# Thermodynamic limit and boundary energy of the spin-1 Heisenberg chain with non-diagonal boundary fields

### Submission summary

 As Contributors: Xiaotian Xu Preprint link: scipost_202106_00001v2 Date submitted: 2021-08-12 20:47 Submitted by: Xu, Xiaotian Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Mathematical Physics Approach: Theoretical

### Abstract

The thermodynamic limit and boundary energy of the isotropic spin-1 Heisenberg chain with non-diagonal boundary fields are studied. The finite size scaling properties of the inhomogeneous term in the T − Q relation at the ground state are analyzed. Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system. These results can be generalized to the SU(2) symmetric high spin Heisenberg model directly.

###### Current status:
Has been resubmitted

Dear Editor,
Thank you very much for your help. We have revised the manuscript (Ref. No. scipost_202106_00001v1) according to the referees' suggestions. Now we are resubmitting our paper. We think that this paper now meets the requirement of SciPost Physics.
Yours Sincerely,
Xiaotian Xu

### List of changes

We have revised the manuscript according to the referees' suggestions, and list the revisions as follows. The page numbers and equation numbers refer to revised version, unless specify.
1. The word "survives" is replaced with "exists" after Eq.(1) in line 28 of P. 2.
2. In line 33 of P. 2, the sentence "the Hamiltonian (1) degenerates into a projector operator" is replaced with "the Hamiltonian (1) degenerates into a projector operator that is in fact the projection onto the sum of the spin-0 and spin-1 subspaces".
3. The word "Antisymmetry" is changed into "Fusion condition" in Eq.(6) in P. 3.
4. We give the most general off-diagonal boundary matrix $K^-(u)$ in (7)-(8) in P. 3 and P.4, which involves three boundary parameters $p_-,\,\alpha_-,\,\phi_-$. The most general off-diagonal boundary matrix $K^+(u)$ is also given in (10) of P. 4, which has three boundary parameters $p_+,\,\alpha_+,\,\phi_+$.
5. The word "generalized" is replaced with "generated" in line 110 in P. 4.
6. We have corrected the misprint of the definition of the model Hamiltonian (15) in P. 5 and give its expression corresponding to the most general off-diagonal boundary $K$-matrices (7)-(8) and (10).
7. The words "hierarchy fusion" are replaced with "fusion hierarchy" in line 115 in P. 5.
8. We have added some sentences "Throughout this paper...for the i-th and j-th ones." (lines 82-87 in P. 3) to introduce our convention," Some remarks are in order....for any finite $N$." (lines 122-128 in P. 6) to give some remarks, and " It should be emphasized that ....(see the follwing parts of the paper)." (lines 135-138 in P. 6 and P. 7).
9. We have added a footnote in P. 6.
10. The word "subfigure" is replaced with "inset" in line 193 and above line 195 in P. 9.
11. The sentence "Meanwhile, two holes $\lambda_1^h$ and $\lambda_2^h$ should be introduced" is replaced with "Note that two holes $\lambda_1^h$ and $\lambda_2^h$ are introduced" in line 230 in P. 11.
12. The words "spin long $z$-direction" are replaced with "spin along the $z$-direction" in line 248, line 253 and line 265 in P. 13.
13. The "$E_b$" is replaced with "$e_b$" in Figure 3 in P. 14.
14. We have changed "...(68) pand the red..." into "...(68) and the red..." in the 2-nd line of the caption of Figure 3 in P. 14.
15. We have added the Figure 4 and the DMRG analysis in lines 271-287 in P. 14.
16. We have rewritten the Conclusions in Section 5 in P. 15.
17. We have deleted some references and added three new ones [33]-[35].
Besides, we have corrected some typos, and words and sentences have also been slightly improved.

### Submission & Refereeing History

Resubmission scipost_202106_00001v4 on 17 November 2021
Resubmission scipost_202106_00001v3 on 14 October 2021

Resubmission scipost_202106_00001v2 on 12 August 2021
Submission scipost_202106_00001v1 on 2 June 2021

## Reports on this Submission

### Anonymous Report 3 on 2021-9-25 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202106_00001v2, delivered 2021-09-25, doi: 10.21468/SciPost.Report.3568

### Strengths

Some new interesting results.

