SciPost Submission Page
Bootstrapping the $R$-matrix
by Zhao Zhang
Submission summary
| Authors (as registered SciPost users): | Zhao Zhang |
| Submission information | |
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| Preprint Link: | scipost_202601_00053v3 (pdf) |
| Code repository: | https://github.com/zz8py/R-matrix-bootstrap |
| Date submitted: | Feb. 6, 2026, 12:17 p.m. |
| Submitted by: | Zhao Zhang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
A bootstrap program is presented for algebraically solving the $R$-matrix of a generic integrable quantum spin chain from its Hamiltonian. The Yang-Baxter equation contains an infinite number of seemingly independent constraints on the operator valued coefficients in the expansion of the $R$-matrices with respect to their spectral parameters, with the lowest order one being the Reshetikhin condition. These coefficients can be solved iteratively in a self consistent way using a lemma due to Kennedy, which reconstructs the $R$-matrix after an infinite number of steps. For a generic Hamiltonian, the procedure could fail at any step, making the conditions useful as an integrability test. However, at least for the most common examples, they always turn out to be satisfied whenever the lowest order condition is. It remains to be understood whether they are indeed implied by the Reshetikhin condition.
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
Author comments upon resubmission
The changes requested in this round a rather specified. So I simply implemented them as requested. The few disputable items have already been communicated in the exchange of comments. Below I will only summarize the additional changes made to equation (47) to (49) concerning the uniqueness of the solution for h'(0), which also involves fixing a mistake in the parameterization of h'(0).
List of changes
This changes slightly the expressions of (48) and (49). It is obvious that in the previous version, (49) has LHS and RHS have opposite symmetry about exchanging indices. This is not the case with the new parameterization, as long as the coefficients are antisymmetric. Correcting this mistake should clarify the uniqueness of the solution to (49).