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Multiband DTRILEX approach to materials with strong electronic correlations
by Matteo Vandelli, Josef Kaufmann, Mohammed ElNabulsi, Viktor Harkov, Alexander I. Lichtenstein, Evgeny A. Stepanov
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Evgeny Stepanov · Matteo Vandelli 
Submission information  

Preprint Link:  https://arxiv.org/abs/2204.06426v3 (pdf) 
Date accepted:  20220726 
Date submitted:  20220707 13:20 
Submitted by:  Vandelli, Matteo 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We present the multiband dual triply irreducible local expansion (DTRILEX) approach to interacting electronic systems and discuss its numerical implementation. This method is designed for a selfconsistent description of multiorbital systems that can also have several atoms in the unit cell. The current implementation of the DTRILEX approach is able to account for the frequency and channeldependent longranged electronic interactions. We show that our method is accurate when applied to small multiband systems such as the HubbardKanamori dimer. Calculations for the extended Hubbard, the twoorbital HubbardKanamori, and the bilayer Hubbard models are also discussed.
Author comments upon resubmission
Reply to the second review of the Referee 1.
The authors have addressed the criticisms raised by the referees substantively. The revised version of the paper includes additional benchmark data which strengthen confidence in the applicability of the proposed DTRILEX method. There are now additional explanations that make the paper more readable, and the main points more easily accessible. The additional comments are written concerning

the meaning of Nimp and the distinction between cluster impurity models and lattices with inequivalent sites

the possibility and the role of including bosonic hybridization functions

systematic improvements towards exact theory (in terms of better reference problems and more complete diagrammatic extensions)

prospects for further generalizations (ordered phases, superconductivity)
The authors also further elaborate on the divergence of bosonic propagators and the strategy to stabilize the calculation. The issue is now presented fairly and in a way that is accessible to a nonspecialist. The added Fig. 2 nicely summarizes the workflow of the method, which greatly improves the readability of the paper. Concerning my request from the previous review:
5  below Eq.2 define →γk,ll′ for the sake of completeness. <<A: This quantity is already defined few lines after Eq.(2).>> Here I meant that one should write the Rashba form of SOC with an equation, rather than to refer to literature. Overall, the revised version of the paper is excellent, and I recommend publication in SciPost Physics.
<<A: We thank the Referee for this nice words and recommendation to accept our work for publication. Concerning the definition of the spinorbit coupling (SOC): In our work the SOC is described by a momentumdependent quantity γk,ll′, which is a Fourier transform of effective spindependent hopping amplitudes between lattice sites. These effective spindependent hoppings can be obtained by diagonalizing a singleparticle part of the Hamiltonian that contains the local term ~L·S, which describes the coupling of the orbital angular momentum L to the spin polarization of electrons S=c†σc (for details see, e.g., Ref. [120] – PRB 52, 10239 (1995)). Since the resulting term γk,ll′ is nonlocal, this form of the SOC is usually referred to as the Rashba form. The precise expression for γk,ll′ depends on the particular system of interest, and for this reason we cannot write down an explicit formula which is valid in general for all the systems that can be studied using our method. To clarify this point, we modified the corresponding sentence in the text of the manuscript. Now it reads: “The nondiagonal contribution in spin space γk,ll′ describes the effect of the external magnetic field and of the spinorbit coupling (SOC), that is usually expressed in the Rashba form [119]. The latter corresponds to a Fourier transform of effective spindependent hopping amplitudes [120].” >>
Reply to the second review of the Referee 2.
The authors added important information that was missing in the previous version showing further limitations of the method, in particular when a nonlocal interaction is added. However, I am not satisfied with the benchmark of the selfenergy results in the case of the dimer. The authors have chosen a set of parameters where the nonlocal component of the self energy is way smaller than the local one. For this set of parameters, DMFT is already a good approximation for the dimer and the benchmark is useless in this case. The authors should use the same parameters that yields a substantial deviation between ED and DMFT results as shown in Fig.3 when U = t.
