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Realspace spectral simulation of quantum spin models: Application to generalized Kitaev models
by Francisco M. O. Brito, Aires Ferreira
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Submission summary
Authors (as registered SciPost users):  Francisco Brito 
Submission information  

Preprint Link:  https://arxiv.org/abs/2110.01494v2 (pdf) 
Date submitted:  20231115 13:25 
Submitted by:  Brito, Francisco 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
The proliferation of quantum fluctuations and longrange entanglement presents an outstanding challenge for the numerical simulation of interacting spin systems with exotic ground states. Here, we present a toolset of Chebyshev polynomialbased iterative methods that provides a unified framework to study the thermodynamical properties, critical behavior and dynamics of frustrated quantum spin models with controlled accuracy. Similar to previous applications of the Chebyshev spectral methods to condensed matter systems, the algorithmic complexity scales linearly with the Hilbert space dimension and the Chebyshev truncation order. Using this approach, we study two paradigmatic quantum spin models on the honeycomb lattice: the KitaevHeisenberg (KH) and the KitaevIsing (KI) models. We start by applying the Chebyshev toolset to compute nearestneighbor spin correlations, specific heat and entropy of the KH model on a 24spin cluster. Our results are benchmarked against exact diagonalization and a popular iterative method based on thermal pure quantum states. The transitions between a variety of magnetic phases, namely ferromagnetic, N\'eel, zigzag and stripy antiferromagnetic and quantum spin liquid phases are obtained accurately and efficiently. We also accurately obtain the temperature dependence of the spin correlations, over more than three decades in temperature, by means of a finite temperature Chebyshev polynomial method introduced here. Finally, we report novel dynamical signatures of the quantum phase transitions in the KI model. Our findings suggest that the efficiency, versatility and lowtemperature stability of the Chebyshev framework developed here could pave the way for previously unattainable studies of quantum spin models in two dimensions.
Author comments upon resubmission
Expert Referee Report 1
We kindly thank the referee for carefully reading our manuscript, pointing out its strengths and shortcomings, and providing valuable suggestions for improvement. Please note that in the original submission, the convergence speed and performance of CPGF were severely underestimated due to a mistake in the input of the spectrum bounds. This has now been corrected, leading to a major boost in computational performance. We apologise for this mistake. We also now consider a broader parameterization of the KitaevHeisenberg model as employed by Chaloupka et al. Phys. Rev. Lett. 110(9), 097204 (2013). We are confident that we addressed the 5 mentioned weaknesses in this resubmission.
First, we reply to the general points made by the Referee (numbered and in quotes):
1) “Unclear, why this method is needed to compute ground state energies or ground state spin correlations”
We agree with the referee: CPGF is not needed to compute ground state energies or spin correlations. The advantage of the CPGF method is in probing arbitrary target energies that need not be close to the ground state (i.e. target energies beyond low lying excitations that are accessible using the standard Lanczos method). Notice that in this resubmission, we also compare CPGF with the microcanonical Lanczos method (MCLM), which is also capable of probing arbitrary target energies. We illustrate CPGF by considering the ground state energy as the target so that we can benchmark it against Lanczos exact diagonalization (ED) and the microcanonical variant of TPQ (which we refer to as MTPQ, as opposed to the canonical variant, referred to as CTPQ). Please note that the MTPQ recursion achieves its maximum accuracy as the ground state is approached, so the latter is a natural target energy to compare all methods. As noted by the referee, another reason to consider the ground state is implementation validation. Since the ground state Lanczos results are well established, matching between CPGF and ED validates our implementation, serving as a consistency check. Notwithstanding, our resubmission goes beyond a simple consistency check, by showing that CPGF compares favourably with both MCLM and MTPQ. Another important remark is that on this resubmission we also introduce two methods that were not included in the original submission: the finite temperature Chebyshev polynomial (FTCP) method for the evaluation of canonical expectations and a hybrid LanczosChebyshev (HLC) method to compute spectral functions. We believe that the development of these methods is a significant contribution. In particular, while ED remains the goto method for groundstate (T = 0) physics, we give compelling arguments in favour of FTCP for finitetemperature calculations.
2) “Only one application, the KitaevHeisenberg model”
We added two subsections, where we apply the newly developed FTCP and HLC approaches to the KitaevIsing (KI) model. Specifically, we use FTCP to study the temperature dependence of relevant quantities, such as the specific heat and the entropy density. We use HLC to study the dynamical spin susceptibility of the KI model. Lanczos ED results for this model are also reproduced in order to validate our implementation.
