SciPost Submission Page
Bootstrapping the $R$-matrix
by Zhao Zhang
Submission summary
| Authors (as registered SciPost users): | Zhao Zhang |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202601_00053v1 (pdf) |
| Code repository: | https://github.com/zz8py/R-matrix-bootstrap |
| Date submitted: | Jan. 21, 2026, 4:19 p.m. |
| Submitted by: | Zhao Zhang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| 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 in all known examples they all turn out to be satisfied whenever the lowest order condition is. It remains to be understood whether they are all 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 manuscript has been significantly enriched following the referee's suggestion to focus more on examples. A new section 4 is now devoted to pursue this direction, as well as a brushed-up appendix B including explicit expression for the first few terms in the R-matrix expansion.
The rest of the major changes are clarifications on the usage of the bootstrap program and integrability test, the difference between conserved charge and Yang-Baxter definitions of integrability, and on what can and cannot be bootstrapped from the Hamiltonian in models like the Hubbard chain.
List of changes
-
The explanation of constant shift following Eq. (1) has been simplified according to the referee's suggestion.
-
Sec. 3 has been mostly rewritten to explain the usage of Eq. (9) from three different perspectives. First as a way to find R-matrix for known integrable Hamiltonians. Then as an integrability test for generic Hamiltonians. And finally as a way to search for integrable points of a parametrized class of Hamiltonians. Also in this section, more details on Kennedy's trick is provided following Eq. (11).
-
The brand-new Sec. 4 further clarifies the procedure described before by parametrizing general local Hamiltonians with n local degrees of freedom. As an example it arrives at the full classification of integrable spin-1/2 chains using the concretized version of the Reshetikhin condition, and explains that anisotropic higher-spin/multicomponent chains are non-integrable.
-
The beginning and ending of Sec. 5 has been modified to make the point that higher order Reshetikhin conditions are not implied by the lowest order one, even if/though three-local charge implies infinite conserved charges. The statement on the correspondence between these two has also been made softer, lacking a derivation from one to the other.
-
Eq. (46) in Sec. 8 has been simplified noticing [h, h']=0, a simple fact that seems also eluded Ref. 35. It has also been made explicit within the framework of Sec. 4. Comments including in footnote 11 has been added to show how h'(0) could be solved from Eq. (46), and that it does not mean much for the purpose of getting h(\mu) or R(\mu). Particular emphasize has also been made on clarifying that Theorem 13 of Ref. 23 is not useful for claiming integrability of models like the Hubbard.
-
The manipulations in Appendix. A has been reformulated for the RHS of (A.1) and added for the LHS.
-
Appendix. B has basically been reworked. B.1 is a rare example that R^(3) can be obtained by hand, which is now done that way using the language of Sec. 4. B.2 now focuses on the Takhtajan-Babujian model, which explains the subtleties of constant c. It also serves as a demonstration of how generic Hamiltonians as given by (B.5) can be put into the form of (B.7) using a complete basis of corresponding local Hilbert space dimension. I used this opportunity to show also things look different if Sec. 4 did not start with a diagonalization of matrix a. Finally, B.3 has been made slightly more general by letting the anisotropic parameter a vector, which gives very similar results. The appendix is complemented by the Mathematica code uploaded to a repository.