<<A: The corresponding results for the nonlocal selfenergy have been added to Fig. 4. These new results show that the nonlocal part of the DTRILEX selfenergy also follows the trend of the exact result and it is by far the largest contribution to the selfenergy for this kind of systems. >>
Also, I am not particularly impressed by the explanation of why this method should work at strong coupling. First, numerical results point to the opposite direction.
<<A: We do not understand what numerical results the Referee is referring to. As we wrote in our reply to a similar question of the Referee in the previous review round: “We want to stress, however, that the [DTRILEX] method is exact in the largeU regime, as it is an expansion around an impurity that represents the exact solution in that limit. Indeed, in work mentioned by the Referee [PRB 103, 245123 (2021)], it is shown that the method performs better when the interaction strength is larger than the bandwidth (U=1012).” As a matter of fact, Fig. 3 of PRB 103, 245123 (2021) clearly demonstrates that the accuracy of DTRILEX method is worst in the regime between U=6 and U=8 and improves at the interaction strengths starting from U=8. >>
Second, even if this is an expansion around the strong coupling limit, it really does not imply that the method would yield accurate results at the twoparticle level at strong coupling. In fact, the papers I have suggested to be referenced address this topic thoroughly: DIFFERENT strong coupling expansions can yield DIFFERENT values of the Neel critical temperature in 3D and sometimes completely destroy antiferromagnetism (AFM). The right approximation schemes are the one that stabilize AFM starting from the atomic limit (where AFM is absent) and yield the right critical temperature. A natural question that arises at this point, would DTrilex yield the correct value of TNeel at strong coupling in 3D? Is out there a reference showing this?
<<A: It is not surprising that different approximations lead to different results for the Néel critical temperature (or for any other quantity of interest) in some intermediate regime between the weakcoupling and the atomic limit. Therefore, we do not fully understand what the Referee means with “the right Néel temperature” and “right approximation schemes”. If we use DMFT as a reference point, we are guaranteed that our expansion will work in some vicinity of the atomic limit. The same holds true for the other choices of the interacting reference problem that are exact in the atomic limit. As any approximation can in principle give a different value for the Néel temperature, we think that comparing DTRILEX results for the susceptibility with nonexact results presented in the two works [PRB 99, 235106 (2019) and PRB 104, 235128 (2021)] mentioned by the Referee does not add any value to our work. Indeed, in these two papers the Néel temperatures are obtained by calculating the DMFTlike susceptibilities. However, in the 3D case considered there, the DMFT susceptibility does not correspond by any means to the exact susceptibility of the problem. Additionally, we would like to stress again that no approximation for the threepoint vertex functions is utilized in DTRILEX, contrarily to what is done in the works [PRB 99, 235106 (2019) and PRB 104, 235128 (2021)].
That said, we would like to provide some arguments on why DTRILEX provides reasonably accurate results also for twoparticle quantities. First of all, in our approach, single and two particle quantities are obtained selfconsistently from the same functional introduced in Eq. (75). As a consequence, one can expect the accuracy of the DTRILEX method to be comparable between the single and twoparticle levels. Secondly, as explicitly demonstrated in Ref. [PRB 103, 245123 (2021)], the D‑TRILEX diagrammatic expansion effectively takes into account only longitudinal fluctuations in the twoparticle BetheSalpeter equation (BSE) considered in dual theories and also when calculating the DMFT susceptibility (see Fig. 1 in Ref. [PRB 103, 245123 (2021)]). These longitudinal fluctuations represent the leading contribution to BSE as we explicitly demonstrated in Ref. [PRB 103, 245123 (2021)] by means of exact diagrammatic Monte Carlo calculations. Therefore, DTRILEX should be in a good agreement with the other dual theories and the DMFT result for the susceptibility when nonlongitudinal fluctuations can be neglected.