3) “Too verbose in some parts”
The revised manuscript uses a more concise and easiertounderstand writing style. Following the referee’s suggestion, section 4 (“Applications”) refrains from discussing the physics of the models and focuses solely on the methods and their advantages and challenges. The exception is Sec. 4.2.3 that introduces new results, for which we provide a physical interpretation.
4) “The method is already known” and 5) “Comparison only to TPQ, no obvious difference”
Even though CPGF was already known prior to this submission, it had never been applied to interacting quantum spin systems. We believe that this resubmission makes it clearer that it has interesting advantages with respect to stateoftheart methods for strongly correlated systems, namely MCLM and TPQ. On this revised version of the manuscript, we also introduce two methods that we developed while working on the points previously made by the referees. Both of these methods show clear strengths. On the one hand, FTCP is found to be two times faster than microcanonical TPQ (MTPQ) and is immune to both the numerical instabilities of canonical TPQ (CTPQ) and the lowtemperature fluctuations that plague FTLM. On the other hand, the newly developed HLC method for spectral functions is more flexible than the continued fraction Lanczos approach and is shown to be 33% faster on average in our extensive tests (about 4000 independent simulations). In this resubmission, we extend the comparison far beyond just the microcanonical variant of TPQ. We now compare our methods to the microcanonical and canonical variants of TPQ, MCLM, the Finite Temperature Lanczos method (FTLM) and the Lanczos method using the continued fraction approach to compute spectral functions.
Regarding the specific issues brought up by the referee, we believe that we have resolved every item:
Items 1) and 2)
We added Sections 4.1. 2 and 4.2.2., where we show that FTCP compares favourably with the other methods. Its memory cost is similar to FTLM, but it avoids lowtemperature statistical fluctuations. While MTPQ also avoids these fluctuations while having 50% lower memory cost, FTCP uses half the computer time of MTPQ thanks to its better convergence properties and the accuracy control it offers. Unfortunately, CTPQ has numerical stability issues at the very low temperatures that are of particular interest in these systems. For these systems, the canonical ensemble is typically more useful. Following the referee’s suggestion, we now include comparisons with the canonical TPQ and finite temperature Lanczos methods. We find that the advantage of the microcanonical ensemble is mainly in MTPQ. Since this method is able to estimate the actual temperature corresponding to each iteration, one can actually compute the temperature dependence of relevant quantities using MTPQ. Unlike CTPQ, MTPQ does not suffer from numerical instabilities at low temperatures. However, this comes at a high computational cost. Moreover, the accuracy increases as more iterations are completed, which in practice means that MTPQ’s hightemperature results are unavoidably less reliable. Our newly introduced method (FTCP) solves both of these problems.
Item 3)
We compare CPGF with MCLM, and find that CPGF is more efficient on average. Moreover, the resolution is given as an input to CPGF. The scaling of the number of polynomials needed for convergence with resolution is known a priori in CPGF. We confirm this expected behaviour in Fig. 14. The advantage of the added control over resolution in CPGF is that one can predict how many iterations will be required to reach a certain accuracy. This is not known a priori in MCLM.
Item 4)
We agree with the referee on this point. Applying Chebyshevbased methods to spectral functions has proven fruitful in the past and they can certainly be used for this class of systems. We found that spectral functions had already been computed with Lanczos for the KitaevHeisenberg model in Phys. Rev. B 95, 024426 (2017). So, we applied a hybrid LanczosChebyshev (HLC) method to recover results of this paper and discovered that our Chebyshevbased method has some advantages in terms of efficiency and flexibility. Additionally, we took another suggestion of the referee into account and applied this method to another model. In Sec. 4.2.2., we study the dynamical spin susceptibility of the KitaevIsing model for the first time to our knowledge and find dynamical signatures of the quantum phase transitions first described in Phys. Rev. Lett. 118, 137203 (2017).
Item 5)
We find that FTCP is advantageous with respect to MTPQ and CTPQ. Similarly to MTPQ, it is immune to the lowtemperature numerical instability of CTPQ. Yet, it is 2 times faster than MTPQ and has better control over the accuracy of the results. This is because the accuracy of MTPQ improves for low temperatures, while in FTCP the accuracy depends only on the number of polynomials used on the Chebyshev expansion at each inverse temperature step (see Eq. (35)).
