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Modified rational six vertex model on a rectangular lattice : new formula, homogeneous and thermodynamic limits
by Matthieu Cornillault, Samuel Belliard
Submission summary
| Authors (as registered SciPost users): | Matthieu Cornillault |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2509.11797v2 (pdf) |
| Date submitted: | Oct. 16, 2025, 2:46 p.m. |
| Submitted by: | Matthieu Cornillault |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
The author(s) disclose that the following generative AI tools have been used in the preparation of this submission:
We used an generative AI (ChatGPT in November 2024) to write a function in Wolfram Mathematica that computes the complete symmetric polynomials. This function helped us to verify our theoretical computations for the inverse matrix of the partial Cauchy matrix (proposition A.2).
Abstract
We continue the work of Belliard, Pimenta and Slavnov (2024) studying the modified rational six vertex model. We find another formula of the partition function for the inhomogeneous model, in terms of a determinant that mix the modified Izergin one and a Vandermonde one. This expression enables us to compute the partition function in the homogeneous limit for the rectangular lattice, and then to study the thermodynamic limit. It leads to a new result, we obtain the first order of free energy with boundary effects in the thermodynamic limit.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
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On the whole, I find the article interesting and worth publishing in SciPost Physics.
Requested changes
- The boundary conditions considered by the authors are not entirely general, since they are restricted to cases in which the twist matrices B and \hat{C} are both not invertible. This is written in the middle of a paragraph (“Let us focus on this last case,” end of page 6), so that it could be not completely clear that this restriction concerns the entire article (and not just the end of this paragraph). It seems to me that the authors should make this clearer, for example by writing: “From now on, we restrict ourselves…”
- More general boundary conditions could be considered: different twists B_i and \hat{C}_j depending on the horizontal or vertical line of the lattice. Can the authors briefly comment about this case? Is it possible to write a generalization of (1.12) or (1.20) in that case?
- Formula (1.20) seems to vanish in some particular cases in which the trace of one (e.g. \hat{C} if n<m) or both twist matrices is zero. Is there some geometric reason for that? Can the authors comment about this cases?
- In Formula (1.6), I suppose that the authors want to indicate that this quantity is equal to 0? The sentence after the formula has also to be corrected.
- The same notation w_1,2 has been used for two different quantities: the Boltzmann weights in Formula (1.1) and the coordinates of the vector w starting from Remark 1.2. One of these notations has to be changed.
- In Figure 4, it is very difficult to make the difference between blue and dark-blue, I suggest to change one of these two colors.
- There should be a misprint in Formula (A.32): l-p is negative there.
- There is no equation (24) in the published version of [FW12] quoted by the authors in Remark B.2.
- Misprints (n instead of N) in Formulas (D.6) and (D.7) should be corrected.
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Publish (meets expectations and criteria for this Journal)
Thank you very much for your report.
Author: Matthieu Cornillault on 2026-01-09 [id 6221]
(in reply to Report 2 on 2026-01-01)Thank you for your detailed report and your feedback on our article. We have made the minor changes you indicated (points 1 and 4 to 9).
For point 2, it is indeed possible to define different twists B_i and \hat{C_j} that depend on the horizontal or vertical line in the lattice. However, to preserve the symmetry properties of the partition function, due to the fact that [R_{ab}, B_a Bt_{b}] = 0, where B and Bt are both twists on rows or both on columns, B and Bt must be proportional to each other. This implies that all vectors on a given edge of the lattice must be proportional to one another. By factoring out the proportionality constants, we return to our original problem.
Otherwise, the formulation in terms of a single family of operators A,B,C,D — which is crucial in our calculations of the modified Izergin determinant — fails. Such models require further research, especially if one seeks a closed-form formula involving a determinant. This point remains unclear to us.
For certain configurations with local twists, we know that a determinant formulation exists (e.g., with the pDWBC model), but in general, we don't know.
Regarding point 3, this is indeed the case. For example, when n is strictly smaller than m, if tr(\hat{C}) = 0 (meaning the vectors on opposite sides are orthogonal), the partition function vanishes. This can be interpreted as the absence of any possible configuration for the lattice under such boundary conditions. However, we do not yet know how to prove this geometrically.
By drawing small rectangular lattices with domain wall boundary conditions, we observe indeed that no valid configuration exists in this scenario.
Anonymous on 2026-01-09 [id 6222]
(in reply to Matthieu Cornillault on 2026-01-09 [id 6221])I would like to rephrase the sentence "For certain configurations with local twists, (...) we don't know.".
For certain configurations involving local twists, it is the partition function of the model comprising all possible boundary conditions (meeting a certain criterion) that can be described by a formula in terms of a determinant (as is the case with the pDWBC model). Indeed, we observe that, across all the configurations considered, there is symmetry in the boundary conditions that appear.