Based on the same argument, we expect our method to give accurate results for the Néel temperature for the singleband Hubbard model on a cubic lattice. In order to support this claim, we performed calculations for the Néel temperature for the halffilled singleband Hubbard model on a cubic lattice at t=1 and U=8. We found a Néel temperature TNéel = 0.33, which is in very good agreement with exact results obtained using Determinant Diagrammatic Monte Carlo (DDMC) and Quantum Monte carlo (QMC) approaches, as well as using the ladder dual fermion method. All these results are nicely collected in a recent work on DiagMC [arXiv:2112.15209 (2021)]. A more thorough analysis of the Néel transition in a cubic lattice will be addressed in a separate work and does not fit within our discussion of the method beyond the singleband Hubbard model. >>
Also, even if DTrilex does not use the fourpoint vertex function, DB technique does, therefore having reliable approximate schemes to calculate the fourpoint vertex function would reduce the computational complexity for numerical calculation of strongly correlated systems in regime where DTrilex does not work. I think these are subtle and important aspects that the authors missed and could leave for future work, but I truly believe that mentioning them in the discussion section and adding the suggested references would enrich their work.
<<A: The current work does not address the DB theory, except as an intermediate step to derive the DTRILEX method. As a consequence, we believe that the discussion of approximations for the calculation of the fourpoint vertex is not strictly related with the content of our work. Going beyond the simple DTRILEX diagrammatics would indeed necessarily involve dealing with the fourpoint vertex functions, which will unavoidably ruin the simplicity of the computational scheme proposed in our work. However, a detailed discussion of the multiband DB theory and/or much more complex extensions of DTRILEX theory is beyond the scope of this work. >>
List of changes
1. We clarified the meaning of the Rashba form of the SOC in the introduction.
2. We introduced two new panels in Fig.4, showing the local and nonlocal selfenergy as a function of interaction and frequency.
3. We added a description of the new content of Fig.4 in Sec. 5.1.
Published as SciPost Phys. 13, 036 (2022)
Reports on this Submission
Report #1 by Anonymous (Referee 5) on 2022721 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2204.06426v3, delivered 20220721, doi: 10.21468/SciPost.Report.5431
Report
The authors have improved the paper by properly answering my points and adding essential information to their manuscript.
However, they did not understand and did not address my criticism about the validity of DTrilex at strong coupling at the twoparticle level.
I think that my criticism could be reformulated in a single simpler question that is:
Is DTrilex exact in the limit of infinite dimensions at the twoparticle level?
If this is the case, DTrilex would most likely yield good results in 3D when compared to other methods, at least qualitatively.
In fact, DMFT already guarantees the correct power law behaviour of the Neel Temperature at strong coupling in infinite dimensions. In three dimensions, the selfconsistency of the DTrilex method will most certainly yield a reduction of the critical temperature value from the one calculated in the limit of infinite dimensions.
If this is not the case, it is not obvious to me wether DTrilex would even yield sound results from a qualitative point of view, e.g. power law behaviour of Neel
temperature at strong coupling. In fact, I brought to the authors' attention approximation schemes based on a strong coupling expansion that completely suppress Neel temperature even in the limit of infinite dimensions.
However, DTrilex could still yield the correct power law behaviour even if it is not exact in infinite dimensions, but it is not obvious and it should be explicitly shown (in future work) as it has been done in PRB 99, 235106 (2019) for an approximation to the vertex function similar to the one the authors used that have been introduced in ref. PRB 100, 155149 (2019).
Having a single point in the phase diagram compared with another method is encouraging but not enough to answer this question.
With this said, and as I already stated in my previous report, I do not expect and I do not ask the authors to do further calculations because such a numerical study could be address in future work.
Anyhow the authors should answer my point and discuss it in their manuscript.
Author: Matteo Vandelli on 20220823 [id 2745]
(in reply to Report 1 on 20220721)We would like to reply to the Referee in order not to leave the questions unanswered.
