# Photoemission spectral functions from the three-body Green's function

### Submission summary

 As Contributors: Arjan Berger Preprint link: scipost_202110_00015v3 Date accepted: 2022-02-09 Date submitted: 2022-01-20 23:51 Submitted by: Berger, Arjan Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical

### Abstract

We present an original strategy 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. To this end we use as a key quantity the three-body Green's function, or, more precisely, its hole-hole-electron and electron-electron-hole parts, and we show how the one-body Green's function, and hence the corresponding spectral function, can be retrieved from it. We show that, contrary to the one-body Green's function, information about satellites is already present in the non-interacting three-body 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 give a proof of principle of our strategy by applying it to the Hubbard dimer, for which the exact self-energy is available.

Published as SciPost Phys. 12, 093 (2022)

### Author comments upon resubmission

Dear Editor-in-charge,

Thank you for sending us the reports of the referees.

We thank the referees for their reading of the manuscript and for their questions and comments.
Reviewer 1 writes "I think the manuscript can be now accepted for publication." and Reviewer 2 writes "The authors have made helpful modifications to their manuscript that provide the clarification that I (and, in my opinion, also the other referee and the Editor) requested." Reviewer 4 writes that "the manuscript meets the acceptance criteria once a few minor corrections have been made".
Finally, the comments made by Reviewer 3 seem to indicate that there is a misunderstanding with respect to the objective of our work. We have therefore made several changes to the manuscript including the title which now reads "Photoemission spectral functions from the three-body Green's function." More details are given below.

In the following we address in detail the remaining points raised by the referees.

We hope that our second revision of the manuscript will be suitable for publication in SciPost Physics.

Sincerely, the authors.

REVIEWER 1:

Reviewer's comment:

"The authors have addressed the points raised in my review.
In particular for the definition of the ARPES spectral function from G3. The role of the basis set and the related diagonal approximation remains to be further explored. Future applications on real materials could possibly clarify this.
I think the manuscript can be now accepted for publication."

Authors' response:

We thank the reviewer for recommending our paper for publication in SciPost Physics.

REVIEWER 2:

Reviewer's comment:

"The authors have made helpful modifications to their manuscript that provide the clarification that I (and, in my opinion, also the other referee and the Editor) requested."

Authors' response:

We are glad that the reviewer is happy with the modifications we made to the manuscript.

REVIEWER 3:

Before addressing the comments of the reviewer we would like to make the following general remark.

The reviewer seems to have understood that we want to use the three-body Green's function as fundamental quantity in the calculation of photoemission spectra to include corrections beyond the sudden approximation. This is probably the reason that the reviewer writes that these corrections in terms of the three-body Green function needs to be derived in a mathematically consistent fashion from fundamental considerations on the photocurrent."
However this is not the main idea of the paper. We are always working within the sudden approximation and the main goal is to obtain the one-body Green's function, and its corresponding spectral function, from the three-body Green's function. Therefore, to avoid any confusion we have changed the title of our work to "Photoemission spectral functions from the three-body Green's function."

Reviewer's comment:

"The direct and inverse photoemission intensity is given, within the sudden approximation, by the one-electron removal and addition spectra that are proportional to the imaginary part of the retarded single particle Green function (e.g. Rev. Mod. Phys. 75, 473 2003, Phys. Rev. B 94, 115119 2016). This is the case for both (strongly) interacting and non-interacting/weakly interacting systems. This results stems from a direct calculation of the photoelectron current using scattering theory. Beyond the sudden approximation corrections to the photocurrent appear. Hedin and coworkers (Phys. Rev. B 58 15565 1998) have shown that whereas the sudden approximation includes intrinsic losses" or satellite structure, adiabatic corrections provide further the extrinsic losses"."

Authors' response:

In this work we are always working within the sudden approximation which is the standard in our field. When we write that we use the 3-body Green's function (3-GF) to calculate photoemission spectra it is implied that we mean photoemission spectra within the sudden approximation. It is not mentioned explicitly since it is standard practice in our field. We have now made this point explicit in the second revision of the manuscript in order to avoid any possible misunderstanding by rewriting the following sentence in the introduction:

"The main reason is that the one-body Green's function (1-GF) can be easily linked to photoemission spectra since its poles are the electron removal and addition energies.

as

"The main reason is that the one-body Green's function (1-GF) can be easily linked to photoemission spectra (within the sudden approximation) since its poles are the electron removal and addition energies.