### Weaknesses

The main analytic results are derived for boundary conditions different from those declared.

### Report

Dear Editor,
The revisited version of the manuscript contains some minor improvements. In particular, some clarification on the numerical (DMRG) results derived by the authors for the case of unparallel boundary conditions, see figure 4. The authors also give some explanation of what the reduced BE are, see their point 8 in the list of changes. Nevertheless, as also remarked by the second referee, they have done very little about the misleading statements and structure of the manuscript. A part the exceptions above remarked, they have mainly leave unattended my previous remarks so that my previous report is largely still actual for the current revision.
The three referees seem all to agree on the fact that the main results of the current manuscript are: a) the analytic computations under the string hypothesis of the boundary energy of the spin 1 open chain under PARALLEL boundary conditions. b) the numerical (DMRG) analysis up to large chains of this quantity for the UNPARALLEL boundary conditions. Then, the comparison of the “analytical/numerical” results in the large chain limit allows to reasonably argue that bulk and boundary energy of the spin 1 open chain should be independent from the specific form of the integrable boundary conditions.
These are interesting and sounding results and this should be what I would expect to understand reading the abstract and all along the manuscript but this is not the case. Instead, to arrive to this conclusion, I have done a careful disentanglement between manuscript statements and analysis there presented and I have the impression that this holds true for the other referees, too. It is enough to look to the abstract:
“The finite size scaling properties of the inhomogeneous term in the T-Q relation at the ground state are analyzed.” This is done only numerically, e.g. by DMRG.
“Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system.” This is done analytically by string hypothesis only for the PARALLEL case. Then, the numerical analysis allows to argue that the same result should hold in the thermodynamic limit also for the UNPARALLEL case.
Let me also cite directly, the opening sentence of section 4:
“In this section, we study the physical effects induced by the unparallel boundary magnetic fields and compute the boundary energy”
Once again, they numerically (DMRG) study the physical effects induced by the UNPARALLEL boundary magnetic fields while they compute analytically by string hypothesis the boundary energy of the PARALLEL case.
As well as the sentence in the conclusion:
“The method provided in this paper can be used to study the thermodynamic properties of other quantum integrable models associated with rational R-matrix.”
It is not clear to me what method? As told, they make analytical analysis of the PARALLEL case, associated to ordinary Bethe equations, while they do numerical (DMRG) analysis for the UNPARALLEL case. Does the statement mean that they expect to be able to make this mixed “analytical/numerical” analysis also for other models and that they expect that also for these models the boundary energy can be argued to be independent from the integrable boundary conditions?
This only to make the point about misleading statements. Let me also recall the problems I cited about the Hamiltonian which are still there as well as an autoreferential attitude about citations.
On this basis I cannot suggest the publication of the manuscript in its current form.

• validity: ok
• significance: ok
• originality: ok
• clarity: low
• formatting: reasonable
• grammar: reasonable

### Author:  Xiaotian Xu  on 2021-10-14  [id 1851]

(in reply to Report 3 on 2021-09-25)

Thank you very much for your many helpful suggestions raised about our paper and we really appreciate your support. We believe that these comments and suggestions have significantly improved our manuscript. To these comments we respond as follows. The page numbers and equation numbers refer to revised version, unless specify.

1) Reviewer: Nevertheless, as also remarked by the second referee, they have done very little about the misleading statements and structure of the manuscript. A part the exceptions above remarked, they have mainly leave unattended my previous remarks so that my previous report is largely still actual for the current revision.

Authors: Thank you very much. According to your kind suggestion, we have carefully modified the abstract and corrected the misleading statements in the new revision. For example, we have revised the statements of lines 3-8 in the abstract. We have revised the statements of lines 138-141 in section 3 of P.6. We added some discussions in lines 218-226 of P.10 at the end of section 3. We revised the statements and added discussion of lines 228-232 in section 4 of P.10.

