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

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

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.