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Excitonic Bloch equations from first principles
by Gianluca Stefanucci, Enrico Perfetto
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Authors (as registered SciPost users):  Gianluca Stefanucci 
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Preprint Link:  https://arxiv.org/abs/2407.17077v1 (pdf) 
Date submitted:  20240725 06:24 
Submitted by:  Stefanucci, Gianluca 
Submitted to:  SciPost Physics 
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Academic field:  Physics 
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Approach:  Theoretical 
Abstract
The ultrafast conversion of coherent excitons into incoherent excitons, as well as the subsequent exciton diffusion and thermalization, are central topics in current scientific research due to their relevance in optoelectronics, photovoltaics and photocatalysis. Current approaches to the exciton dynamics rely on {\em model} Hamiltonians that depend on already screened electronelectron and electronphonon couplings. In this work, we subject the stateoftheart methods to scrutiny using the {\em ab initio} Hamiltonian for electrons and phonons. We offer a rigorous and intuitive proof demonstrating that the exciton dynamics governed by model Hamiltonians is affected by an overscreening of the electronphonon interaction. The introduction of an auxiliary exciton species, termed the irreducible exciton, enables us to formulate a theory free from overscreening and derive the excitonic Bloch equations. These equations describe the timeevolution of coherent, irreducible, and incoherent excitons during and after the optical excitation. They are applicable beyond the linear regime, and predict that the total number of excitons is preserved when the external fields are switched off.
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This work presents a consistent treatment for the dynamics of bound electronhole pairs out of equilibrium, in terms of a set of “excitonic” Bloch equations driven by the coupling with lattice vibrations. These equations account for both the coherent and incoherent dynamics of the excitons, as well as the conversion between excitonic polarizations and populations. In addition, the authors work out the microscopic forms of the excitonphonon couplings discussing the complex role played by electronic screening.
I think this work represents a relevant contribution to both the theories of nonequilibrium carrier dynamics and of excitonphonon interaction, presenting very timely results leading to a step forward in the theoretical framing of these processes. Thus, in my opinion the manuscript should appear on SciPost Physics.
That being said, I do have several questions and comments. I do not think the work is always easy to follow, and sometimes it is not especially clear what has been deduced via derivations and what constitutes a somewhat arbitrary approximation. I would ask the authors to please clarify these points and, whenever they feel the discussion can be beneficial to the manuscript, to consider amending the text.
1. The BSE has a spin structure exhibiting different channels: excitons (singlet and triplet) plus magnons. In the case of collinear spin polarization, singlet and triplet excitons are mixed. In the case of spinorbit interaction, excitons and magnons are mixed. [See Bechstedt, ManyBody Approach to Electronic Excitations, Chapter 18.2] The theory presented here is most likely applicable to the exciton channel (singlet+triplet) for all kinds of spin polarizations and including spinorbit coupling, while neglecting the magnon channel. The authors could explicitly remark on this.
2. The authors could remark on the connection between the exciton polarization in Eq. (15), which is derived from the disconnected part of the twoparticles Green’s function, and photoabsorption spectra. This could be useful since in most textbook derivations of the BSE, which are in the context of linear response and Hedin’s equations, photoabsorption spectra are obtained starting from the Dyson’s equation for L instead of GG. [See e.g. Strinati, Rivista del Nuovo Cimento, 11, 12 (1988)]
3. The theory is formulated in the TammDancoff approximation, in which the Coulomb couplings between resonant and antiresonant valenceconduction transitions are neglected in the BSE kernel. It is presently unclear how much these terms are important out of equilibrium (even for a semiconductor), but I suspect that if they are included in the phononassisted kernel, they will also lead to the explicit appearance of the interband valenceconduction (screened) electronphonon couplings inside additional excitonphonon contributions. Do the authors have any comments on this?
4. The authors distinguish between the coherent exciton regime, where the excitonphonon coupling is “irreducible”, and the incoherent regime, where it is “reducible”. Excitonic polarizations are then converted to populations again via the irreducible exciton coupling with phonons.