2) Reviewer: These are interesting and sounding results and this should be what I would expect to understand reading the abstract and all along the manuscript but this is not the case. Instead, to arrive to this conclusion, I have done a careful disentanglement between manuscript statements and analysis there presented and I have the impression that this holds true for the other referees, too. It is enough to look to the abstract:

"The finite size scaling properties of the inhomogeneous term in the T-Q relation at the ground state are analyzed." This is done only numerically, e.g. by DMRG.

"Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system." This is done analytically by string hypothesis only for the PARALLEL case. Then, the numerical analysis allows to argue that the same result should hold in the thermodynamic limit also for the UNPARALLEL case.

Authors: Thank you for your recognition of our work and helpful suggestions! We have modified the abstract.

We have changed the sentence in abstract lines 3-5 "The finite size scaling properties of the inhomogeneous term in the $T-Q$ relation at the ground state are analyzed" to "The finite size scaling properties of the inhomogeneous term in the $T-Q$ relation at the ground state are calculated by the density matrix renormalization group".

We have modify the sentence in abstract lines 5-8 "Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system" to "Based on our findings, the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations of a related model with parallel boundary fields".

3) Reviewer: Let me also cite directly, the opening sentence of section 4: "In this section, we study the physical effects induced by the unparallel boundary magnetic fields and compute the boundary energy". Once again, they numerically (DMRG) study the physical effects induced by the UNPARALLEL boundary magnetic fields while they compute analytically by string hypothesis the boundary energy of the PARALLEL case.

Authors: In this paper, we study the physical effects induced by the boundary magnetic fields and compute the boundary energy in the thermodynamic limit, where the system size $N \to \infty$. From the finite size scaling analysis by using the density matrix renormalization group (DMRG) method, we find that the inhomogeneous term indeed can be neglected at the ground state when the system size $N$ tends to infinity. Then we can analytically calculate the boundary energy in the thermodynamic limit based on the string hypothesis of the reduced BAEs (29), which is equal to the result induced by the parallel boundary magnetic fields in the thermodynamic limit. In figure 3, the numerical extrapolation results ($N \to \infty$) of DMRG data also prove the correctness of the analytical results.

We add the related discussion of lines 229-232 of P.10 after the first sentence in section 4.

4) Reviewer: As well as the sentence in the conclusion: "The method provided in this paper can be used to study the thermodynamic properties of other quantum integrable models associated with rational R-matrix." It is not clear to me what method? As told, they make analytical analysis of the PARALLEL case, associated to ordinary Bethe equations, while they do numerical (DMRG) analysis for the UNPARALLEL case. Does the statement mean that they expect to be able to make this mixed “analytical/numerical” analysis also for other models and that they expect that also for these models the boundary energy can be argued to be independent from the integrable boundary conditions?

Authors: Thank you for your helpful suggestion! We modified the related statements in the conclusion as follows.

In this paper, we have studied the thermodynamic limit and boundary energy of the isotropic spin-1 Heisenberg chain with generic integrable non-diagonal boundary reflections. It is shown that the contribution of the inhomogeneous term in the associated $T-Q$ relation (18) (due to the unparallel boundary fields) at the ground state can be neglected when the system size $N$ tend to infinity. Then we calculate the analytical expression of boundary energy (68) in the thermodynamic limit based on the string hypothesis of the reduced BAEs (29).

5) Reviewer: Let me also recall the problems I cited about the Hamiltonian which are still there as well as an autoreferential attitude about citations.

Authors: Thank you very much. According to your suggestion, we have rewritten the Hamiltonian (15) in a more symmetric form, and we have corrected the problems in the citations. We deleted some self-citations and added five new references:

[39] L. Mezincescu, R. I. Nepomechie and V. Rittenberg, Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms, Phys. Lett. A 147, 70 (1990).

[40] T. Inami, S. Odake and Y.-Z. Zhang, Reflection K-matrices of the 19-vertex model and XXZ spin-1 chain with general boundary terms, Nucl. Phys. B 470, 419 (1996).