Reviewer's comment:

"The authors state that the three-body Green function contribute to photoemission intensity, but do not derive this statement from a calculation of the photocurrent."

Authors' response:

Nowhere in the manuscript have we made this statement.

Reviewer's comment:

"Instead the existing literature (some of which referred to above) seems to agree that within the sudden approximation (and apart from matrix-element effects), the one-particle Green function contains all spectral information relevant for photoemission. In this context it is not clear how the authors challenge the present status quo and can justify their statement that the three-body Green function is a fundamental quantity to the calculation photoemission spectra. Possibly the implication is that the three-body Green function embodies corrections beyond the sudden approximation. If so, these corrections in terms of the three-body Green function needs to be derived in a mathematically consistent fashion from fundamental considerations on the photocurrent."

Authors' response:

We agree with the reviewer that the one-particle Green function (within the sudden approximation) contains all spectral information relevant for photoemission. However, that is if one has the exact one-body Green's function (1-GF). The main idea of this work is still to calculate the 1-GF, but using the 3-GF to improve the approximations to the 1-GF and, in particular, to capture satellites. This point is explained in detail in the Introduction and Results sections of the paper. In a nutshell, to capture satellites using the standard 1-body approach one requires a dynamical self-energy since the non-interacting 1-GF only contains information about quasi-particles. It is well-known that it is difficult to obtain good dynamical approximations for the self-energy. For example, the $GW$ approximation does not yield very accurate satellites. Instead, when using a 3-body approach, information about satellites is already contained in the non-interacting 3-GF and, therefore, a simpler static 3-body self-energy is sufficient to capture satellites. Once the 3-GF is obtained we contract (according to Eq. (17)) to obtain the 1-GF and therefore the photoemission spectrum including the satellites.
The strategy is similar to that of the Bethe-Salpeter equation. In principle, the one-body polarisation is sufficient to calculate optical spectra but in practice it is more convenient to use a two-body polarisation.
To make this point clearer in the second revision of the manuscript we modified the following sentence in the Introduction:

"Therefore, we will study here the three-body Green's function (3-GF) as the fundamental quantity from which to calculate photoemission spectra.

to

"Therefore, we will study here the three-body Green's function (3-GF) as the fundamental quantity from which to calculate the 1-GF and, hence, photoemission spectra."

REVIEWER 4:

Reviewer's comment:

"It is somewhat unclear to me inhowfar the method which is presented
really is helpful for more complicated systems than a Hubbard dimer.
In partiular i wonder if in the case of the inverse photoemission spectrum in
the quarter filled ground state - i.e. with one electron in the dimer - taking
all states with one added electron and a 'particle hole excitation' of the
electron present initially is not equivalent to an exact solution?"

Authors' response:

This work presents the first step towards an approach based on the 3-GF to calculate accurate spectral functions, namely a proof of principle that with a static 3-body self-energy one can obtain satellites in these spectra.
We show this both by looking at the fundamental equations and by applying our strategy to the Hubbard dimer.
This finding is true for any system, not only for the Hubbard dimer.
As mentioned in the Conclusions, we are currently working on the second step which is to derive a general static approximation for the three-body self-energy.
This could be achieved, for example, by using the equation of motion of the 3-GF.
We agree with the reviewer that only when the second step has finished we will be able to fully assess whether our method is helpful for more complicated systems than a Hubbard dimer.
We think it will almost certainly be the case because, since the 3-GF contains more information than the 1-GF, we can put less information into the 3-body self-energy than in the 1-body self-energy.
This point is indeed nicely illustrated with the Hubbard dimer at 1/4 filling.
Since one cannot have more than three particles in the system, i.e. the added electron and the electron-hole pair which it creates, the exact 3-body self-energy is static since no extra excitations have to be created.
Instead, the exact one-body self-energy is a more complicated dynamical quantity.
We note that our strategy is also similar to that of the Bethe-Salpeter equation. In principle, the one-body polarisation is sufficient to calculate optical spectra but in practice it is more convenient to use a two-body polarisation because it contains more information.