First of all, we would like to emphasize that the strong coupling limit and the limit of infinite dimensions are two different limits. For this reason, the question about the «validity of DTRILEX at strong coupling at the twoparticle level» cannot be reformulated as «Is DTRILEX exact in the limit of infinite dimensions at the twoparticle level?»
In our replies to similar questions raised by the Referee in the two previous review rounds, we have already explained why the DTRILEX method is exact in the weak and strong coupling limits (independently on dimension). To complement our previous arguments, we would like to mention that the diagrammatic expansion in the DTRILEX theory is performed in terms of the threepoint vertex functions that are connected by the interaction line W, and the bare dual Green’s functions. In the weak and strong coupling limits this diagrammatic expansion is a perturbative expansion. Indeed, at weak coupling the small parameter in the diagrammatic expansion is the interaction. In the strong coupling limit the bare dual Greens function, which is purely nonlocal, is small, because electrons in the system are strongly localized.
As we said previously, we agree with the Referee that different strong coupling expansions can yield different results for the twoparticle quantities. However, we believe that the diagrammatic expansion performed in DTRILEX is consistent, because it is formulated on the basis of the partially bosonized dual action that is derived by means of the exact transformations or controlled approximations. In fact, there are only two approximations that are used when deriving the action for the DTRILEX theory, namely we neglect the local vertex functions that are higher order than the twoparticle (threepoint and fourpoint) ones and we replace the fourpoint vertex function by its partiallybosoniszed approximation. The first approximation is a standard approximation that is widely used in many DMFTbased diagrammatic expansions including dual fermion/boson theories [Phys. Rev. B 77, 033101 (2008); Ann. Phys. 327(5), 1320 (2012)], DГA method [Phys. Rev. B 75, 045118 (2007)], and TRILEX approach [Phys. Rev. B 100, 205115 (2019); PRB 103, 245123 (2021)]. The second approximation has been tested in detail in a singleorbital case, which showed its validity in a wide range of interaction strengths from weak to strong coupling regimes [PRB 103, 245123 (2021)]. Regarding the validity of the DTRILEX method at strong coupling at the two particle level, it is worth noting that the leading term (the secondorder in the dual Green’s function) that contributes to the susceptibility in the strong coupling limit is exactly the polarization operator of DTRILEX. More elaborate diagrammatic contributions already have four dual Green’s function in their stricture. In addition, as we point out in the current work, the DTRILEX polarization operator has the same structure as the expression for the exchange interaction between magnetic densities, and this expression gives the correct result for the exchange interaction ~t2/U in the atomic limit [PRL 121, 037204 (2018)]. This observation also shows that the DTRILEX diagrammatic expansion is correct at strong coupling at the twoparticle level.
Now, let us comment on the dinfinity limit. In this limit, the DTRILEX method is exact at the singleparticle level if the DMFT reference problem is considered. However, DTRILEX is not exact at the two particle level, because it uses a partially bosonized approximation instead of the exact fourpoint vertex function. However, we would like to point out that there is no correlation between the exactness of the theory in the limit of infinite dimensions and the accuracy of the theory in finite dimensions. Let us consider DMFT as a particular example proposed by the Referee. DMFT is exact in dinfinity limit at both single and twoparticle levels. However, it does not predict the AFM phase boundary accurately enough for the case of a simple cubic lattice. For instance, at t=1, U=8, where DTRILEX and the exact Determinant Diagrammatic Monte Carlo give the TNéel = 0.33 value for the Neel temperature (see discussion in the previous review round), DMFT predicts a much higher value TNéel = 0.45 for the AFM phase boundary (see, e.g., Refs. [PRB 92, 144409 (2015); PRB 94, 115117 (2016)]). The t=1, U=8 point is very close to the top of the AFM dome, which means that DTRILEX accurately predicts the Neel temperature in the most challenging regime of very strong magnetic fluctuations in addition to the exact results that the theory provides in the weak and strongcoupling limits as argued above.
This discussion about the different limits of the theory has been added to main text after Eq. (19) and at the end of Section. 3.2.