Real-space spectral simulation of quantum spin models: Application to the Kitaev-Heisenberg model

1. This paper tries to show the numerical reliability of the Chebyshev Polynomial Green Function (CPGF) to investigate the ground-state properties of the Kitaev-Heisenberg model.

2. This paper shows that the CPGF method can detect the signature of the ground-state phase transition in the Kitaev-Heisenberg model with comparable accuracy as the results obtained by the conventional exact diagonalization method.

3. The paper also points out that the CPGF method can be used to access even the excited energy states, which are useful for studying the non-equilibrium systems.

1. Clear advantage is not clarified in using the CPGF method instead of the conventional Lanczos method to investigate the ground-state properties of the quantum frustrated magnets although the paper investigates the numerical reliability only in the ground-state of the Kitaev-Heisenberg model.

2. The paper recommends using the TPQ method to get the ground-state energy instead of the ED method. However, the clear advantage is not clarified to use the TPQ method in terms of the required memory cost, the needed number of (TPQ/Lanczos) iteration, and the number of initial random vectors.

3. The numerical results in Fig. 6, the left and right panel data should match in the end, but not so. This inconsistency easily violates the reliability of the CPGF method, but the paper does not explain the reason.

The current paper investigates mainly the reliability and usefulness of the Chebyshev Polynomial Green Function (CPGF) to study the ground-state properties of the Kitaev-Heisenberg model as an alternative numerical method to the quantum Monte Carlo (QMC) method and the exact diagonalization (ED) method. By careful comparison to numerical data by the conventional ED Lanczos method and the thermal pure quantum state (TPQ) method, some numerical reliabilities are clarified, and the computational costs in the CPGF method are also partly reported. However, the current manuscript does not seem to meet the acceptance criteria of the SciPost Physics because of the lack of the clear advantage in using the CPGF method instead of the ED method.

To improve the paper further, I would like to authors to comment on the following points.

1. Please clarify the computational memory costs in the CPGF, TPQ, and Lanczos methods to compute the energy and the correlation functions in the ground state. I think these methods require the almost same memory cost ($\sim$ 2 or 3 $D$). But the TPQ and the CPGF methods require few initial random vectors to get the same level of accuracy as the results obtained by the ED Lanczos method.

The authors may answer that only a few or 1 realization is enough to use the TPQ method, but that is the case in larger enough systems because the error scales as $\sim 1/\sqrt{D}$.
(Can we believe this scale also in the CPGF method?)

The Lanczos method does not need to take the average over the initial random vectors and does not require $\sim$ $10^4$ iteration (typically, $10^2$~$10^3$ in the Lanczos method).

From the viewpoint of the memory costs, the number of initial realizations, and the number of iteration, for me, the conventional Lanczos method is the best way among these three methods to investigate the ground state physics.

In the conclusion of this manuscript, the authors suggest a hybrid TPQ-CPGF approach. However, based on the above reasons, the only ED Lanczos approach seems to be better for the computational costs to study the ground-state physics.

And it is not also shown that the TPQ-CPGF approach is better than the ED-CPGF approach to study the microcanonical excited state physics.

2. To use the TPQ and CPGF methods, not only the lowest energy $E_{\rm min}$ but also the maximum energy $E_{\rm max}$ are needed in advance. The authors suggest using the ED or the TPQ method in getting the lowest energy state, but the general way to get the $E_{\rm max}$ is not written. Of course, in some simple cases, we can know the value easily. For example, in the simple antiferromagnet, we can use the energy of the fully polarized state as the $E_{\rm max}$. But in general, could we avoid using the Lanczos-like method to know the $E_{\rm max}$?

3. Please explain explicitly the inconsistency between the left and right panels in Fig. 6. These calculation results should match in the same parameter $\alpha$ in the large enough iteration. And please use the same scale in the vertical axis to make the comparison easier for readers.

4. The authors say "CPGF is a potential alternative to DMRG because it poses no restrictions on dimensionality and its accuracy can be precisely controlled by ensuring statistical convergence and by adjusting the spectral resolution." on page 13. However, I think that the current CPGF method is also suffered from the huge memory cost which increases exponentially as in the Lanczos and the TPQ methods. I don't think that the current CPGF method can be used in three-dimensional systems from this point.

5. When we have the degeneracy in the ground states or the excited states, can we use the CPGF method as a reliable method? There are often degenerated ground states and excited states in the quantum frustrated systems.

1-The authors provide a combined benchmark of the TPQ and CPGF approach for a specific frustrated quantum spin model.
2-The methods are applied here to a relevant and timely quantum spin model: the Kitaev-Heisenberg model.

1-No new physics results are reported.
2-The considered methods have been presented in detail in earlier publications.
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.
4-There are several minor deficits in the presentation, see list below.

The authors perform a systematic benchmark and comparison of the TPQ and CPGF method as applied to a specific frustrated quantum spin model, the Kitaev-Heisenberg model. This type of analysis is certainly interesting and timely, and the authors also address a timely and relevant frustrated quantum spin model.