The overscreening discussion in the coherent case, along with the identification of the correct electronphonon selfenergy and irreducible exph coupling, is equivalent to the findings in Ref. [50] starting from linear response. The discussion of the excitonphonon BSE kernel and relative reducible exph coupling for the incoherent case is equivalent to Refs. [75] and [Cudazzo, PRB 102, 045136 (2020)] starting from an extension of the static BSE kernel (here I propose that the latter reference may be cited in the text). However, while these cited papers assumed that either the irreducible or reducible couplings should be used in all cases, the present paper makes the case for both formulations being correct, albeit in different regimes. It could be useful to explicitly state this in the text in order to make the points of convergence and departure with previous works clearer. I do have some additional comments on this.
4.1 The authors emphasize that the coherent part of the theory could be expressed in terms of reducible excitons if bare elph couplings were used. In the text they write “$g_S \tilde{L} = g L^{(v)}$”, but the implication seems to be – by looking at Appendix B – that actually the $v$reducible $L$ could be replaced with $L^{HSEX}$, i.e. only the first term in Eq. (67). Is this correct?
This means that the excitonphonon selfenergy obtained with the irreducible exph coupling and the irreducible $\tilde{L}$ should be equivalent to the one obtained with the bare exph coupling and the reducible – actually HSEX – $L$. But $L^{HSEX}$ and $\tilde{L}$ have different poles. How can Eq. (88) still be valid (including in the lowdensity regime) in the $g$*$ L^{HSEX}$ formulation? In particular, in Ref. [50] it is pointed out that the $g_S$ * $\tilde{L}$ treatment gives rise to intrinsic nonzero phononmediated linewidths also for the lowestlying excitonic state. I don’t understand how a treatment with $g$*$ L^{HSEX}$ could give the same.
4.2 About the incoherent excitons. In this case, the authors do not start from an electronic selfenergy, but rather from the BSE for $L$ (the correlated part of the twoparticles Green’s function) with a general interaction kernel, which is then approximated to HSEX for the Coulomb part, and to first order in $D$ (the phonon Green’s function) for the phonon part.
Are these choices compelled by consistency with the previous coherent treatment (such as: approximating $\tilde{L}$ as $L^{SEX}$ requires $L$ to become $L^{HSEX}$) and/or compliance with stateofthe art simulations (since BSE calculations are usually done in the HSEX approximation)?
Is it not possible to obtain an electronic selfenergy whose functional derivative with respect to the Green’s function would yield this HSEX+ph kernel? And if not, can there be consistency issues (such as missed cancellations) between terms arising from the dressing of the electron Green’s functions (quasiparticle corrections) and those appearing in the incoherent excitonphonon kernel?
4.3 On a related note: does Eq. (66) for the conservation of the total exciton number remain valid independently of the approximation chosen for $N^{inc}$ in Eq. (65), that is, the type of incoherent kernel employed?
5. Does $N^{inc}(t)$ tend to relax to a BoseEinstein distribution after long times? What about $\tilde{N}(t)$? In other words, can this theory provide some hints at the form of the “exciton” occupation function after they have relaxed to the bottom of their dispersion curves, before recombination?
Spotted typos:
 Eq. (7): the last $z^+$ should be $z^{\prime +}$
 Before Eq. (50): $N^{HSEX}$ in the text should be $\tilde{N}^{SEX}$
Recommendation
Ask for minor revision
Report
In this manuscript, starting from basic equations of many body Green's function theory, the authors provide a derivation of the excitonic Bloch equations beyond that based on model Hamiltonians. In this way they provide a theoretical tool for the description of the exciton dynamics that can be interfaced with modern abinitio calculations of electronic structure and lattice vibrations. In addition their derivation suggests that the exciton dynamics governed by model Hamiltonians is affected by an overscreening of the electronphonon interaction. The problem of the overscreening seems to be removed in their formulation.
In my opinion the manuscript sounds physically clear, well explained and I am in favor of its publication. However I found that there are some issues in the formulation of the theory and some questions the author should answer before the paper is published.
