# 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:
Has been resubmitted

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.

### 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 2021-11-11 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202106_00001v3, delivered 2021-11-11, doi: 10.21468/SciPost.Report.3832

### 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 taken into account my earlier suggestions. I therefore
recommend publication of the manuscript if the following correction is applied:

In the newly added paragraph on page 10 the authors write

"The finite size correction O(N^1) for the bulk and O(N^0) term for the
boundaries to the ground state energy do not depend on the orientations of the
boundary fields."

The terminology "finite size correction O(N^1)" is inappropriate. Please write

"The O(N^1) bulk term and the O(N^0) boundary term for the ground
state energy do not depend on the orientations of the boundary fields."

### Requested changes

see above

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

### Anonymous Report 2 on 2021-11-6 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202106_00001v3, delivered 2021-11-06, doi: 10.21468/SciPost.Report.3804

### Strengths

Some new and interesting results on spin 1 XXX chain by numeric (DMRG) for the non-parallel case and by analytic analysis of the Bethe Ansatz equations for the parallel case.

### Weaknesses

Still some text to improve in particular about references

### Report

Dear Editor,
The authors have finally taken into account some of the main remarks presented by the other referees and myself. The manuscript then states more frankly what are the results there derived and the way they are derived, i.e. numerically for the non-parallel case and by analytic analysis of the Bethe Ansatz equations for the parallel case. So, the manuscript can meet the minimal requirements for its publication in SciPost and I will propose its publication once the authors will implement the following further improvements.
The Author, should make their modifications compatible with the remaining text. Here, the main point is about the terminology “reduced BAEs”. In line 66 they introduce this terminology without any explanation. I suggest to define there what reduced Bethe equations are, i.e. the Bethe equations of an associated parallel XXX spin 1 model.
While some pertinent references are finally included, I think that the manuscript is still deficient of important ones.
i) First of all, one should recall that the so-called TQ-equations, which are heavily used in the manuscript, have been originated by the work of Baxter, reference to the original papers must be added, e.g.
“R. J. BAXTER, Stud. Appl. Math. (Mass. Inst. of Technology) 50 (1971), 51-69.”
“Baxter,R.J.: Partition function of the eight-vertex lattice model. Ann.Phys.70 193-228 (1972)”
ii) The Ansatz functional analysis of the eigenvalue spectrum by TQ-equations has been pioneered and systematically applied to a large class of integrable models by Reshetikhin and it goes under the name of “Analytic Bethe Ansatz”, e.g.:
“Reshetikhin,N.Yu.: The functional equation method in the theory of exactly soluble quantum systems. Sov.Phys.JETP 57 691-696 (1983)”
The analysis developed in the cited papers [28-33] is a natural development of the Reshetikhin’s approach (based on the fusion equations to introduce an Ansatz) once applied to the models under consideration, the same is definitively the case for the so-called ODBA method.
iii) The authors have missed to cite another Ansatz approach, the so-called “Modified Algebraic Bethe Ansatz”:
“S. Belliard and N. Crampé, Heisenberg XXX model with general boundaries: Eigenvectors
from algebraic Bethe ansatz, SIGMA 9, 072 (2013),”
“S. Belliard, Modified algebraic Bethe ansatz for XXZ chain on the segment - I - Triangular
cases, Nucl. Phys. B 892, 1 (2015),”
which allows to overcome the absence of reference states and to have access to an algebraic Bethe Ansatz form of the transfer matrix eigenstates.
iv) Discussing about the analysis of quantum integrable models without U(1) symmetry and without a proper reference states, one is obliged to refer to the seminal works of Sklyanin and its non-Ansatz method of quantum separation of variables:
“E. K. Sklyanin, The quantum Toda chain, In N. Sanchez, ed., Non-Linear Equations in Classical
and Quantum Field Theory, pp. 196–233. Springer Berlin Heidelberg, Berlin, Heidelberg,
ISBN 978-3-540-39352-8”
“E. K. Sklyanin, Functional Bethe Ansatz, In B. Kupershmidt, ed., Integrable and Superintegrable
Systems, pp. 8–33. World Scientific, Singapore, doi:10.1142/9789812797179_0002
(1990)”
Already in the 1985, Sklyanin has derived an algebraic approach to define the spectrum (eigenvalues and wave-functions) of models like the XXX spin ½ chain with general quasi-periodic boundary conditions and the Toda model. Both models for which a reference state is missing and ordinary algebraic Bethe Ansatz does not apply. This SoV approach has since been largely developed in the literature with further recent developments. This is, in particular, the case for the XXX/XXZ/XYZ spin chains with non-periodic boundary condition, see e.g.:
“H. Frahm, A. Seel and T. Wirth, Separation of variables in the open XXX chain, Nucl. Phys.
B 802, 351 (2008)”
“H. Frahm, J. H. Grelik, A. Seel and T. Wirth, Functional Bethe ansatz methods for the open XXX chain, J. Phys A: Math. Theor. 44, 015001 (2011).”