I request several changes, as specified below, to be performed on the manuscript, before approving that the general acceptance criteria for SciPost Physics are met by this manuscript. Given that no new physics results are reported here and the considered methods have in fact been put forward (applied to different models) already in earlier publications, I don't find one of the expectations for a publication in SciPost Physics to be particularly met by this submission.

1-The sentence in the introduction, "The severity of the problem depends on the nature of the correlations in the model." is not correct: it is a well known fact and by now documented by several example cases, that the severeness of the sign-problem depends on the computational basis used in a QMC approach. The quoted statement thus needs to be corrected and augmented by appropriate references.
2-The abbreviation "STE" at the bottom of page 4 needs to be introduced. Also not defined at the point of first usage explicitly are $N_{rd.vec.}$, $D$.
3-It is not clear what is meant by the $xy-plane$ on page 5.
4-The meaning of the coefficients ${c_i}$ on page 6 is not clear.
5-Please provide the range $n=0,1,2,...$ when the Chebyshev polynomials are first defined (after eq. (5)).
6-The bond lines in Fig. 2 are hardly seen, the arrows are too long. In later figures, the boundary boxes are too faint and the font sizes too small.
7-At the end of the introduction to Sec. 4 please motivate, why the n.n. correlations will be considered only. What about longer ranged correlations?
8-Please reverse the order in the legend in Fig. 6, for easier readability.
9-While DMRG is discussed as an competing method, the authors should also mention other tensor-network-based approaches, such as iPEPS, where remarkable advances have been demonstrated recently in exploring the thermodynamics of frustrated 2D quantum spin systems.
10-Extend the current analysis to additional models, in particular such models, for which larger-scale benchmark data is available, i.e., for system sizes that extend beyond those accessible to ED.

1) rigorous consistency checks of their method
2) well-written and clear structure of the manuscript

1) Unclear, why this method is needed to compute ground state energies or ground state spin correlations
2) Only one application, the Kitaev-Heisenberg model
3) Too verbose in some parts
4) The method is already known
5) Comparison only to TPQ, no obvious difference

The authors apply a Chebychev Polynomial method to study spectral properties of the Kitaev-Heisenberg model. The manuscript is well written, and the need for proper numerical methods is motivated well. The method is based on previous works, which have applied it to study spectral functions. Here, it appears as if the authors are using the method to study static quantities in the microcanonical ensemble, which is a valid application. The method is benchmarked against the TPQ method, in its microcanonical form, and Exact Diagonalization. The Kitaev-Heisenberg model is taken as a benchmark problem for the method. No new insight into its physics is added. There are several issues, that need to be addressed before the manscript can be considered for publication.

1) The method is only applied to show consistency with ground state Exact Diagonalization. While this is a good consistency check, this can clearly not be the purpose of this method. We see how ground state energy and ground state spin-spin correlations agree in Figs. 3 and 4. However, computing ground state expectation values are clearly better done using ED alone. When the authors go to N_{poly} = 40000 to achieve convergence of the spectral resolution, this is around 200 times more expensive than a simple ground state computation. The method would need to be compared at finite energies, where it is a valid application. The paper cannot be published if it only shows consistency checks at T=0.

2) The use of the microcanonical ensemble is poorly motivated. It would be good to explain, where the advantages lie in this approach and compare this to the canonical TPQ (Sugiura, Shimizu, Phys. Rev. Lett. 111, 010401 (2013)) or the finite-temperature Lanczos method (see e.g. Jaklic, Prelovsek, Phys. Rev. B 49, 5065(R) (1994))

3) A well-known method to study spectral properties in the microcanonical ensemble is the microcanonical Lanczos algorithm, see e.g. Long, Prelovšek, Shawish, Karadamoglou, Zotos, Phys. Rev. B 68, 235106 (2003) or https://www.cond-mat.de/events/correl17/manuscripts/prelovsek.pdf. A comparison to this method would greatly improve the manuscript.

4) Static observables in the microcanonical ensemble are somewhat interesting, but since previous studies have already applied this method to spectral functions, it would be good, if the authors could also attempt this. This has already been done in the paper that is cited by the authors: Braun, Schmitteckert, Phys. Rev. B 90, 165112 (2014), but applying it to the Kitaev Heisenberg model would be a novel step.

5) It is not clear from their manuscript whether the Chebychev approach is advantageous to TPQ. The authors need to compare results at finite energy, where Exact Diagonalization with e.g. a simple Lanczos algorithm is not feasible anymore.

6) Since the Heisenberg-Kitaev model is only used as a benchmark, the description of its physics can be shortened. After all, there are no new insights added to the understanding of this model.

7) Since this paper is benchmarking the Chebychev method, it would be advisable to not just have one model it is applied to, but rather two or three.

While both the method and the application are interesting, the above-stated problems hinder me from recommending this manuscript for publication in its present form. However, upon addressing these major points, the manuscript might be suitable for publication in SciPost.