Photoemission spectroscopy from the three-body Green's function

Dear Editor,

Thank you for sending us the reports of the referees.

We thank the referees for their careful reading of the manuscript and for their questions and comments.
We are pleased to read that the referees consider the manuscript as "well written and potentially of strong interest for the community" and "an important addition to the literature".

Both referees ask for some points to be clarified.
In the following we do so and we provide a list of changes.
We also address the comments of the editor-in-charge.


We hope that our revised manuscript will be suitable for publication in SciPost Physics.

Sincerely, the authors.


REVIEWER 1:

Reviewer 1 considers our manuscript as ``well written and potentially of strong interest for the community".
Reviewer 1 would like us to address the following two points.

Reviewer's comment:

"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."

Our reply:

We agree with the referee that in the ARPES spectral function the satellite amplitudes are non-zero only when the interaction is switched on. This spectral function is defined as the imaginary part of the one-body Green's function (1-GF), for which satellite amplitudes are zero when the interaction is switched off. The confusion stems from the fact that in figures 1-3 we had also reported the imaginary part of the three-body Green's function (3-GF), i.e., without the contraction to get the 1-GF. The imaginary part of the 3-GF is not equal to the ARPES spectral function. The 3-GF contains more information than the 1-GF. Therefore the imaginary part of the 3-GF has non-vanishing satellite amplitudes also in the non-interacting case. The 1-GF are obtained using the contractions in equations (34) and (35) and from the 1-GF the ARPES spectral function can be obtained. Nevertheless, for analysis purposes, it is convenient to introduce a 3-body spectral function as the imaginary part of the 3-GF.
To clarify these points in the revised manuscript we have modified section II.D by adding the following two paragraphs:

Since the spectral representation of $G_3^{e+h}(\omega)$ given in Eq. (12) is similar to the one of $G_1$
it is convenient to introduce a 3-body spectral function for $G_3^{e+h}(\omega)$ that is similar to the spectral function corresponding to $G_1$.
The latter is defined as
\begin{equation}
A(x_1,x_{1'};\omega)=\frac{1}{\pi}\text{sign}(\mu-\omega)\text{Im} G_1(x_1,x_{1'};\omega).
\end{equation}
We can thus define the spectral function $A_3(\omega)$ corresponding to $G_3^{e+h}(\omega)$ according to
\begin{equation}
A_3(\omega)=\frac{1}{\pi}\text{sign}(\mu-\omega)\text{Im} G_3^{e+h}(\omega),
\end{equation}
where, for notational convenience, the spin-position arguments are omitted.

and

We note that the 3-body spectral function is not the spectral function that corresponds to photoemission spectroscopy. Both spectral functions have the same poles but the corresponding amplitudes are different.
In particular, in the non-interacting case the amplitudes of satellites can be non-zero in the 3-body spectral function.
To retrieve the spectral function that corresponds to photoemission spectra Eq. (17) has to be used.

Moreover, figures 1-3 have been modified to emphasize more clearly the differences between the 1- and 3-body spectral functions. These figures are now divided into two panels; in the upper panel we report the 1-body spectral function (the ARPES spectral function), while the 3-body spectral function is reported in the bottom panel.
We included this latter case to show that the 3-GF has the poles at the same position as the 1-GF and that only the amplitudes differ.

Reviewer's comment:

"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."

Our reply:

We thank the referee for raising this very interesting point. Indeed the diagonal approximation to the self-energy is an important practical tool to calculate the poles of the one-body Green's function. If the self-energy is dynamical both quasi-particle energies and satellites can be obtained with this approximation, although in practice mainly quasi-particle energies are calculated. Instead, if the self-energy is static only quasi-particles can be calculated.
A diagonal approximation can also be made for the three-body self-energy and this could be very interesting because it would reduce the numerical cost of the calculations significantly. Moreover, from a static three-body self-energy both quasi-particles and satellites can be obtained. Of course, the quality of the quasi-particle energies and satellites not only depend on the diagonal approximation but also on the approximation to the self-energy. The latter approximation is probably the most crucial. As alluded to by the referee we can not test the diagonal approximation on the Hubbard dimer since the three-body self-energy is already diagonal in the basis that diagonalizes the non-interacting 3-GF. The referee mentions the contraction of the external indices in eqs. 33-34 (eqs. 34-35 in the revised manuscript), i.e., $m=i$. However, the equivalent of the diagonal approximation in the three-body case would be to set $m=i$, $o=j$ and $k=l$.

