception, such as for the s-wave 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 intra-cluster correlation reduces the ordering tendency of the system w.r.t. the pure mean-field, while they find (which is quite reasonable) that the intercluster mean-field self-consistency partially "re-enhances 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 (C-DMFT) and (C-DMFT + 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 metal-insulator 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 phase-separation shrinks with increasing on-site repulsion U. However, while this is intuitively understandable in the weak-coupling 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 on-site 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 non-local interactions (even long-range 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 phase-separation.
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 four-site 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.

1- The authors provide a detailed and well-cited 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 non-local (inter-cluster) 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 inter-cluster 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 spin-orbit interactions).

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.

The manuscript investigates the phase diagram of the Extended Hubbard Model (EHM) using Cluster Dynamical Mean-Field Theory (CDMFT) + Hartree-Fock mean-field Decoupling (HFD). This method is advantageous because it treats the interactions within a cluster non-perturbatively and it further goes beyond that by accounting for inter-cluster interactions in a mean-field 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 nearest-neighbor 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 nearest-neighbor interactions and strong local repulsion a first-order transition from an antiferromagnetic to a charge density wave state is observed. Upon hole-doping a $d$-wave superconducting dome is observed. For attractive nearest-neighbor interactions, at the particle-hole symmetric chemical potential, a first-order transition takes place from antiferromagnetism to $d$-wave superconductivity, accompanied by phase separation in the vicinity of half-filling. Away from half-filling, regions of $d_{x^2-y^2}$-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 non-perturbative treatment of local and short-range correlations with the inclusion of spatial fluctuations, in a mean-field 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).

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 4-site 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 quarter-filling, 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.