Item 6)
We agree that our work does not produce new insights into the physics of the KitaevHeisenberg model. This model is chosen to benchmark our Chebyshevbased methods. We do introduce some insight into the physics of the KitaevIsing model, in the revised version. These models are well described, respectively, in Phys. Rev. B 95, 024426 (2017) and Phys. Rev. Lett. 118, 137203 (2017). So, we keep our descriptions to a minimum and only discuss these models to explain the conventions we used to parametrize energy scales.
Item 7)
We now apply the Chebyshev methods to the KitaevHeisenberg (Sec. 4.1) and KitaevIsing models (Sec. 4.2).
Expert Referee Report 2
We kindly thank the referee for carefully reading our manuscript, pointing out its strengths and shortcomings, and providing valuable suggestions for improvement. Please note that in the original submission, the convergence speed and performance of CPGF were severely underestimated due to a mistake in the input of the spectrum bounds. This has now been corrected, leading to a major boost in computational performance. We apologise for this mistake. We also now consider a broader parameterization of the KitaevHeisenberg model as employed by Chaloupka et al. Phys. Rev. Lett. 110(9), 097204 (2013).
First, we reply to the general points made by the Referee (numbered and in quotes):
1) “No new physics results are reported.”
In this resubmission, we report new results, i.e. dynamical signatures of the quantum phase transitions in the KitaevIsing model (see Phys. Rev. Lett. 118, 137203 (2017)).
2) “The considered methods have been presented in detail in earlier publications.”
We introduce two novel methods that were not included in the original submission: the finite temperature Chebyshev polynomial (FTCP) method for computations of canonical expectations and a hybrid LanczosChebyshev (HLC) method to compute spectral functions. We believe that the introduction of these novel methods is a significant contribution.
3) “No comparison is performed for models for which, e.g., QMC or iPEPS results are available, which would allow the authors to test the proposed method for larger systems sizes than those available to ED.”
In our current implementation, these Chebyshev methods cannot reach the system sizes of QMC or tensor networks. The limitation is the relatively high memory cost of these Chebyshev methods, which is approximately the same as Lanczos ED applied to the ground state. Both Chebyshev methods and “ground state” Lanczos ED allow larger systems than full exact diagonalization, i.e. the brute force approach, where all eigenvalues and eigenvectors are obtained. This is because their memory requirements are much smaller than full ED. Lanczos, TPQ and Chebyshev all share the following feature. When carrying out matrixvector multiplications involving the Hamiltonian “onthefly”, which is the most memoryefficient technique that is currently available, one still has to store a few vectors of the size of the Hilbert space. In this regard, QMC has a clear advantage because it relies on the concept of importance sampling to probe only the relevant part of the Hilbert space, and thus bypasses the need to store such large vectors in memory. The advantage of the Chebyshev methods with respect to Lanczos or TPQbased methods is better control of accuracy, more flexibility and higher efficiency. Chebyshev methods are also generally applicable, signproblemfree and work equally well, irrespective of the number of spatial dimensions, model complexity and type of boundary conditions.
4) “There are several minor deficits in the presentation, see list below.”
We believe that we have addressed all the minor deficits in our presentation, as detailed below.
We went through the list of requested changes and included them in our resubmission. Below, we point the referee to the location of these changes in the manuscript:
1) We correct the statement about QMC in page 3: “ The severity of the problem depends on the computational basis used to tackle the specific model” and provided more references about this topic: Annual Review of Condensed Matter Physics 10 (1), 337 (2019), Phys. Rev. Lett. 126(21), 216401 (2021), Phys. Rev. B 105 (16), 165124 (2022), Phys. Rev. E 99 (3),033306 (2019) and Phys. Rev. B 105(19), 195130 (2022).
2) We introduce the abbreviation “STE” in page 5: “This technique, dubbed stochastic trace evaluation (STE), is ubiquitous in the study of condensed phases and is used in ED methods…”; we also define the other quantities, respectively in “The rationale in the STE is to approximate the trace of an operator by an average of expectation values using $N_\mathrm{rd.vec.}$ random vectors” and “The relative error scales favorably with the Hilbert space dimension, $D$”.
3) We now clarify the meaning of these variables on footnote 3.
4) The coefficients are now defined on footnote 2.
5) We now define the range of the Chebyshev polynomials in page 10, after the sentence “As customary, we work with Chebyshev polynomials of the first kind…”
6) We improved the readability of the figures throughout the manuscript.