[45] H.J. de Vega and F. Woynarovich, Method for calculating finite size corrections in Bethe ansatz systems: Heisenberg chain and six-vertex model, Nucl. Phys. B 251, 439 (1985).

[46] A. Kl\"umper, M. T. Batchelor and P. A. Pearce, Central charges of the 6- and 19-vertex models with twisted boundary conditions, J. Phys. A: Math. Gen. 24, 3111 (1991).

[47] J. Suzuki, Spinons in magnetic chains of arbitrary spins at finite temperatures, J. Phys. A: Math. Gen. 32, 2341 (1999).

### Author:  Xiaotian Xu  on 2021-10-14  [id 1850]

(in reply to Report 3 on 2021-09-25)

Thank you very much for your many helpful suggestions raised about our paper and we really appreciate your support. We believe that these comments and suggestions have significantly improved our manuscript. To these comments we respond as follows. The page numbers and equation numbers refer to revised version, unless specify. 1. Reviewer: Nevertheless, as also remarked by the second referee, they have done very little about the misleading statements and structure of the manuscript. A part the exceptions above remarked, they have mainly leave unattended my previous remarks so that my previous report is largely still actual for the current revision. Authors: Thank you very much. According to your kind suggestion, we have carefully modified the abstract and corrected the misleading statements in the new revision. For example, we have revised the statements of lines 3-8 in the abstract. We have revised the statements of lines 138-141 in section 3 of P.6. We added some discussions in lines 218-226 of P.10 at the end of section 3. We revised the statements and added discussion of lines 228-232 in section 4 of P.10. 2. Reviewer: These are interesting and sounding results and this should be what I would expect to understand reading the abstract and all along the manuscript but this is not the case. Instead, to arrive to this conclusion, I have done a careful disentanglement between manuscript statements and analysis there presented and I have the impression that this holds true for the other referees, too. It is enough to look to the abstract: "The finite size scaling properties of the inhomogeneous term in the T-Q relation at the ground state are analyzed." This is done only numerically, e.g. by DMRG. "Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system." This is done analytically by string hypothesis only for the PARALLEL case. Then, the numerical analysis allows to argue that the same result should hold in the thermodynamic limit also for the UNPARALLEL case. Authors: Thank you for your recognition of our work and helpful suggestions! We have modified the abstract. We have changed the sentence in abstract lines 3-5 "The finite size scaling properties of the inhomogeneous term in the $T-Q$ relation at the ground state are analyzed" to "The finite size scaling properties of the inhomogeneous term in the $T-Q$ relation at the ground state are calculated by the density matrix renormalization group". We have modify the sentence in abstract lines 5-8 "Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system" to "Based on our findings, the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations of a related model with parallel boundary fields". 3. Reviewer: Let me also cite directly, the opening sentence of section 4: "In this section, we study the physical effects induced by the unparallel boundary magnetic fields and compute the boundary energy". Once again, they numerically (DMRG) study the physical effects induced by the UNPARALLEL boundary magnetic fields while they compute analytically by string hypothesis the boundary energy of the PARALLEL case. Authors: In this paper, we study the physical effects induced by the boundary magnetic fields and compute the boundary energy in the thermodynamic limit, where the system size $N \to \infty$. From the finite size scaling analysis by using the density matrix renormalization group (DMRG) method, we find that the inhomogeneous term indeed can be neglected at the ground state when the system size $N$ tends to infinity. Then we can analytically calculate the boundary energy in the thermodynamic limit based on the string hypothesis of the reduced BAEs (29), which is equal to the result induced by the parallel boundary magnetic fields in the thermodynamic limit. In figure 3, the numerical extrapolation results ($N \to \infty$) of DMRG data also prove the correctness of the analytical results. We add the related discussion of lines 229-232 of P.10 after the first sentence in section 4. 4. Reviewer: As well as the sentence in the conclusion: "The method provided in this paper can be used to study the thermodynamic properties of other quantum integrable models associated with rational R-matrix." It is not clear to me what method? As told, they make analytical analysis of the PARALLEL case, associated to ordinary Bethe equations, while they do numerical (DMRG) analysis for the UNPARALLEL case. Does the statement mean that they expect to be able to make this mixed “analytical/numerical” analysis also for other models and that they expect that also for these models the boundary energy can be argued to be independent from the integrable boundary conditions? Authors: Thank you for your helpful suggestion! We modified the related statements in the conclusion as follows. In this paper, we have studied the thermodynamic limit and boundary energy of the isotropic spin-1 Heisenberg chain with generic integrable non-diagonal boundary reflections. It is shown that the contribution of the inhomogeneous term in the associated $T-Q$ relation (18) (due to the unparallel boundary fields) at the ground state can be neglected when the system size $N$ tend to infinity. Then we calculate the analytical expression of boundary energy (68) in the thermodynamic limit based on the string hypothesis of the reduced BAEs (29). 5. Reviewer: Let me also recall the problems I cited about the Hamiltonian which are still there as well as an autoreferential attitude about citations. Authors: Thank you very much. According to your suggestion, we have rewritten the Hamiltonian (15) in a more symmetric form, and we have corrected the problems in the citations. We deleted some self-citations and added five new references: [39] L. Mezincescu, R. I. Nepomechie and V. Rittenberg, Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms, Phys. Lett. A 147, 70 (1990). [40] T. Inami, S. Odake and Y.-Z. Zhang, Reflection K-matrices of the 19-vertex model and XXZ spin-1 chain with general boundary terms, Nucl. Phys. B 470, 419 (1996). [45] H.J. de Vega and F. Woynarovich, Method for calculating finite size corrections in Bethe ansatz systems: Heisenberg chain and six-vertex model, Nucl. Phys. B 251, 439 (1985). [46] A. Kl\"umper, M. T. Batchelor and P. A. Pearce, Central charges of the 6- and 19-vertex models with twisted boundary conditions, J. Phys. A: Math. Gen. 24, 3111 (1991). [47] J. Suzuki, Spinons in magnetic chains of arbitrary spins at finite temperatures, J. Phys. A: Math. Gen. 32, 2341 (1999).

