Rigorous Yang–Baxter formulation for models Y2 and Y3

Construct solutions of the Yang–Baxter equation whose associated transfer matrices generate the spin-1/2 Hamiltonians H_Y2 and H_Y3, thereby rigorously establishing integrability for these models by producing an infinite commuting family of local charges. Model Y2 is the PT-symmetric three-site Hamiltonian with two-step hopping and an Ising term parameterized by γ, and model Y3 is the SU(2)-symmetric medium-range Hamiltonian built from permutation operators with density h_{Y3}(j)=(P_{j,j+3}-1)(P_{j+1,j+2}-1)−(P_{j,j+2}-1).

Background

The paper derives Bethe Ansatz solutions and provides numerical checks for two previously unsolved integrable-looking spin chains: model Y2 (a PT-symmetric, left–right noninvariant, three-site interaction chain) and model Y3 (an SU(2)-symmetric, medium-range chain with four-site interactions built from permutation operators).

While the Bethe Ansatz and exact diagonalization strongly suggest integrability, the authors note that a rigorous proof in the Yang–Baxter framework is still missing because no R-matrix generating these Hamiltonians has been found. Establishing such an R-matrix would confirm integrability in the standard sense by producing a commuting family of local conserved charges.