7) We added a sentence, motivating the use of n.n. correlations in page 19: “The NN spin–spin correlation is used for our benchmark for two reasons. Firstly, in Phys. Rev. B 95, 024426 (2017), the authors show that longerrange correlations vanish in the vicinity of the spin liquid phases. Given that we are particularly interested in this region of the phase diagram, it is reasonable to focus on NN correlations. Secondly, the steplike behavior of the NN spinspin correlation coincides with quantum critical points (see Phys. Rev. B 95, 024426 (2017) ). Moreover, the behavior of the NN correlation with temperature is intimately connected to peaks in the specific heat (see Phys. Rev. Research 2 (4), 043015 (2020) ) — that we also compute using Lanczos, TPQ and Chebyshev  and that are particularly relevant experimentally (see Phys. Rev. B 99(9),094415 (2019) ).”
8) The data shown in Fig. 6 of the original manuscript was reobtained in the context of the broader parameterization of the KitaevHeisenberg model in Phys. Rev. Lett. 110(9), 097204 (2013). In the new revised manuscript the analogous comparisons are contained in figure 5 and 7. In these figures, we improved the readability by having the same scales on the left and right panels and by using the same colour scheme for the range of parameters of the model where the methods are being compared.
9) We mention Phys. Rev. Research 2 (4), 043015 (2020), where the authors use an exponential tensor network approach to study the Kitaev model, and show that our results match those of this approach in Sec. 4.1.2. iPEPS is also mentioned in page 3, with some extra references added.
10) As mentioned above, unfortunately the current (also the first) implementation of our methods to quantum spin models do not go beyond the system size limitations of ED because of their similar memory cost. We compared our results for the Kitaev model with Phys. Rev. Research 2 (4), 043015 (2020) and confirmed that they agree with their largerscale benchmark data. We also applied our newly introduced HLC to compute the dynamical spin susceptibility of the KitaevIsing model and put forward new results concerning these dynamical signatures of the quantum phase transitions in this model.
Expert Referee Report 3
We kindly thank the referee for carefully reading our manuscript, pointing out its strengths and shortcomings and requesting the improvement of important points that improved our manuscript considerably. Please note that in the original submission, the convergence speed and performance of CPGF were severely underestimated due to a mistake in the input of the spectrum bounds. This has now been corrected, leading to a major boost in computational performance. We apologise for this mistake. We also now consider a broader parameterization of the KitaevHeisenberg model as employed by Chaloupka et al. Phys. Rev. Lett. 110(9), 097204 (2013).
We are confident that we satisfactorily addressed all the 5 points mentioned by the referee in this resubmission:
1) In this resubmission, we introduce two methods that were not included in the original submission: the Finite Temperature Chebyshev Polynomial (FTCP) method and the hybrid LanczosChebyshev (HLC) method to compute spectral functions. We believe that the introduction of these novel methods is a significant contribution. In particular, while Lanczos remains the goto method for the ground state, we give compelling arguments as to why it might be preferable to use FTCP when one desires to go beyond T=0.
The computational costs are indeed those mentioned by the referee. Specifically, memorywise, the dominant contributions are O(2D) for the TPQ methods and O(3D) for the Lanczos and Chebyshevbased methods. So, at the first sight, all other factors being the same, TPQ may seem preferable to go beyond T=0. However, in this updated version of our paper, we show that this is not actually the case. While the memory cost of the FTCP is about 50% higher than TPQ (that is, O(3D) instead of O(2D)), FTCP is 2 times faster than TPQ in terms of computer time. We attribute this to the better convergence properties of the Chebyshev expansion and the modest number of operations per iteration that is required in Chebyshev. In conclusion, while we agree with the referee in saying that Lanczos is superior for T=0, in this updated version of the manuscript, we show that Chebyshevbased methods can be advantageous for studies of: target energies away from the ground state and lowlying excitations, temperature dependence of experimentally relevant quantities such as spin correlations, specific heat and entropy, and dynamics, for example the dynamical spin susceptibility, studied in Sec. 4.2.3. In this last section, we present novel results, showing dynamical signatures of the quantum phase transitions in the KitaevIsing model (see Phys. Rev. Lett. 118, 137203 (2017)).
2) In general, the most reliable method to obtain the energy bounds is Lanczos. For the specific case of the KitaevHeisenberg model, we noticed that a useful symmetry allows one to avoid the computation of the maximum energy (see Sec. 4.1.1.). This is because there is a mapping between minimum and maximum energies for different values of the parameter of the KitaevHeisenberg model.
