# 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_00001v3 Date submitted: 2021-10-14 12:18 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 calculated by the density matrix renormalization group. 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 can be generalized to the $SU(2)$ symmetric high spin Heisenberg model directly.

###### Current status:
Editor-in-charge assigned

Dear Editor,

Thank you very much for your help. We have revised the manuscript (Ref. No. scipost_202106_00001v2) 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. We have modified the sentence lines 3-5 in Abstract "The finite size scaling ... are analyzed" into "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".
2. We have modified the sentence lines 5-8 in Abstract "Based on the reduced Bethe ansatz equations (BAEs), we obtain the boundary energy of the system." 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.".
3. We have deleted the word "unparallel" in line 69 in P.3.
4. We have rewritten Eq.(15) in P.5 into a more symmetric form.
5. We have changed the sentence "the $\Lambda_{hom}(u)$ is not the eigenvalue $\Lambda(u)$ for any finite ... of the paper)." in lines 138-141 in 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).".
6. We have added some discussions in lines 218-226 in P.10 at the end of section 3 and added five references [39], [40], [45], [46] and [47].
7. We have deleted the word "unparallel" in line 228 in P.10 and added "in the thermodynamic limit" at the end of the first sentence in lines 228-229 in P.10.
8. We have added some discussions in lines 229-232 in P.10 after the first sentence in Section 4.
9. We have changed the exponent "b" to "$\beta$" in caption of Figure 4 and lines 297, 298 in P.15.
10. We have rewritten the Conclusions in Section 5 in P. 15.
11. We have polished the English.