Polariton condensation into vortex states in the synthetic magnetic field of a strained honeycomb lattice

In this manuscript, the authors consider theoretically Bose Einstein condensation in a strained honeycomb lattice. The strained honeycomb lattice was previously considered in several previous works. The effect of the strain can be modelled as an emergent magnetic field acting on the lattice particles as if they were charged. The sign of the field is opposite in both valleys. The resulting eigen states are well described as artificial Landau levels. On top of this, the authors use a condensation model supposed to describe polariton condensation. They show that by using an appropriate pump profile in real space (pumping only the B-atoms), condensation preferentially occurs in the n=0 Landau levels because these states have the best spatial overlap with the pump. The intrinsic angular momentum carried by these states results in the formation of quantum vortices. The authors show some regimes where condensation occurs in a single valley, which results in the formation of a vortex lattice with all vortices having the same sign.
I found the manuscript very interesting, and I certainly recommend publication. I have a few comments and questions below.

1 A key aspect is that the n=0 Landau levels are B polarized. May be it was discussed in details in previous publications on strained honeycomb lattices, but it is not evident why it is so. It is not completely evident why the A-B symmetry is broken.
2 The n=0 Landau level is favored because it has the best spatial overlap with the pump. Other important aspects which are not really taken into account are the state life-time and energy relaxation. The life time might vary because of the change of the exciton-photon fraction, but also because of the profile of the wave function. Antisymmetric states are more localized in pillars and show longer lifetime than symmetric ones which are more in contact with the edges of the pillar. This type of effect is beyond the tight binding limit used, but it could even more favor condensation in this n=0 state. May be the authors could discuss a little bit more these questions.

3 The next point is about the symmetry breaking effect, namely the choice of a specific valley. This is the key point of the manuscript, and the explanation given is a bit elusive. “On the contrary, interactions play instead a key role in the case of wider incoherent pump profiles, where they make condensation in a single Dirac valley—as opposed to in both valleys simultaneously—more likely. In standard BECs at thermal equilibrium, it is well known [46] that repulsive interactions tend to suppress fragmentation of a BEC. Here, a similar mechanism hinders the condensate from occupying both valleys at the same time.”
This is not clear at all for me. Repulsive interactions favor the homogeneity of the condensate density in real space. If we consider a Kibble Zurek mechanism for the condensation, domains of homogeneous phase should show up when the condensate form. These domains are separated by domain walls associated with a finite kinetic energy cost. These domain walls can then disappear or not. The domain size depends on the healing length and therefore on interactions. In order to have all vortices having the same size, one needs to form domains larger in size than the pump. Another aspect is that interactions lead to the formation of vortices which are topological defect. So they can annihilate, but they cannot scatter in the other valley, which might be the case in the linear regime I guess ? In general I would suggest that the authors, could, may be, elaborate a bit more on these questions. I would suggest to cite two works which might be relevant:
arXiv:2103.06009 where condensation (at the gamma point) in a polariton graphene lattice (but without strain) is considered. This leads to the formation of vortex anti-vortex pairs following the Kibble Zurek scenario.
Nat. Com. 9, 3991, 2018. which considers a staggered honeycomb lattice. Here, it is shown that the filling of the lattice by a BEC makes that Valley excitations are quantum vortices. This is suppressing inter-valley scattering because of the topological protection associated with of the quantum vortex winding number.

4 In the abstract and outlook, it is said that this work paves the way toward the fractional quantum Hall regime. It sounds nice. It is an attractive goal for many of us, and this paper can be useful in that perspective. On the other hand, I personally don’t so clearly see the paving of the way. Here we are approaching the integer quantum hall effect, in a way which makes it extremely difficult to observe experimentally. Going toward the fractional requires much more, and it is not even clear for me if it is possible at all. So except if some clear arguments, explanations can be brought, I would suggest to tone down a bit the fact that this is a goal easy to achieve on a short time scale.