### Anonymous Report 2 on 2021-9-3 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202106_00001v2, delivered 2021-09-03, doi: 10.21468/SciPost.Report.3489

### Report

In their revision the authors have taken into account the technical points raised in the referees' reports. With respect to their numerical work they have added Fig.~4 showing that the $O(N^0)$ contribution to the ground state energy of the system converges to that computed from the Bethe ansatz equations (31), (32). Note, however, that the latter are based on the string hypothesis (i.e. strings with exponential accuracy of their components) which is known not to capture the $O(1/N)$ contributions to the energy correctly.
Still, this gives further support to their finding that bulk and boundary energy are correctly obtained using established Bethe ansatz techniques - even for non-parallel boundary fields.Together with the explicit calculation of the boundary energy of the integrable spin-1 chain this is a result which is worth to be published in SciPost Physics.

The referees' objections concerning the partly ambiguous discussion of the nature of boundary conditions considered in the paper, however, have not been addressed satisfactorily.
Specifically, I would suggest changes along the following lines

Abstract, lines 5,6 (and similar in the Conclusion, lines 293-296):
... are analyzed. Based on our findings the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations (BAEs) of a related model with parallel boundary fields. These results ...

Introduction, line 67: delete "unparallel"

Section 3, lines 134-137: change to
... is not the eigenvalue $\Lambda(u)$ for any finite $N$ but rather that of the transfer matrix with parallel boundary fields of the same strength. In the limit $N\to\infty$ it will give, however, the correct boundary energy (...

Section 4, line 216/7:
delete "unparallel" and add "in the thermodynamic limit" at the end of the first sentence.

With such changes implemented the paper will meet the general acceptance criteria of SciPost Physics.

