Long-range entanglement and topological excitations

We thank the referee for her/his work, but we are very surprised by her/his judgment, in particular regarding the lack of results of general interest in our paper. We thought that it was clear that indeed we are proposing two novel messages with this article: that the Disconnected Entanglement Entropy can detect more than standard topological orders, but rather also other types of long-range entanglement (while providing a concrete example of that fact) and therefore it is an even more valuable tool toward the study of exotic states of matter beyond local order. Additionally, we also proved that the simple combination of geometrical frustration and quantum effects gives rise to a phase that, while defying standard topological classifications, showcases most of the properties characteristic of usual symmetry-protected topological orders.
The referee’s report signals that, in our original submission, we have not been successful in conveying these messages effectively. We apologize and we addressed this issue with this resubmission: we rewrote both the abstract and part of the introduction, and we added a few sentences in the body of our manuscript in the hope that our motivations are clearer.
Our findings uncover new features in the field of non-traditional quantum phases of matter and might usher in a new class of models with different topological properties. Thus, we ask to still be considered for publication in the flagship Scipost journal, since we believe that our work meets its highest standard. We acknowledge that more work is needed to arrive at a comprehensive and complete picture, but our results are conclusive and their impact in motivating the community grants them a general interest.
We thank the referee for spotting several typos, which we corrected.
We stand behind the fact that eq. (9) provides a quantization for the DEE in models with single delocalized excitations and we explained this fact better in the new version of the manuscript when we derived eq. (9). Namely, while in symmetry-protected and standard topological phases the DEE is quantized to a value that depends on the ground state degeneracy/number of the edge states, in the case we consider in this paper we have a delocalized excitation whose contribution to the DEE is determined by the choice of partitioning done to calculate the DEE, whose purely geometrical nature supports our claim that this value constitutes a quantization figure.

Coming to the referee's questions:

1) In Fig.5 S_2^D(t) goes to much higher values after a global quench than in the equilibrium state in Fig. 3. Could the authors comment on this discrepancy?

There is no discrepancy here: when a system is taken out of equilibrium it develops higher entanglement over time. As it reaches a stationary state, the cancellations inherent in the definition of DEE might eventually reduce it, but in the transient regime, it is expected that the DEE can grow in time as a reflection of the entanglement increase. The same behaviour was observed also in references 35,36, where DEE was analyzed for models with SMTP. What is important in Fig. 5 is that for times that grow and asymptotically diverge with the system size the DEE stays constant and fixed to the initial value because, as it happens also for SMTPs, the growth of local entanglement cannot modify DEE and also correlation spreading across the whole chain can affect it. Hence, proving that the signal we observe in our model has indeed a non-local nature.

2) Eq.(13) contains S_2^D(0) in the denominator. But S_2^D(0) in the left panel in Fig.5 is zero. It seems to me that Eq.(13) should be somehow corrected. We would like to thank the referee for highlighting this point. We agree that the text is not sufficiently clear regarding the procedure we used since some information is missing. In the original submission, we employed two different definitions of $t_c$ for the two plots in Fig. 5, with Eq. (13) being valid only for the frustrated-unfrustrated evolution. For the other plot, we used an alternative one:

t_c := \min_{t} \left\lbrace\vert S_2^D(0) - S_2^D(t) \vert > 0.1\right\rbrace.

This latter definition does not suffer from the problem correctly noticed by the referee. The different choices of $t_c$ were physically motivated and also justified by the fact that different definitions do not alter significantly the conclusions, since the growth of S_D, when it starts, is very rapid. Following the referee’s remark, we decided to improve our analysis by employing a common definition for both cases, namely: t_c :=\min_{t} \left\lbrace\vert S_2^D(0) - S_2^D(t) \vert > 0.1\right\rbrace ,

and by adding additional points to the insets (these points refer to evolutions not shown in the main plots in order not to clutter them). Thanks to these additional points spanning larger sizes it has been possible to conclude that there is a linear relation between t_c and the chain length, strengthening our claim that the critical time diverges in the thermodynamic limit. We are grateful because the referee’s remark pushed us to improve our analysis and our paper.

3) S_2^D in Eq.(9) is valid in the classical point h=0. However, since it is a topological invariant it should be constant in the topological phase. Can it be proved at least for small fields?

As explained in the text, Eq. (9) is strictly not valid for h=0, since there the ground state is massively degenerate and the DEE would depend on the ground state choice. To the contrary, Eq. (9) has been derived in perturbation theory for small h. Then in section III.B, we argue why this expression should remain valid in the whole phase and in sec. III.C we proceed in proving it by numerical comparison.

4) In Fig. 6 the authors show S_2^D as a function of the bond disorder strength δJ, and the field disorder strength δh. For large values of the disorder parameters, S_2^D seems to tend to the same asymptotic value. I am curious if it is possible to find this value analytically, analogously as it was done in Eq. (9).

This is an excellent question, but we do not have an answer to it yet. We are working on it and we'll report the result in a future work.

We are submitting a pdf in which we marked in red every change made