Shift-invert diagonalization of large many-body localizing spin chains

Dear editor,

thank you for arranging the review of our manuscript. We would like to thank the referees for their comprehensive reports which have helped us to further improve it, and we are happy that our work is perceived as interesting and useful.

To address one of the remarks raised in report 2, we specifically improved the introduction to make it more pedagogical, performing the changes requested by the referee. Please note that this paper is also accompanied by a well-commented and documented code, which we trust will make it especially clear and helpful for researchers aiming to learn this numerical method.

In the updated version of our manuscript we have carefully addressed all the suggested changes; below we reply in detail to the various points raised in the three referee reports highlighting the changes made. We also performed very minor reformulations in the body of the text.

We hope that with these improvements, you will judge our manuscript suitable for publication in SciPost Physics.

Yours sincerely, The authors

Reply to Referees

Reply to Referee 1

We thank the referee for his/her positive appreciation of our work.

Ref. 1: On page 4 a sector Sz=1 is mentioned. I believe this should be the Sz=1/2 sector for odd length chains Our reply: The referee is correct and we have changed $S_z=1$ to $1/2$.

Ref. 1: on page 13, Fig. 5 suffers from an impaired font on my screen and in print Our reply: We have improved Figure 5 with respect to the fonts and colors used.

Ref. 1: on page 8 there is a type "bot" $\to$ "not" Our reply: We thank the referee and have corrected this typo.

Reply to Referee 2

We thank the referee for his/her overall positive appreciation of our work, and for the suggestions for improving our manuscript. We discuss them in detail below. Regarding the pedagogical nature of this work and besides the improvements discussed below, we would like to point out that our manuscript is accompanied by a well-commented and documented code, which we trust will make it especially clear and helpful for researchers aiming to learn this numerical method.

Regarding the specific changes requested:

• Ref. 2: Eq. (1): Here the Hamiltonian ``H" is introduced with a different font than later in the text. If this was done intentionally, a comment would be appropriate, otherwise this typesetting error should be corrected.

  Our reply: We thank the referee, this was indeed a typesetting error. We now use the same symbol everywhere for the Hamiltonian.

• Ref. 2: Two lines below Eq. (1): the authors immediately specialize the anisotropy parameter $\Delta$ to the value $\Delta=1$. Without the local fields, i.e., for hj=0, this would be a very special case with enhanced SU(2) symmetry. Hence, I think that a comment would be useful whether the authors expect the value $\Delta=1$ to correspond to a special or rather a generic situation for the many-body localization problem.

  Our reply: We have clarified here that $\Delta>0$ is a generic representative of a system with an MBL transition. Also please note that any non-zero value for the random field breaks the SU(2) symmetry, as well as the integrability of the clean XXZ chain, as can be seen e.g. in the statistical properties of energy eigenvalues.

• Ref. 2: Likewise, three lines below Eq. (1), the authors specialize the local fields to be drawn from a box distribution. It would again be interesting to know if the authors expect this distribution to be "universal", i.e., to exhibit behavior representative of generic field distributions.

  Our reply: The specific choice of disorder distribution is not crucial for the purpose of this paper, which mainly focuses on numerical methods. We added a comment to this effect. We note however that other distributions (e.g. quasiperiodic or binary) may lead to different physical properties.

• Ref. 2: The comment on the Jordan-Wigner transformation at the top of page 4 is meaningful only to a specialist reader. I am aware that a more detailed explanation goes beyond the scope of the present manuscript, but at least a reference would probably be helpful for the non-specialist reader. Note also that it may be useful to state that the "spinless fermions" are interacting.

  Our reply: We removed the mention to the Jordan-Wigner transformation since it is not central here and would be meaningful only to specialists.

• Ref. 2: Section 3.1 has only one reference and this is Anderson's original work on the "Anderson" localization. I think that the authors should add some references for diagonalization algorithms. Actually, it is clear from the Discussion in chapter 6 that the authors are aware of relevant references. However, they should not postpone references to chapter 6, but at least cite Refs. [42-47] also in section 3.1.

  Our reply: We have reorganized the citations, which now appear in section 3 rather than only in section 6.

• Ref. 2: A typesetting detail: why does footnote 3 not appear before page 10?

  Our reply: We have fixed this typesetting anomaly.

• Ref. 2: End of page 8: since the authors present CPU times, they need to specify the compute system exactly, most notably quote the CPU model and network interconnect.

  Our reply: We agree with the referee and we now mention the hardware of both computing systems used.

• Ref. 2: In the line below Fig. 2 there is a reference to "Table 4.1", but no such table exists (same at the end of page 11).

  Our reply: We have corrected these references to correctly point to table 2.

• Ref. 2: Figure 3: Firstly, panel labels "(a)" and "(b)" should be added. Secondly, I did not managed to identify "the thick line" mentioned in the caption.

  Our reply: We added the (a) and (b) labels and further marked the `thick line' in black.

• Ref. 2: Figure 5: Black font on a dark blue background is difficult to read. Maybe the authors should use a white font for "16".

  Our reply: We have improved the fonts in figure 5 (see also Reply to Referee 1).

• Ref. 2: Figure 7: Clarity of the left panel would probably be improved if the data for h=1 and 100 would be presented in separate panels.

  Our reply: We understand the referee's motivation for this comment. However, it is our opinion that separating the left panel of figure 7 in two panels (like in the right panel) would decrease the readability of the plot in this case. Also we believe it helps the reader realizing that the ETH (at $h=1$) and MBL ($h=100$) phases have clear distinct behaviors for physical observables.

• Ref. 2: Last paragraph of section 5.1: without consulting Ref. [41], it is not clear with respect to which variable the "slope" of the entropy S is computed. In a presentation that pretends to be a review, I think that the authors should add the necessary explanation rather than sending the reader to the literature.

  Our reply: We have added a clear definition of the entanglement entropy slope with respect to subsystem size, to make this section self explanatory without consulting the literature.

• Ref. 2: Figure 10: Firstly, this figure seems not to be cited at all in the text. Secondly, the caption refers to "3 eigenstates" whereas according to the legend, the actual number seems to be 10.

  Our reply: The referee is right and we have corrected the caption. Additionally, we cited this figure in the text (section 5.2).

• Ref. 2: The last sentence of the Discussion (chapter 6) did not make sense to me. Maybe this is because the term "Matrix Market" would need explanation.

  Our reply: The Matrix Market exchange format is a standard for files storing matrices (see https://en.wikipedia.org/wiki/Matrix_Market_exchange_formats ). We modified the corresponding sentence and added a footnote in order to make it clear that Matrix Market refers to this exchange file format.

• Ref. 2: In the "Numerical libraries", the authors specify version 3.8.2 of PETSc and SLEPc. However, a little further down in Appendix A.1, they use versions 3.7.7 and 3.7.4, respectively. This should be harmonized.

  Our reply: After proper testing, we now mention versions 3.8.2 as the reference PETSc and SLEPc versions in the appendix.

• Ref. 2: References need to be proof-read carefully. Proper names such as "Hamilton(ians)", "Schur", "Anderson", and "Chebyshev" should be in upper case (most of the time they are not), there is garbage in the initials of the second author of Ref. [19], and "XXZ" in the title of Ref. [19] should probably also be upper case.

  Our reply: We corrected the wrong capitalizations in the bibliography style.

• Ref. 2: In addition, I have one suggestion that probably goes beyond the scope of the present work. If single-precision arithmetics performs so well (section 4.4), one might ask if this could be combined with double-precision arithmetics in a post-processing step. In particular, since eigenvectors are computed, one could double-check if the off-diagonal matrix elements of H between states that cannot be separated in single precision are non-zero in double precision, and if necessary perform a rotation of the basis vectors to diagonalize H in this quasi-degenerate subspace. Exploration of such alternatives to the two heuristic approaches proposed at the end of page 14 might be an interesting topic for future work.

  Our reply: We thank the referee for this interesting suggestion. Following it, we have performed numerical experiments which appear to indicate that the knowledge of single-precision eigenvectors spanning the quasi-degenerate subspace is not sufficient to accurately reconstruct the double-precision non-zero off-diagonal elements, making it impossible to find the rotation yielding the correct eigenvectors. For that reason, to determine the correct eigenvectors from single-precision data, we still resort to the minimum of entropy method exposed in the article.

Reply to Referee 3

We thank the referee for his/her positive appreciation of our work. We reviewed the draft carefully for typos and corrected them.

Besides minor typos, all the changes correspond to precisions asked by the referees, in particular Ref. 2.
See above for a precise list of the changes made.