Real-space spectral simulation of quantum spin models: Application to generalized Kitaev models

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 Kitaev-Heisenberg 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 Lanczos-Chebyshev (HLC) method to compute spectral functions. We believe that the development of these methods is a significant contribution. In particular, while ED remains the go-to method for ground-state (T = 0) physics, we give compelling arguments in favour of FTCP for finite-temperature calculations.

2) “Only one application, the Kitaev-Heisenberg model”

We added two subsections, where we apply the newly developed FTCP and HLC approaches to the Kitaev-Ising (K-I) 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 K-I 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 easier-to-understand 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 state-of-the-art 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 low-temperature 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 low-temperature 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 high-temperature 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 Chebyshev-based 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 Kitaev-Heisenberg model in Phys. Rev. B 95, 024426 (2017). So, we applied a hybrid Lanczos-Chebyshev (HLC) method to recover results of this paper and discovered that our Chebyshev-based 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 Kitaev-Ising 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 low-temperature 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 Kitaev-Heisenberg model. This model is chosen to benchmark our Chebyshev-based methods. We do introduce some insight into the physics of the Kitaev-Ising 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 Kitaev-Heisenberg (Sec. 4.1) and Kitaev-Ising 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 Kitaev-Heisenberg 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 Kitaev-Ising 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 Lanczos-Chebyshev (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 matrix-vector multiplications involving the Hamiltonian “on-the-fly”, which is the most memory-efficient 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 TPQ-based methods is better control of accuracy, more flexibility and higher efficiency. Chebyshev methods are also generally applicable, sign-problem-free 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 bench-mark for two reasons. Firstly, in Phys. Rev. B 95, 024426 (2017), the authors show that longer-range 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 step-like behavior of the NN spin--spin 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 re-obtained in the context of the broader parameterization of the Kitaev-Heisenberg 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 larger-scale bench-mark data. We also applied our newly introduced HLC to compute the dynamical spin susceptibility of the Kitaev-Ising 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 Kitaev-Heisenberg 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 Lanczos-Chebyshev (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 go-to 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, memory-wise, the dominant contributions are O(2D) for the TPQ methods and O(3D) for the Lanczos and Chebyshev-based 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 Chebyshev-based methods can be advantageous for studies of: target energies away from the ground state and low-lying 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 Kitaev-Ising 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 Kitaev-Heisenberg 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 Kitaev-Heisenberg 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 resolution-independent, 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 Chebyshev-based 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, well-defined) 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, Chebyshev-based 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 low-temperature fluctuations and two-fold increased computer time compared with FTCP; c) computing spectral functions, in which case we considered a particular range of parameters in the Kitaev-Heisenberg model, performed 4000 independent simulations and found that on average, our newly introduced hybrid Lanczos-Chebyshev method was 33% faster than the continued fraction Lanczos approach. Lastly, we carried out a novel calculation of the dynamical spin susceptibility for the Kitaev-Ising 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 Chebyshev-based methods can be advantageous with respect to both Lanczos and TPQ-based 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.

- The whole paper has been re-written, 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 Kitaev-Heisenberg model in the revised version. Figures 3-7 of the new manuscript contain results that are analogous to those of figures 4-6 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 Kitaev-Heisenberg model.
- Figure 10 is a consistency check for our implementation of the methods for the Kitaev-Ising model. Figure 11 presents a comparison between two particularly relevant finite temperature methods in the context of the Kitaev-Ising model. Figure 12 displays novel dynamical signatures of the quantum phase transitions in the Kitaev-Ising 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.