\begin{enumerate}
\item I am not really convinced that the definition of the excitonphonon coupling provided in previous works such as in Ref. [73,75] of the present manuscript is affected by overscreening. From basic equations of many body Green's function theory, the BetheSalpeter equation (BSE) in presence of electronphonon interaction can be obtained once an approximation for the electron selfenergy in terms of the electronphonon coupling is given. In particular, the functional derivative of the electron selfenergy respect to the electron Green's function gives the kernel of the BSE from which we can extract an effective exciton selfenergy that provides also the definition of the excitonphonon coupling as discussed in Ref. [50] of the present manuscript. Still in that work the authors have shown that a suitable approximation for the electron selfenergy is that reported in Fig. 1 (b) of the present manuscript. Indeed in this expression, as mentioned by the authors, the electronphonon coupling is not overscreened in contrast with the expression reported in Fig.1 (a) where the overscreening is clear. Thus, I totally agree with authors that the correct expression of the electron selfenergy is that reported in Fig. 1 (b). At this point, it is important to note that the structure of the exciton selfenergy (and hence the excitonphonon coupling) depends from the way in which the functional derivative of the electron selfenergy is performed. In Ref. [50] the functional derivative is done neglecting the explicit dependence from the electron Green's function of $K^{(r),SEX}$ and the electronphonon matrix elements $g^s$. This leads to an expression of the exciton selfenergy in terms of the proper part of the towparticle correlation function $\tilde{L}^{SEX}$ instead of the full $L^{HSEX}$ as in Ref. [73,75]. This is the origin of the asymmetric structure of the excitonphonon coupling $\tilde{\mathcal{G}}$ in Eq. (52). However, in principle there is no reason to neglect the functional derivative of $g^s$ which depends from the electron Green's function through the screening. This leads to the appearance of additional diagrams in the exciton selfenergy. In particular, I suspect that, when the screening is evaluated using $L^{HSEX}$ (i.e. consistently with our treatment of vertex corrections), the inclusion of the terms arising from $\frac{\delta g^s}{\delta G}$ would lead to the natural appearance of $L^{HSEX}$ in the exciton selfenergy and thus to a symmetric excitonphonon coupling $\mathcal{G}$ as in Eq. (44).
In the present manuscript, an expression of the excitonphonon matrix element formally equivalent to that of Ref. [50] has been obtained evaluating the collision integral in Eq. (43) using the ansatz in Eq. (46) and taking only terms linear in the offdiagonal part of $G$. However in doing this, the authors consider only offdiagonal $G$ appearing in the external lines. But what about the linear terms where the offdiagonal $G$ appear inside $g^s$? I suspect that neglecting these terms is in some way equivalent to neglect $\frac{\delta g^s}{\delta G}$ in the derivation of Ref. [50].
I suggest to think about this point before resubmitting the manuscript.
\item The eigenvalue problem in Eq. (2) describes excitons in absence of population. Thus it can be used only for optical absorption. In the present case the authors are treating excitons outofequilibrium. What about the Pauli blocking factors. Is there a reason to neglect them in Eq. (2)? Is this related to some assumption concerning the laser pulse (frequency of the laser pulse close to the exciton energy, small population etc)?
The author should clarify this point in the manuscript.
\item The Bloch equations obtained in this work should describe how photoexcited coherent excitons are converted into incoherent excitons and how the latter propagate and eventually thermalize. Thus when the quasiequilibrium is reached, these equations predict a photoexcited system consisting of a thermalized incoherent exciton gas. However, in general we expect a configuration in which incoherent excitons coexist with an electronhole plasma [see for example: phys. stat. sol. (b) 131, 151 (1985)]. What about the electronhole plasma? What is the regime in which the theory developed by the authors is applicable?
I think the author should discuss these aspects in the article.
\item In Eq. (2) I do not see any index associated to the spin degrees of freedom. Thus, I suppose that spin variables have been summed up in some way. This requires the introduction of two decoupled BSEs. One for the singlet channel and one for the triplet channel [see: RIVISTA DEL NUOVO CIMENTO
VOL. 11, N. 12]. If this is the case Eq. (2) refer to the singlet channel [see Phys. Rev. B 62, 4927 (2000)]. As a consequence, the matrix elements of the bar Coulomb potential in Eq. (3) should be multiplied for a factor 2.
\item In the manuscript the author call $v$ "direct electronhole interaction" and $W$ "exchange electronhole interaction". In the literature, on the contrary, direct and exchange are usually used to indicate $W$ and $v$, respectively.
\end{enumerate}
Recommendation
Ask for minor revision
Author: Gianluca Stefanucci on 20241021 [id 4883]
(in reply to Report 1 on 20240920)We provide a reply to the Referee's comments in the attached file
Author: Gianluca Stefanucci on 20241021 [id 4884]
(in reply to Report 2 on 20241009)We provide a reply to the Referee's comments in the attached file
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