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The open XYZ spin 1/2 chain: Separation of Variables and scalar products for boundary fields related by a constraint
by G. Niccoli, V. Terras
Submission summary
| Authors (as registered SciPost users): | Giuliano Niccoli |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2402.04112v3 (pdf) |
| Date accepted: | Aug. 25, 2025 |
| Date submitted: | July 30, 2025, 8:55 a.m. |
| Submitted by: | Giuliano Niccoli |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We consider the open XYZ spin chain with boundary fields. We solve the model by the new Separation of Variables approach introduced in arXiv:1904.00852. In this framework, the transfer matrix eigenstates are obtained as a particular sub-class of the class of so-called separate states. We consider the problem of computing scalar products of such separate states. As usual, they can be represented as determinants with rows labelled by the inhomogeneity parameters of the model. We notably focus on the special case in which the boundary parameters parametrising the two boundary fields satisfy one constraint, hence enabling for the description of part of the transfer matrix spectrum and eigenstates in terms of some elliptic polynomial Q-solution of a usual TQ-equation. In this case, we show how to transform the aforementioned determinant for the scalar product into some more convenient form for the consideration of the homogeneous and thermodynamic limits: as in the open XXX or XXZ cases, our result can be expressed as some generalisation of the so-called Slavnov determinant.
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Author comments upon resubmission
We would like to thanks both the referees for their attentive reading and analysis of our manuscript which has allowed us to improve it, answering to their requests of clarifications, verifications and typos. We have taken into account mainly all the requirements of the two referees and we detail them in the following
List of changes
1- We have added an appendix to address the trigonometric limit on the elliptic case and we have verified that in this limit the results and formulae derived in the XYZ case reproduces those of the XXZ case. The limit from the XXZ case to the XXX is then easy and it works as well.
2- We have added a footnote at page 7 to clarify the meaning of “almost any”. The main point is that the linear independence of the $2^N$ covectors defined in (3.17), is proven by showing that the determinant of the matrix of coefficients of these covectors is nonzero. This determinant is a polynomial in the inhomogeneity parameters and in the coordinates of the reference convector. So, we just need to prove that it is nonzero for some special values of these parameters to have that it is not identically zero. So that the “almost any” means that these covectors are independent and so form a basis for any values of these parameters with the exceptions of the zeros of this multivariable polynomial, which is a codimension 1 hypersurface in the full space of inhomogeneities and reference convector coordinates.
3- We have corrected the typo evidenced by the referee.
4- We have corrected the typo evidenced by the referee.
5- One should point out that the analysis developed in [93,94] to advance their conjecture is mainly a numerical analysis on small chains. While this study is developed without further restrictions on the boundary parameters a part (3.35), the absence so far in the literature of an analytic study does not allow to answer if completeness fails for specific parameter ranges. So, we have changed our last sentence and removed the statement "at least for some range of the boundary parameters" and stated explicitly that the conjecture is based only on numerical analysis.
Modifications done to take into account the second referee’s report:
1- In eq (3.7)-(3.10), $det_q$ stays for quantum determinant, it is central, i.e. just a number and it was defined in eq (2.22) and it is the product of the coefficients $a(\lambda)$ and $d(\lambda)$ defined in (2.23).
2- We have clarified that the condition (3.35), in Proposition 3.2, is required to have that the entire function $\tau(\lambda)$ is indeed an elliptic polynomial of degree 2N+6 once it satisfied the TQ functional equation (3.37), which is one of the requirements that it has to satisfy to be a transfer matrix eigenvalue.
3- We have corrected the misprint remarked by the referee in equation (4.5).
Published as SciPost Phys. 19, 090 (2025)
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