On integrable matrix product operators with bond dimension D=4

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We construct and study a two-parameter family of matrix product operators of bond dimension D=4. The operators M(x,y) act on (C₂)⊗ ⁿ, i.e., the space of states of a spin-1/2 chain of length N. For the particular values of the parameters: x=1/3 and y=1/√3, the operator turns out to be proportional to the square root of the reduced density matrix of the valence-bond-solid state on a hexagonal ladder. We show that M(x,y) has several interesting properties when (x,y) lies on the unit circle centered at the origin: x² + y²=1. In this case, we find that M(x,y) commutes with the Hamiltonian and all the conserved charges of the isotropic spin-1/2 Heisenberg chain. Moreover, M(x₁,y₁) and M(x₂,y₂) are mutually commuting if x²ᵢ + y²ᵢ=1 for both i=1 and 2. These remarkable properties of M(x,y) are proved as a consequence of the Yang-Baxter equation.