where the functional version (i.e. eigenvalues and wave-functions) of SoV has been derived and see e.g.

“G. Niccoli, Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables:
Complete spectrum and matrix elements of some quasi-local operators, J. Stat. Mech. 2012,
P10025 (2012)”

where the SoV bases have been constructed in the Hilbert space of the quantum spin chain, so allowing for the construction of the eigenstates in the same Hilbert space and providing the tools for a simple proof of the completeness of the spectrum description.

Indeed, one should stress the non-Ansatz nature of the Sklyanin’s SoV approach for which the completeness of the spectrum description is mainly self-contained in it. In fact, it is in the SoV framework that TQ-equations and associated Bethe equations are rederived without any Ansatz and naturally proven to be complete.
This is the case for the inhomogeneous Bethe equations whose Ansatz has been introduced in the ODBA framework:
“J. Cao,W.-L. Yang, K. Shi, and Y.Wang. Off-diagonal Bethe ansatz solutions of the anisotropic
spin-1/2 chains with arbitrary boundary fields. Nucl. Phys. B, 877:152–175, 2013.”
and in reference [32], while a proof of the completeness of the spectrum has been naturally presented in the SoV framework:
“N. Kitanine, J. M. Maillet and G. Niccoli, Open spin chains with generic integrable boundaries:
Baxter equation and Bethe ansatz completeness from separation of variables, J. Stat.
Mech. 2015, P05015 (2014)”
As well as for the simplest example of models without U(1) symmetry, i.e. the XXZ chain with anti-periodic boundary conditions, where the Ansatz on the TQ-equation have been introduced in
“M. T. Batchelor, R. J. Baxter, M. J. O’Rourke, and C. M. Yung, Exact solution and interfacial tension of the six-vertex model with anti-periodic boundary conditions, J. Phys. A: Math. Gen. 28 (1995), 2759.”
“C. M. Yung and M. T. Batchelor, Exact solution for the spin-s XXZ quantum chain with non-diagonal twists, Nucl. Phys. B 446 (1995), 461–484.”
conjecturing the existence of the Q-operator, while proven in the SoV framework in
“G. Niccoli and V. Terras, Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate
construction by functional equations in separation of variables, Lett. Math. Phys. 105, 989
(2015)”

v) Finally in the lines 42-43, where the authors refer to results for periodic and parallel boundary conditions, one should mention the fundamental results on correlation functions for spin 1/2 of the Kyoto and then of the Lyon school. If the authors restrict their discussion to the higher spin case only then it is anyhow worth citing:
“O. A. Castro-Alvaredo and J. M. Maillet, Form factors of integrable Heisenberg (higher) spin chains, J. Phys. A: Math. Theor. 40 (2007), 7451.”
as first results toward the dynamics of the higher spin models.

### Requested changes

See report

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

### Anonymous Report 1 on 2021-10-30 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202106_00001v3, delivered 2021-10-30, doi: 10.21468/SciPost.Report.3761

### Report

The authors have implemented the changes requested by the referees. In the present version the abstract and the statements in the main text are in line with the actual results of the paper so that the general acceptance criteria of SciPost Phys are met.

The calculation of the surface energy's dependence on the boundary fields is new. Potentially these results allow for the identification of the boundary conformal field theory describing the continuum limit of the open boundary spin-1 chain (at least for the model with parallel boundary fields).

I recommend to accept the paper for publication in SciPost Phys in its present form.

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