We have now clarified this point in the revised manuscript by adding the following sentences to the outlook given in section 4.

Moreover, we can reduce the numerical cost of the calculations by applying a diagonal approximation to the three-body self-energy, in similar manner as is often done for the one-body self-energy, to calculate only the poles of $G_3^{e+h}$. While a static one-body self-energy can only yield poles that correspond to quasi-particles, a diagonal static three-body self-energy would yield the poles corresponding to both quasi-particles and satellites.

We have also corrected the indices in Eq. (30) (Eq. (31) of the revised manuscript).


REVIEWER 2:

Referee 2 considers our manuscript ``an important addition to the literature" and referee 2 recommends publication after two points are clarified.

Reviewer's comment:

"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."

Our reply:

We thank the referee for this question. This point was indeed not clear.
In the revised manuscript we have modified figures 1-3 to emphasize more clearly the differences between the 1- and 3-body spectral functions. These figures are now divided into two panels; in the upper panel we report the 1-body spectral function (the one that corresponds to photoemission spectra), while the 3-body spectral function is reported in the bottom panel. We included this latter case to show that the 3-GF has the poles at the same position as the 1-GF and that only the amplitudes differ. In figures 4 and 5 only spectral functions corresponding to the 1-GF are shown.

Reviewer's comment:

"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?"

Our reply:

The exact 3-body self-energy is defined by the Dyson equation given in Eq. (18) and it is indeed a non-local quantity. We do not yet have an expression for the 3-body self-energy in which it is given as an explicit functional of $G_3$ and the interaction. Therefore, in this work we are not doing many-body perturbation theory. For our application to the Hubbard dimer we have derived the exact $G_3^{e+h}$ and $G_{03}^{e+h}$. Therefore, we know the exact $\Sigma_3$ because we can solve the Dyson equation (Eq. (18)) and we can take the static approximation by setting $\omega=0$. To obtain the corresponding $G_3^{e+h}$ we solve once more the Dyson equation in Eq. (18). As a consequence self-consistency is not an issue here. The main goal of this manuscript is to show that, with a static self-energy, the 3-GF has information about satellites, contrary to the 1-GF. We are now trying to make the theory applicable to real systems, by looking for approximations to $\Sigma_3$ as explicit functionals of $G_3^{e+h}$ and the interaction. This interaction, which could be, for example, the bare Coulomb interaction or the screened Coulomb interaction, will then be the small parameter. This is, however, beyond the scope of this work.
We have now clarified these points in the conclusions by adding the following paragraph,

For the specific case of the Hubbard dimer we were able to obtain the exact $G_3^{e+h}$. Therefore we could obtain an exact three-body self-energy by solving a Dyson equation. However, in general, the exact three-body self-energy is unknown. Therefore, our next goal 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 for $G^{e+h}_3$ along the same lines as has been done for $G_1$ or by using a similar strategy as in Ref. [27], where a practical scheme to calculate $G_3$ for the description of Auger spectra is proposed.

We also added the following sentences to the outlook given in section 4 briefly discussing a possible strategy to reduce the numerical cost of the calculations, namely by using a diagonal approximation to the three-body self-energy, as was mentioned by Reviewer 1.

Moreover, we can reduce the numerical cost of the calculations by applying a diagonal approximation to the three-body self-energy, in similar manner as is often done for the one-body self-energy, to calculate only the poles of $G_3^{e+h}$. While a static one-body self-energy can only yield poles that correspond to quasi-particles, a diagonal static three-body self-energy would yield the poles corresponding to both quasi-particles and satellites.


