Topological transitions with continuously monitored free fermions

Dear Editor,

We thank the referees for their comments and critiques. We feel that addressing these issues has greatly improved the manuscript.

Both referees acknowledge that the manuscript deals with an interesting and timely topic and recognize the importance of the new ideas and perspectives presented in the manuscript. However, both also comment on some shortcomings of the presentation. Indeed, one one referee comments that our central results are not elaborated on, and that the paper only “scratches the surface” of the potential of the introduced ideas.

While we acknowledge some shortcomings of the presentation, and we deal with these comments individually in the responses to the Referees' reports, we feel our work goes beyond scratching the surface and incorporates some new and innovative features that will yield many more fruitful results. Moreover, the potential for further research in no way minimizes the impact of this work. Indeed we think this is why the manuscript meets the criteria: “Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work”. Of course, we could always expand upon every aspect of our work. However, we feel that given the many avenues open to us from here, it is important that we first present these ideas and key initial findings as concisely and accurately as possible.

In the new iteration of the manuscript we have taken on-board the suggestions and criticisms to better highlight the main messages of the manuscript. For clarity we stress here the significance of our findings. Our work shows for the first time that competing continuous non-projective measurements in lattice fermionic systems can drive a topological entanglement transition between two area-law phases. In particular:

1. We show that weak measurements drive a topological transition of a different universality class compared to projective measurements. This follows from the different critical behavior of the two models (see our detailed comment to referee 1 below).

2. In the full (unitary + measurement) dynamical evolution, the model demonstrates an extended critical phase with a logarithmic entanglement scaling. In the new manuscript we show that the combination of S^top_L and S_L more clearly marks this critical scaling transition.

3. Our analysis shows that the entanglement transition in the previously studied charge conserving model [Ref 45 (arXiv:2005.09722 )] ($\alpha= 0$ line) is non generic. In particular our numerical data does not point to a transition at finite measurement rate.

4. We introduce a simple-to-apply model of partial post-selection that recovers the full post-selected dynamics (for which we can map the full phase diagram and associated topological indices) as a continuum limit of a family of models with stochastic dynamics. This allows one to interpolate between the fully stochastic dynamics and the post-selected models. We feel this approach will be invaluable in future studies.

We are confident that the new version can be published in Scipost Physics.

Sincerely,

Graham Kells, Dganit Meidan, Alessandro Romito

- rewritten part of the abstract and changed part of the introduction to clarify the significance of our findings and polish the writing
- rephrased part of section II
- added paragraphs in section III to improve clarity
- added paragraphs to section IIIA to expand our comments on the results therein
- rewritten section IIIB to present in a new way the combined role of the entanglement entropy and topological entanglement entropy
- rephrased parts of section IIIC to clarify the role of the partial post-selection and and post-selection models
- rewritten part of the conclusions
- added new appendix B
- replaced Eq. (1) more more general formulation
-changed Eq. (7) for clarity
- rearranged Fig. 2
- changed Fig. 3 (c-d) to present the combined results of topological entanglement entropy and half-cut entanglement entropy
- added new Fig. 8 and Fig. 9 in Appendix B
- corrected typos
- added/updated references