3) We improved the readability of the figures overall. Namely, we used the same scale in all of these comparisons (see figures 5 and 7). The inconsistency that was pointed out by the referee was due to the fact that the original CPGF simulations were not fully converged, i.e. a yet finer resolution was needed. In the revised version, we show that the MTPQ and CPGF results do match (see right panels of figures 5 and 7). In order to ensure that CPGF was fully converged, we systematically considered finer resolutions until the results were resolutionindependent, whilst increasing the number of iterations accordingly. Then, we compared the coarsest resolution CPGF result that had already converged with TPQ.
4) We realised that this sentence was misleading. Indeed, like the referee pointed out, CPGF has similar memory cost to Lanczos and TPQ. What we meant was that CPGF and the other Chebyshev methods do not require any specific boundary conditions as the dimensionality is varied. For example, in the case considered in our work, we considered a 2D hexagonal cluster with periodic boundary conditions. DMRG typically considers open boundary conditions along at least one of the spatial dimensions. We changed the sentence in question to: “The Chebyshevbased methods used throughout this paper are a potential alternative to DMRG because, unlike the latter, they pose no restrictions on boundary conditions and their accuracy can be precisely controlled by ensuring statistical convergence and, in the case of CPGF, by adjusting the spectral resolution.” (page 33)
5) CPGF considers target energies contained within a (uniform, welldefined) resolution. Thus, the state that results from the iterative process will always be a linear combination of the degenerate states in question. This applies to Lanczos and TPQ as well. Unfortunately, Chebyshevbased methods do not have any obvious advantages over Lanczos and TPQ as far as degeneracies are concerned.
In conclusion, we believe that in the revised manuscript, we have shown that while Lanczos ED remains the preferable method for ground state studies, Chebyshev methods can be advantageous for other applications, namely: a) studying target energies, in which case we found CPGF to be more efficient than the micro canonical Lanczos method (MCLM) and TPQ; b) finite temperature studies, where the finite temperature Lanczos method (FTLM) and TPQ have shortcomings, respectively, large lowtemperature fluctuations and twofold increased computer time compared with FTCP; c) computing spectral functions, in which case we considered a particular range of parameters in the KitaevHeisenberg model, performed 4000 independent simulations and found that on average, our newly introduced hybrid LanczosChebyshev method was 33% faster than the continued fraction Lanczos approach. Lastly, we carried out a novel calculation of the dynamical spin susceptibility for the KitaevIsing model and found dynamical signatures of the quantum phase transitions in this model. We provided a brief physical interpretation for these results. All summed up, we believe that Chebyshevbased methods can be advantageous with respect to both Lanczos and TPQbased methods, except to obtain the ground state, in which case Lanczos is the best method. In particular, FTCP has a clear advantage over the state of the art method for finite temperature studies (TPQ), achieving the same results 2 times faster, as shown in Sections 4. 1. 2. and 4. 2. 2., with only a minor increase in memory cost mentioned on the first point of this reply.
List of changes
 The whole paper has been rewritten, including the abstract. The title also had a minor change to reflect the inclusion of another model in our study. These changes were made in order to accommodate all the suggestions made by the referees. The underlying structure has been kept, but there are more subsections in Sec. 3, where the additional considered methods are discussed. Crucially, Sec. 4 suffered a major overhaul, with Sec. 4. 2. added. Also, Sec. 4.1.2 is completely new. Changes to Secs. 1 and 2 are relatively minor.
 In the revised manuscript, all figures are completely new, except for Figure 1 and Figure 13. This is because we used a broader parametrization of the KitaevHeisenberg model in the revised version. Figures 37 of the new manuscript contain results that are analogous to those of figures 46 of the old manuscript, with some additional material.
 Figures 8 and 9 show the results of an extensive comparison of finite temperature methods for the KitaevHeisenberg model.
 Figure 10 is a consistency check for our implementation of the methods for the KitaevIsing model. Figure 11 presents a comparison between two particularly relevant finite temperature methods in the context of the KitaevIsing model. Figure 12 displays novel dynamical signatures of the quantum phase transitions in the KitaevIsing model.
 Sec. 5 has been extensively updated to include the conclusions of our additional studies of finite temperature and dynamics.
 Appendix B has been added. This serves the purpose of illustrating important advantages of the CPGF method that were not apparent in the original manuscript.
