A convenient basis for the Izergin-Korepin model

We propose a convenient orthogonal basis of the Hilbert space for the Izergin-Korepin model (or the quantum spin chain associated with the $A^{(2)}_{2}$ algebra). It is shown that the monodromy-matrix elements acting on the basis take relatively simple forms (c.f. acting on the original basis ), which is quite similar as that in the so-called F-basis for the quantum spin chains associated with $A$-type (super)algebras. As an application, we present the recursive expressions of Bethe states in the basis for the Izergin-Korepin model.

💡 Research Summary

The paper addresses the long‑standing problem of constructing a convenient basis for the Izergin‑Korepin (IK) quantum spin chain, which is associated with the twisted affine algebra (A^{(2)}_{2}). While the algebraic Bethe Ansatz (ABA) provides a powerful framework for solving integrable models, the standard computational basis (the tensor product of local three‑state vectors) leads to highly non‑local actions of the monodromy‑matrix elements. This non‑locality makes the explicit evaluation of correlation functions and the construction of Bethe eigenstates extremely cumbersome.

In the first part of the work the authors recall the definition of the IK model. The local space (V) is three‑dimensional with orthonormal basis ({|1\rangle,|2\rangle,|3\rangle}). The R‑matrix (R(u)) is given explicitly in terms of trigonometric functions (a(u),b(u),c(u),d(u),e(u),\dots) (eq. 2.1‑2.2) and satisfies the quantum Yang‑Baxter equation. The monodromy matrix (T(u)=R_{0N}(u-\theta_N)\dots R_{01}(u-\theta_1)) obeys the RTT relation (2.6). The transfer matrix (t(u)=\operatorname{tr}0 T_0(u)) generates a family of commuting operators, and the Hamiltonian with periodic boundary conditions follows from (\partial_u\ln t(u)\big|{u=0}).

The core contribution begins in Section 3, where the authors construct a new orthogonal basis that dramatically simplifies the action of the monodromy matrix. They first decompose the monodromy matrix into operators \