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arXiv:1408.4385

R-matrices for integrable axially symmetric S=1 spin chains

P. N. Bibikov, А. Г. Нураматов

Published Aug 19, 2014• 0 citations
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Abstract

The Reshetikhin condition for the general Hamiltonian density matrix of the $S=1$ axially symmetric spin chain is completely solved. 16 new integrable models and corresponding $R$-matrices are presented.

DOI
10.48550/arxiv.1408.4385

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The Reshetikhin condition for the general Hamiltonian density matrix of the S=1S=1S=1 axially symmetric spin chain is completely solved. 16 new integrable models and corresponding RRR-matrices are presented.
A bootstrap program is presented for algebraically solving the univariate R 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 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 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.
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Quantum integrability is a cornerstone of the exact theory of interacting quantum spin chains. In its standard formulation, however, one starts from R-matrices satisfying the Yang--Baxter equation, rather than from the Hamiltonian itself. It has therefore remained unclear how Yang--Baxter solvability can be characterized directly at the Hamiltonian level, and how it is related to the existence of local conservation laws. Here we prove that, in a broad standard setting, the Reshetikhin condition is not only necessary but also sufficient for Yang--Baxter integrability, thereby reducing the hidden algebraic structure of integrability to a Hamiltonian-level conservation law. Since the Reshetikhin condition is equivalent to conservation of the total energy current, this Hamiltonian-level criterion is also experimentally accessible. This result establishes a quantum counterpart of the Liouville--Arnold theorem for isotropic spin chains, stating that Yang--Baxter solvability is equivalent to an infinite hierarchy of local conserved quantities. Our result also simplifies substantially the search for integrable spin chains by replacing the search for R-matrices with a direct criterion on local Hamiltonians.