Multi-band D-TRILEX approach to materials with strong electronic correlations

The authors present an extension of the D-TRILEX numerical method for many body systems introduced in Ref. [91] for a single-band model to the more generic case of multi-orbital systems. After introducing the formalism in Sections 2 and 3 (with the support of the Appendices) and discussing the details of the numerical implementation in Sec. 4, the authors showed applications of the method to several cases, benchmarking it when possible to exact solutions or less approximated methods.
Although there are no conceptual advances with respect to the single orbital case, the article reports technical advances due to numerical difficulties for the theoretical treatment of more complex multi-orbital systems.

The benchmark with the Kanamori dimer is particularly interesting. In fact, given the extreme simplicity of it, the dimer has an exact solution that can be used to validate the method. In section 5.1., it is shown how DMFT fails to predict the correct value of the imaginary part of the local self-energy at low frequency while the D-TRILEX gives the correct result.
However, the benchmark with the exact solution is not complete. In fact, the authors should show how the non-local components of the self-energy, that are missing in DMFT, compare with the exact solution. At the same time, it would be important to check that also the susceptibilities calculated with D-TRILEX compare well with the exact solution.
Furthermore, since the method can treat non-local interactions, it would be instructive to show a comparison with the dimer in presence of non-local interactions as well.

In section 5.2 the authors apply the D-TRILEX to the extended Hubbard model on a square lattice and compare the results to the ones obtained using DIAGMC@DB in Ref.[82]. While for weak-to-intermediate values of the onsite repulsion U the agreement between the two methods is good, their predictions start to deviate for larger values of the Hubbard U. While the authors report this problem they do not really discuss in too much detail why it is happening and how this could be overcome.
In these works Phys. Rev. B 99, 235106 (2019) and Phys. Rev. B 104, 235128 (2021) efficient ways for calculating vertex functions in the strong coupling limit have been put forward reproducing the asymptotic power law behavior of the Neel Temperature in three dimensions for large values of U. The authors should cite these papers and comment if it is possible to use approximate forms for the vertices to obtain more accurate results at stronger couplings.

As a suggestion to improve the readability of the paper, I recommend the authors to add a figure where they show schematically the workflow of the D-TRILEX discussed in section 4.

I find the manuscript interesting enough and I recommend its publication after my comments have been properly addressed.

1 - presentation of the formalism is thorough
2 - numerical benchmarks are convincing
3 - the method presented truly appears to have good prospects for application

1 - it is difficult to extract the main points from the paper (introduction and conclusion sections are very short compared to the overall length)
2 - the paper presents very little new physical insight
3 - the paper presents very few new technical and conceptual ideas (the generalization of the D-TRILEX to the multiband case appears rather straightforward)

The authors give a thorough account of the D-TRILEX formalism, developed for a rather general, multi-band lattice model. Details of the implementation are discussed, and the method is tested on several examples, and compared directly to other methods (DMFT, DiagMC@DB). In cases where no benchmark data is available, the physical picture emerging in the tests is analyzed in some detail and is found to be consistent with the findings of previous works.

The subject of the paper is important. The development of methods that capture non-local correlations in 2D lattice models falls under central tasks in condensed matter theory. Added to that, the ability to treat multiband models may prove essential for the description of numerous materials of interest. In my view, the work presented in this paper is very well motivated.

The numerical results appear very convincing: Figs.2-4 demonstrate that D-TRILEX performs consistently better than DMFT and gives results which are comparable to (I imagine) much more expensive DiagMC@DB. The amount of data presented also indicates that the computations are, indeed, feasible and relatively cheap.

However, I find that some important points are not sufficiently emphasized. As far as I understand, one of the big advantages of the D-TRILEX method is that it does not require retarded interactions in the impurity problem (unlike TRILEX). There are no mentions of this in neither the introduction or the abstract. In the conclusions, it is stated that even frequency-dependent interactions in the original model would not change this desirable property of the D-TRILEX method. This should be more elaborated as it may not be immediately obvious to a casual reader.

I also find that the benchmark in the case of the extended Hubbard model (Section 5.2) is out of place. There have been previous implementations of the D-TRILEX method which could have treated this same model (e.g. Ref. 91). I do not understand why the multiband generalization is needed for this benchmark. It would have made more sense to focus on actual multiband models in the main text. At least, the authors should have emphasized that Fig.4 does not actually test any of the new generality of D-TRILEX developed in this work.

I also find a general lack of discussion regarding the systematic errors in the theory. One of the main touted strengths of D-TRILEX is that the expression for the self-energy and polarization in D-TRILEX satisfy symmetries that are not satisfied in standard TRILEX. In standard TRILEX, the expressions $\Sigma=\lambda G W \Lambda$ and $P=\lambda GG \Lambda$ (featuring only one renormalized vertex $\Lambda$) allow for a straightforward formulation of systematic corrections to the theory: In the cluster TRILEX, the expressions for $\Sigma$ and $P$ are of the same form as in the single-site theory, and as $N_\mathrm{imp}\rightarrow \infty$, one recoveres the exact result . It is not immediately clear that the single-site D-TRILEX can be systematically improved towards the exact theory. What is the $N_\mathrm{imp}\rightarrow \infty$ limit of D-TRILEX? I think that in the introductions and conclusion, the properties of D-TRILEX should be discussed in terms of control parameters and sources of systematic error. Is there a series of systematic improvements to D-TRILEX which leads to the exact solution of a model? Can the extended Hubbard model (discussed in Section 5.2) be rewritten in a 2-site supercell notation and then solved straightforwardly by D-TRILEX as a multiband model? If possible, such benchmark would certainly be of great interest for the readers, and would greatly strenghthen the paper. In fact, in the introduction it is stated that the authors formulate D-TRILEX "in a multi-orbital and multi-site framework", yet no actual benchmarks with $N_\mathrm{imp}>1$ were given, to my understanding.

In general, I would suggest that the authors expand the introduction and conclusions so as to rely less on the reader's ability to sift through very long formalism. Main points should be accessible in a quick reading of the paper.

1 - in the introduction and conclusions, emphasize more and elaborate on the properties of the impurity problem. If no retarded interactions are needed in the impurity problem, why is that? Under what conditions is this no longer true?

2 - explain better the motivation for the benchmark in the case of the extended Hubbard model. What does this have to do with the multiband generalization of D-TRILEX? What about the multi-site ($N_\mathrm{imp}>1$) solution for the extended Hubbard model?

3 - discuss more the systematic errors, and possibility for systematic improvements. How does the formalism change if we improve the dual boson starting point? Explain what is the $N_\mathrm{imp}\rightarrow \infty$ limit of the method.

4 - Discuss more the relationship of D-TRILEX to TRILEX and other methods. Citation of Phys. Rev. B 96, 104504 (2017) in the context of TRILEX and GW+DMFT is due, as it showcases the versatility of those methods. Can D-TRILEX be formulated in Nambu space to treat superconductivity (it is, in a way, a 2-orbital calculation)?

5 - Discuss more the stability of the Dyson equation in the second paragraph of page 11. Have you encountered problems with diverging bosons? Is there a general problem in reaching low temperatures? Can one suppress ordering instabilities? Can one apply D-TRILEX in ordered (say, AFM) phases?

5 - below Eq.2 define $\vec{\gamma}_{\mathbf{k},ll'}$ for the sake of completeness.

6 - last paragraph before Section 6, fix "the complex physics in the considered this model"