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Author:  Xiaotian Xu  on 2021-10-14  [id 1852]

(in reply to Report 2 on 2021-09-03)

Thank you very much for your many helpful points raised about our paper and we really appreciate your support. We believe that these comments and suggestions have significantly improved our manuscript. To these comments we respond as follows. The page numbers and equation numbers refer to revised version, unless specify.

1) Reviewer: Abstract, lines 5,6 (and similar in the Conclusion, lines 293-296): ... are analyzed. Based on our findings the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations (BAEs) of a related model with parallel boundary fields. These results ...

Authors: According to your suggestion, we modified the sentence lines 5-8 in Abstract "Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system. These results ..." into "Based on our findings, the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations of a related model with parallel boundary fields. These results ...".

2) Reviewer: Introduction, line 67: delete "unparallel"

Authors: Thank you for your kind advice. We deleted the word "unparallel".

3) Reviewer: Section 3, lines 134-137: change to ... is not the eigenvalue $\Lambda(u)$ for any finite $N$ but rather that of the transfer matrix with parallel boundary fields of the same strength. In the limit $N\rightarrow\infty$ it will give, however, the correct boundary energy ...

Authors: According to your advice, we changed the sentence "the $\Lambda_{hom}(u)$ is not the eigenvalue $\Lambda(u)$ for any finite... of the paper)." in lines 138-141 of P. 6 to "the $\Lambda_{hom}(u)$ is not the eigenvalue $\Lambda(u)$ for any finite $N$ but rather that of the transfer matrix with parallel boundary fields of the same strength. In the limit $N\rightarrow\infty$ it will give, however, the correct boundary energy (see the following parts of the paper)."

4) Reviewer: Section 4, line 216/7: delete "unparallel" and add "in the thermodynamic limit" at the end of the first sentence.

Authors: Thank you for your kind advice. We deleted the word "unparallel" in line 228 and added "in the thermodynamic limit" at the end of the first sentence in lines 228-229 of P.10.

### Author:  Xiaotian Xu  on 2021-10-14  [id 1848]

(in reply to Report 2 on 2021-09-03)

Thank you very much for your many helpful points raised about our paper and we really appreciate your support. We believe that these comments and suggestions have significantly improved our manuscript. To these comments we respond as follows. The page numbers and equation numbers refer to revised version, unless specify. 1. Reviewer: Abstract, lines 5,6 (and similar in the Conclusion, lines 293-296): ... are analyzed. Based on our findings the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations (BAEs) of a related model with parallel boundary fields. These results ... Authors: According to your suggestion, we modified the sentence lines 5-8 in Abstract "Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system. These results ..." into "Based on our findings, the boundary energy of the system in the thermodynamic limit can be obtained from Bethe ansatz equations of a related model with parallel boundary fields. These results ...". 2. Reviewer: Introduction, line 67: delete "unparallel" Authors: Thank you for your kind advice. We deleted the word "unparallel". 3. Reviewer: Section 3, lines 134-137: change to ... is not the eigenvalue $\Lambda(u)$ for any finite $N$ but rather that of the transfer matrix with parallel boundary fields of the same strength. In the limit $N\rightarrow\infty$ it will give, however, the correct boundary energy ... Authors: According to your advice, we changed the sentence "the $\Lambda_{hom}(u)$ is not the eigenvalue $\Lambda(u)$ for any finite... of the paper)." in lines 138-141 of P. 6 to "the $\Lambda_{hom}(u)$ is not the eigenvalue $\Lambda(u)$ for any finite $N$ but rather that of the transfer matrix with parallel boundary fields of the same strength. In the limit $N\rightarrow\infty$ it will give, however, the correct boundary energy (see the following parts of the paper)." 4. Reviewer: Section 4, line 216/7: delete "unparallel" and add "in the thermodynamic limit" at the end of the first sentence. Authors: Thank you for your kind advice. We deleted the word "unparallel" in line 228 and added "in the thermodynamic limit" at the end of the first sentence in lines 228-229 of P.10.

### Anonymous Report 1 on 2021-8-31 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202106_00001v2, delivered 2021-08-31, doi: 10.21468/SciPost.Report.3472

### Strengths

1) the vanishing of the contribution of the inhomogeneous term to the ground-state energy is shown

2) new results for the surface energies are derived.

