Photoemission spectroscopy from the three-body Green's function

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.

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?

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.

See clarifications above.

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

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.

The manuscript is well written and potentially of strong interest for the community. However, the two weaknesses identified should be clarified. In particular W1.