Quasi-equilibrium polariton condensates in the non-linear regime and beyond

- The authors were able to formulate a meanfield theory for electron-hole photon condensate without introducing UV cutoff.
- This formalism enabled the authors to show that the BCS regime is enhanced for Rytova-Keldysh potential compared to Coulomb or contact interaction.

- The overall picture of what the authors obtained is not very different from the known results.
- The treatment of dissipation is problematic even at the phenomenological level.

In this work, the authors present a mean-field theory of an electron-hole-photon mixture in a two-dimensional semiconductor microcavity. While there have been several works that performed a similar analysis to the model considered in this manuscript, the novelty here is that they have employed an appropriate renormalization scheme. They considered three types of interactions: Coulomb interaction, contact interaction, and Rytova-Keldysh potential, and found that the Rytova-Keldysh potential offers the largest parameter regime of the BCS regime. This treatment allowed the authors to eliminate the ultraviolet cutoff dependence from the model.
The results are non-trivial, and the manuscript is well-written. I, therefore, recommend its publication in SciPost Physics once the following comments are addressed.

1. Is there an intuitive understanding of why the Rytova-Keldysh potential provides a more significant BCS regime than the Coulomb interaction? I would have thought that the shorter range nature of the Rytova-Keldysh potential would suppress the band renormalization that made it possible to exhibit the BCS regime.


2. In Sec. IV, the authors consider the effect of dissipation by introducing a gain and a loss to the exciton and photon component, respectively, in a phenomenological manner. However, I find some of their treatment problematic.

The chemical potential μ must not have an imaginary part since its presence implies damping or gain of the condensate. Note that, in the canonical ensemble, the condensate has an oscillating phase determined by the chemical potential, as Delta(t) = Delta_0 exp[I 2μt]; see Ref. [49]. This contradicts the assumption that the system is in a steady-state. In general, there would be a nonlinear imaginary term that gives rise to the saturation effect (as done in [Wouters and Carusotto, PRL 99, 140402 (2007)] for the one-component case and Ref. [49] and [Hanai and Littlewood, PRR 2, 033018 (2020)] for the two-component case) that automatically makes the steady-state condition (=μ being real) satisfied in the long-time limit.
This can be achieved by interpreting their gamma (RX in Ref. [49]) to include these nonlinear effects. In particular, gamma should be considered as a parameter that is determined by demanding μ to be real, as done in Ref. [49] (instead of assuming gamma = kappa as done in the manuscript).

On the other hand, there is no need to add an imaginary part to the electron density Eq. (37). In the Keldysh formalism, adding dissipation to the system would make the spectrum have a Lorentz distribution rather than the delta-function of the mean-field approximation but would never give rise to an imaginary part of the density (See, e.g., Ref. [14].). Therefore, solving Eq. (49) is not necessary. What one should solve instead is the requirement that the chemical potential μ is real, WITHOUT assuming by hand that gamma = kappa, as mentioned above.

One of the reasons that I strongly recommend the authors to perform the above analysis is that assuming gamma = kappa would lead to a somewhat misleading conclusion that one always goes through an exceptional point by tuning the density. Instead, as pointed out in Ref. [49] (and more recently in [Fruchart, Hanai, Littlewood, Vitelli, Nature 592, 363 (2021)] from a symmetry perspective), one needs to fine-tune TWO parameters to go through an exceptional point in a U(1)-broken system like exciton-polariton condensates.

3. I am confused by the authors’ comment “Reference [49] has proposed that … due to Pauli blocking or phase space filling effects [2, 3, 54]. However, we observe no such decrease of the Rabi splitting with increasing density ...” I am pretty sure that the Pauli blocking effect already appears at a meanfield level; see Eq. (S81) of the SI of Ref. [49]. Could the authors comment on why they could not see such effects in more detail?