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Photoemission spectroscopy from the three-body Green's function
by Gabriele Riva, Timothée Audinet, Matthieu Vladaj, Pina Romaniello and J. Arjan Berger
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Submission summary
Authors (as registered SciPost users): | Arjan Berger |
Submission information | |
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Preprint Link: | scipost_202110_00015v1 (pdf) |
Date submitted: | 2021-10-11 22:30 |
Submitted by: | Berger, Arjan |
Submitted to: | SciPost Physics |
Ontological classification | |
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Academic field: | Physics |
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Approach: | Theoretical |
Abstract
We present an original approach for the calculation of direct and inverse photo-emission spectra from first principles. The main goal is to go beyond the standard Green's function approaches, such as the $GW$ method, in order to find a good description not only of the quasiparticles but also of the satellite structures, which are of particular importance in strongly correlated materials. Our method uses as a key quantity the three-body Green's function, or, more precisely, its hole-hole-electron and electron-electron-hole parts. We show that, contrary to the one-body Green's function, satellites are already present in the corresponding non-interacting Green's function. Therefore, simple approximations to the three-body self-energy, which is defined by the Dyson equation for the three-body Green's function and which contains many-body effects, can still yield accurate spectral functions. In particular, the self-energy can be chosen to be static which could simplify a self-consistent solution of the Dyson equation. We also show how the one-body Green's function can be retrieved from the three-body Green's function. We illustrate our approach by applying it to the symmetric Hubbard dimer.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 7) on 2021-11-15 (Invited Report)
- Cite as: Anonymous, Report on arXiv:scipost_202110_00015v1, delivered 2021-11-15, doi: 10.21468/SciPost.Report.3847
Strengths
The authors consider the calculation of electron-addition spectra (e.g. as measured in photoemission). The exact 1-particle Green's function G1 of many-body perturbation theory is closely related to the measured photoemission spectrum, and it generally includes plasmon satellites and a spectral background, in addition to quasiparticle peaks that are the shifted, renormalized counterparts of the eigenvalues of a corresponding non-interacting Hamiltonian.
1. Reliably accurate, unbiased methods for calculating G1 (including the non-quasiparticle features) are not available in practice. In this paper the authors consider the 3-particle Green's function G3, on the face of it a much more complex object than G1; however, they show that a rather simple G3 (even one calculated using a omega-dependent self-energy) already contains information about the non-quasiparticle features (e.g. if G3 is used to calculate G1).
2. The authors test their idea on an extremely simple multi-electron system, the symmetric Hubbard dimer. The Hilbert space of the many-electron wavefunction is very small, which means that the exact G1 and G3 can be calculated, and also a simplified G3 in which the self-energy that appears in the Dyson equation for G3 is constrained to be omega-independent, simplifying the calculations. The G1 calculated from this simplified G3 already contains an impressive account of the non-quasiparticle features.
3. The paper is clearly written and the analysis of the connections between G3 and G1 in the general case, and for the Hubbard dimer in particular, is impressive.
4. The paper raises the important and interesting idea of a tradeoff between the degree of the Green's functions used and the level of approximation needed in calculated intermediate quantities.
Weaknesses
The following clarifications are needed, in my view:
1. In Figs. 1-5 can it be clarified whether a curve labeled G3, or Sigma3, is the 3-particle spectral function or the 1-particle spectral function obtained from a calculation of G3? Hopefully the latter.
2. Can the actual steps in the calculation of the spectral function using their simplified G3 (static Sigma3) be set out in the language of a general many-electron system and of many-body perturbation theory? E.g. as a flowchart. This would help the reader assess how realistic the Hubbard model is here as a prototype many-electron system. Is Sigma3 spatially non-local? What is the extent of self-consistency imposed by the calculation? To what extent can the recommended procedure be regarded as a true perturbation theory, and if it can, what is the small quantity?
Report
This is an important addition to the literature, even though I suspect an ab initio calculation using a G3 is many years away. I recommend publication in SciPost after clarifications 1-2 above are addressed.
Requested changes
See clarifications above.
Report #1 by Davide Sangalli (Referee 1) on 2021-11-12 (Invited Report)
- Cite as: Davide Sangalli, Report on arXiv:scipost_202110_00015v1, delivered 2021-11-12, doi: 10.21468/SciPost.Report.3835
Strengths
S1 - defines a new approach to compute photoemission within many-body perturbation theory based on the three-body green function G3. Via G3 a static self-energy could capture satellites and more in general features which would require a dynamical self-energy when using the more standard approach based on G1. This has the potential of solving many open issues in systems where approaches based on G1 fail
S2 - there is a physical and well-done discussion of how time orderings are selected
S3 - the application on the Hubbard dimer, which is exactly solvable, gives some further insight
Weaknesses
There are two aspects that need further clarification
W1 - a satellite in photoemission arise from the interaction between electrons. In the non-interacting limit, only single-particle poles exist. It is ok that the G3 contains extra poles also in the non-interacting case, but such poles should not contribute to the ARPES spectral function, i.e. they should have zero intensity. Only interaction, i.e. a static self-energy, could give finite intensity to these poles. The authors find these poles in the analytical description (eq. 24), and they seem to suggest that they contribute to ARPES also in the non-interacting limit. Indeed there is a pole, which they call satellite, in Fig. 1. This point should be clarified. I think the case is similar to quantum chemistry approaches. When calculations such as CI-SD or CC-SD are performed extra poles (i.e. double excitations) arise. However, they can be seen in photoemission only if they mix via interaction with single excitations (unless one accounts for two-photons processes).
W2 - An important approximation of standard approaches based on G1 (especially in the ab inito community) is to take the diagonal only component of the self-energy, i.e. only the poles are corrected with respect to the zero-order (usually DFT) simulation. G3 has six indexes, and it is not easy to understand where such approximation would enter. This is also related to how demanding would be the inversion of G3 in the QP basis set (appendix C), and which kind of satellites the approach could give. I would expect that, the contraction of the external indexes (indexes im in eqs. 33-34) could give the QP approximation. However, for correlated satellites (plasmons, excitons, magnons, etc .. ) the internal indexes should be allowed to mix. Side comment, something is wrong in the indexes of eq. 30. The Hubbard dimer does not help much here. At least a general discussion in this direction would be useful.
Report
The manuscript is well written and potentially of strong interest for the community. However, the two weaknesses identified should be clarified. In particular W1.
Claudio Attaccalite on 2021-11-12 [id 1938]
Dear authors
I have some remarks on your manuscripts that I put here as a comment, in such a way to make the referee/editorial process more transparent. Consider this comment as an additional referee report.
1) The solution of the three-body green's function was already used in the literature: for the Auger problem by A. Marini and M. Cini in Journal of Electron Spectroscopy and Related Phenomena 127 (2002) 17–28 and by C. Calandra and F. Manghi in Phys. Rev. B 50, 2061 to study satellite structures and the occurrence of the metal-insulator transition. I think the authors should mention these two works in their manuscript
2) Another physical phenomenon that could be studied with the present approach is probably "trions". May the author comment on this possibility? Do they expect trions will be well described by solving the G3 problem?
3) I think in Eq. 54 you wrote "G′′2(ω)" instead of "G′′1(ω)"
best regards Editor-in-charge Dr Attaccalite