Weak integrability breaking and level spacing distribution

We sincerely thank the referees for their very helpful and constructive comments which were very useful in improving our presentation. We revised the paper accordingly, cf. the list of changes.

Invited referee 1

Changes requested:

We included the paper by Jung et al. as we found it directly relevant to the issue of integrability breaking.

1. We have included the statement that Q3 is the energy current and inserted the appropriate citation.
2. We re-labelled the vertical axis in Figure 2.1 for clarification.
3. The systematics are explained in the text. Since the current explicitly depends on the anisotropy parameter Delta, its norm shows such a dependence as well, contrary to NNNI. In addition, we make it clear that these norms grow extensively with the volume, with some fluctuations due to the local extension of the individual terms. Further details are not so relevant since the norms themselves are dependent of the conventions in the definition of the operators. Indeed, this is one of the motivations of introducing the effective coupling, as stated now explicitly after (2.13).
4. We added axes labels and the requested clarifications in Figs. 3.1 and 3.2.
5. Sector Sz=0 is invariant an additional symmetry under spin-flip (see footnote added on page 6), which would have been necessary to project out. It is possible to do that, but there is no added value in the data
6. We have expanded the discussion for two of the figures, now labelled 3.2 and 3.3. Old figures 4 and 5 have been replaced by 3.4 and 3.5, following suggestion 8.
7. We implemented the referee’s suggestion. To do that, we brought forward the description of our peak finding procedure from Subsection 3.4 to 3.2
8. We explain in Subsection 3.4 of the revised version that due to the smooth crossover there is no unique way to define where it happens. The position of the peak is very natural since it moves from the origin to the position of about 0.8. Previous works used other definitions. Note that our procedure is also justified by recovering the 1/L^3 scaling obtained in the previous works.
9. Typo corrected (together with some others we found in the text).
10. We added a more detailed description. Indeed, in the light of the detailed argument this is not surprising, so we reduced the figure (now 3.6) to just 4 subplots on one line. We still decided to keep it as it shows actual data and provides a visual illustration.
11. We removed the line from Fig 6 (now 3.6) as it has no role there. Figs 7 and 8 are now in Fig. 3.3 and include the description of the line and other clarifications.
12. We changed J perturbation to current perturbation everywhere.
13. We expanded the captions of the figures (now 3.7 and 3.8) as suggested.
14. Typo corrected.
15. We added a short explanation and references to the beginning of Subsection 3.4.2.

Invited referee 2

Changes requested:

1. We have extensively rewritten the text, paying attention to make the wording clear.
2. We made it clear that the only fit parameter both for exponential and Wigner-Dyson statistics is the overall normalization (which should match the total number of level spacings in the histograms).
3. The difference between the volume dependences is independent of the rescaling of the couplings. We added a sentence to the conclusions. The rescaling is primarily motivated by eliminating the dependence on multiplicative redefinition of the perturbing operators. It also makes their couplings more directly comparable, but not in a really strict sense. It is the volume dependence of the crossover coupling which really shows the difference in the strength of integrability breaking.
4. We have added clarifications about the notations and hope it’s more readable now.
5. We have added significant new discussion to the introduction putting our results in context (the 4th paragraph). We also modified slightly the abstract and added material in the conclusions to relate our results to the topic of quantum chaos. As emphasized in the conclusions, what we find here is in a sense a “weak quantum chaos”. This phrase has already been used in the literature in connection with out-of-time-order correlators, but the sense we use it here is novel and refers to a distinct class of integrability breaking operators, as emphasized now in the introduction and the conclusions.

Weaknesses addressed:

1. We have gone through the paper to improve the text.
2. We included a new paragraph in the introduction.
3. Cf. point 5 above.
4. We agree with the referee that the work of Vassuer’s group is relevant, and included the appropriate reference ([17]).
5. –
6. Cf. point 2 above.
7. The new paragraph in the introduction includes a clarification of the concept of weak breaking of integrability.
8. Cf. point 7 above.
9. We included the description of the functions in the argument in Subsection 3.3.
10. The radius used for the Gaussian filter is now included in the text. Note that due to a rearrangement (referee 1, point 8) this text is now in Section 3.2.

Contributed review

Answers to major comments:

• This is indeed an interesting issue. We could in principle take the next current, but explicit construction of these currents is quite tedious and error prone. In fact, when we took the current J3 we had to correct typos in our sources which we did by explicitly checking its required properties. In GHD they are more of a theoretical concept than objects of explicit computations, although there has been great progress in giving an algebraic construction for them (https://arxiv.org/abs/2005.06242).
We have found that while in principle we know infinitely many such operators, explicit construction would require quite a large effort. However, we agree that this is an interesting problem for the future.
Note that – at least to our knowledge – there is no theoretical explanation for the 1/L^3 law either. We think this is an interesting problem, as we state at the end of our conclusions.
• We changed the terminology “critical strength” to “crossover coupling”.
• Power law scaling in gapless systems was observed in previous literature. In fact, the 1/L^3 law was found quite universal. It is not a great surprise that finite size effects scale with a power when the correlation length is infinite. On the other hand, we found exponential decay in the gapped regime, which is not surprising given the finite correlation length (cf. the beginning of Subsection 3.4.2).

Answers to minor comments:

• There are simply not enough states for L<16 to construct a meaningful statistics (see the inserted footnote 4 on page 7). We agree that this is not optimal, but previous studies had this limitation as well and we found no way on improving this situation.
• The Delta>1 and Delta<-1 phases are physically different (Ising antiferromagnet vs. Ising ferromagnet). We now clarify this after eqn. (2.2).
• We have a larger set of data (all showing the same effects), and we simply did not maintain consistency of our selections. Now we changed the coupling everywhere to the same values (-1.6, 0.2 and 1.6).