### Weaknesses

1) the finite size correction term const.N^beta is not evaluated (analytically)

### Report

The authors have improved their manuscript. I therefore recommend publication
of the manuscript.

Please write (15) in a more symmetric form where the left and the right
boundary terms appear on equal footing.

Figure 4: please change the exponent b to \beta as in Figure 2.

The discussion should contain information on the status of the finite-size
analysis:

-- the O(N^1) bulk term and the O(N^0) boundary terms to the ground state
energy do not depend on the orientation of the boundary fields,

-- the true finite size terms are probably of order O(N^-1) and are out of
reach for the inhomogeneous/off-diagonal case,

-- due to higher order terms the effective exponents determined in the
manuscript differ somewhat from -1

and maybe

-- the diagonal case is possibly tractable along the lines of A. Klumper et
al. and J Suzuki

Please remove as many language problems as possible. There are a couple of
them.

• validity: high
• significance: good
• originality: ok
• clarity: good
• formatting: good
• grammar: reasonable

### Author:  Xiaotian Xu  on 2021-10-14  [id 1853]

(in reply to Report 1 on 2021-08-31)

Thank you very much for your many helpful points raised about our paper and we really appreciate your support. We believe that these comments and suggestions have significantly improved our manuscript. To these comments we respond as follows. The page numbers and equation numbers refer to revised version, unless specify.

1) Reviewer: Please write (15) in a more symmetric form where the left and the right boundary terms appear on equal footing.

Authors: Thank you for your suggestion. We rewrote the Hamiltonian (15) in a more symmetric form.

2) Reviewer: Figure 4: please change the exponent b to $\beta$ as in Figure 2.

Authors: Thank you for your careful reading. We changed the exponent "b" to "$\beta$" in caption of Figure 4 and lines 297, 298.

3) Reviewer: The discussion should contain information on the status of the finite-size analysis:

-- the $O(N^1)$ bulk term and the $O(N^0)$ boundary terms to the ground state energy do not depend on the orientation of the boundary fields,

-- the true finite size terms are probably of order $O(N^{-1})$ and are out of reach for the inhomogeneous/off-diagonal case,

-- due to higher order terms the effective exponents determined in the manuscript differ somewhat from -1.

and maybe

-- the diagonal case is possibly tractable along the lines of A. Klumper et al. and J Suzuki.

Authors: We added the discussion at the end of section 3 in P.10, and added three references [45], [46] and [47] mentioned in the report.

Due to the existence of inhomogeneous term in BAEs.(24), it is hard to analytically calculate the finite size correction for the present off-diagonal boundary reflections along the lines given in references [45-47]. We shall note that the diagonal case is tractable along the lines of A. Klumper et al. [46] and J. Suzuki [47]. The finite size correction $\mathcal{O}(N^1)$ for the bulk and $\mathcal{O}(N^0)$ term for the boundaries to the ground state energy do not depend on the orientations of the boundary fields. The true finite size correction terms are probably of order $\mathcal{O}(N^{-1})$ and are out of reach for the inhomogeneous/off-diagonal case. Due to higher order correction terms, the effective exponents $\beta$ determined in the paper differ from $-1$.

4) Reviewer: Please remove as many language problems as possible. There are a couple of them.

Authors: Thank you very much for your careful reading. We have corrected the language problems and polished the English.

