1) In Eq(11) they mean "Re" instead of "Im"?
2) The argument that leads to Eq.(16) overlooks one important fact: that threading one flux per surface unit cell is an extremely large perturbation, so adiabaticity is far from guaranteed. In fact it is known that when half-flux quantum is threaded per unit cell strong topological insulators develop a kind of 1D metallic wire along the flux tube, in the form of a pair of gapless conter-propagating 1D gapless modes that penetrates into the bulk (see e.g. Phys. Rev. B 82, 041104(R) (2010)). Perhaps a safer argument can be made by assuming an enlarged unit cell. Can the authors should clarify or remove these arguments?
3) The authors state: "An effective magnetoelectric susceptibility can only be defined in ... where the net Hall response is zero." (notice missing word ...="systems"). This is an important point, and I kind of see why (my view: if it is non-zero the system might have net charge accumulation from Streda formula after threading magnetic field, also surface cannot be fully insulating and hence charge might flow). But the authors should explain why this makes the effective magnetoelectric susceptibility ill defined.
4) The authors state: "The hybrid Wannier function representation makes explicit the fact that one cannot create Wannier functions in such a topological systems despite the fact that the eigenstates of Hamiltonian have the Bloch form." I guess the authors mean that one cannot create Wannier functions that are localized and strictly respect the symmetry? (see e.g. Phys. Rev. B 83, 035108 (2011), Phys. Rev. B 93, 035453 (2016)).
5) Related limitations to the measurement of the apparent monopole at the surface of a TI described here were also discussed in Phys. Rev. Lett. 111, 016801 (2013).
6) The authors write: "Again by way of analogy with the 1D chain, this suggests a
way of looking at inversion symmetric insulators as overlapping e^2/h and -e^2
/h layers. As shown in Fig. 8, one can conceive of conventional insulators as being materials these conducting layers are centered on top of each other and spatially overlap and cancel, whereas a TI is where layers of them are displaced from each other by half a unit cell, giving 1/2e^2/h on the surface." How literal should this picture be taken? e.g. how is time reversal supposed to act in this hypothetical system of displaced quantum Hall layers? or, are the authors then imagining a system with large breaking of TRS throughout the bulk? if so, how to think then about TR invariant TI's?
7) In view of the comments in report above, the authors might want to revisit/rephrase statements such as:
"Although the development of systems that realize this conguration is very important from a materials perspective, we do not believe it warrants any particular consideration as anything special or fundamental. Both scenarios have the same formal ME susceptibility. As shown in Fig. 10, the two configurations should just be considered as different experimental conditions and realize fundamentally the same thing."
"However, as is hopefully clear from this discussion there is no intrinsic dierence from one scenario the other. They are all just different demonstrations of the same underlying physics and both experiments are measures of the formal ME susceptibility."