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Many-body perturbation theory for strongly correlated effective Hamiltonians using effective field theory methods
by Raphaël Photopoulos, Antoine Boulet
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Submission summary
| Authors (as registered SciPost users): | Antoine Boulet · Raphaël Photopoulos |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2402.17627v1 (pdf) |
| Date submitted: | 2024-03-05 23:26 |
| Submitted by: | Boulet, Antoine |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Introducing low-energy effective Hamiltonians is usual to grasp most correlations in quantum many-body problems. For instance, such effective Hamiltonians can be treated at the mean-field level to reproduce some physical properties of interest. Employing effective Hamiltonians that contain many-body correlations renders the use of perturbative many-body techniques difficult because of the overcounting of correlations. In this work, we develop a strategy to apply an extension of the many-body perturbation theory, starting from an effective interaction that contains correlations beyond the mean field level. The goal is to re-organize the many-body 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 Rayleigh-Schr\"odinger perturbation theory by including free parameters adjusted to reproduce the appropriate limits. In particular, the expansion in the bare weak-coupling regime and the strong-coupling 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 beyond-mean-field strategies for the description of many-body 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 many-body perturbation theory. Finally, our method suggests interpreting the formal results obtained as an effective field theory using the proposed reorganization of the many-body calculation. The results, like ground state energies, are improved systematically by considering higher orders in the extended many-body perturbation theory while maintaining a straightforward polynomial expansion.
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The authors propose a many-body 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 (strongly-correlated) limit. The goal is to set up an approach which can interpolate between the weak- and strong-correlation limits. The main problem is to avoid double counting of correlations when going beyond lowest-order 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 "non-perturbative methods express the problem with multidimensional integrals". It is not clear what is referred to here. Slightly below, they identify first-order perturbation theory with mean-field theory, which I find slightly confusing. Can BCS mean-field theory be considered as a first-order 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 second-order 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 many-body 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 two-particle Green functions?
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Author: Antoine Boulet on 2024-05-17 [id 4491]
(in reply to Report 1 on 2024-04-16)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 point-by-point responses.
Sincerely,
R. Photopoulos and A. Boulet
Attachment:
responsev1.pdf