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Manybody perturbation theory for strongly correlated effective Hamiltonians using effective field theory methods
by Raphaël Photopoulos, Antoine Boulet
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Authors (as registered SciPost users):  Antoine Boulet 
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Preprint Link:  https://arxiv.org/abs/2402.17627v1 (pdf) 
Date submitted:  20240305 23:26 
Submitted by:  Boulet, Antoine 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
Introducing lowenergy effective Hamiltonians is usual to grasp most correlations in quantum manybody problems. For instance, such effective Hamiltonians can be treated at the meanfield level to reproduce some physical properties of interest. Employing effective Hamiltonians that contain manybody correlations renders the use of perturbative manybody techniques difficult because of the overcounting of correlations. In this work, we develop a strategy to apply an extension of the manybody perturbation theory, starting from an effective interaction that contains correlations beyond the mean field level. The goal is to reorganize the manybody calculation to avoid the overcounting of correlations originating from the introduction of correlated effective Hamiltonians in the description. For this purpose, we generalize the formulation of the RayleighSchr\"odinger perturbation theory by including free parameters adjusted to reproduce the appropriate limits. In particular, the expansion in the bare weakcoupling regime and the strongcoupling limit serves as a valuable input to fix the value of the free parameters appearing in the resulting expression. This method avoids double counting of correlations using beyondmeanfield strategies for the description of manybody systems. The ground state energy of various systems relevant for ultracold atomic, nuclear, and condensed matter physics is reproduced qualitatively beyond the domain of validity of the standard manybody perturbation theory. Finally, our method suggests interpreting the formal results obtained as an effective field theory using the proposed reorganization of the manybody calculation. The results, like ground state energies, are improved systematically by considering higher orders in the extended manybody perturbation theory while maintaining a straightforward polynomial expansion.
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The authors propose a manybody perturbation theory for strongly correlated systems using effective field theory methods. The main idea is to start from an effective Hamiltonian which allows one to reproduce the perturbation expansion in some interaction parameter while recovering exactly the known (exact) result in some (stronglycorrelated) limit. The goal is to set up an approach which can interpolate between the weak and strongcorrelation limits. The main problem is to avoid double counting of correlations when going beyond lowestorder perturbative expansion.
The paper is clearly written. I would like to mention the following points for the authors' consideration.
1) In the introduction, the authors point out that "nonperturbative methods express the problem with multidimensional integrals". It is not clear what is referred to here. Slightly below, they identify firstorder perturbation theory with meanfield theory, which I find slightly confusing. Can BCS meanfield theory be considered as a firstorder perturbation theory?
2) The main point of the authors is to show that, starting from Hamiltonian (4), one can reproduce the perturbation expansion order by order while satisfying the exact result $E_\infty/E_0=\xi_0$. To do so, one has to introduce an unknown parameter, $\beta$, to second order; two parameters $\beta_1$ and $\beta_2$ to third order order, etc. The procedure followed to second order, Eq.(26), seems rather arbitrary. Could the authors justify it? Is it the only possible way to introduce the two parameters $\beta_1$ and $\beta_2$ and, if not, why choosing this one?
3) I do not understand the meaning of the sentence "which is again independent of $\beta_1$ and $\beta_2$" following Eq.(26).
4) It is shown how to reproduce the perturbation expansion order by order. I understand, although it is not said explicitly, that this is equivalent to avoiding double counting of correlations. A short discussion would be welcome.
5) At the top of page 4, "an energy operator $\hat\omega\Psi_0\rangle$" should be replaced by "an energy operator $\hat\omega$".
6) The various examples considered in the manuscript are quite convincing except the 1D Hubbard model. In the case $U/t>0$, it seems that the secondorder perturbation theory results are better than the $l=0$ and $l=1$ results. Moreover I do not understand what the model with $l=2$ and $l=3$, mentioned in the caption of Fig.4, refer to.
7) The authors discuss only the calculation of the ground state energy. In manybody systems, correlation functions are also of prime interest. Is the method proposed in the manuscript restricted to thermodynamic quantities or would it be possible to also compute one and twoparticle Green functions?
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Author: Antoine Boulet on 20240517 [id 4491]
(in reply to Report 1 on 20240416)Dear Editor, Dear Reviewers,
We would like to thank you for your time in reviewing our paper and providing valuable comments that led to possible improvements in the current version. We have carefully considered the comments and tried our best to address every one of them. We hope that the manuscript after careful revisions, will meet your high standards. We welcome further constructive comments if any. In the file attachment, we provide the pointbypoint responses.
Sincerely,
R. Photopoulos and A. Boulet
Attachment:
responsev1.pdf