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Thermodynamic limit and boundary energy of the spin-1 Heisenberg chain with non-diagonal boundary fields
by Zhihan Zheng, Pei Sun, Xiaotian Xu, Tao Yang, Junpeng Cao, Wen-Li Yang
This is not the current version.
|As Contributors:||Xiaotian Xu|
|Date submitted:||2021-08-12 20:47|
|Submitted by:||Xu, Xiaotian|
|Submitted to:||SciPost Physics|
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.
Author comments upon resubmission
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.
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 -.
Besides, we have corrected some typos, and words and sentences have also been slightly improved.
Submission & Refereeing History
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- Report 3 submitted on 2021-09-25 23:16 by Anonymous
- Report 2 submitted on 2021-09-03 11:18 by Anonymous
- Report 1 submitted on 2021-08-31 19:55 by Anonymous
Reports on this Submission
Anonymous Report 3 on 2021-9-25 (Invited Report)
Some new interesting results.
The main analytic results are derived for boundary conditions different from those declared.
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.
Anonymous Report 2 on 2021-9-3 (Invited 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.
Anonymous Report 1 on 2021-8-31 (Invited Report)
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.
1) the finite size correction term const.N^beta is not evaluated (analytically)
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
-- 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
-- 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