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Exact physical quantities of a competing spin chain in the thermodynamic limit

by Pengcheng Lu, Yi Qiao, Junpeng Cao, Wen-Li Yang

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Yi Qiao
Submission information
Preprint Link: scipost_202211_00041v2  (pdf)
Date accepted: 2023-05-31
Date submitted: 2023-04-11 11:09
Submitted by: Qiao, Yi
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
Approach: Theoretical


We study the exact physical quantities of a competing spin chain which contains many interesting and meaningful couplings including the nearest neighbor, next nearest neighbor, chiral three spins, Dzyloshinsky-Moriya interactions and unparallel boundary magnetic fields in the thermodynamic limit. We obtain the density of zero roots, surface energies and elementary excitations in different regimes of model parameters. Due to the competition of various interactions, the surface energy and excited spectrum show many different pictures from those of the Heisenberg spin chain.

Author comments upon resubmission

Dear editor,

Thank you for arranging a timely review for our manuscript submitted to SciPost.

We have substantially revised our manuscript after reading the comments provided by the reviewers. Those comments are all valuable and very helpful for improving our paper. Replies to the referee comments are attached as PDF files.

If you have any questions about this paper, please don't hesitate to let me know.

Yours sincerely
Yi Qiao

List of changes

We have revised the manuscript according to the referee's suggestions, and list the revisions as follows. The page numbers and equation numbers refer to revised version, unless specify.

1. Added the explain of the index $j$ ``where the index $j$ is the summation index in $H_{bulk}$ (2.2)" after Eq.(2.6).

2. Replaced ``In the derivation, we have used the relation $\sigma(\theta)=\delta(\theta)$." with ``From now on, we use $\sigma(\theta)=\delta(\theta)$." after (4.3).

3. Replaced the words ``charity" with ``chiral three spin" in the manuscript.

4. Added ``for $\{\bar{\theta}_j= 0|j=1,\cdots,2N\}$" in the captions of Figs. 3, 5, 6 and 7.

5. Replaced the summation indices $j$ with $l$ and $k$ in the first and last of Eq.(4.1), respectivly.

6. Redrawn Fig.7(a) and incorporated it with the root configuration of the ground state. Corresponding descriptions have also been added in the context of Sect. 6.

7. Added some discussions of previous results obtained using conventional Bethe ansatz methods in line 7 on page 10 and after Eq.(5.2).

8. Added some arguments about the finite size after Eq.(2.24) in line 2.

9. Added a simplified and explicit form of the elementary excitation energy in Eq.(5.2) and Eq.(6.2).

10. Modified the caption of Fig.7(a) to ``(a) The distribution of $\bar{z}$-roots for $\{\bar{\theta}_j= 0|j=1,\cdots,2N\}$ with $2N=8$, $a=0.66i$, $p=0.1$, $\bar{q}=1.2$ and $\xi=1.2$. Here the blue asterisks represent the pattern of zero roots at the ground state and the red circles denote those at the excited state with boundary string $i(\frac{1}{2}-|p|)$".

11. Added a new section 7 to present the surface energies in ferromagnetic regime.

12. Added a new Appendix A to present a clear and simple way recommended by an anonymous referee.

13. Added references:

[30] M. T. Grisaru, L. Mezincescu and R. I. Nepomechie, {\it J. Phys. A: Math. Gen.} {\bf 28} 1027 (1995).

[31] A. Kapustin and S. Skorik, {\it J. Phys. A: Math. Gen.} {\bf 29} 1629 (1996)

[32] H. Frahm and C. R{\"o}denbeck, {\it Europhys. Lett.} {\bf 33} 47-52 (1996).

Besides, some words and sentences have also been slightly improved.

Published as SciPost Phys. 15, 060 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-4-24 (Invited Report)


The authors have addressed the issues raised by the referees. Using the solution of the functional equation (2.22) for states close to the antiferromagnetic vacuum in the thermodynamic limit by direct Fourier transform their calculation of bulk and boundary properties has become more transparent. Some of the final results have been simplified.

The manuscript should be accepted for publication in SciPost Physics.

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

Anonymous Report 1 on 2023-4-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202211_00041v2, delivered 2023-04-22, doi: 10.21468/SciPost.Report.7088


see report


see report: necessary clarification of (7.4) and (7.5)


I found the replies by the authors to both reports satisfactory. The
manuscript has been amended and has gained significantly.

The authors are correctly pointing out that the elegant calculation presented
in Appendix A is not applicable to the ferromagnetic regime, because there the
eigenvalue function has an extensive number of zeros between the lines Re(z) =
0 and Re(z) = −1 at the ground state. However, for the ferromagnetic regime
many calculations simplify drastically, because in the conventional Bethe
Ansatz no Bethe roots appear for the ground state. Of course in the case of
general boundary fields this argument does not hold for the ODBA equations,
but the authors focus on the bulk O(N) and O(1) terms and ignore O(1/N) terms.

I am convinced that (7.4) must simplify considerably to something like N times
(J_1+J_2). Also, the explicit result in (7.5) resembles the terms in (2.3) and
(2.4), but there are differences. At this point the physical intuition tells
us that the bulk interactions favour a highly degenerate ground state of fully
polarized spins (in arbitrary direction). The calculations become simple and
can be done by elementary means. However, a fully polarized state will "see"
the differently oriented boundary fields and the result of the boundary energy
should depend non-trivially on the parameter \xi. However \xi dropped out in
the authors' calculation or has been set to 0 from the beginning.

Provided the authors clarify the last issue, I recommend the manuscript for
publication in SciPost.

Requested changes

necessary clarification of (7.4) and (7.5)

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

Author:  Yi Qiao  on 2023-05-09  [id 3660]

(in reply to Report 1 on 2023-04-22)

Please see the attachment.



Anonymous on 2023-05-26  [id 3689]

(in reply to Yi Qiao on 2023-05-09 [id 3660])

Thank you very much for your answers/explanations. Now the physical situation is clear to me.

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