Speaker
Description
Quantum integrability is usually characterized from two perspectives. One is Yang-Baxter solvability, formulated in terms of an R-matrix satisfying the Yang-Baxter equation, or a transfer matrix constructed from it. The other is conservation laws, formulated in terms of infinitely many local conserved quantities commuting with the Hamiltonian.
From either viewpoint, however, it is generally difficult to decide whether a given Hamiltonian is integrable. In the Yang-Baxter framework, one usually starts by finding, often through nontrivial insight, a solution of a matrix-valued functional equation, and the Hamiltonian is then recovered as only part of the information encoded in it. Reversing this procedure, namely finding the corresponding R-matrix from a given Hamiltonian, is difficult in general. On the other hand, directly testing the existence of local conserved quantities requires solving large systems of linear equations, and is therefore impractical. Thus, determining integrability directly from the Hamiltonian remains a difficult problem.
In this talk, I will report that, for isotropic nearest-neighbor spin chains, this problem admits an extremely efficient criterion [1,2]. More specifically, we consider Hamiltonians of the form $H=\sum_i\sum_{n=1}^{2S}J_n(S_i\cdot S_{i+1})^n$ for arbitrary S. We prove that, in this class, the condition known as the Reshetikhin condition is equivalent to integrability, both in the sense of Yang-Baxter solvability and in the sense of local conserved quantities. Namely, if the Reshetikhin condition holds, the system is Yang-Baxter solvable, the corresponding R-matrix can be constructed algorithmically, and infinitely many local conserved quantities can be generated. Conversely, if the Reshetikhin condition does not hold, the system is not Yang-Baxter solvable and has no nontrivial local conserved quantities. This result shows that, within the class of isotropic chains, integrable and non-integrable systems fall into two sharply distinct classes, and that the Reshetikhin condition provides a necessary and sufficient criterion distinguishing them.
Finally, I will explain that some of the key lemmas used in the proof hold beyond the isotropic case. This suggests a new direction in the study of quantum integrable systems: to search the space of local Hamiltonians efficiently and exhaustively for integrable systems, and then solve them constructively.
References
[1] N. Shiraishi and M. Yamaguchi, Dichotomy theorem separating complete integrability and non-integrability of isotropic spin chains, Phys. Rev. B (2026).
[2] N. Shiraishi, M. Yamaguchi, and F. Ishii, in preparation.