Reviewer's comment:

"I somewhat resent the use of the term 'satellites' in this manuscript.
For example in the noninteracting three-particle Green's function I would
expect that these 'satellites' really are structureless continua and have
little to do with the features called satellites in photoemission spectra
of correlated electron systems, which are more something like
Hubbard bands."

Authors' response:

The poles of the 3-GF (its electron-electron-hole and hole-hole-electron parts, to be precise) are the same as those of the 1-GF but the amplitudes corresponding to these poles are not.
The amplitudes of both quasi-particles and satellites found in photoemission spectra are then obtained by using the contractions in Eq. (17) to obtain the 1-GF from the 3-GF, and thus the one-body spectral function.
Therefore, although the 3-GF does not directly yield information about the amplitudes of the satellites (one has to do the contraction first) it contains all the necessary information about the satellites.
For this reason we have also used the term "satellite" when discussing the 3-GF.
We have made several modifications in the second revision of the manuscript to be more careful when using the term "satellite".
In particular we now write that the noninteracting 3-GF contains information about satellites.

Reviewer's comment:

"I do not understand why the authors are using a Green's function of
6 Fermion operators. Would the most natural extension not be a Green's function
that has three fermions at time $t_1$ and one Fermion at time $t_2$,
i.e. the type of Green's function which shows up in the equation
of motion of the single particle Green's function?"

Authors' response:

The referee is right that using a 2-body Green's function of the form

$\langle\Psi_0^N| (\psi^{\dagger}\psi\psi)_{t_1}|\Psi_n^{N+1}\rangle\langle\Psi_n^{N+1}|(\psi^{\dagger})_{t_2}|\Psi_0^N\rangle$ we have indeed information about addition energies (or removal energies if we consider another order of the field operators). However the non-interacting two-body Green's function (which is the product of two non-interacting 1-GF) corresponding to these times would only have information about quasi-particles, but not about satellites.
Therefore, a static 2-body self-energy would yield a photoemission spectrum without satellites.

Reviewer's comment

"Three-particle Green's functions are being studied for a very long time.
For example the well-known Hubbard-operators are nothing but products of
three Fermion operators. More generally, composite operators have been used
for a long time, see PHYSICAL REVIEW B104, 155128 (2021) for a recent example.
It would appear to me that these works are more physically motivated than
the rather technical approach of the authors and in any way should be mentioned."

Authors' response

We have cited the work mentioned by the reviewer as well as another recent work (Physics Reports 929, 1 (2021))

Reviewer's comment

"The derivation of the spectral representation is rather unpleasant to
read because some equations are only in section 2.1., others only in Appendix A,
so that a lot of back-and-forth scrolling is necessary if one wants to follow
the calculation. I would suggest to change this."

Authors' response

We have added all the relevant equations to Appendix A to avoid the back-and-forth scrolling.

### List of changes

1) We changed the title to "Photoemission spectral functions from the three-body Green's function"

2) We rewrote the following sentence in the introduction as

"The main reason is that the one-body Green's function (1-GF) can be easily linked to photoemission spectra (within the sudden approximation) since its poles are the electron removal and addition energies.

3) We rewrote the following sentence in the introduction as

"Therefore, we will study here the three-body Green's function (3-GF) as the fundamental quantity from which to calculate the 1-GF and, hence, photoemission spectra."

4) We have made several modifications in the second revision of the manuscript to be more careful when using the term "satellite". In particular we now write that the noninteracting 3-GF contains information about satellites.

5) We have cited the work mentioned by Reviewer 4 (PHYSICAL REVIEW B104, 155128 (2021)) as well as another recent work (Physics Reports 929, 1 (2021))

6) We have added all the relevant equations to Appendix A to avoid the back-and-forth scrolling.

7) We corrected eqs.(21-24) that presented an additional imaginary number in the denominator

8) We added the complex conjugate symbols that were missing in eq.(31).

### Submission & Refereeing History

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Resubmission scipost_202110_00015v3 on 20 January 2022
Resubmission scipost_202110_00015v2 on 27 November 2021
Submission scipost_202110_00015v1 on 11 October 2021

## Reports on this Submission

### Report

In their reply the authors have clarified the context of their work and adjusted the manuscript where appropriate. I support publication of the manuscript in SciPost.

• validity: good
• significance: good
• originality: good
• clarity: good
• formatting: excellent
• grammar: excellent

### Report

With the revisions performed by the authors I think the acceptance criteria now are met.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -