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Shiftinvert diagonalization of large manybody localizing spin chains
by Francesca Pietracaprina, Nicolas Macé, David J. Luitz, Fabien Alet
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Submission summary
As Contributors:  Fabien Alet · David J. Luitz · Nicolas Macé · Francesca Pietracaprina 
Arxiv Link:  https://arxiv.org/abs/1803.05395v2 (pdf) 
Date submitted:  20180614 02:00 
Submitted by:  Alet, Fabien 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
We provide a pedagogical review on the calculation of highly excited eigenstates of disordered interacting quantum systems which can undergo a manybody localization (MBL) transition, using shiftinvert exact diagonalization. We also provide an example code at https://bitbucket.org/dluitz/sinvert_mbl/. Through a detailed analysis of the simulational parameters of the random field Heisenberg spin chain, we provide a practical guide on how to perform efficient computations. We present data for midspectrum eigenstates of spin chains of sizes up to $L=26$. This work is also geared towards readers with interest in efficiency of parallel sparse linear algebra techniques that will find a challenging application in the MBL problem.
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Anonymous Report 3 on 2018823 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1803.05395v2, delivered 20180823, doi: 10.21468/SciPost.Report.562
Strengths
1 nice pedagogical review for the shiftinevert method for the exact diagonalization
2 detailed examination of the efficiency of the shiftinvert method
Weaknesses
no weakness
Report
In this paper, the authors explain the efficient shiftinvert technique to calculate the excited states for the random Heisenberg model.
They examine the efficiency of the shiftinvert computations by using the several different solvers for linear equations. They also apply the method for large MBL systems and obtain insight in the MBL problems.
The example code and the detailed examination of the
shiftinvert method are useful for the researchers in
the field of the condensed matter physics and the computational science.
Thus, I recommend the publication of the paper
in the SciPost Physics.
Requested changes
1 several typos in the manuscript. e.g.,
page. 3 denotes the strenght > denotes the strength
Anonymous Report 2 on 2018815 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1803.05395v2, delivered 20180815, doi: 10.21468/SciPost.Report.556
Strengths
1 Technical details on the shiftinvert method.
2 Benchmarks of the shiftinvert method applied to the manybody localization problem.
3 Some new data for bigger systems than published previously.
Weaknesses
1 No new physics.
2 Not always as pedagogical as intended.
3 A number of minor oversights in the presentation.
Report
The manybody localization problem, i.e., the phenomenon of a transition to localization in highly excited states of a quantum manybody system, has attracted a significant amount of interest over the past decade. The
present authors have made important contributions to the field using in particular "exact diagonalization". An important challenge in the manybody problem is the exponential growth of the Hilbert space dimension with the number of sites $L$ under consideration. For a spin1/2 model with more than $L=18$ sites, the authors advocate the use of the "shiftinvert" method in order to compute highly excited states for up to $L=26$ spins 1/2.
In the present manuscript, the authors give technical details on the application of the shiftinvert method to the manybody localization problem and provide benchmarks. They also present some results on bigger systems than studied previously, specifically $L=24$ (section 5.1) and even
some partial results for $L=25$ and $26$ (section 5.2). However, the authors postpone a detailed analysis of these new results to future work (see, e.g., first paragraph of the Discussion, chapter 6). Consequently, there is no real new physics in the present paper. Rather, the present
manuscript is intended to be a "pedagogical review" (first line of abstract).
While such a technical background is interesting and deserves publication in principle, I feel that the manuscript requires a bit more work. In particular, the authors do not always live up to their pedagogical ambitions. Some concrete comments and suggestions for improving the manuscript are contained in the list below under "Requested changes" (order corresponding to order of appearance in the manuscript, by no means to importance).
In addition, I have one suggestion that probably goes beyond the scope of the present work. If singleprecision arithmetics performs so well (section 4.4), one might ask if this could be combined with doubleprecision arithmetics in a postprocessing step. In particular, since eigenvectors
are computed, one could doublecheck if the offdiagonal matrix elements of ${\sf H}$ between states that cannot be separated in single precision are nonzero in double precision, and if necessary perform a rotation of the basis vectors to diagonalize ${\sf H}$ in this quasidegenerate 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.
Requested changes
1 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.
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 $h_j=0$, 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 manybody localization problem.
3 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.
4 The comment on the JordanWigner 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 nonspecialist reader. Note
also that it may be useful to state that the "spinless fermions" are interacting.
5 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. [4247] also in section 3.1.
6 A typesetting detail: why does footnote 3 not appear before page 10 ?
7 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.
8 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).
9 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.
10 Figure 5: Black font on a dark blue background is difficult to read. Maybe the authors should use a white font for "16".
11 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.
12 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.
13 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.
14 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.
15 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.
16 References need to be proofread 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.
Anonymous Report 1 on 201885 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1803.05395v2, delivered 20180805, doi: 10.21468/SciPost.Report.549
Strengths
1 Very accessible introduction to state of the art exact diagonalization numerics
for the fielddisordered Heisenberg chain.
2 This paper explains in detail how the impressive simulations contained in previous publications by some of the authors have been achieved, and enables a broader range of scientist to use these methods for similar problems, disordered or not.
3 Link to code is contained in the manuscript
Weaknesses
no particular weaknesses
Report
This manuscript explains in detail how the impressive simulations contained in previous publications by some of the authors have been achieved, and also pushes them to the currently possible limit of L=24 to L=26 spins depending on the precision. The reader is able to understand what libraries have been used, what has been tested and what the authors recommend to tackle these largescale diagonalizations, which constitute the current state of the art in unbiased numerics for the MBL problem.
Requested changes
1 On page 4 a sector Sz=1 is mentioned. I believe this should be the Sz=1/2 sector for odd length chains
2 on page 8 there is a type "bot" > "not"
3 on page 13, Fig. 5 suffers from an impaired font on my screen and in print.