Microscopic theory of fractional excitations in gapless quantum Hall states: semi-quantized quantum Hall states

We thank the referees very much for their effort in reviewing our paper, and the constructive suggestions they made. Below, we address the issues raised by the referees one by one. Most importantly, we extended the introduction to address a broader readership and highlighted the advantages of our coupled-wire construction. In addition, we reviewed the presentation of the references in the introduction and made additional minor changes to the text. We are confident that our replies and modifications will allow publication of our manuscript in SciPost.

Referee 1
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We thank the referee for their constructive feedback, and for the very positive assessment of our paper. In response to their helpful suggestions, we have made the following modifications in the introductory section 1.

- We added a new paragraph starting with "In a more general perspective" that puts our work into a bigger picture in order to address a broader readership.

- We explain why a partially gapless system can exhibit quantized transport. For this to happen, it is key that the gapless sector is electrically disconnected. As we now write, "We discuss that the persistence of quantized responses arises only if the layer associated with the gapless sector is electrically disconnected. As one might expect, driving a current through that layer would instead lead to a non-quantized response."

- It is correct that the present approach is an extension of the composite fermion theory developed in Ref 54 using the complementary wire-construction approach, which we now state explicitly.

- We also clarify the advantages of using the wire construction approach: "As the main advantage as compared to a continuum theory as the one in Sec. 2, our coupled-wire construction not only provides a microscopic model for semi-quantized quantum Hall states but also facilitates an exact translation of observables between the languages of the effective low-energy field theory and the original electronic operators".

To iterate, coupled-wire constructions are to be seen as an alternative technical implementation of flux attachment and the controlled derivation on the universal low-energy theory, not as an opposing physical picture. The general logic is that coupled-wire constructions describe anisotropic limits of topological states of matter. The states described by these constructions should be adiabatically connected to the isotropic states realized in quantum Hall systems for the approach to make sense. The anisotropic limit in then "buys" more explicit calculations, which in turn facilitates, for example, easier identification of the various operators in the low-energy theory in terms of the original degrees of freedom.

- We have corrected the typos found by the referee. Note that we do not have an interaction between the \down-electrons. Adding one would be straightforward and not affect the result of our heuristic argument, but would render the equations heavier. We now state this explicitly in the text.

Referee 2
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We thank the referee for his assessment of our paper as being beautiful, sophisticated, and pedagogical at the same time, and stating that he feels like it will generate future interest in semi-quantized quantum Hall states. We apologize for the poor referencing in the earlier version of the paper and have corrected this now.