EDITOR-IN-CHARGE:

We now address the comments and questions of the editor-in-charge.

Editor's comment:

``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".

Our reply:

We thank the editor for these two important references, which we have now added to the manuscript in the Introduction, as:

We note that the three-body Green's function has been employed to describe Auger spectra [27] and to study satellite structures and the occurrence of the metal-insulator transition. [28]


Editor's comment:

"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?"

Our reply:

We agree with the editor that trions would be another interesting applications of our approach. We had briefly mentioned this possibility in the conclusions. We have now added more references there. We cannot foresee the performance of our approach to describe these excitations because it will depend on the quality on the approximations to the 3-body self-energy. It will be interesting to explore this problem in the future.

Editor's comment:

"I think in Eq. 54 you wrote "$G_2(\omega)$" instead of "$G_1(\omega)$"."

Our reply:

We thank the editor for noticing this typo. We have now corrected it.

1.

We have modified section II.D by adding the following two paragraphs:

Since the spectral representation of $G_3^{e+h}(\omega)$ given in Eq. (12) is similar to the one of $G_1$
it is convenient to introduce a 3-body spectral function for $G_3^{e+h}(\omega)$ that is similar to the spectral function corresponding to $G_1$.
The latter is defined as
\begin{equation}
A(x_1,x_{1'};\omega)=\frac{1}{\pi}\text{sign}(\mu-\omega)\text{Im} G_1(x_1,x_{1'};\omega).
\end{equation}
We can thus define the spectral function $A_3(\omega)$ corresponding to $G_3^{e+h}(\omega)$ according to
\begin{equation}
A_3(\omega)=\frac{1}{\pi}\text{sign}(\mu-\omega)\text{Im} G_3^{e+h}(\omega),
\end{equation}
where, for notational convenience, the spin-position arguments are omitted.

and

We note that the 3-body spectral function is not the spectral function that corresponds to photoemission spectroscopy. Both spectral functions have the same poles but the corresponding amplitudes are different.
In particular, in the non-interacting case the amplitudes of satellites can be non-zero in the 3-body spectral function.
To retrieve the spectral function that corresponds to photoemission spectra Eq. (17) has to be used.

Moreover, figures 1-3 have been modified to emphasize more clearly the differences between the 1- and 3-body spectral functions. These figures are now divided into two panels; in the upper panel we report the 1-body spectral function (the ARPES spectral function), while the 3-body spectral function is reported in the bottom panel.
We included this latter case to show that the 3-GF has the poles at the same position as the 1-GF and that only the amplitudes differ.

2.

We have added the following sentences to the outlook given in section 4.

Moreover, we can reduce the numerical cost of the calculations by applying a diagonal approximation to the three-body self-energy, in similar manner as is often done for the one-body self-energy, to calculate only the poles of $G_3^{e+h}$. While a static one-body self-energy can only yield poles that correspond to quasi-particles, a diagonal static three-body self-energy would yield the poles corresponding to both quasi-particles and satellites.

We have also corrected the indices in Eq. (30) (Eq. (31) of the revised manuscript).

3.

We have added the following paragraph to the conclusions,

For the specific case of the Hubbard dimer we were able to obtain the exact $G_3^{e+h}$. Therefore we could obtain an exact three-body self-energy by solving a Dyson equation. However, in general, the exact three-body self-energy is unknown. Therefore, our next goal 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 for $G^{e+h}_3$ along the same lines as has been done for $G_1$ or by using a similar strategy as in Ref. [27], where a practical scheme to calculate $G_3$ for the description of Auger spectra is proposed.

4.

We have added the following sentence to the Introduction,

We note that the three-body Green's function has been employed to describe Auger spectra [27] and to study satellite structures and the occurrence of the metal-insulator transition. [28]


5.

We have added some references of works in the literature that involve the 3-GF

6.

in Eq. 54 we modified "$G_2(\omega)$" to "$G_1(\omega)$"."