### Author:  Xiaotian Xu  on 2021-10-14  [id 1849]

(in reply to Report 1 on 2021-08-31)

Thank you very much for your many helpful points raised about our paper and we really appreciate your support. We believe that these comments and suggestions have significantly improved our manuscript. To these comments we respond as follows. The page numbers and equation numbers refer to revised version, unless specify. 1. Reviewer: Please write (15) in a more symmetric form where the left and the right boundary terms appear on equal footing. Authors: Thank you for your suggestion. We rewrote the Hamiltonian (15) in a more symmetric form. 2. Reviewer: Figure 4: please change the exponent b to $\beta$ as in Figure 2. Authors: Thank you for your careful reading. We changed the exponent "b" to "$\beta$" in caption of Figure 4 and lines 297, 298. 3. Reviewer: The discussion should contain information on the status of the finite-size analysis: -- the $O(N^1)$ bulk term and the $O(N^0)$ boundary terms to the ground state energy do not depend on the orientation of the boundary fields, -- the true finite size terms are probably of order $O(N^{-1})$ and are out of reach for the inhomogeneous/off-diagonal case, -- due to higher order terms the effective exponents determined in the manuscript differ somewhat from -1. and maybe -- the diagonal case is possibly tractable along the lines of A. Klumper et al. and J Suzuki. Authors: We added the discussion at the end of section 3 in P.10, and added three references [45], [46] and [47] mentioned in the report. Due to the existence of inhomogeneous term in BAEs.(24), it is hard to analytically calculate the finite size correction for the present off-diagonal boundary reflections along the lines given in references [45-47]. We shall note that the diagonal case is tractable along the lines of A. Klumper et al. [46] and J. Suzuki [47]. The finite size correction $\mathcal{O}(N^1)$ for the bulk and $\mathcal{O}(N^0)$ term for the boundaries to the ground state energy do not depend on the orientations of the boundary fields. The true finite size correction terms are probably of order $\mathcal{O}(N^{-1})$ and are out of reach for the inhomogeneous/off-diagonal case. Due to higher order correction terms, the effective exponents $\beta$ determined in the paper differ from $-1$. 4. Reviewer: Please remove as many language problems as possible. There are a couple of them. Authors: Thank you very much for your careful reading. We have corrected the language problems and polished the English.

### Author:  Xiaotian Xu  on 2021-10-14  [id 1847]

(in reply to Report 1 on 2021-08-31)

Thank you very much for your many helpful points raised about our paper and we really appreciate your support. We believe that these comments and suggestions have significantly improved our manuscript. To these comments we respond as follows. The page numbers and equation numbers refer to revised version, unless specify.

1. Reviewer: Please write (15) in a more symmetric form where the left and the right boundary terms appear on equal footing.

Authors: Thank you for your suggestion. We rewrote the Hamiltonian (15) in a more symmetric form.

1. Reviewer: Figure 4: please change the exponent b to $\beta$ as in Figure 2.

Authors: Thank you for your careful reading. We changed the exponent "b" to "$\beta$" in caption of Figure 4 and lines 297, 298.

1. Reviewer: The discussion should contain information on the status of the finite-size analysis:

-- the $O(N^1)$ bulk term and the $O(N^0)$ boundary terms to the ground state energy do not depend on the orientation of the boundary fields,

-- the true finite size terms are probably of order $O(N^{-1})$ and are out of reach for the inhomogeneous/off-diagonal case,

-- due to higher order terms the effective exponents determined in the manuscript differ somewhat from -1.

and maybe

-- the diagonal case is possibly tractable along the lines of A. Klumper et al. and J Suzuki.

Authors: We added the discussion at the end of section 3 in P.10, and added three references [45], [46] and [47] mentioned in the report.

Due to the existence of inhomogeneous term in BAEs.(24), it is hard to analytically calculate the finite size correction for the present off-diagonal boundary reflections along the lines given in references [45-47]. We shall note that the diagonal case is tractable along the lines of A. Klumper et al. [46] and J. Suzuki [47]. The finite size correction $\mathcal{O}(N^1)$ for the bulk and $\mathcal{O}(N^0)$ term for the boundaries to the ground state energy do not depend on the orientations of the boundary fields. The true finite size correction terms are probably of order $\mathcal{O}(N^{-1})$ and are out of reach for the inhomogeneous/off-diagonal case. Due to higher order correction terms, the effective exponents $\beta$ determined in the paper differ from $-1$.

1. Reviewer: Please remove as many language problems as possible. There are a couple of them.

Authors: Thank you very much for your careful reading. We have corrected the language problems and polished the English.