Current status:
Reports on this Submission
Anonymous Report 2 on 20231215 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.01494v2, delivered 20231215, doi: 10.21468/SciPost.Report.8283
Strengths
1. This paper introduced several wellknown nonbiased numerical (EDlike) methods in pedagogical manner that have the potential to surpass the Lanczos method to investigate excited states and finitetemperature physics.
2. This paper developed three methods (CPGF/FTCP/HLC methods) based on Chebyshev technique which can be superior to the previous methods in some aspects (such as in computational time and in accuracy).
3. Using HLC method, the paper found the signatures of quantum phase transition in a KitaevIsing model via spin dynamics.
Weaknesses
1. This paper did not check the performance of the CPGF method for excited states even though this method was developed for excited state as an alternative method to the traditional Lanczos method.
2. There may be inconsistency between the computational results of specific heat and entropy obtained by CTPQ method.
3. No explicit reason was explained why the FTCP method can avoid convergence problem at lower temperature regime.
Report
First of all, I would like to express a respect for the author's two years of effort. Compared to their 1st paper from two years ago, this paper begins with an pedagogical introduction to commonly used nonbiased methods in this field. This paper also then provides a wealth of numerical data demonstrating the merits of improvements based on the Chebyshev method. Additionally, the paper successfully calculates spin dynamics using the HLC method and captures the signatures of quantum phase transitions in a Kitaev model. While there may not be groundbreaking discoveries in physics, I believe that the series of methods presented in this paper hold value for publication as they have potential applications to other models, making it a worthwhile methodological contribution.
Requested changes
1. It would be better to write the reason why the FTCP method can avoid the problem to evaluate the specific heat (in figure 9) in contrast to the CTPQ method. I guess, the reason is because you use \tilde{h} in stead of h.
2. It would be better to cite some references for your FTCP method. In your FTCP method, you use Chebyshev polynomial expansion instead of Taylor expansion to express \exp{\beta/2 h} in the CTPQ method. There are several references which already used Chebyshev polynomial expansion in the CTPQ method such as, https://arxiv.org/abs/1503.06111v1, https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.220601. Of course, these papers didn't show the method details and benchmark as you showed. But it would be nice to cite their papers, and if there were difference in their methods and your method, you can also write the difference in your paper.
3. Could you explain why you chose the ground state to investigate the benchmark of the CPGF method (in Fig.6)?
I think for the ground state, the traditional Lanczos method is the best. I also think that some readers are interested in if the FCPT method is better than the MCLM method also to see the excited states.
Anonymous Report 1 on 20231211 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2110.01494v2, delivered 20231211, doi: 10.21468/SciPost.Report.8235
Report
The authors provide a substantially revised, extended and improved version of their manuscript. In particular, they now introduce two new methods, the finite temperature Chebyshev polynomial (FTCP) method for computations of canonical expectation values and a hybrid LanczosChebyshev (HLC) method to compute spectral functions. The authors provide a detailed analysis of the performance of the various methods and highlight subtle advantages. I find some of their discussion, such as the energycutoffdependence of the accuracy of the MTPQ particularly interesting. The authors also extended substantially on the physics side, and in particular now provide some dynamical data for the KI model. Unfortunately, they do not go beyond the system with 24 sites. Nevertheless, I think that the paper may now be considered appropriate for SciPost Physics, but I would still request a few minor changes, as given below.
Requested changes
1 Address the limitations on system size more explicitly: which new steps could be taken to go well beyond those feasible thus far with your methods?
2 Explain the meaning of the white space in Fig. 12 or adapt the the color bar, in case that white means 0.
3 I do not find evidence in the paper regarding the second part of the statement "While Lanczos I found...", the last sentence of the first paragraph in Sec. 5. If you want to keep this statement, please provide explicit data for excited state targeting.
4 The list of references regarding the basisdependence of the sign in QMC is incomplete. Cite also: T. Nakamura, Phys. Rev. B 57, R3197 (1998); A. Honecker, S. Wessel, R. Kerkdyk, T. Pruschke, F. Mila, and B. Normand, Phys. Rev. B 93, 054408 (2016); F. Alet, K. Damle, and S. Pujari, Phys. Rev. Lett. 117, 197203 (2016); L. Weber, A. Honecker, B. Normand, P. Corboz, F. Mila, and S. Wessel, SciPost Phys. 12, 54 (2022).